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**Validating Computer System and Network Trustworthiness**

Prof. William H. Sanders Department of Electrical and Computer Engineering and Coordinated Science Laboratory University of Illinois at Urbana-Champaign ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Combinatorial Modeling Stochastic Activity Network Concepts **

Course Outline Issues in Model-Based Validation of High-Availability Computer Systems/Networks Combinatorial Modeling Stochastic Activity Network Concepts Analytic/Numerical State-Based Modeling Case Study: Embedded Fault-Tolerant Multiprocessor System Solution by Simulation Symbolic State-space Exploration and Numerical Analysis of State-sharing Composed Models Case Study: Security Evaluation of a Publish and Subscribe System The Art of System Trust Evaluation /Conclusions ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**What is Validation? Definition:**

Valid (Webster’s Third New International Dictionary) “Able to effect or accomplish what is designed or intended” Two basic notions: Specification - A description of what a system is supposed to do. Realization - A description of what a system is and does. Definition (for class): Validation - the process of determining whether a realization meets its specification. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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What’s a System? Many things, but in the context of this session, an embedded system consisting of hardware networks operating systems, and application software that is intended to be dependable, secure, survivable or have predictable performance. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**What is Validated? -- Dependability**

Dependability is the ability of a system to deliver a specified service. System service is classified as proper if it is delivered as specified; otherwise it is improper. System failure is a transition from proper to improper service. System restoration is a transition from improper to proper service. The “properness” of service depends on the user’s viewpoint! Reference: J.C. Laprie (ed.), Dependability: Basic Concepts and Terminology, Springer-Verlag, 1992. failure proper service improper service restoration ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Basic Validation Terms**

Measures -- What you want to know about a system. Used to determine if a realization meets a specification Models -- Abstraction of the system at an appropriate level of abstraction and/or details to determine the desired measures about a realization. Dependability Model Solution Methods -- Method by which one determines measures from a model. Models can be solved by a variety of techniques: Combinatorial Methods -- Structure of the model is used to obtain a simple arithmetic solution. Analytical/Numerical Methods -- A system of linear differential equations or linear equations is constructed, which is solved to obtain the desired measures Simulation -- The realization of the system is executed, and estimates of the measures are calculated based on the resulting executions (known also as sample paths or trajectories.) Möbius supports performance/reliability/availability validation by analytical/numerical and simulation-based methods. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Dependability Measures: Availability**

Availability - quantifies the alternation between deliveries of proper and improper service. A(t) is 1 if service is proper at time t, 0 otherwise. E[A(t)] (Expected value of A(t)) is the probability that service is proper at time t. A(0,t) is the fraction of time the system delivers proper service during [0,t]. E[A(0,t)] is the expected fraction of time service is proper during [0,t]. P[A(0,t) > t*] (0 t* 1) is the probability that service is proper more than 100t*% of the time during [0,t]. A(0,t)t is the fraction of time that service is proper in steady state. E[A(0,t)t], P[A(0,t)t > t*] as above. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Other Dependability Measures**

Reliability - a measure of the continuous delivery of service R(t) is the probability that a system delivers proper service throughout [0,t]. Safety - a measure of the time to catastrophic failure S(t) is the probability that no catastrophic failures occur during [0,t]. Analogous to reliability, but concerned with catastrophic failures. Time to Failure - measure of the time to failure from last restoration. (Expected value of this measure is referred to as MTTF - Mean time to failure.) Maintainability - measure of the time to restoration from last experienced failure. (Expected value of this measure is referred to as MTTR - Mean time to repair.) Coverage - the probability that, given a fault, the system can tolerate the fault and continue to deliver proper service. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Non-State-space-based**

How is Validation Done? Möbius supports model-based validation of italicized (red) items. Validation Measurement Modeling Passive (no fault injection) Active (Fault Injection on Prototype) Analysis/ Numerical Simulation Without Contact With Contact Deterministic Non-Deterministic Discrete Event (state) Continuous State Hardware- Implemented Software- Implemented Probabilistic Non-Probabilistic Sequential Parallel Stand-alone Systems Networks/ Distributed Systems Non-State-space-based (Combinatorial) State-space-based ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Integrated Validation Procedure**

Requirement Decomposition S P Q Functional Model of the System (Probabilistic or Logical) Functional Model of the Relevant Subset of the System ModuleA ModuleB … ModuleZ Assumptions AA1 AA2 AA3 AP1 AP2 Low-level mechanisms can support assumptions directly, or can support middle-level mechanisms. Supporting Logical Arguments and Experimentation M1 M2 M3 M4 M5 M6 L1 L2 L3 ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Model Solution Issues (Many More Details will Follow)**

In general: Use “tricks” from probability theory to reduce complexity of model Choose the right solution method Simulation: Result is just an estimator based on a statistical experiment Estimation of accuracy of estimate essential Use confidence Intervals! Analytic/Numerical model solution: Avoid state space explosion Limit model complexity Use structure of model (symmetries) to reduce state space size Understand accuracy/limitations of chose numerical method Transient Solution (Iterative or Direct) Steady-state solution ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Probability Review: Exponential Random Variables**

An exponential random variable X with parameter l has the CDF P[X t] = Fx(t) = The density function is given by fx(t) = The exponential random variable is the only continuous random variable that is “memoryless.” To see this, let X be an exponential random variable representing the time that an event occurs (e.g., a fault arrival). Important Fact 1: (memoryless property)! { 0 t 0 1-e-lt t > 0 . { 0 t 0 le-lt t > 0 ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Probability Review: Exponential Event Rate**

The fact that the exponential random variable has the memoryless property indicates that the “rate” at which events occur is constant, i.e., it does not change over time. Often, the event associated with a random variable X is a failure, so the “event rate” is often called the failure rate or the hazard rate. The event rate of X is defined as the probability that the event associated with X occurs within the small interval [t, t + Dt], given that the event has not occurred by time t, per the interval size Dt: This can be thought of as looking at X at time t, observing that the event has not occurred, and measuring the number of events (probability of the event) that occur per unit of time at time t. Important Fact 2: The exponential random variable has a constant failure rate! ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Probability Review: Minimum of Two Independent Exponentials**

Another interesting property of exponential random variables is that the minimum of two independent exponential random variables is also an exponential random variable. Let A and B be independent exponential random variables with rates a and b respectively. Let us define X = min{A,B}. What is FX(t)? FX(t) = P[X t] = P[min{A,B} t] = P[A t OR B t] = 1 - P[A > t AND B > t] = 1 - P[A > t] P[B > t] = 1 - (1 - P[A t])(1 - P[B t]) = 1 - (1 - FA(t))(1 - FB(t)) = 1 - (1 - [1 - e-at])(1 - [1 - e-bt]) = 1 - e-ate-bt = 1 - e-(a + b)t Important Fact 3: The minimum of two independent exponential random variables is itself exponential with rate the sum of the two rates! ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Probability Review: Competition of Two Independent Exponentials**

If A and B are independent and exponential with rate a and b respectively, and A and B are competing, then we know that one will “win” with an exponentially distributed time (with rate a + b). But what is the probability that A wins? Important Fact 4: If A and B are independent, competing exponentials, with rates a and b respectively, the probability that A occurs before B is a/(a + b)! ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Combinatorial Modeling Stochastic Activity Network Concepts **

Course Outline Issues in Model-Based Validation of High-Availability Computer Systems/Networks Combinatorial Modeling Stochastic Activity Network Concepts Analytic/Numerical State-Based Modeling Case Study: Embedded Fault-Tolerant Multiprocessor System Solution by Simulation Symbolic State-space Exploration and Numerical Analysis of State-sharing Composed Models Case Study: Security Evaluation of a Publish and Subscribe System The Art of System Trust Evaluation /Conclusions ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Combinatorial Methods**

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Introduction to Combinatorial Methods**

Combinatorial validation methods are the simplest kind of analytical/numerical techniques and can be used for reliability and availability modeling under certain assumptions. Assumptions are that component failures are independent, and for availability, repairs are independent. When these assumptions hold, simple formulas for reliability and availability exist. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Review definition of reliability Failure rate System reliability **

Lecture Outline Review definition of reliability Failure rate System reliability Maximum Minimum k of N Reliability formalisms Reliability block diagrams Fault trees Reliability graphs Reliability modeling process ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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Reliability One key to building highly available systems is the use of reliable components and systems. Reliability: The reliability of a system at time t (R(t)) is the probability that the system operation is proper throughout the interval [0,t]. Probability theory and combinatorics can be directly applied to reliability models. Let X be a random variable representing the time to failure of a component. The reliability of the component at time t is given by RX(t) = P[X > t] = 1 - P[X t] = 1 - FX(t). Similarly, we can define unreliability at time t by UX(t) = P[X t] = FX(t). ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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Failure Rate What is the rate that a component fails at time t? This is the probability that a component that has not yet failed fails in the interval (t, t + Dt), as Dt 0. Note that we are not looking at P[X (t, t + Dt)] = fX(t). Rather, we are seeking P[X (t, t + Dt)| X > t]. rX(t) is called the failure rate or hazard rate. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Typical Failure Rate Break in Normal operation Wear out rX(t) time**

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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System Reliability While FX can give the reliability of a component, how do you compute the reliability of a system? System failure can occur when one, all, or some of the components fail. If one makes the independent failure assumption, system failure can be computed quite simply. The independent failure assumption states that all component failures of a system are independent, i.e., the failure of one component does not cause another component to be more or less likely to fail. Given this assumption, one can determine: 1) Minimum failure time of a set of components 2) Maximum failure time of a set of components 3) Probability that k of N components have failed at a particular time t. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Maximum of n Independent Failure Times**

Let X1, , Xn be independent component failure times. Suppose the system fails at time S if all the components fail. Thus, S = max{X1, , Xn} What is Fs(t)? Fs(t) = P[S t] = P[X1 t AND X2 t AND AND Xn t] = P[X1 t] P[X2 t] P[Xn t] By independence = By definition = ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Minimum of n Independent Component Failure Times**

Let X1, , Xn be independent component failure times. A system fails at time S if any of the components fail. Thus, S = min{X1, , Xn}. What is FS(t)? FS(t) = P[S t] = P[X1 t OR X2 t OR OR Xn t] This is an application of the law of total probability (LOTP). A1 A2 A3 ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Minimum cont. Fs(t) = P[X1 t OR X2 t OR . . . OR Xn t]**

= 1 - P[X1 > t AND X2 > t AND AND Xn > t] By trick = 1 - P[X1 > t] P[X2 > t] P[Xn > t] By independence = 1 - (1 - P[X1 t])(1 - P[X2 t]) (1 - P[Xn t]) By LOTP = ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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k of N Let X1, , Xn be component failure times that have identical distributions (i.e., = ). The system fails at time S if k of the N components fail. FS(t) = P[at least k components failed by time t] = P[k failed OR k + 1 failed OR OR N failed] = P[k failed] + P[k + 1 failed] P[N failed] What is P[exactly k failed]? = P[k failed and (N - k) have not] = where FX(t) is the failure distribution of each component. Thus, - by independence and axiom of probability. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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k of N in General For non-identical failure distributions, we must sum over all combinations of at least k failures. Let Gk be the set of all subsets of {X1, , XN} such that each element in Gk is a set of size at least k, i.e., Gk = {gi {X1, , XN} : |gi| k}. The set Gk represents all the possible failure scenarios. Now FS is given by ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Component Building Blocks**

Complex systems can be analyzed hierarchically. Example: A computer fails if both power supplies fail or both memories fail or the CPU fails. FS(t) = 1 - (1 - FP1(t)FP2(t))(1- FM1(t)FM2(t))(1 - FC(t)) ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**identical distributions**

Summary A system comprises N components, where the component failure times are given by the random variables X1, , XN. The system fails at time S with distribution FS if: Condition: all components fail one component fails k components fail, identical distributions general case Distribution: ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Reliability Formalisms**

There are several popular graphical formalisms to express system reliability. The core of the solvers is the methods we have just examined. In particular, we will examine Reliability Block Diagrams Fault Trees Reliability Graphs There is nothing particularly special about these formalisms except their popularity. It is easy to implement these formalisms, or design your own, in a spreadsheet, for example. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Reliability Block Diagrams**

Blocks represent components. A system failure occurs if there is no path from source to sink. Series: System fails if any component fails. Parallel: System fails if all components fail. k of N: System fails if at least k of N components fail. C1 C2 C3 source sink C1 C2 C3 source sink C1 C2 C3 source sink 2 of 3 ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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Example A NASA satellite architecture under study is designed for high reliability. The major computer system components include the CPU system, the high-speed network for data collection and transmission, and the low-speed network for engineering and control. The satellite fails if any of the major systems fail. There are 3 computers, and the computer system fails if 2 or more of the computers fail. Failure distribution of a computer is given by FC. There is a redundant (2) high-speed network, and the high-speed network system fails if both networks fail. The distribution of a high-speed network failure is given by FH. The low-speed network is arranged similarly, with a failure distribution of FL. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**RBG Example HSN LSN source sink HSN LSN 2 of 3 computer computer**

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Fault Trees Components are leaves in the tree**

A component fails = logical value of true, otherwise false. The nodes in the tree are boolean AND, OR, and k of N gates. The system fails if the root is true. AND gates true if all the components are true (fail). OR gates true if any of the components are true (fail). k of N gates true if at least k of the components are true (fail). AND C1 C2 C3 OR C1 C2 C3 C1 C3 C2 2 of 3 ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Fault Tree Example OR AND AND 2 of 3 C1 C2 C3 H1 H2 L1 L2**

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Combinatorial Methods: Review**

A system comprises N components, where the component failure times are given by the random variables X1, , XN. The system fails at time S with distribution FS if: Condition: all components fail one component fails k components fail, identical distributions general case Distribution: ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Reliability Formalisms**

There are several popular graphical formalisms to express system reliability. The core of the solvers is the methods we have just examined. In particular, we will examine Reliability Block Diagrams Fault Trees Reliability Graphs There is nothing particularly special about these formalisms except their popularity. It is easy to implement these formalisms, or design your own, in a spreadsheet, for example. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Reliability Block Diagrams**

Blocks represent components. A system failure occurs if there is no path from source to sink. Series: System fails if any component fails. Parallel: System fails if all components fail. k of N: System fails if at least k of N components fail. C1 C2 C3 source sink C1 C2 C3 source sink C1 C2 C3 source sink 2 of 3 ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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Example A NASA satellite architecture under study is designed for high reliability. The major computer system components include the CPU system, the high-speed network for data collection and transmission, and the low-speed network for engineering and control. The satellite fails if any of the major systems fail. There are 3 computers, and the computer system fails if 2 or more of the computers fail. Failure distribution of a computer is given by FC. There is a redundant (2) high-speed network, and the high-speed network system fails if both networks fail. The distribution of a high-speed network failure is given by FH. The low-speed network is arranged similarly, with a failure distribution of FL. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**RBG Example HSN LSN source sink HSN LSN 2 of 3 computer computer**

