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Introduction To Simulation

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2 Overview Simulation: Key Questions Common Mistakes in Simulation Other Causes of Simulation Analysis Failure Checklist for Simulations Terminology Types of Models

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3 Simulation: Key Questions What are the common mistakes in simulation and why most simulations fail? What language should be used for developing a simulation model? What are different types of simulations? How to schedule events in a simulation? How to verify and validate a model? How to determine that the simulation has reached a steady state? How long to run a simulation?

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4 Simulation: Key Questions (cont ’ d) How to generate uniform random numbers? How to verify that a given random number generator is good? How to select seeds for random number generators? How to generate random variables with a given distribution? What distributions should be used and when?

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5 Common Mistakes in Simulation 1. Inappropriate Level of Detail: More detail More time More Bugs More CPU More parameters More accurate 2. Improper Language General purpose More portable, More efficient, More time 3. Unverified Models: Bugs 4. Invalid Models: Model vs. reality 5. Improperly Handled Initial Conditions 6. Too Short Simulations: Need confidence intervals 7. Poor Random Number Generators: Safer to use a well-known generator 8. Improper Selection of Seeds: Zero seeds, Same seeds for all streams

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6 Other Causes of Simulation Analysis Failure 1. Inadequate Time Estimate 2. No Achievable Goal 3. Incomplete Mix of Essential Skills (a) Project Leadership (b) Modeling and (c) Programming (d) Knowledge of the Modeled System 4. Inadequate Level of User Participation 5. Obsolete or Nonexistent Documentation 6. Inability to Manage the Development of a Large Complex Computer Program Need software engineering tools 7. Mysterious Results

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7 Checklist for Simulations 1. Checks before developing a simulation: (a) Is the goal of the simulation properly specified? (b) Is the level of detail in the model appropriate for the goal? (c) Does the simulation team include personnel with project leadership, modeling, programming, and computer systems backgrounds? (d) Has sufficient time been planned for the project? 2. Checks during development: (a) Has the random number generator used in the simulation been tested for uniformity and independence? (b) Is the model reviewed regularly with the end user? (c) Is the model documented?

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8 Checklist for Simulations (cont ’ d) 3.Checks after the simulation is running: (a) Is the simulation length appropriate? (b) Are the initial transients removed before computation? (c) Has the model been verified thoroughly? (d) Has the model been validated before using its results? (e) If there are any surprising results, have they been validated? (f) Are all seeds such that the random number streams will not overlap?

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9 Terminology Introduce terms using an example of simulating CPU scheduling Study various scheduling techniques given job characteristics, ignoring disks, display… State Variables: Define the state of the system Can restart simulation from state variables E.g., length of the job queue. Event: Change in the system state E.g., arrival, beginning of a new execution, departure

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10 Terminology: Types of Models Continuous Time Model State is defined at all times Discrete Time Models State is defined only at some instants

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11 Terminology: Types of Models (cont ’ d) Continuous State Model State variables are continuous Discrete State Models State variables are discrete

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12 Terminology: Types of Models (cont ’ d) Discrete state = Discrete event model Continuous state = Continuous event model Continuity of time Continuity of state Four possible combinations 1. discrete state/discrete time 2. discrete state/continuous time 3. continuous state/discrete time 4. continuous state/continuous time

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13 Terminology: Types of Models (cont ’ d) Deterministic and Probabilistic Models Deterministic - If output predicted with certainty Probabilistic - If output different for different repetitions

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14 Terminology: Types of Models (cont ’ d) Static and Dynamic Models Static - Time is not a variable Dynamic - If changes with time Linear and nonlinear models Linear - Output is linear combination of input Nonlinear - Otherwise Output Input (Linear) Output Input (Non-Linear)

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15 Terminology: Types of Models (cont ’ d) Open and closed models Open - Input is external and independent Closed - Model has no external inputs Ex: if same jobs leave and re-enter queue then closed, while if new jobs enter system then open cpu open cpu closed

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16 Terminology: Types of Models (cont ’ d) Stable and unstable Stable - Model output settles down Unstable - Model output always changes

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17 Computer System Models Continuous time Discrete state Probabilistic Dynamic Nonlinear Open or closed Stable or unstable

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18 Selecting a Language for Simulation Four choices 1. Simulation language 2. General purpose 3. Extension of a general purpose language 4. Simulation package

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19 Selecting a Language for Simulation (cont ’ d) Simulation language – built in facilities for time steps, event scheduling, data collection, reporting General-purpose – known to developer, available on more systems, flexible The major difference is the cost tradeoff (SL vs. GPL) SL+: save development time (if you know it), more time for system specific issues, more readable code SL-: requires startup time to learn GPL+: Analyst's familiarity, availability, quick startup GPL-: may require more time to add simulation flexibility, portability, flexibility Recommendation may be for all analysts to learn one simulation language so understand those “costs” and can compare

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20 Selecting a Language for Simulation Extension of general-purpose – collection of routines and tasks commonly used. Often, base language with extra libraries that can be called Simulation packages – allow definition of model in interactive fashion. Get results in one day Tradeoff is in flexibility, where packages can only do what developer envisioned, but if that is what is needed then is quicker to do so Examples: GASP (for FORTRAN) Collection of routines to handle simulation tasks Compromise for efficiency, flexibility, and portability. Examples: QNET4, and RESQ Input dialog Library of data structures, routines, and algorithms Big time savings Inflexible Simplification

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21 Types of Simulation Languages Continuous Simulation Languages CSMP, DYNAMO Differential equations Used in chemical engineering Discrete-event Simulation Languages SIMULA and GPSS Combined SIMSCRIPT and GASP Allow discrete, continuous, as well as combined simulations.

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22 Types of Simulations 1. Emulation: Using hardware or firmware 2. Monte Carlo Simulation 3. Trace-Driven Simulation 4. Discrete Event Simulation

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23 Types of Simulations (cont ’ d) Emulation Simulation that runs on a computer to make it appear to be something else Examples: JVM, NIST Net Operating System Hardware Process Java program Java VM

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24 Types of Simulation (cont ’ d) Monte Carlo method [Origin: after Count Montgomery de Carlo, Italian gambler and random-number generator ( ).] A method of jazzing up the action in certain statistical and number-analytic environments by setting up a book and inviting bets on the outcome of a computation. - The Devil's DP Dictionary McGraw Hill (1981)

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25 Monte Carlo Simulation A static simulation has no time parameter Runs until some equilibrium state reached Used to model physical phenomena, evaluate probabilistic system, numerically estimate complex mathematical expression Driven with random number generator So “Monte Carlo” (after casinos) simulation Example, consider numerically determining the value of Area of circle = for radius 1

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26 Monte Carlo Simulation (cont ’ d) Imagine throwing dart at square Random x (0,1) Random y (0,1) Count if inside sqrt(x 2 +y 2 ) < 1 Compute ratio R in / (in + out) Can repeat as many times as needed to get arbitrary precision Unit square area of 1 Ratio of area in quarter to area in square = R = 4R

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27 Monte Carlo Simulation (cont ’ d) Evaluate the following integral 1. Generate uniformly distributed x ~ Uniform(0,2) 2. Density function f(x)=1/2 iff 0 x 2 3. Compute:

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28 Monte Carlo Simulation (cont ’ d) Expected value