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Fault Trees Components are leaves in the tree**

A component fails = logical value of true, otherwise false. The nodes in the tree are boolean AND, OR, and k of N gates. The system fails if the root is true. AND gates true if all the components are true (fail). OR gates true if any of the components are true (fail). k of N gates true if at least k of the components are true (fail). AND C1 C2 C3 OR C1 C2 C3 C1 C3 C2 2 of 3 ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Fault Tree Example OR AND AND 2 of 3 C1 C2 C3 H1 H2 L1 L2**

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Reliability Graphs FC1 FC2 1 2 3 FC1 FC2 1 2 FC3**

The arcs represent components and have failure distributions. A failure occurs if there is no path from source to sink. Can implement series: Can implement parallel: FC1 FC2 1 2 3 source sink FC1 FC2 1 2 source sink FC3 ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Reliability Graph Example**

Reliability graphs can implement more complex interactions. For example, a telephone network “fails” if there is no path from source to sink. How do we solve this? 2 A D 1 4 C source sink B E 3 ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Solving by Conditioning**

E F ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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2 A D 1 C 4 source B sink E 3 First, condition the system on link C being failed. Then the system becomes the series AD in parallel with the series BE. 2 A D 1 4 source B sink E 3 ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**1 2,3 4 Second, condition the system on link C being up. A D B E**

source sink B E ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Conditioning Fault Trees**

It is also possible to use conditioning to solve more complex fault trees. If the same component appears more than once in a fault tree, it violates the independent failure assumption. However, a conditioned fault tree can be solved. Example: A component C appears multiple times in the fault tree. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Reliability/Availability Point Estimates**

Frequently, the desired measure of a reliability model is the reliability at some time t. Thus, the distribution of the system reliability is superfluous; R(t) is the only thing of interest. This condition simplifies computation because all that is necessary for solution is the reliability of the components at time t. Solution then becomes a straightforward computation. If a system is described in terms of the availability of components at time t, then we may compute the system availability in the same way that reliability is computed. The restriction is that all component behaviors must be independent of one another. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Reliability/Availability Tables**

A system comprises N components. Reliability of component i at time t is given by RXi(t), and the availability of component i at time t is given by AXi(t). Condition System Reliability System Availability system fails if all components fail system fails if one component fails system fails if at least k components fail, identical distribution system fails if at least k components fail, general case ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Estimating Component Reliability**

For hardware, MIL-HDBK-217 is widely used. Not always current with modern components. Lacks distributions; it only contains failure rates. While not perfect, it seems to be the best source that exists. However, numbers from MIL-HDBK-217 should be used with caution. Due to the nature of software, no accepted mechanism exists to predict software reliability before the software is built. Best guess is the reliability of previously built similar software. In all cases, numbers should be used with caution and adjusted based on observation and experience. No substitute for empirical observation and experience! ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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Modeling Process Reliability models are built only after proper service is specified. Reliability models are built to answer the question “What subsystem or components must be proper for the system to be proper?” Build models hierarchically out of subsystems. Estimation and guesses are acceptable, but state them explicitly. If unsure, do sensitivity analysis to see how much it matters. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Reliability Modeling Process**

Realistic systems result in large RBDs and must be managed hierarchically. RBD Process(system) Define the system Define “proper service” Create RBD out of components for each component if component is simple obtain reliability data of component else Do RBD Process(component) end if Compute reliability of system Do results meet specification? Modify design and repeat as necessary ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Reliability: review of definition Failure rate System reliability **

Summary Reliability: review of definition Failure rate System reliability Independent failure assumption Minimum, maximum, k of N Reliability block diagrams, fault trees, reliability graphs Reliability modeling process ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Stochastic Activity Network Concepts**

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Session Outline Stochastic Petri nets**

Places, tokens, input / output arcs, transitions Readers / Writers example Stochastic activity networks Input / output gates, cases, instantaneous and timed activities Marking dependent behavior, well-specified, general distributions Simple database server model Reward variables Reward structures Reward variable classification Predicate / function implementation in Möbius Fault-tolerant computer example Composed models Fault-tolerant computer revisited ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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Introduction Stochastic activity networks, or SANs, are a convenient, graphical, high-level language for describing system behavior. SANs are useful in capturing the stochastic (or random) behavior of a system. Examples: The amount of time a program takes to execute can be computed precisely if all factors are known, but this is nearly impossible and sometimes useless. At a more abstract level, we can approximate the running time by a random variable. Fault arrivals almost always must be modeled by a random process. We begin by describing a subset of SANs: stochastic Petri nets. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Stochastic Petri Net Review**

One of the simplest high-level modeling formalisms is called stochastic Petri nets. A stochastic Petri net is composed of the following components: Places: which contain tokens, and are like variables tokens: which are the “value” or “state” of a place transitions: which change the number of tokens in places input arcs: which connect places to transitions output arcs: which connect transitions to places ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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Firing Rules for SPNs A stochastic Petri net (SPN) executes according to the following rules: A transition is said to be enabled if for each place connected by input arcs, the number of tokens in the place is the number of input arcs connecting the place and the transition. Example: Transition t1 is enabled. P1 P2 t1 ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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Firing Rules, cont. A transition may fire if it is enabled. (More about this later.) If a transition fires, for each input arc, a token is removed from the corresponding place, and for each output arc, a token is added to the corresponding place. Example: Note: tokens are not necessarily conserved when a transition fires. P1 P3 t1 fires t1 P2 P4 ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Specification of Stochastic Behavior of an SPN**

A stochastic Petri net is made from a Petri net by Assigning an exponentially distributed time to all transitions. Time represents the “delay” between enabling and firing of a transition. Transitions “execute” in parallel with independent delay distributions. Since the minimum of multiple independent exponentials is itself exponential, time between transition firings is exponential. If a transition t becomes enabled, and before t fires, some other transition fires and changes the state of the SPN such that t is no longer enabled, then t aborts, that is, t will not fire. Since the exponential distribution is memoryless, one can say that transitions that remain enabled continue or restart, as is convenient, without changing the behavior of the network. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**SPN Example: Readers/Writers Problem**

There are at most N requests in the system at a time. Read requests arrive at rate lra, and write requests at rate lwa. Any number of readers may read from a file at a time, but only one writer may write at a time. A reader and writer may not access the file at the same time. Locks are obtained with rate lL (for both read and write locks); reads and writes are performed at rates lr and lw respectively. Locks are released at rate lrel. Note: N (N arcs) . . . ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**SPN Representation of Reader/Writers Problem**

lra lL lr lrel N N N N lwa lL lw lrel ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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Notes on SPNs SPNs are much easier to read, write, modify, and debug than Markov chains. SPN to Markov chain conversion can be automated to afford numerical solutions to Markov chains. Most SPN formalisms include a special type of arc called an inhibitor arc, which enables the SPN if there are zero tokens in the associated place, and the identity (do nothing) function. Example: modify SPN to give writes priority. Limited in their expressive power: may only perform +, -, >, and test-for-zero operations. These very limited operations make it very difficult to model complex interactions. Simplicity allows for certain analysis, e.g., a network protocol modeled by an SPN may detect deadlock (if inhibitor arcs are not used). More general and flexible formalisms are needed to represent real systems. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Stochastic Activity Networks**

The need for more expressive modeling languages has led to several extensions to stochastic Petri nets. One extension that we will examine is called stochastic activity networks. Because there are a number of subtle distinctions relative to SPNs, stochastic activity networks use different words to describe ideas similar to those of SPNs. Stochastic activity networks have the following properties: A general way to specify that an activity (transition) is enabled A general way to specify a completion (firing) rule A way to represent zero-timed events A way to represent probabilistic choices upon activity completion State-dependent parameter values General delay distributions on activities ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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SAN Symbols Stochastic activity networks (hereafter SANs) have four new symbols in addition to those of SPNs: Input gate: used to define complex enabling predicates and completion functions Output gate: used to define complex completion functions Cases: (small circles on activities) used to specify probabilistic choices Instantaneous activities: used to specify zero-timed events ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**SAN Enabling Rules An input gate has two components:**

enabling_function (state) boolean; also called the enabling predicate input_function(state) state; rule for changing the state of the model An activity is enabled if for every connected input gate, the enabling predicate is true, and for each input arc, the number of tokens in the connected place number of arcs. We use the notation MARK(P) to denote the number of tokens in place P. Note that in Mobius, the this would be written as P-> MARK ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Example SAN Enabling Rule**

IG1 Predicate: if((MARK(P1)>0 && MARK(P2)==0)|| (MARK(P1)==0 && MARK(P2)>0)) return 1; else return 0; Activity a1 is enabled if IG1 predicate is true (1) and MARK(P3) > 0. (Note that in Möbius, “1” is used to denote true.) ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

70
Cases Cases represent a probabilistic choice of an action to take when an activity completes. When activity a completes, a token is removed from place P1, and with probability a, a token is put into place P2, and with probability 1 - a, a token is put into place P3. Note: cases are numbered, starting with 1, from top to bottom. a 1-a ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

71
Output Gates When an activity completes, an output gate allows for a more general change in the state of the system. This output gate function is usually expressed using pseudo-C code. Example OG Function MARK(P) = 0; 1 - c c ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

72
**Instantaneous Activities**

Another important feature of SANs is the instantaneous activity. An instantaneous activity is like a normal activity except that it completes in zero time after it becomes enabled. Instantaneous activities can be used with input gates, output gates, and cases. Instantaneous activities are useful when modeling events that have an effect on the state of the system, but happen in negligible time, with respect to other activities in the system, and the performance/dependability measures of interest. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

73
**SAN Terms 1. activation - time at which an activity begins**

2. completion - time at which activity completes 3. abort - time, after activation but before completion, when activity is no longer enabled 4. active - the time after an activity has been activated but before it completes or aborts. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

74
**Illustration of SAN Terms**

activity time enabled activation completion t enabled activation aborted activity time t activity time enabled activation completion and activation ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

75
Completion Rules When an activity completes, the following events take place (in the order listed), possibly changing the marking of the network: 1. If the activity has cases, a case is (probabilistically) chosen. 2. The functions of all the connected input gates are executed (in an unspecified order). 3. Tokens are removed from places connected by input arcs. 4. The functions of all the output gates connected to the chosen case are executed (in an unspecified order). 5. Tokens are added to places connected by output arcs connected to the chosen case. Ordering is important, since effect of actions can be marking-dependent. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

76
**Marking Dependent Behavior**

Virtually every parameter may be any function of the state of the model. Examples of these are rates of exponential activities parameters of other activity distributions case probabilities An example of this usefulness is a model of three redundant computers where the coverage (probability that a single computer crashing crashes the whole system) increases after a failure. 1 - c c ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

77
Example Problem A database server is composed of a compute server and three file servers, and can queue up to Nc requests at a time (including the one in service). Requests arrive at rate la and spend on average 1/lCPU time at the compute server being processed. The request is then forwarded to the file server that has the fewest outstanding requests. Requests are processed at a rate of lD1, lD2, and lD3 for file servers D1, D2, and D3 respectively. File server buffers may hold at most Nf requests (including requests in service); if all buffers are full, the request is discarded. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

78
**SAN Representation of Example Database Problem**

lD1 lD2 lD3 ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

79
Gate Functions for SAN ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

80
**General Delay Distributions**

SANs (and their implementation in Möbius) support many activity time distributions, including: All distribution parameters can be marking-dependent The obvious implication of general delay distributions is that there is no conversion to a CTMC. Hence, no solutions to CTMCs are applicable. However, simulation is still possible. Analytical/numerical solution is possible for certain mixes of exponential and deterministic activities. See the Möbius manual for details. See [Kececioglu 91], for example, for appropriate use of some of these distributions. Exponential Hyperexponential Deterministic Weibull Conditional Weibull Normal Erlang Gamma Beta Uniform Binomial Negative Binomial ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

81
**Fault-Tolerant Computer Failure Model Example**

A fault-tolerant computer system is made up of two redundant computers. Each computer is composed of three redundant CPU boards. A computer is operational if at least 1 CPU board is operational, and the system is operational if at least 1 computer is operational. CPU boards fail at a rate of 1/106 hours, and there is a 0.5% chance that a board failure will cause a computer failure, and a 0.8% chance that a board will fail in a way that causes a catastrophic system failure. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

82
**SAN computer for Computer Failure Model**

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

83
**Activity Case Probabilities and Input Gate Definition**

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

84
**Output Gate Definitions**

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

85
Reward Variables Reward variables are a way of measuring performance- or dependability-related characteristics about a model. Examples: Expected time until service System availability Number of misrouted packets in an interval of time Processor utilization Length of downtime Operational cost Module or system reliability ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

86
**Reward Structures Reward may be “accumulated” two different ways:**

A model may be in a certain state or states for some period of time, for example, “CPU idle” states. This is called a rate reward. An activity may complete. This is called an impulse reward. The reward variable is the sum of the rate reward and the impulse reward structures. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

87
**Reward Structure Example**

A web server failure model is used to predict profits. When the web server is fully operational, profits accumulate at $N/hour. In a degraded mode, profits accumulate at Repairs cost $K. m is a fully functioning marking m is a degraded-mode marking otherwise a is an activity representing repair otherwise By carefully integrating the reward structure from 0 to t, we get the profit at time t. This is an example of an “interval-of-time” variable. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

88
Reward Variables A reward variable is the sum of the impulse and rate reward structures over a certain time. Let [t, t + l] be the interval of time defined for a reward variable: If l is 0, then the reward variable is called an instant-of-time reward variable. If l > 0, then the reward variable is called an interval-of-time reward variable. If l > 0, then dividing an interval-of-time reward variable by l gives a time-averaged interval-of-time reward variable. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

89
**Reward Variable Specification**

Reward Structure Interval-of-Time Instant-of-Time Time-Average Interval-of-Time [t, t + l] [t, t + l] [t, t + l] lim as l goes to infinity t lim as t goes to infinity lim as t goes to infinity [t, t + l] [t, t + l] [t, t + l] lim as t goes to infinity lim as l goes to infinity ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

90
**Reward Variables are Random Variables**

Note that since the behavior of a SAN is a stochastic process, then a reward variable is a measure defined on the stochastic process, and therefore a reward variable is a random variable. A tool can solve for the reward variables, but solving for the distribution in many cases can be difficult. It is often much simpler to solve for the mean or variance of the reward variable, especially when using numerical techniques. Example reward variables: A(0,t) - Fraction of time the system delivers proper service during [0,t]. Hard to compute. E[A(0,t)] - Expected value of A(0,t). Easier to compute. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

91
**Specifying Reward Variables in Möbius**

When specifying a rate portion of a reward structure in Möbius, you must define a predicate and function. predicate: while true (i.e., integer greater than 0 in C), accumulate the reward function: the value (i.e., double in C) to accumulate Note that both the predicate and function may be any C statement or expression. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

92
**Reward Variables for Computer Failure Model**

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

93
**Reward Variables for Computer Failure Model**

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

94
Model Composition A composed model is a way of connecting different SANs together to form a larger model. Model composition has two operations: Replicate: Combine 2 or more identical SANs and reward structures together, holding certain places common among the replicas. Join: Combine 2 or more different SANs and reward structures together, combining certain places to permit communication. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

95
**Composed Model Specification**

Join two or more submodels together Certain places in different submodels can be made common Replicate submodel a certain number of times Hold certain places common to all replicas ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

96
**Rationale There are many good reasons for using composed models.**

Building highly reliable systems usually involves redundancy. The replicate operation models redundancy in a natural way. Systems are usually built in a modular way. Replicates and Joins are usually good for connecting together similar and different modules. Tools can take advantage of something called the Strong Lumping Theorem that allows a tool to generate a Markov process with a smaller state space (to be described in Session 7). ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

97
**Rules for Building Composed Models**

(as Implemented in Möbius) Places that are joined together must have the same name and initial marking. Places that are common at a certain level of the tree must be common at all lower levels. Places that are common cannot be connected to the input side of an instantaneous activity. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

98
**Computer Failure Model Revisited: Single computer Model**

(Note initial marking of NumComp is two since there will be two computers in the composed model.) ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

99
**Composed Model for Computer Failure Model**

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

100
**Reward Variables for Composed Model**

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

101
**Reward Variables for Composed Model**

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

102
Composed Model How does adding an additional computer affect reliability? In the composed model, change number of replications to 3 and change various reward variables - easy (Use a global variable if you think suspect you may want to do this.) In “flat” model, add another computer - hard In composed model, the number of states in the underlying Markov chain is much smaller, especially for large numbers of replications. (Details will be given in Session 7.) ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

103
**Analytic/Numerical State-Based Modeling**

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

104
**Session Outline Review of Markov process theory and fundamentals**

Methods for constructing state-level models from SANs Analytic/numerical solution techniques Transient solution Standard uniformization (instant-of-time variables) Adaptive uniformization (instant-of-time variables) Interval-of-time uniformization (interval-of-time variables) Steady-state solution (steady-state instant-of-time variables) Direct solution Iterative solution ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

105
**Weaknesses of Simulation**

Simulation relies on good pseudo-random number generation, sufficient observations, and good statistical techniques to produce an approximate solution Increasing accuracy by a factor of n requires on the order of n2 more work, which can be prohibitively expensive. For example, a 5-Nines system reliability model will require approximately 100,000 observations to observe one failure. One digit of accuracy can easily require over 1,000,000 observations! (For many models, 1,000,000 observations can be generated quickly, but as system failure becomes even rarer, standard simulation quickly becomes infeasible.) ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

106
**The Case for Analytical/Numerical Techniques**

If you can model using exponential delays and your model is sufficiently small, continuous time Markov chains (CTMCs) offer some advantages. These include: Typically faster solution time for systems with rare events Typically takes less time to get more accurate answers Typically more confidence in the solution In order to understand when we get these advantages, we must better understand the methods of obtaining solutions to CTMCs. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

107
**Random Variable Review**

It is often convenient to assign a (real) number to every element in . This assignment, or rule, or function, is called a random variable. X(w) w -1 1 X : W ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

108
Random Process Review Random processes are useful for characterizing the behavior of real systems. A random process is a collection of random variables indexed by time. Example: X(t) is a random process. Let X(1) be the result of tossing a die. Let X(2) be the result of tossing a die plus X(1), and so on. Notice that time (T) = {1,2,3, . . .}. One can ask: ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

109
**Random Process Review, cont.**

If X is a random process, X(t) is a random variable. Remember that a random variable Y is a function that maps elements in W to elements in . Therefore, a random process X maps elements in the two-dimensional space W T to elements in . When we fix t, then X becomes a function of W to . However, if we fix w, then X becomes a function of T to . By fixing w and observing X as a function of T, we are observing a sample path of X. This is extremely useful for simulation. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

110
**Describing a Random Process**

Recall that for a random variable X, we can use the cumulative distribution FX to describe the random variable. In general, no such simple description exists for a random process. However, a random process can often be described succinctly in various different ways. For example, if Y is a random variable representing the roll of a die, and X(t) is the sum after t rolls, then we can describe X(t) by X(t) - X(t - 1) = Y, P[X(t) = i|X(t - 1) = j] = P[Y = i - j], or X(t) = Y1 + Y Yt, where the Yi’s are independent. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

111
**Classifying Random Processes: Characteristics of T**

If the number of time points defined for a random process, i.e., |T|, is finite or countable (e.g., integers), then the random process is said to be a discrete-time random process. If |T| is uncountable (e.g., real numbers) then the random process is said to be a continuous-time random process. Example: Let X(t) be the number of fault arrivals in a system up to time t. Since t T is a real number, X(t) is a continuous-time random process. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

112
**Classifying Random Processes: State Space Type**

Let X be a random process. The state space of a random process is the set of all possible values that the process can take on, i.e., S = {y: X(t) = y, for some t T}. If X is a random process that models a system, then the state space of X can represent the set of all possible configurations that the system could be in. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

113
**Random Process State Spaces**

If the state space S of a random process X is finite or countable (e.g., S = {1,2,3, . . .}), then X is said to be a discrete-state random process. Example: Let X be a random process that represents the number of bad packets received over a network. X is a discrete-state random process. If the state space S of a random process X is infinite and uncountable (e.g., S = ), then X is said to be a continuous-state random process. Example: Let X be a random process that represents the voltage on a telephone line. X is a continuous-state random process. We examine only discrete-state processes in this lecture. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

114
**Stochastic-Process Classification Examples**

Time Continuous Discrete State Continuous Discrete ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

115
Markov Process A special type of random process that we will examine in detail is called the Markov process. A Markov process can be informally defined as follows. Given the state (value) of a Markov process X at time t (X(t)), the future behavior of X can be described completely in terms of X(t). Markov processes have the very useful property that their future behavior is independent of past values. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

116
Markov Chains A Markov chain is a Markov process with a discrete state space. We will always make the assumption that a Markov chain has a state space in {1,2, . . .} and that it is time-homogeneous. A Markov chain is time-homogeneous if its future behavior does not depend on what time it is, only on the current state (i.e., the current value). We make this concrete by looking at a discrete-time Markov chain (hereafter DTMC). A DTMC X has the following property: (1) (2) ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

117
**DTMCs Notice that given i, j, and k, is a number!**

can be interpreted as the probability that if X has value i, then after k time-steps, X will have value j. Frequently, we write to mean ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

118
Markov Chains A Markov chain is a Markov process with a discrete state space. We will always make the assumption that a Markov chain has a state space in {1,2, . . .} and that it is time-homogeneous. A Markov chain is time-homogeneous if its future behavior does not depend on what time it is, only on the current state (i.e., the current value). We make this concrete by looking at a discrete-time Markov chain (hereafter DTMC). A DTMC X has the following property: (1) (2) ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

119
**DTMCs Notice that given i, j, and k, is a number!**

can be interpreted as the probability that if X has value i, then after k time-steps, X will have value j. Frequently, we write to mean ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

120
**State Occupancy Probability Vector**

Let p be a row vector. We denote pi to be the i-th element of the vector. If p is a state occupancy probability vector, then pi(k) is the probability that a DTMC has value i (or is in state i) at time-step k. Assume that a DTMC X has a state-space size of n, i.e., S = {1, 2, , n}. We say formally pi(k) = P[X(k) = i] Note that for all times k. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

121
**Computing State Occupancy Vectors: A Single Step Forward in Time**

If we are given p(0) (the initial probability vector), and Pij for i, j = 1, , n, how do we compute p(1)? Recall the definition of Pij. Pij = P[X(k+1) = j | X(k) = i] = P[X(1) = j | X(0) = i] Since ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

122
**Transition Probability Matrix**

Notice that this resembles vector-matrix multiplication. In fact, if we arrange the matrix P = {Pij}, that is, if P = then pij = Pij, and p(1) = p(0)P, where p(0) and p(1) are row vectors, and p(0)P is a vector-matrix multiplication. The important consequence of this is that we can easily specify a DTMC in terms of an occupancy probability vector p and a transition probability matrix P. p1n p11 pn1 pnn , ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

123
**Transient Behavior of Discrete-Time Markov Chains**

Given p(0) and P, how can we compute p(k)? We can generalize from earlier that p(k) = p(k - 1)P. Also, we can write p(k - 1) = p(k - 2)P, and so p(k) = [p(k - 2)P]P = p(k - 2)P2 Similarly, p(k - 2) = p(k - 3)P, and so p(k) = [p(k - 3)P]P2 = p(k - 3)P3 By repeating this, it should be easy to see that p(k) = p(0)Pk ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

124
A Simple Example Suppose the weather at Urbana-Champaign, Illinois can be modeled the following way: If it’s sunny today, there’s a 60% chance of being sunny tomorrow, a 30% chance of being cloudy, and a 10% chance of being rainy. If it’s cloudy today, there’s a 40% chance of being sunny tomorrow, a 45% chance of being cloudy, and a 15% chance of being rainy. If it’s rainy today, there’s a 15% chance of being sunny tomorrow, a 60% chance of being cloudy, and a 25% chance of being rainy. If it’s rainy on Friday, what is the forecast for Monday? ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

125
**Simple Example, cont. Clearly, the weather model is a DTMC.**

1) Future behavior depends on the current state only 2) Discrete time, discrete state 3) Time homogeneous The DTMC has 3 states. Let us assign 1 to sunny, 2 to cloudy, and 3 to rainy. Let time 0 be Friday. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

126
**Simple Example Solution**

The weather on Saturday p(1) is that is, 15% chance sunny, 60% chance cloudy, 25% chance rainy. The weather on Sunday p(2) is The weather on Monday p(3) is p(3) = p(2)P = (.4316, .42, .1484), that is, 43% chance sunny, 42% chance cloudy, and 15% chance rainy. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

127
**Solution, cont. Alternatively, we could compute P3 since we found**

p(3) = p(0)P3. Working out solutions by hand can be tedious and error-prone, especially for “larger” models (i.e., models with many states). Software packages are used extensively for this sort of analysis. Software packages compute p(k) by (. . . ((p(0)P)P)P. . .)P rather than computing Pk, since computing the latter results in a large “fill-in.” ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

128
**Graphical Representation**

It is frequently useful to represent the DTMC as a directed graph. Nodes represent states, and edges are labeled with probabilities. For example, our weather prediction model would look like this: .45 2 .3 .15 1 = Sunny Day 2 = Cloudy Day 3 = Rainy Day .4 .6 .1 .6 1 3 .15 .25 ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

129
**“Simple Computer” Example**

3 1 2 Pidle Pr Pff Pfi Pcom Parr Pfb Pbusy X = 1 computer idle X = 2 computer working X = 3 computer failed ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

130
**Limiting Behavior of DTMCs**

It is sometimes useful to know the time-limiting behavior of a DTMC. This translates into the “long term,” where the system has settled into some steady-state behavior. Formally, we are looking for To compute this, what we want is There are various ways to compute this. The simplest is to calculate p(n) for increasingly large n, and when p(n + 1) p(n), we can believe that p(n) is a good approximation to steady-state. This can be rather inefficient if n needs to be large. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

131
Classifications It is much easier to solve for the steady-state behavior of some DTMC’s than others. To determine if a DTMC is “easy” to solve, we need to introduce some definitions. Definition: A state j is said to be accessible from state i if there exists an n 0 such that We write i j. Note: recall that If one thinks of accessibility in terms of the graphical representation, a state j is accessible from state i if there exists a path of non-zero edges (arcs) from node i to node j. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

132
**State Classification in DTMCs**

Definition: A DTMC is said to be irreducible if every state is accessible from every other state. Formally, a DTMC is irreducible if i j for all i,j S. A DTMC is said to be reducible if it is not irreducible. It turns out that irreducible DTMC’s are simpler to solve. One need only solve one linear equation: p = pP. We will see why this is so, but first there is one more issue we must confront. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

133
**Periodicity 1 2 Consider the following DTMC: 1 1**

However, does exist; it is called the time-averaged steady-state distribution, and is denoted by p*. Definition: A state i is said to be periodic with period d if only when n is some multiple of d. If d = 1, then i is said to be aperiodic. A steady-state solution for an irreducible DTMC exists if all the states are aperiodic. A time-averaged steady-state solution for an irreducible DTMC always exists. 1 1 2 1 ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

134
**Steady-State Solution of DTMCs**

The steady-state behavior can be computed by solving the linear equation p = pP, with the constraint that For irreducible DTMC’s, it can be shown that this solution is unique. If the DTMC is periodic, then this solution yields p*. One can understand the equation p = pP in two different ways. In steady-state, the probability distribution p(n + 1) = p(n)P, and by definition p(n + 1) = p(n) in steady-state. “Flow” equations. Flow equations require some visualization. Imagine a DTMC graph, where the nodes are assigned the occupancy probability, or the probability that the DTMC has the value of the node. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

135
**. . . . . . Flow Equations i Probability must be conserved, i.e.,**

Let piPij be the “probability mass” that moves from state j to state i in one time-step. Since probability must be conserved, the probability mass entering a state must equal the probability mass leaving a state. Prob. mass in = Prob. mass out Written in matrix form, p = pP. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

136
**Continuous Time Markov Chains (CTMCs)**

For most systems of interest, events may occur at any point in time. This leads us to consider continuous time Markov chains. A continuous time Markov chain (CTMC) has the following property: A CTMC is completely described by the initial probability distribution p(0) and the transition probability matrix P(t) = [pij(t)]. Then we can compute p(t) = p(0)P(t). The problem is that pij(t) is generally very difficult to compute. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

137
CTMC Properties This definition of a CTMC is not very useful until we understand some of the properties. First, notice that pij(t) is independent of how long the CTMC has previously been in state i, that is, There is only one random variable that has this property: the exponential random variable. This indicates that CTMCs have something to do with exponential random variables. First, we examine the exponential r.v. in some detail. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

138
**Exponential Random Variables**

Recall the property of the exponential random variable. An exponential random variable X with parameter l has the CDF P[X t] = Fx(t) = The distribution function is given by fx(t) = The exponential random variable is the only random variable that is “memoryless.” To see this, let X be an exponential random variable representing the time that an event occurs (e.g., a fault arrival). We will show that { 0 t 0 1-e-lt t > 0 . { 0 t 0 le-lt t > 0 ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

139
**Memoryless Property Proof of the memoryless property:**

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

140
Event Rate The fact that the exponential random variable has the memoryless property indicates that the “rate” at which events occur is constant, i.e., it does not change over time. Often, the event associated with a random variable X is a failure, so the “event rate” is often called the failure rate or the hazard rate. The event rate of X is defined as the probability that the event associated with X occurs within the small interval [t, t + Dt], given that the event has not occurred by time t, per the interval size Dt: This can be thought of as looking at X at time t, observing that the event has not occurred, and measuring the number of events (probability of the event) that occur per unit of time at time t. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

141
**In the exponential case,**

Observe that: In the exponential case, This is why we often say a random variable X is “exponential with rate l.” ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

142
**Minimum of Two Independent Exponentials**

Another interesting property of exponential random variables is that the minimum of two independent exponential random variables is also an exponential random variable. Let A and B be independent exponential random variables with rates a and b respectively. Let us define X = min{A,B}. What is FX(t)? FX(t) = P[X t] = P[min{A,B} t] = P[A t OR B t] = 1 - P[A > t AND B > t] see comb. methods section = 1 - P[A > t] P[B > t] = 1 - (1 - P[A t])(1 - P[B t]) = 1 - (1 - FA(t))(1 - FB(t)) = 1 - (1 - [1 - e-at])(1 - [1 - e-bt]) = 1 - e-ate-bt = 1 - e-(a + b)t Thus, X is exponential with rate a + b. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

143
**Competition of Two Independent Exponentials**

If A and B are independent and exponential with rate a and b respectively, and A and B are competing, then we know that one will “win” with an exponentially distributed time (with rate a + b). But what is the probability that A wins? ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

144
**Competing Exponentials in CTMCs**

1 a b X(0) = 1 P[X(0) = 1] = 1 2 3 Imagine a random process X with state space S = {1,2,3}. X(0) = 1. X goes to state 2 (takes on a value of 2) with an exponentially distributed time with parameter a. Independently, X goes to state 3 with an exponentially distributed time with parameter b. These state transitions are like competing random variables. We say that from state 1, X goes to state 2 with rate a and to state 3 with rate b. X remains in state 1 for an exponentially distributed time with rate a + b. This is called the holding time in state 1. Thus, the expected holding time in state 1 is The probability that X goes to state 2 is The probability X goes to state 3 is This is a simple continuous-time Markov chain. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

145
**Competing Exponentials vs. a Single Exponential With Choice**

Consider the following two scenarios: 1. Event A will occur after an exponentially distributed time with rate a. Event B will occur after an independent exponential time with rate b. 2. After waiting an exponential time with rate a + b, event A occurs with probability and event B occurs with probability These two scenarios are indistinguishable. In fact, we frequently interchange the two scenarios rather freely when analyzing a system modeled as a CTMC. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

146
**State-Transition-Rate Matrix**

A CTMC can be completely described by an initial distribution p(0) and a state-transition-rate matrix. A state-transition-rate matrix Q = [qij] is defined as follows: qij = Example: A computer is idle, working, or failed. When the computer is idle, jobs arrive with rate a, and they are completed with rate b. When the computer is working, it fails with rate lw, and with rate li when it is idle. rate of going from i j, state i to state j i = j. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

147
**“Simple Computer” CTMC**

a 1 2 b li lw 3 Let X = 1 represent “the system is idle,” X = 2 “the system is working,” and X = 3 a failure. If the computer is repaired with rate m, the new CTMC looks like a 1 2 b li lw m 3 ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

148
**Analysis of “Simple Computer” Model**

Some questions that this model can be used to answer: What is the availability at time t? What is the steady-state availability? What is the expected time to failure? What is the expected number of jobs lost due to failure in [0,t]? What is the expected number of jobs served before failure? What is the throughput of the system (jobs per unit time), taking into account failures and repairs? ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

149
**State-Space Generation from SANs**

If the activity delays are exponential, it is straightforward to convert a SAN to a CTMC. We first look at the simple case, where there is no composed model. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

150
**State Space (Generated by Möbius)**

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

151
**Underlying Markov Model (State Transition Rates Not Shown)**

3 2 1 5 6 4 10 12 7 9 8 11 14 15 13 16 17 20 19 18 22 21 ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

152
**Reduced Base Model Construction**

“Reduced Base Model” construction techniques make use of composed model structure to reduce the number of states generated. A state in the reduced base model is composed of a state tree and an impulse reward. During reduced base model construction, the use of state trees permits an algorithm to automatically determine valid lumpings based on symmetries in the composed model. The reduced base model is constructed by finding all possible (state tree, impulse reward) combinations and computing the transition rates between states. Generation of the detailed base model is avoided. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

153
**Example Reduced Base Model State Generation**

Composed Model computer ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

154
**Example Reduced Base Model States and Transitions**

R (NumComp = 2) (state 1) 2 computer (CPUboards = 3) covered catastrophic uncovered R (NumComp = 2) R (NumComp = 1) R (NumComp = 0) 1 1 1 1 1 1 computer (CPUboards = 3) computer (CPUboards = 2) computer (CPUboards = 3) computer (CPUboards = 0) computer (CPUboards = 3) computer (CPUboards = 0) (state 2) (state 3) (state 4) ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

155
**Markov Chain of Reduced Base Model (State Transition Rates not Shown)**

1 2 3 4 5 6 8 7 9 10 11 12 13 ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

156
State-Space Generation in Möbius (For generating random process representations of models with all exponential or exponential/deterministic timed activities) Print out states and reward variables Print out absorbing states. Useful to detect problems when attempting a steady-state solution. Place comments, as specified by edit comments, in file. State-space generation must be done before all analytic/numerical solutions are done. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

157
**Numerical/Analytical Solution Techniques**

1) Transient Solution Standard Uniformization (instant-of-time variables) Adaptive Uniformization (instant-of-time variables) Interval-of-time Uniformization (expected value, interval-of-time variables) 2) Steady-state Solution Direct Solution (instant-of-time steady-state variables) Iterative Solution (instant-of-time steady-state variables) ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

158
**CTMC Transient Solution**

We have seen that it is easy to specify a CTMC in terms of the initial probability distribution p(0) and the state-transition-rate matrix. Earlier, we saw that the transient solution of a CTMC is given by p(t) = p(0)P(t), and we noted that P(t) was difficult to define. Due to the complexity of the math, we omit the derivation and show the relationship Solving this differential equation in some form is difficult but necessary to compute a transient solution. where Q is the state transition rate matrix of the Markov chain. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

159
**Transient Solution Techniques**

Solutions to can be done in many (dubious) ways*: Direct: If the CTMC has N states, one can write N2 PDEs with N2 initial conditions and solve N2 linear equations. Laplace transforms: Unstable with multiple “poles” Nth order differential equations: Uses determinants and hence is numerically unstable Matrix exponentiation: P(t) = eQt, where Matrix exponentiation has some potential. Directly computing eQt by performing can be expensive and prone to instability. If the CTMC is irreducible, it is possible to take advantage of the fact that Q = ADA-1, where D is a diagonal matrix. Computing eQt becomes AeDtA-1, where * See C. Moler and C. Van Loan, “Nineteen Dubious Ways to Compute the Exponential of a Matrix,” SIAM Review, vol. 20, no. 4, pp , October 1978. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

160
**Standard Uniformization**

Starting with CTMC state transition rate matrix (Q) construct Probability of k transitions in time t k-step state transition probability Choose truncation point to obtain desired accuracy Compute p(k) iteratively, to avoid fill-in ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

161
**Error Bound in Uniformization**

Answer computed is a lower bound, since each term in summation is positive, and summation is truncated. Number of iterations to achieve a desired accuracy bound can be computed easily. Error for each state Choose error bound, then compute Ns on-the-fly, as uniformization is done. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

162
**A Simple Example (In the Reverse Direction)**

Start with: Discrete time Markov chain Poisson process with rate l = 2 Generate a CTMC: Q = l(P - I) Make sense? Look at sum of a geometric number of exponentials (geometric with parameter r) Result: exponential with rate rl. Holding time in state 1 has mean 1/1.4, holding time in state 2 has mean 1. Matches that for CTMC. 1 2 .3 .7 .5 1 2 1.4 ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

163
**Transient Uniformization Solver (for transient solution of instant-of-time variables)**

Instant-of-time variable time points of interest. Multiple time points may be specified, separated by spaces. Number of digits of accuracy in the solution. Solution reported is a lower bound. Volume of intermediate results reported. “1” gives the greatest volume, greater numbers less. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

164
**Adaptive Uniformization**

Instead of uniformizing at the highest departure rate among all states, uniformize at a rate that changes, and is highest among the “reached” states after particular numbers of transitions. Then In actual computation: with p(k + 1) = p(k)Pk. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

165
**Adaptive Uniformization Solver (atrs) (for transient solution of instant-of-time variables)**

Instant-of-time variable time points of interest. Multiple time points may be specified. Number of digits of accuracy in the solution. Solution reported is a lower bound. Volume of intermediate results reported. “1” gives the greatest volume, greater numbers less. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

166
**Hints for Effective Use of Uniformization and Adaptive Uniformization**

The computation time of trs is primarily determined by the number of iterations. The number of iterations is proportional to time point times highest departure rate of a state. Models with high rate transitions relative to the time point of interest will take a long time to solve. E.g., reliability model with slow failure, fast repair, evaluated at large time points. Adaptive uniformization is more time-efficient than standard uniformization when high-rates are not encountered immediately. Use this solver to get transient solutions in this case. See [van Moorsel 94, van Moorsel 97] for details. For large values of t the result becomes identical to the steady-state result, and will not change any longer if t increases. Use the iss solver to detect when this occurs. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

167
Accumulated Reward Solver (ars) (solves for expected values of interval-of-time and time-averaged interval-of-time variables on intervals [t0, t1] when both t0 and t1 are finite) Number of digits of accuracy in the solution. Solution reported is a lower bound. Series of time intervals for which solution is desired. Intervals are separated by spaces. Each interval can be specified as t1:t2. Volume of intermediate results reported. “1” gives the greatest volume, greater numbers less. The accumulated reward solver is based on uniformization, so the hints given for the transient solver apply here as well. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

168
**Steady-State Behavior of CTMCs**

since P(t) 0 ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

169
**Steady-State Behavior of CTMCs via Flow Equations**

Another way to arrive at the equation p*Q = 0, where is to use the flow equations. The “flow” of probability mass into a state i must equal the “flow” of probability mass out of state i. The “flow” of probability mass from state i to state j is simply piqij, which is the probability of being in state i times the rate at which transitions from i to j take place. In matrix form, for all i, we obtain pQ = 0. (1) (3) (2) (4) ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

170
**Steady-State Behavior of CTMCs, cont.**

This yields the elegant equation p*Q = 0, where the steady-state probability distribution. If the CTMC is irreducible, then p* can be computed with the constraint that Definition: A CTMC is irreducible if every state in the CTMC is reachable from every other state. If the CTMC is not irreducible, then more complex solution methods are required. Notice that for irreducible CTMCs, the steady-state distribution is independent of the initial-state distribution. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

171
**Direct Steady-State Solution**

One steady-state solver in Möbius is the direct steady-state solver. This solver solves the augmented matrix using a form of Gaussian elimination. Pros: Cons: Recommendation: Use for small CTMCs (tens of states) or medium-sized and stiff CTMCs (hundreds to a few thousands), or when high accuracy is required. Reminder: High accuracy in solution does not mean high accuracy in prediction. Use accuracy to do relative comparisons. Can get a very accurate solution in a fixed amount of time; “stiffness” (described later) does not affect solution time. Solution complexity is O(n3), so does not scale well to large models; memory requirements are high due to fill-in and are not known a priori. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

172
**Direct Steady-State Solver (dss) (for steady-state solution of instant-of-time variables)**

Stopping criterion used in iterative refinement phase, after direct solution is done. Number of rows to search for the “best” pivot when performing LU decomposition “Grace” factor by which elements may become pivots Volume of intermediate results reported. “1” gives the greatest volume, greater numbers less. Value that, when multiplied by smallest matrix element, is threshold at which elements may be dropped in LU decomposition. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

173
**Hints for Effective Use of the Direct Steady-State Solver**

dss can be used for steady-state distributions if the Markov model consists of a single class of recurrent non-null states. I.e., dss cannot be applied to a model with multiple absorbing states. The message invmnorm: zero diagonal element may indicate multiple closed communicating classes. Set the flag Flag Absorbing States in the state-space generator to help determine this. Fill-in can be a serious problem for lager Markov chains. Therefore, dss is usually not used except for smaller Markov chains (10s or 100s, perhaps states). As memory consumption increases, computation time increases as O(n3). ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

174
**Iterative Solution Methods**

The simplest iterative solution methods are called stationary iterative methods, and they can be expressed as p(k + 1) = p(k)M, where M is a constant (stationary) matrix. Computing p(k + 1) from p(k) requires one vector-matrix multiplication, or one iteration, which on modern workstations is extremely fast. The simplest stationary iterative method for CTMCs is called the power method. Recall p*Q = 0. Let M = Q + I. p(M - I) = 0 pM - p = 0 pM = p p(k + 1) = p(k)(Q + I) The power method typically converges (gets close to the answer) slowly. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

175
**Iterative Solution Characteristics**

Stationary iterative solution methods have the following characteristics: Low memory usage (no fill-in); predictable memory usage Low time per iteration, proportional to the number of non-zero entries Fast solution time for non-stiff matrices (tens or hundreds of iterations) Stop when sufficiently accurate Slow solution time for stiff matrices Difficult to quantify accuracy, especially for stiff matrices Easy to implement ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

176
**Convergence of Iterative Methods**

We say that an iterative solution method converges if Convergence is of course an important property of any iterative solution method. The rate at which p(k) converges to p* is an important problem, but a very difficult one. Loosely, we say method A converges faster than method B if the smallest k such that is less for A than for B. Which iterative solution method is fastest depends on the Markov chain! ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

177
**Stopping Criteria for Iterative Methods**

An important consideration in any iterative method is knowing when to stop. Computing the solution exactly is often wasteful and unnecessary. There are two popular methods of determining when to stop. The residual norm is usually better, but is sometimes a little more difficult to compute. Both norms do have a relationship with although that relationship is complex. The unfortunate fact is that the more iterations necessary, the smaller e must be to guarantee the same accuracy. called the difference norm called the residual norm ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

178
Gauss-Seidel One of the most widely used stationary iterative methods is called Gauss-Seidel. The algorithm appears as follows: An intuitive explanation for this algorithm: flow out of node i flow into node i ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

179
SOR There is an extension to Gauss-Seidel called successive over-relaxation, or SOR, that sometimes gives better performance. Choosing w is a hard problem in general. Automatic techniques for choosing w exist but are not implemented in Möbius. Note: w = 1 is the same as Gauss-Seidel. Recommendation: Leave w = 1 unless you are solving a similar system many times and the matrix is stiff. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

180
**Iterative Steady-State Solver (iss) (for steady-state solution of instant-of-time variables)**

Stopping criterion, expressed as 10-x, where x is given. The criterion used is the infinity difference norm. SOR weight factor. Values < 1 guarantee convergence, but slow it. Values >= 1 speed convergence, but may not converge. Maximum number of iterations allowed. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

181
**Hints for Effective Use of Iterative Steady-State Solver**

The iss solver will not work with models with models with absorbing states. It will print the message iss_solver: zero on the diagonal and quit Use Flag Absorbing States to determine if / which states are absorbing. The algorithm used in iss stops when the difference norm is less than ten to the power of negative weight. Normalization is done after stopping, so the actual difference norm could be much less (or more). A value of 9 is typically sufficient. As a rule of thumb, the additional time to get an n-times-as-accurate result is of the order log10n. Hence, increased accuracy tends to be not too costly. A weight equal to 1 is usually sufficient. Weight less than 1 (e.g., .99) guarantees convergence, but typically slows it down. Weights greater than 1 can increase the convergence rate. Weights a only slightly larger than “optimum” can cause divergence. Weight must be between 0 and 2. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

182
Tips for Using iss A simple indicator of stiffness is the ratio of the highest rate transition to the lowest rate transition. About 104 or 105 may make a problem stiff. This is only a rule of thumb. Observed fast_repair and slow_repair have the same stiffness ratio but considerably different convergence characteristics. Over-relaxation can be a real time-saver, but only if you can invest the time. For example, w = 1.4 would work well for all experiments for both studies. For some other model, however, it could diverge. Use “verbosity” to observe the stopping criterion A “verbosity” of 1000 prints every 1000 iterations. Use a stopping criterion at least as large as the accuracy required. Better, take the required accuracy and add the stiffness ratio. E.g., a desired accuracy of 10-4 with a stiffness ratio of 105 means using a stopping criterion of at least 9. This is also just a rule of thumb. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

183
**Möbius Analytical Solvers**

a if only rate rewards are used, the time-averaged interval-of-time steady-state measure is identical to the instant-of-time steady-state measure (if both exist). b provided the instant-of-time steady-state distribution is well-defined. Otherwise, the time-averaged interval-of-time steady-state variable is computed and only results for rate rewards should be derived. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

184
**Case Study: Fault-Tolerant Embedded Multiprocessor System**

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

185
**Session Outline Problem description Problem solution Choice of SANs**

Choice of activities Choice of places Tricks of the trade Discussion of constructed model Composed model SAN models Model solution ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

186
Problem Origin This problem was originally posed in 1992 as a reliability model of a large, embedded fault-tolerant computer, presumably for space-borne applications. It was posed as a hierarchical model with non-perfect coverage at each level, with the purpose of showing the inadequacy of existing techniques. Combinatorial methods were incapable of including coverage at all levels of the hierarchy, thus grossly overstating the reliability. Markov- or SPN-based methods create far too many states to solve. Monte-Carlo simulation works, but provides only an estimate (which is often not good enough). A specialized tool was developed to do numerical integration of a semi-Markov process to solve this and similar problems. In Möbius, we solve a smaller version of the same architecture “exactly” using Markov models generated by SANs. This is made possible by automatic state lumping using composed models. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

187
**Problem Description System consists of 2 computers**

Each computer consists of 3 memory modules (2 must be operational) 3 CPU units (2 must be operational) 2 I/O ports (1 must be operational) 2 error-handling chips (non-redundant) Each memory module consists of 41 RAM chips (39 must be operational) 2 interface chips (non-redundant) A CPU consists of 6 non-redundant chips An I/O port consists of 6 non-redundant chips 10 to 20 year operational life ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

188
**Diagram of Fault-Tolerant Multiprocessor System**

.. 41 RAMs 2 int. ch. .. 41 RAMs 2 int. ch. .. 41 RAMs 2 int. ch. 2 ch. memory module memory module memory module errorhandlers .. 6 CPU chips 6 I/O interface bus CPU module CPU module CPU module I/O port I/O port computer . . . computer computer ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

189
**Definition of “Operational”**

The system is operational if at least one computer is operational A computer is operational if all the modules are operational A memory module is operational if at least 39 RAM chips and both interface chips are operational. A CPU unit is operational if all 6 CPU chips are operational An I/O port is operational if all 6 I/O chips are operational The error-handling unit is operational if both error-handling chips are operational Failure rate per chip is 100 failures per 1 billion hours ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

190
Coverage This system could be modeled using combinatorial methods if we did not take coverage into account. Coverage is the chance that the failure of a chip will not cause the larger system to fail even if sufficient redundancy exists. I.e., coverage is the probability that the fault is contained. The coverage probabilities are given in the following table: For example, if a RAM chip fails, there is a 0.2% chance the memory module will fail even if sufficient redundancy exists. If the memory module fails, there is a 5% chance the computer will fail. If a computer fails, there is a 5% chance the system will fail. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

191
**Outline of Solution: List of SANs**

The model is composed of four SANs: 1. memory_module 2. cpu_module 3. errorhandlers 4. io_port_module Each SAN models the behavior of the module in the event of a module component failure. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

192
**List of Places Seven places represent the state of the system:**

1. cpus – the number of operational CPU modules 2. ioports – the number of operational I/O modules 3. errorhandlers – whether the two error-handler chips are operational 4. computer_failed – the number of failed computers 5. memory_failed – the number of failed memory modules 6. memory_chips – number of operational RAM chips 7. interface_chips – number of operational interface chips ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

193
**List of Activities Five activities represent failures in the system**

1. cpu_failure – the failure of any CPU chip 2. ioport_failure – the failure of any I/O chip 3. errorhandling_chip_failure – the failure of either error-handler chip 4. memory_chip_failure – the failure of a RAM chip 5. interface_chip_failure – the failure of a memory interface chip Cases on these activities represent behavior based on coverage or non-coverage. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

194
Tricks of the Trade Since we intend to solve this model analytically, we want the fewest number of states possible. We don’t care which component failed or what particular failed state the model is in. Therefore, we lump all failure states into the same state. We don’t care which computer or which module is in what state. Therefore, we make use of replication to further reduce the number of states. We use marking-dependent rates to model RAM chip failure, making use of the fact that the minimum of independent exponentials is an exponential. We use cases to denote coverage probabilities, and adjust the probabilities depending on the state of the system. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

195
Composed Model ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

196
cpu_modules SAN ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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cpu_modules SAN, cont. cpu_modules input gate predicates and functions: cpu_modules activity time distributions: ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

198
**cpu_modules SAN, cont. cpu_modules case probabilities for activities:**

case 1: chip failure covered case 2: chip failure causes computer failure case 3: chip failure causes system (catastrophic) failure ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

199
**cpu_modules SAN, cont. cpu_modules output gate functions:**

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

200
errorhandlers SAN ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

201
**errorhandlers SAN cont.**

Input gate definitions for SAN model errorhandlers: Activity time distributions for SAN model errorhandlers: Activity case probabilities for SAN model errorhandlers: case 1: chip failure causes computer failure case 2: chip failure causes system failure ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

202
**errorhandlers SAN cont.**

Output gate definitions for SAN model errorhandlers: ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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memory_module SAN Note: memory_module is replicated 3 times, computer_failed and memory_failed held in common. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

204
**memory_chip_failure of memory_modules SAN**

Input gate definition for SAN model memory_module: Activity time distributions for SAN model memory_module: ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

205
**memory_chip_failure of memory_modules SAN, cont.**

Activity case probabilities for SAN model memory_module: case 1: chip failure, sufficient redundancy case 2: chip failure causes memory_module failure case 3: chip failure causes computer failure case 4: chip failure causes system failure ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

206
**memory_chip_failure of memory_modules SAN, cont.**

Output gate definitions for SAN model memory_module: ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

207
**interface_chip_failure of memory_modules SAN**

Input gate definitions for SAN model memory_module: Activity time distributions for SAN model memory_module: Activity case probabilities for SAN model memory_module: case 1: chip failure causes memory module failure case 2: chip failure causes computer failure case 3: chip failure causes system failure ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

208
**interface_chip_failure of memory_modules SAN, cont.**

Output gate definitions for SAN model memory_module: ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

209
io_port_modules SAN ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

210
**io_port_modules SAN, cont.**

I/O port modules input gate predicates and functions: I/O port modules activity time distributions: ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

211
**io_port_modules SAN, cont.**

I/O port modules case probabilities for activities: case 1: chip failure causes I/O port failure case 2: chip failure causes computer failure case 3: chip failure causes system failure ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

212
**io_port_modules SAN, cont.**

I/O port modules output gate functions: ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

213
Model Solution The modeled two-computer system with non-perfect coverage at all levels (i.e., the model as described), the state space contains 10,114 states. The 10 year mission reliability was computed to be ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

214
Impact of Coverage Coverage can have a large impact on reliability and state-space size. Various coverage schemes were evaluated with the following results. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

215
Conclusion Because there are no fast rate transitions, this model affords efficient solution using uniformization. Rep / Join are a natural way to model redundancy and offer great state-space savings. Möbius is able to provide an accurate and efficient solution where previous solutions required simulation or approximations. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

216
**Solution by Simulation**

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

217
**Review of simulation fundamentals **

Session Outline Advantages and disadvantages of simulation, relative to other model solution methods Review of simulation fundamentals Estimating measures: Estimators and confidence intervals Simulation in Möbius ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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Motivation High-level formalisms (like SANs) make it easy to specify realistic systems, but they also make it easy to specify systems that have unreasonably large state spaces. State-of-the-art tools (like Mobius) can handle state-level models with a few ten’s of million states, but not more. When state spaces become too large, discrete event simulation is often a viable alternative. Discrete-event simulation can be used to solve models with arbitrarily large state spaces, as long as the desired measure is not based on a “rare event.” When “rare events” are present, variance reduction techniques can sometimes be used. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Advantages of Simulation**

Simulation can be applied to any SAN model. The most prominent difference, compared with analytic solvers, is that generally distributed activities can be used. Simulation does not require the generation of a state space and therefore does not require a finite state space. Therefore, much more detailed models can be solved. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

220
**Disadvantages of Simulation**

Simulation only provides an estimate of the desired measure. An approximate confidence interval is constructed that contains the actual result with some user-specified probability. Higher desired accuracy dramatically increases the necessary simulation time. As a rule, to make the confidence interval n times narrower, the simulation has to be run n2 times as long. The “rare event problem” may arise. If simulation is used to estimate a small probability, such as the reliability of a highly-reliable system, extremely long simulations may have to be performed to encounter the particular event often enough. Complicated models can require long simulation times, even if the rare event problem is not an issue. The simulators in Möbius perform the necessary event scheduling very efficiently, but it should be realized that simulation is not a panacea. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

221
**Simulation as Model Experimentation**

State-based methods (such as Markov chains) work by enumerating all possible states a system can be in, and then invoking a numerical solution method on the generated state space. Simulation, on the other hand, generates one or more trajectories (possible behaviors from the high-level model), and collects statistics from these trajectories to estimate the desired performance/dependability measures. Just how this trajectory is generated depends on the: nature of the notion of state (continuous or discrete) type of stochastic process (e.g., ergodic, reducible) nature of the measure desired (transient or steady-state) types of delay distributions considered (exponential or general) We will consider each of these issues in this module, as well as the simulation of systems with rare events. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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Types of Simulation Continuous-state simulation is applicable to systems where the notion of state is continuous and typically involves solving (numerically) systems of differential equations. Circuit-level simulators are an example of continuous-state simulation. Discrete-event simulation is applicable to systems in which the state of the system changes at discrete instants of time, with a finite number of changes occurring in any finite interval of time. Since we will focus on validating end-to-end systems, rather than circuits, we will focus on discrete-event simulation. There are two types of discrete-event simulation execution algorithms: Fixed-time-stamp advance Variable-time-stamp advance ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

223
**Fixed-Time-Stamp Advance Simulation**

Simulation clock is incremented a fixed time Dt at each step of the simulation. After each time increment, each event type (e.g., activity in a SAN) is checked to see if it should have completed during the time of the last increment. All event types that should have completed are completed and a new state of the model is generated. Rules must be given to determine the ordering of events that occur in each interval of time. Example: Good for all models where most events happen at fixed increments of time (e.g., gate-level simulations). Has the advantage that no “future event list” needs to be maintained. Can be inefficient if events occur in a bursty manner, relative to time-step used. 2Dt Dt 3Dt 4Dt 5Dt e1 e2 e5 e4 e3 e6 ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

224
**Variable-Time Step Advance Simulation**

Simulation clock advanced a variable amount of time each step of the simulation, to time of next event. If all event times are exponentially distributed, the next event to complete and time of next event can be determined using the equation for the minimum of n exponentials (since memoryless), and no “future event list” is needed. If event times are general (have memory) then “future event list” is needed. Has the advantage (over fixed-time-stamp increment) that periods of inactivity are skipped over, and models with a bursty occurrence of events are not inefficient. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

225
**Basic Variable-Time-Step Advance Simulation Loop for SANs**

A) Set list_of_active_activities to null. B) Set current_marking to initial_marking. C) Generate potential_completion_time for each activity that may complete in the current_marking and add to list_of_active_activities. D) While list_of_active_activities null: 1) Set current_activity to activity with earliest potential_completion_time. 2) Remove current_activity from list_of_active_activities. 3) Compute new_marking by selecting a case of current_activity, and executing appropriate input and output gates. 4) Remove all activities from list_of_active_activities that are not enabled in new_marking. 5) Remove all activities from list_of_active_activities for which new_marking is a reactivation marking. 6) Select a potential_completion_time for all activities that are enabled in new_marking but not on list_of_active_activities and add them to list_of_active_activities. E) End While. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

226
**Types of Discrete-Event Simulation**

Basic simulation loop specifies how the trajectory is generated, but does not specify how measures are collected, or how long the loop is executed. How measures are collected, and how long (and how many times) the loop is executed depends on type of measures to be estimated. Two types of discrete-event simulation exist, depending on what type of measures are to be estimated. Terminating - Measures to be estimated are measured at fixed instants of time or intervals of time with fixed finite point and length. This may also include random but finite (in some sense) times, such as a time to failure. Steady-state - Measures to be estimated depend on instants of time or intervals whose starting points are taken to be t . ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

227
**Issues in Discrete-Event Simulation**

1) How to generate potential completion times for events 2) How to estimate dependability measures from generated trajectories Transient measures Steady-state measures 3) How to implement the basic simulation loop Sequential or parallel ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

228
**Generation of Potential Completion Times**

1) Generation of uniform [0,1] random variates Used as a basis for all random variate samples Types Linear congruential generators Tausworthe generators Other types of generators Tests of uniform [0,1] generators 2) Generation of non-uniform random variates Inverse transform technique Convolution technique Composition technique Acceptance-rejection technique Technique for discrete random variates 3) Recommendations/Issues ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

229
**Generation of Uniform [0,1] Random Number Samples**

Goal: Generate sequence of numbers that appears to have come from uniform [0,1] random variable. Importance: Can be used as a basis for all random variates. Issues: 1) Goal isn’t to be random (non-reproducible), but to appear to be random. 2) Many methods to do this (historically), many of them bad (picking numbers out of phone books, computing p to a million digits, counting gamma rays, etc.). 3) Generator should be fast, and not need much storage. 4) Should be reproducible (hence the appearance of randomness, not the reality). 5) Should be able to generate multiple sequences or streams of random numbers. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

230
**Linear Congruential Generators (LCGs)**

Introduced by D. H. Lehmer (1951). He obtained xn = an mod m xn = (axn - 1) mod m Today, LCGs take the following form: xn = (axn b) mod m, where xn are integers between 0 and m - 1 a, b, m non-negative integers If a, b, m chosen correctly, sequence of numbers can appear to be uniform and have large period (up to m). LCGs can be implemented efficiently, using only integer arithmetic. LCGs have been studied extensively; good choices of a, b, and m are known. See, e.g., Law and Kelton (1991), Jain (1991). ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

231
**Tausworthe Generators**

Proposed by Tausworthe (1965), and are related to cryptographic methods. Operate on a sequence of binary digits (0,1). Numbers are formed by selecting bits from the generated sequence to form an integer or fraction. A Tausworthe generator has the following form: bn = cq - 1bn - 1 cq - 2bn - 2 c0bn - q where bn is the nth bit, and ci (i = 0 to q - 1) are binary coefficients. As with LCGs, analysis has been done to determine good choices of the ci. Less popular than LCGs, but fairly well accepted. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

232
**Generation of Non-Uniform Random Variates**

Suppose you have a uniform [0,1] random variable, and you wish to have a random variable X with CDF FX. How do we do this? All other random variates can be generated from uniform [0,1] random variates. Methods to generate non-uniform random variates include: Inverse Transform - Direct computation from single uniform [0,1] variable based on observation about distribution. Convolution - Used for random variables that can be expressed as sum of other random variables. Composition - Used when the distribution of the desired random variable can be expressed as a weighted sum of the distributions of other random variables. Acceptance-Rejection - Uses multiple uniform [0,1] variables and a function that “majorizes” the density of the random variate to be generated. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

233
**Inverse Transform Technique**

Suppose we have a uniform [0,1] random variable U. If we define X = F-1(U), then X is a random variable with CDF FX = F. To see this, FX(a) = P[X a] = P[F-1(U) a] = P[U F(a)] = F(a) Thus, by starting with a uniform random variable, we can generate virtually any type of random variable. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

234
**Example of Inverse Transform**

Let X be an exponentially distributed random variable with parameter l. Let U be a uniform [0,1] random variable generated by a pseudo-random number generator. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

235
**Convolution Technique**

Technique can be used for all random variables X that can be expressed as the sum of n random variables X = Y1 + Y2 + Y Yn In this case, one can generate a random variate X by generating n random variates, one from each of the Yi, and summing them. Examples of random variables: Sum of n Bernoulli random variables is a binomial random variable. Sum of n exponential random variables is an n-Erlang random variable. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

236
**Composition Technique**

Technique can be used when the distribution of a desired random variable can be expressed as a weighted sum of other distributions. In this case F(x) can be expressed as The composition technique is as follows: 1) Generate random variate i such that P[I = i] = pi for i = 0, 1, . . . (This can be done as discussed for discrete random variables.) 2) Return x as random variate from distribution Fi(x), where i is as chosen above. A variant of composition can also be used if the density function of the desired random variable can be expressed as weighted sum of other density functions. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

237
**Acceptance-Rejection Technique**

Indirect method for generating random variates that should be used when other methods fail or are inefficient. Must find a function m(x) that “majorizes” the density function f(x) of the desired distribution. m(x) majorizes f(x) if m(x) f(x) for all x. Note: If random variates for m(x) can be easily computed, then random variates for f(x) can be found as follows: 1) Generate y with density m(x) 2) Generate u with uniform [0,1] distribution 3) ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

238
**Generating Discrete Random Variates**

Useful for generating any discrete distribution, e.g., case probabilities in a SAN. More efficient algorithms exist for special cases; we will review most general case. Suppose random variable has probability distribution p(0), p(1), p(2), on non-negative integers. Then a random variate for this random variable can be generated using the inverse transform method: 1) Generate u with distribution uniform [0,1] 2) Return j satisfying ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Recommendations/Issues in Random Variate Generation**

Use standard/well-tested uniform [0,1] generators. Don’t assume that because a method is complicated, it produces good random variates. Make sure the uniform [0,1] generator that is used has a long enough period. Modern simulators can consume random variates very quickly (multiple per state change!). Use separate random number streams for different activities in a model system. Regular division of a single stream can cause unwanted correlation. Consider multiple random variate generation techniques when generating non-uniform random variates. Different techniques have very different efficiencies. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

240
**Estimating Dependability Measures: Estimators and Confidence Intervals**

An execution of the basic simulation loop produces a single trajectory (one possible behavior of the system). Common mistake is to run the basic simulation loop a single time, and presume observations generated are “the answer.” Many trajectories and/or observations are needed to understand a system’s behavior. Need concept of estimators and confidence intervals from statistics: Estimators provide an estimate of some characteristic (e.g., mean or variance) of the measure. Confidence intervals provide an estimate of how “accurate” an estimator is. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

241
**Typical Estimators of a Simulation Measure**

Can be: Instant-of-time, at a fixed t, or in steady-state Interval-of-time, for fixed interval, or in steady-state Time-averaged interval-of-time, for fixed interval, or in steady-state Estimators on these measures include: Mean Variance Interval - Probability that the measure lies in some interval [x,y] Don’t confuse with an interval-of-time measure. Can be used to estimate density and distribution function. Percentile - 100bth percentile is the smallest value of estimator x such that F(x) b. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Different Types of Processes and Measures Require Different Statistical Techniques**

Transient measures (terminating simulation): Multiple trajectories are generated by running basic simulation loop multiple times using different random number streams. Called Independent Replications. Each trajectory used to generate one observation of each measure. Steady-State measures (steady-state simulation): Initial transient must be discarded before observations are collected. If the system is ergodic (irreducible, recurrent non-null, aperiodic), a single long trajectory can be used to generate multiple observations of each measure. For all other systems, multiple trajectories are needed. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

243
**Confidence Interval Generation: Terminating Simulation**

Approach: Generate multiple independent observations of each measure, one observation of each measure per trajectory of the simulation. Observations of each measure will be independent of one another if different random number streams are used for each trajectory. From a practical point of view, new stream is obtained by continuing to draw numbers from old stream (without resetting stream seed). Notation (for subsequent slides): Let F(x) = P[X x] be measure to be estimated. Define m = E[X], s2 = E[(X - m)2]. Define xi as the ith observation value of X (ith replication, for terminating simulation). Issue: How many trajectories are necessary to obtain a good estimate? ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

244
**Terminating Simulation: Estimating the Mean of a Measure I**

Wish to estimate m = E[X]. Standard point estimator of m is the sample mean To compute confidence interval, we need to compute sample variance: ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Terminating Simulation: Estimating the Mean of a Measure II**

Then, the (1 - a) confidence interval about x can be expressed as: Where N is the number of observations. Equation assumes xn are distributed normally (good assumption for large number of xi). The interpretation of the equation is that with (1 - a) probability the real value (m) lies within the given interval. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

246
**Terminating Simulation: Estimating the Variance of a Measure I**

Computation of estimator and confidence interval for variance could be done like that done for mean, but result is sensitive to deviations from the normal assumption. So, use a technique called jackknifing developed by Miller (1974). Define Where ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Terminating Simulation: Estimating the Variance of a Measure II**

Now define (where s2 is the sample variance as defined for the mean) And Then is a (1 - a) confidence interval about s2. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

248
**Terminating Simulation: Estimating the Percentile of an Interval About an Estimator**

Computed in a manner similar to that for mean and variance. Formulation can be found in Lavenberg, ed., Computer Performance Modeling Handbook, Academic Press, 1983. Such estimators are very important, since mean and variance are not enough to plan from when simulating a single system. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

249
**Confidence Interval Generation: Steady-State Simulation**

Informally speaking, steady-state simulation is used to estimate measures that depend on the “long run” behavior of a system. Note that the notion of “steady-state” is with respect to a measure (which has some initial transient behavior), not a model. Different measures in a model will converge to steady state at different rates. Simulation trajectory can be thought of as having two phases: the transient phase and the steady-state phase (with respect to a measure). Multiple approaches to collect observations and generate confidence intervals: Replication/Deletion Batch Means Regenerative Method Spectral Method Which method to use depends on characteristics of the system being simulated. Before discussing these methods, we need to discuss how the initial transient is estimated. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

250
**Estimating the Length of the Transient Phase**

Problem: Observations of measures are different during so-called “transient phase,” and should be discarded when computing an estimator for steady-state behavior. Need: A method to estimate transient phase, to determine when we should begin to collect observations. Approaches: Let the user decide: not sophisticated, but a practical solution. Look at long-term trends: take a moving average and measure differences. Use more sophisticated statistical measures, e.g., standardized time series (Schruben 1982). Recommendation: Let the user decide, since automated methods can fail. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

251
**Methods of Steady-State Measure Estimation: Replication/Deletion**

Statistics similar to those for terminating simulation, but observations collected only on steady-state portion of trajectory. One or more observations collected per trajectory: Compute as ith observation, where Mi is the number of observations in trajectory i. xi are considered to be independent, and confidence intervals are generated. Useful for a wide range of models/measures (the system need not be ergodic), but slower than other methods, since transient phase must be repeated multiple times. transient phase O11 O12 O21 O22 O31 O32 O33 O34 O23 O24 O13 O14 trajectory 1 trajectory 2 trajectory n . . . ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Methods of Steady-State Measure Estimation: Batch Means**

Similar to Replication/Deletion, but constructs observations from a single trajectory by breaking it into multiple batches. Example Observations from each batch are combined to construct a single observation; these observations are assumed to be independent and are used to construct the point estimator and confidence interval. Issues: How to choose batch size? Only applicable to ergodic systems (i.e., those for which a single trajectory has the same statistics as multiple trajectories). Initial transient only computed once. In summary, a good method, often used in practice. O11 O12 O13 ... O21 O22 initial transient O23 ... O31 O32 ... ... ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Other Steady-State Measure Estimation Methods I**

Regenerative Method (Crane and Iglehart 1974, Fishman 1974) Uses “renewal points” in processes to divide “batches.” Results in batches that are independent, so approach used earlier to generate confidence intervals applies. However, usually no guarantee that renewal points will occur at all, or that they will occur often enough to efficiently obtain an estimator of the measure. Autoregressive Method (Fishman 1971, 1978) Uses (as do the two following methods) the autocorrelation structure of process to estimate variance of measure. Assumes process is covariance stationary and can be represented by an autoregressive model. Above assumption often questionable. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Other Steady-State Measure Estimation Methods II**

Spectral Method (Heidelberger and Welch 1981) Assumes process is covariance stationary, but does not make further assumptions (as previous method does). Efficient method, if certain parameters chosen correctly, but choice requires sophistication on part of user. Standardized Time Series (Schruben 1983) Assumes process is strictly stationary and “phi-mixing.” Phi-mixing means that Oi and Oi + j become uncorrelated if j is large. As with spectral method, has parameters whose values must be chosen carefully. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

255
**Summary: Measure Estimation and Confidence Interval Generation**

1) Only use the mean as an estimator if it has meaning for the situation being studied. Often a percentile gives more information. This is a common mistake! 2) Use some confidence interval generation method! Even if the results rely on assumptions that may not always be completely valid, the methods give an indication of how long a simulation should be run. 3) Pick a confidence interval generation method that is suited to the system that you are studying. In particular, be aware of whether the system being studied is ergodic. 4) If batch means is used, be sure that batch size is large enough that batches are practically uncorrelated. Otherwise the simulation can terminate prematurely with an incorrect result. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

256
**Summary/Conclusions: Simulation-Based Validation Techniques**

1) Know how random variates are generated in the simulator you use. Make sure: A good uniform [0,1] generator is used Independent streams are used when appropriate Non-uniform random variates are generated in a proper way. 2) Compute and use confidence intervals to estimate the accuracy of your measures. Choose correct confidence interval computation method based on the nature of your measures and process ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

257
**Simulator Statistics Editor**

Variable Type and Times for Terminating Simulation Batch Size and Initial Transient in Steady-State Simulation ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Simulator Statistics Editor**

Estimator Types Variable Type in Terminating Simulation ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Simulator Statistics Editor**

Confidence Interval Width and Level ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

260
**Setting Initial Transient and Batch Size in Steady-State Simulation**

Set initial transient large enough so transient has “settled down” Think about characteristics of model Make long enough that any one-time events have occurred For events that occur in a roughly cyclic manner, with a certain period, make initial transient a large (say, 1000) time multiple of the period, so markings related to these events will reach steady state Make batch size similar in size to initial transient, using above guidelines ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

261
**Simulator Editor Maximum and Minimum Number of Replications to Run**

Number of Batches between each calculation of the variance Trace-Level for Debugging File Name of Output File ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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**Batch and Replication Outputs (Variable Output Option)**

Typical batch output: Typical replication output: ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

263
**Möbius Simulation Techniques**

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

264
**Hints for Successful Simulation**

Use “Trace Level” option to look at sequence of completion of activities. The batch size in the steady-state simulator must be large enough to assure independence of batches. Enough batches must be collected to assure that the sample variance computed is an accurate reflection of real variance of the measure. Setting the minimum number of batches too low can yield an artificially low (incorrect) confidence interval width. To determine expected run length, monitor the width of the confidence interval using the “Variable Output” option. As a rule of thumb, if the confidence interval is observed after k batches or replications, the simulation will take kn2 additional batches or replications to decrease the width of the confidence interval by a factor of n. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

265
**Symbolic State-space Exploration and Numerical Analysis of State-sharing Composed Models**

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

266
**Costly generation and representation of SS (space and time) **

Motivation State-space (SS) explosion or largeness problem in discrete-state systems Costly generation and representation of SS (space and time) Costly representation of CTMC (space) Costly representation of solution vector (space) and costly iteration/solution time (time) Typical solutions: Largeness avoidance, e.g., using lumping techniques CTMC level Model level Largeness tolerance using BDD, MDD, MTBDD, Kronecker, or Matrix Diagrams (MD) discrete-state systems Previous works have overcome the first two bottlenecks with fairly few model-level restrictions ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

267
**What Is New? Our approach combines**

Model-level lumping induced by structural symmetries Number of states solution vector size Number of states iteration time MDD and Matrix Diagram (MD) data structures Enables us to represent lumped CTMCs not possible using sparse matrix An order of magnitude faster than unlumped sparse representation although it induces slowdown in solution time compared to lumped sparse representation State-sharing composed models as opposed to action-synchronization Maintain almost the same generality ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

268
**State-sharing Composed Models**

Join and Replicate operators Any atomic model formalism that can share state variables E.g., SAN, PEPAk, and Buckets and Balls Replicate induces symmetry Global and local actions Join M1 M2 SV1 M1 M2 SV1 Join M1 Rep (3) M1 I’ll give some necessary definitions on the state-sharing composition formalism Join and Replicate are closed under the set of models ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

269
**Introduction to MDD Represents function where**

Special case : n = 1, f represents a set of vectors 1 2 1 1 1 1 2 1 2 1 I assume you’re already familiar with MDD data structure however I’ll give a very quick overview Used to represent a function from a finite set of tuples to a finite set. Implemented as a directed acyclic graph ordered in levels When n=1 the function will be binary and so it can be used to represent a set of tuples or a set of states of a discrete-state model if states are represented as tuples To represent states as tuples we partition the set of state variables of a model. For each block of the partition we enumerate the possible values that the state variables in that block can take. A state can now be represented as a vector of indices. {(0,0,1), (0,0,2), (0,1,1), (0,1,2), (1,0,1), (1,0,2), (1,1,0), (1,1,1), (2,0,0), (2,0,1), (2,1,1), (2,1,2)} ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

270
**Representation of a set of states of a discrete-state model**

Introduction to MDD Represents function where Special case : n = 1, f represents a set of vectors 1 2 1 1 1 Representation of a set of states of a discrete-state model Partition set of SVs Assign index to unique value assignment of variables of each block Vector of indices represents a state 1 2 1 2 1 I assume you’re already familiar with MDD data structure however I’ll give a very quick overview Used to represent a function from a finite set of tuples to a finite set. Implemented as a directed acyclic graph ordered in levels When n=1 the function will be binary and so it can be used to represent a set of tuples or a set of states of a discrete-state model if states are represented as tuples To represent states as tuples we partition the set of state variables of a model. For each block of the partition we enumerate the possible values that the state variables in that block can take. A state can now be represented as a vector of indices. {(0,0,1), (0,0,2), (0,1,1), (0,1,2), (1,0,1), (1,0,2), (1,1,0), (1,1,1), (2,0,0), (2,0,1), (2,1,1), (2,1,2)} ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

271
**Representation of a set of states of a discrete-state model**

Introduction to MDD Represents function where Special case : n = 1, f represents a set of vectors 1 2 4 8 12 1 2 {(0,0,1), (0,0,2), (0,1,1), (0,1,2), (1,0,1), (1,0,2), (1,1,0), (1,1,1), (2,0,0), (2,0,1), (2,1,1), (2,1,2)} Representation of a set of states of a discrete-state model Partition set of SVs Assign index to unique value assignment of variables of each block Vector of indices represents a state I assume you’re already familiar with MDD data structure however I’ll give a very quick overview Used to represent a function from a finite set of tuples to a finite set. Implemented as a directed acyclic graph ordered in levels When n=1 the function will be binary and so it can be used to represent a set of tuples or a set of states of a discrete-state model if states are represented as tuples To represent states as tuples we partition the set of state variables of a model. For each block of the partition we enumerate the possible values that the state variables in that block can take. A state can now be represented as a vector of indices. Augment by state offsets ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

272
**MDD data structure by example**

Partitioning SVs based on composition structure Maximizing efficiency of local SS exploration Simplifying global SS exploration Dependability model for multicomputer system Join IO port error handler cpu Rep1 (M) memory Rep2 (N) Rep2 Join mem Rep1 outer replicate MDD level assignment inner replicate 1 2 3 2+M Unfold replicate nodes first Partitioning: 1)For atomic models: All variables of an atomic model except shared ones go into the level corresponding to that atomic model 2)For join nodes: Variables shared by the join node and not shared by its parent goes to a level corresponding to the join node If a set of state variables for one level is empty we remove that level such as cpu, error_handler, and IO/port in the figure above The intuition behind such partitioning is to maximize local SS exploration in atomic model levels and simplify global SS exploration Fix M = 3 ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

273
**Algorithm Overview Generate MDD representation of unlumped SS**

Build MD representation of CTMC Convert unlumped SS to lumped SS Solve CTMC by iterating through MD data structure - Unique parts are written in Red ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

274
**Symbolic Generation of Unlumped SS**

set of visited states set of unexplored states expands using sequences of firings of local actions expands using single action firing of global actions Never generate potential or unreachable states Creating necessary matrices and data structures to construct MD of the CTMC at a later stage No consideration of lumping properties “At a high-level” it’s similar to other SSG algorithm the difference is that ….. Alternatively running GlobalSSE and LocalSSE until all states are explored Firing a global action is much more expensive than firing a local one. That’s why we chose to fire a single action in each run. Unlike some other approaches, our algorithm does not work based on potential states. Neither does it generate unreachable states. Unique parts of our algorithm is in Red ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

275
**Symbolic SSG (Local Actions)**

Restriction: immediate actions are local On-the-fly elimination of vanishing states Local SS expansion in levels corresponding to atomic models. No assumption of knowing the local state space in advance Online computation of transitive closure based on Ibaraki and Katoh’s algoritm Avoids costly computation of tr. closure from scratch i j A B local transition i to j A B If an atomic model can make a local transition from i to j, that transition can be done regardless of what state the other parts of the model are in. Therefore, all the currently reachable states of the system with substate i replaced by j are also reachable. All global actions have to be timed otherwise the algorithm can not do local SS exploration in the general case and that drastically degrades the performance of the algorithm. The restriction that all immediate actions are global We do not assume knowing the local state spaces in advance. Since our atomic model has minor restrictions, properties like invariants can not be computed for the model. Therefore the local state space can not be computed in advance and we should employ an online algorithm to compute the transitive closure ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

276
**Symbolic SSG (Global Actions)**

Global action a in component c affects more than one level No “product-form”-like restriction Effect of a on each level need not be determined locally More difficult to handle than synchronizing actions Expensive operation Since we wanted to have as few restrictions as possible on the atomic formalisms, we didn’t impose product-form restriction on the global actions. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

277
**Lumping Redundant states (paths) 1 2 x Rep AM**

Often in practice the lumped state space is less structured, and therefore, its representation takes more space which is still negligible compared to available memory in modern computers and also compared to the size of the solution vector ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

278
**Lumping Redundant states (paths)**

1 2 x Rep AM 1 x 2 Redundant states (paths) Rep node c implies equivalence relation Rc Often in practice the lumped state space is less structured, and therefore, its representation takes more space which is still negligible compared to available memory in modern computers and also compared to the size of the solution vector ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

279
**Redundant states (paths) Rep node c implies equivalence relation Rc**

Lumping 1 2 x Rep AM 1 x 2 Redundant states (paths) Rep node c implies equivalence relation Rc Overall equivalence relation Canonical representative state in each class min(v) Often in practice the lumped state space is less structured, and therefore, its representation takes more space which is still negligible compared to available memory in modern computers and also compared to the size of the solution vector ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

280
**Redundant states (paths) Rep node c implies equivalence relation Rc**

Lumping 1 2 x Rep AM 1 x 2 Redundant states (paths) Rep node c implies equivalence relation Rc Overall equivalence relation Canonical representative state in each class min(v) may become exponentially large break it up into many extremely smaller MDDs faster computation of Often in practice the lumped state space is less structured, and therefore, its representation takes more space which is still negligible compared to available memory in modern computers and also compared to the size of the solution vector ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

281
**where is the set of all states v where min(v) =v **

Lumping where is the set of all states v where min(v) =v may become huge break up into extremely smaller MDDs is often less structured than and therefore larger in number of nodes ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

282
**SSG and Lumping Performance**

Worst case example: No local behavior Drastic decrease in number of states in the lumped SS (up to 6 orders of magnitude) Increase in number of nodes in the lumped state space but still small compared to other entities Very small unlumped and lumped SS representation Point out: 1) how lumped SS is larger than unlumped SS 2) how lumping reduces the number of states by up to 6 orders of magnitude ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

283
**CTMC Generation and Enumeration**

Use Matrix Diagrams (MD) (Ciardo/Miner) CTMC of largest example has <40000 nodes and takes <3MB of memory We don’t get into details of building the matrix diagrams ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

284
**CTMC Generation and Enumeration**

Use Matrix Diagrams (MD) (Ciardo/Miner) CTMC of largest example has <30000 nodes and takes <5MB of memory Projection of the MD on the lumped SS? Problem: some needed transitions are deleted wrong We don’t get into details of building the matrix diagrams correct ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

285
**CTMC Generation and Enumeration**

Use Matrix Diagrams (MD) (Ciardo/Miner) CTMC of largest example has <40000 nodes and takes <3MB of memory and at most a few seconds to build Projection of the MD on the lumped SS? Problem: some needed transitions are deleted Project rows on lumped SS and columns on unlumped SS Redirect transitions on-the-fly DFS-based enumeration of MD using “sorting” MDD wrong correct We don’t get into details of building the matrix diagrams ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

286
**CTMC Enumeration Performance**

Although half of the overhead is due to transition redirection which is necessary because of lumping, lumping reduces the number of states by a factor of 4 to This justifies the slowdown we experience by combining MDs and lumping techniques. Fairly fast iteration: less than 6 times slower than lumped sparse matrix Solving larger CTMCs ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

287
**Integration into Möbius**

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

288
**Case Study: Survivability Evaluation**

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

289
**Defending Against a Wide Variety of Attacks**

HIGH LOW INNOVATION PLANNING STEALTH COORDINATION Economic intelligence Military spying Nation-states, Terrorists, Multinationals Information terrorism Disciplined strategic cyber attack Civil disobedience Selling secrets Serious hackers Embarrassing organizations Harassment Collecting trophies Stealing credit cards Script kiddies Curiosity Copy-cat attacks Thrill-seeking ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

290
**Intrusion Tolerance: A New Paradigm for Security**

Prevent Intrusions (Access Controls, Cryptography, Trusted Computing Base) 1st Generation: Protection Cryptography Trusted Computing Base Access Control & Physical Security Multiple Security Levels Detect Intrusions, Limit Damage (Firewalls, Intrusion Detection Systems, Virtual Private Networks, PKI) 2nd Generation: Detection But intrusions will occur Firewalls Intrusion Detection Systems Boundary Controllers VPNs PKI But some attacks will succeed Tolerate Attacks (Redundancy, Diversity, Deception, Wrappers, Proof-Carrying Code, Proactive Secret Sharing) 3rd Generation: Tolerance Intrusion Tolerance Big Board View of Attacks Real-Time Situation Awareness & Response Graceful Degradation Hardened Operating System ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

291
**Validation of Computer System/Network Survivability**

Security is no longer absolute Trustworthy computer systems/networks must operated through attacks, providing proper service in spite of possible partially successful attacks Intrusion tolerance claims to provide proper operation under such conditions Validation of security/survivability must be done: During all phases of the design process, to make design choices During testing, deployment, operation, and maintenance, to gain confidence that the “amount” of intrusion tolerance provided is as advertised. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

292
**Validating Computer System Security: Research Goal**

CONTEXT: Create robust software and hardware that are fault-tolerant, attack resilient, and easily adaptable to changes in functionality and performance over time. GOAL: Create an underlying scientific foundation, methodologies, and tools that will: Enable clear and concise specifications, Quantify the effectiveness of novel solutions, Test and evaluate systems in an objective manner, and Predict system assurance with confidence. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

293
**Existing Security/Survivability Validation Approaches**

Most traditional approaches to security validation have focus on avoiding intrusions (non-circumventability), or have not been quantitative, instead focusing on and specifying procedures that should be followed during the design of a system (e.g., the Security Evaluation Criteria [DOD85, ISO99]). When quantitative methods have been used, they have typically either been based on formal methods (e.g., [Lan81]), aiming to prove that certain security properties hold given a specified set of assumptions, or been quite informal, using a team of experts (often called a “red team,” e.g. [Low01]) to try to compromise a system. Both of these approaches have been valuable in identifying system vulnerabilities, but probabilistic techniques are also needed. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

294
**Example Probabilistic Validation Study**

Evaluation of DPASA-DV Project design Designing Protection and Adaptation into a Survivability Architecture: Demonstration and Validation Design of a Joint Battlespace Infosphere Publish, Subscribe and Query features (PSQ) Ability to fulfill its mission in the presence of attacks, failures, or accidents Uses Multiple, synergistic validation techniques ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

295
**JBI Design Overview Protection Domains**

JBI Management Staff Executive Zone Crumple Operations JBI Core Quad 1 Quad 2 Quad 3 Quad 4 Network Protection Domains Isolation among selected functions on individual core hosts and on clients Access Proxy (Isolated Process Domains in SE-Linux) Domain6 First Restart Domains Eventually Restart Host Local Controller RMI STCP TCP PS Sensor Rpts UDP IIOP PSQImpl DC Eascii Domain1 Domain2 Domain3 Domain4 Domain5 Forward/ Ratelimit Proxy Logic Inspect / Forward / Rate Limit ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

296
**Survivability/Security Validation Goal**

Provide convincing evidence that the design, when implemented, will provide satisfactory mission support under real use scenarios and in the face of cyber-attacks. More specifically, determine whether the design, when implemented will meet the project goals: This assurance case is supported by: Rigorous logical arguments Experimental evaluation A detailed executable model of the design ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

297
**Goal: Design a Publish and Subscribe Mechanism that …**

Provides 100% of critical functionality when under sustained attack by a “Class-A” red team with 3 months of planning. Detects 95% of large scale attacks within 10 mins. of attack initiation and 99% of attacks within 4 hours with less than 1% false alarm rate. Displays meaningful attack state alarms. Prevent 95% of attacks from achieving attacker objectives for 12 hours. Reduces low-level alerts by a factor of 1000 and display meaningful attack state alarms. Shows survivability versus cost/performance trade-offs. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

298
**Integrated Survivability Validation Procedure**

Requirement Decomposition S P Q Functional Model of the System (Probabilistic or Logical) Functional Model of the Relevant Subset of the System Model for Client Model for Access Proxy … Model for PSQ Server Assumptions AA1 AA2 AA3 AP1 AP2 Low-level mechanisms can support assumptions directly, or can support middle-level mechanisms. Supporting Logical Arguments and Experimentation M1 (Network Domains) M2 M3 M4 M5 M6 L1 (ADF) L2 L3 ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

299
**Integrated Survivability Validation Procedure**

Steps R A precise statement of the requirements S P Q High-level functional model description: Data and alerts flows for the processes related to the requirements, Assumed attacks and attack effects [Threat/vulner-ability analysis; whiteboarding] Functional Model of the Relevant Subset of the System Model for Client Model for Access Proxy … Model for PSQ Server AA1 AA2 AA3 AP1 AP2 Low-level mechanisms can support assumptions directly, or can support middle-level mechanisms. M1 (Network Domains) M2 M3 M4 M5 M6 L1 (ADF) L2 L3 ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

300
**Integrated Survivability Validation Procedure**

Steps R S P Q Detailed descriptions of model component behaviors representing 2a and 2b, along with statements of underlying assumptions made for each component. [Probabilistic modeling or logical argumentation, depending on requirement] Functional Model of the Relevant Subset of the System Model for Client Model for Access Proxy … Model for PSQ Server AA1 AA2 AA3 AP1 AP2 Low-level mechanisms can support assumptions directly, or can support middle-level mechanisms. M1 (Network Domains) M2 M3 M4 M5 M6 L1 (ADF) L2 L3 ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

301
**Integrated Survivability Validation Procedure**

Steps R S P Q Construct executable functional model [Probabilistic modeling, if model constructed in 3 is probabilistic] Functional Model of the Relevant Subset of the System Model for Client Model for Access Proxy … Model for PSQ Server In Parallel a) Verification of the modeling assumptions of Step 3 [Logical argumentation] and, b) where possible, justification of model parameter values chosen in Step 4. [Experimentation] AA1 AA2 AA3 AP1 AP2 Low-level mechanisms can support assumptions directly, or can support middle-level mechanisms. M1 (Network Domains) M2 M3 M4 M5 M6 L1 (ADF) L2 L3 ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

302
**Integrated Survivability Validation Procedure**

Steps R S P Q Run the executable model for the measures that correspond to the requirements of Step 1. [Probabilistic modeling] Functional Model of the Relevant Subset of the System Model for Client Model for Access Proxy … Model for PSQ Server AA1 AA2 AA3 AP1 AP2 Low-level mechanisms can support assumptions directly, or can support middle-level mechanisms. M1 (Network Domains) M2 M3 M4 M5 M6 L1 (ADF) L2 L3 ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

303
**Integrated Survivability Validation Procedure**

Steps R ? Comparison of results obtained in Step 6, noting in particular the configurations and parameter values for which the requirements of Step 1 are satisfied. S P Q Functional Model of the Relevant Subset of the System Model for Client Model for Access Proxy … Model for PSQ Server AA1 AA2 AA3 AP1 AP2 Low-level mechanisms can support assumptions directly, or can support middle-level mechanisms. Note that if the requirement being addressed is not quantitative, steps 4 and 6 are skipped. M1 (Network Domains) M2 M3 M4 M5 M6 L1 (ADF) L2 L3 ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

304
**Step 1: Requirement Specification**

Expressed in an argument graph: JBI critical mission objectives JBI critical functionality JBI mission Detection / Correlation Requirements Initialized JBI provides essential services JBI properly initialized IDS objectives Authorized publish processed successfully Authorized subscribe processed successfully Authorized query processed successfully Authorized join/leave processed successfully Unauthorized activity properly rejected Confidential info not exposed ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

305
**Argument Graph for the Design**

Requirements decomposition Executable model Model assumptions Supporting arguments ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

306
**Step 2: System and Attack Assumption Definition**

Example High level description … Steps 4-5 Access proxy verifies if the client is in valid session by sending the session key accompanying the IO to the Downstream Controller for verification Step 6 Access Proxy forwards the IO to the PSQ Server in its quadrant. .... ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

307
**Attack Model Description**

Definitions Intrusion, prevented intrusion, tolerated intrusion New vulnerabilities Assumptions Outside attackers only Attacker(s) with unlimited resources Consider successful (and harmful) attacks only No patches applied for vulnerabilities found during the mission/scenario execution ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

308
**Attack Model Description**

Attack propagation MTTD: mean time to discovery of a vulnerability MTTE: mean time to exploitation of a vulnerability 3 types of vulnerabilities: Infrastructure-Level Vulnerabilities attacks in depth OS vulnerability Non-JBI-specific application-level vulnerability pcommon : common-mode failure Data-Level Vulnerabilities attacks in breadth Using the application data of JBI software Across process domains flaw in protection domains ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

309
**Attack Model Description**

Attack effects Compromise Launching pad for further attacks Malicious behavior Crash Attack propagation stopped (DoS) Distinction between OSes with and without protection domains ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

310
**Attack Model Description**

Intrusion Detection pdetect=0 if the sensors are compromised pdetect > 0 otherwise. Attack Responses Restart Processes Secure Reboot Permanent Isolation ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

311
**Infrastructure Attacks Example**

Access Proxy, Quad 1, OS 1 DC, Quad 1, OS 1 Policy Server, Quad 1, OS 1 Quadrant 1 T=85 min.: discovery of a vulnerability on the Main PD, OS1 AP IO Outside DC ADF NIC PS AP Hb Se ADF NIC all quad components AP Alert ADF NIC Ac Se LC Ac Outside PSQ Server, Quad 1, OS 1 Guardian, Quad 1, OS 1 LC PSQ Gu Se ADF NIC Se ADF NIC Publishing Client, OS1 SM, Quad 1, OS 1 Ac Ac SD ADF NIC LC LC SM Se ADF NIC Ac Correlator, Quad 1, OS 1 LC ADF NIC Co Crumple Zone Operations Zone Executive Zone Access Proxy, Quad 2, OS 2 Access Proxy, Quad 3, OS 3 AP IO Se Ac LC AP Hb AP Alert ADF NIC PSQ Server, Quad 2, OS 2 Access Proxy, Quad 4, OS 4 AP IO Se Ac LC AP Hb AP Alert ADF NIC PSQ Server, Quad 3, OS 3 SM SM, Quad 1, OS 2 ADF NIC PSQ Se Ac LC ADF NIC AP IO PSQ Server, Quad 4, OS 4 SM SM, Quad 1, OS 3 ADF NIC PSQ Se Ac LC ADF NIC SM, Quad 1, OS 4 AP Hb PSQ ADF NIC ADF NIC Se ADF NIC AP Alert SM Se Ac Outside Ac LC ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author. LC

312
**Step 3: Detailed descriptions of model component behaviors and Assumptions (Access Proxy)**

Model Description AM1: If a process domain in the DJM proxy is not corrupted, it forwards the traffic it is designated to handle from the Quadrant isolation switch to core quadrant elements and vice versa. All traffic being forwarded is well-formed (if the proxy is correct). The following kinds of traffic are handled: 1. IOs (together with tokens) sent from publishing clients to the core (we do not distinguish between IOs sent via different protocols such as RMI or SOAP/HTTP). …. AM2: Attacks on access proxy: attacks on an access proxy are enabled if either/both 1. Quadrant isolation switch is ON, and one or more clients are corrupted, leading to: a) Direct attacks: can cause the corruption of the process domain corresponding to the domain of the attacking process on the compromised client. AM3: If an attack occurs on the access proxy, it can have the following effects: 1. Direct attacks leading to process corruption: a) Enable corruption of other process domains on the host. ….. 4.4.2 Facts and Simplifications AF1: Each access proxy runs on a dedicated host machine. AF2: DoS attacks result in increased delays. Model of Access Proxy Sensors modeled separately appropriate core components (only the components with which the processes in the corrupted domain(s) can communicate). 4.4.3 Assumptions AA1: Only well-formed traffic is forwarded by a correct access proxy. AA2: The access proxy cannot access cryptographic keys used to sign messages that pass through it. AA3: Access proxy cannot access the contents on an IO if application-level end-to-end encryption is being used. AA4: Attacks on an access proxy can only be launched from compromised clients, or from corrupted core elements that interact with the access proxy during the normal course of a mission. …. Assumptions ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

313
**Step 4: Construct Executable Functional Model**

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

314
**Step 5: Supporting Logical Arguments**

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

315
**Logical Argument Sample**

Functional Model PSQ Server Model Access Proxy Model SA3: IO Integrity in PSQ Server SA4: Client Confidentiality in PSQ Server AA2: AP Application-layer Integrity AA3: AP Application-layer Confidentiality Model Assumptions Supporting Arguments Private Key Confidentiality No Cryptography in Access Proxy No Unauthorized Direct Access No Unauthorized Indirect Access Not Preconfigured Not Reconfigurable ADF NIC services protected Keys Protected from Theft Keys Not Guessable Physical Protection of CAC device Protection of CAC Authentication Data No Compromise of Authorized Process Accessing CAC DoD Common Access Card (CAC) Algorithmic Framework Key Length Key Lifetime PKCS #11 Compliance Tamperproof ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

316
**Steps 6 and 7: Measures and Results**

Assumptions: CPUB is the conjunction of C1PUB= the publishing client is successfully registered with the core C2PUB= the publishing client's mission application interacts with the client as intended Definition of a successful publish: EPUB is the conjunction of E1PUB = the data flow for the IO is correct E2PUB = the time required for the publish operation is less than tmax E3PUB = the content of the IO received by the subscriber has the same essential content as that assembled by the publisher Measure: P[EPUB|CPUB] Fraction of successful publishes in a 12 hour period Between clients that cannot be compromised Objective P[EPUB|CPUB] ≥ pPUB for a 12-hour mission ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

317
**Vulnerability Discovery Rate Study**

Fraction of successful publishes versus MTTD Number of successful intrusions versus MTTD ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

318
**Varying the number of OS and OS w/ process domains**

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

319
**Autonomic Distributed Firewall (ADF) NIC policies**

Fraction of successful publishes Total number of intrusions Per-pd policies considerably increase the performance (10% unavailability vs. 1.5% at MTTD=100 minutes) ADF NICs can handle per-port policies => should take advantage of this feature, implying to set the communication ports in advance ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

320
**Design and Implementation Oriented Validation of Survivable Systems**

A. Agbaria, T. Courtney, M. Ihde, W. H. Sanders, M. Seri, and S. Singh Design Phase Validation Implementation Phase Validation A study of the design reveals that integrity and confidentiality can be regarded as probability-1 events. We obtain the following logical decomposition: PUB1: P[E1 E2| E3 E4 C] ≥ p PUB2: P[E3| C] = 1 PUB3: P[E4| C] = 1 It can be shown that: (PUB1 PUB2 PUB3) PUB Let PUB be the requirement of “successfully process a publish request”. Let C be the preconditions. Let E be the desired event, i.e., the successful of a request to publish. E is a conjunction of: E1 = the data flow of the publish is correct E2 = timeliness E3 = integrity E4 = confidentiality The requirement: PUB: P[E|C] ≥ p Attack Tree Step 1: Formulate a precise statement of R. Requirement Sub-requirements Step 2: If R is logically decomposable, decompose it iteratively. Logical Decomposition Yes Decomposable? Step 3: For every atomic requirement Ra Quantitative? Functional Model Access Proxy Model No Logical Argumentation Automatic construction Attack Graph Model Assumptions AA2: AP Application-layer Integrity AA3: AP Application-layer Confidentiality Yes Data Flow Step 4: Detailed description of components Supporting Arguments Private Key Confidentiality No Cryptography in AP Build high-level description of System and its operational environment No Unauthorized Direct Access No Unauthorized Indirect Access Not Preconfigured Not Reconfigurable ADF NIC services protected Keys Protected from Theft Keys Not Guessable Not valid DoD (CAC) Alg. Framework Key Length Key Lifetime Physical Protection of CAC device Protection of CAC Authentication Data No Compromise of Authorized Process Accessing CAC Verify assumptions & parameter values Step 5: Justify the modeling assumptions of Step 4 PKCS #11 Compliance Tamperproof Step 6: Construct a simulation model Probabilistic measures Infrastructure-level attacks Probabilistic model of the system and its operational environment Survivable Publish Subscribe System Management Staff Compare with requirement Core Executive Zone Quad 1 Quad 2 Quad 3 Quad 4 Step 7: Evaluation and comparing Operations Zone System not valid System valid w.r.t. the requirement Crumple Zone Client Zone Network Access Proxy (Isolated Process Domains in SE-Linux) Local Controller Domain6 First Restart Domains Eventually Restart Host Domain1 Domain2 Domain3 Domain4 Domain5 Forward/Rate limit Proxy Logic Inspect / Forward / Rate Limit PS Sensor Rpts DC PSQImpl PSQImpl TCP Eascii UDP TCP IIOP RMI TCP IIOP TCP ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author. University of Illinois at Urbana-Champaign I N F O R M A T I O N T R U S T I N S T I T U T E

321
**The Art of Dependability Evaluation / Conclusions**

©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

322
**Course Outline Revisited**

Issues in Model-Based Validation of High-Availability Computer Systems/Networks Stochastic Activity Network Concepts Analytic/Numerical State-Based Modeling Case Study: Embedded Fault-Tolerant Multiprocessor System Solution by Simulation The Art of System Dependability /Conclusions ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

323
**Model Solution Issues In general:**

Use “tricks” from probability theory to reduce complexity of model Choose the right solution method Simulation: Result is just an estimator based on a statistical experiment Estimation of accuracy of estimate essential Use confidence Intervals! Analytic/Numerical model solution: Avoid state space explosion Limit model complexity Use structure of model (symmetries) to reduce state space size Understand accuracy/limitations of chose numerical method Transient Solution (Iterative or Direct) Steady-state solution ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

324
**The “Art” of Performance and Dependability Validation**

Performance and dependability validation is an art because: There is no recipe for producing a good analysis, The key is knowing how to abstract away unimportant details, while retaining important components and relationships, This intuition only comes from experience, Experience comes from making mistakes. There are many ways to make mistakes. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

325
**Doing it Right: Model Construction**

Understand the desired measure before you build the model. The desired measure determines the type of model and the level of detail required. No model is universal! Steps in constructing a model: 1. Choose the desired measures: Choice of measures form a basis for comparison. It’s easy to choose wrong measure and see patterns where none exist. Measures should be refined during the design and validation process. 2. Choose the appropriate level of detail/abstraction for model components. Key is to represent model at the right level of detail for the chosen measures. It is almost never possible or practical to include all system aspects. Model the system at the highest level possible to obtain a good estimate of the desired measures. 3. Build the model. Decide how to break up the model into modules, and how the modules will interact with one another. Test the model as you build it, to ensure it executes as intended. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

326
**Doing it Right: Model Solution**

Use the appropriate model solution technique: Just because you have a hammer doesn’t mean the world is a nail. There is no universal model solution technique (not even simulation!) The appropriate model solution technique depends on model characteristics. Use representative input values: The results of a model solution are only as good as the inputs. The inputs will never be perfect. Understand how uncertainty in inputs affects measures. Do sensitivity analysis. Include important points in the design/parameter space: Parameterize choices when design or input values are not fixed. A complete parametric study is usually not possible. Some parameters will have to be fixed at “nominal” values. Make sure you vary the important ones. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

327
**Doing it Right: Model Interpretation/Documentation**

Make all your assumptions explicit: Results from models are only as good as the assumptions that were made in obtaining them. It’s easy to forget assumptions if they are not recorded explicitly. Understand the meaning of the obtained measures: Numbers are not insights. Understand the accuracy of the obtained measures, e.g., confidence intervals for simulation. Keep social aspects in mind: Performance and dependability analysts almost always bring bad news. Bearers of bad news are rarely welcomed. In presentations, concentrate on results, not the process. ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

328
**There are many places to go for further information: **

Next Steps You have: Learned theory related to reliability, availability, and performance validation using SANs and Möbius Learned about the advantages and disadvantages of various (analytical/numerical and simulation-based) solution algorithms. There are many places to go for further information: Möbius Software Web pages (www.mobius.uiuc.edu) Performability Engineering Research Group Web pages (www.perform.csl.uiuc.edu) ©2005 William H. Sanders. All rights reserved. Do not duplicate without permission of the author.

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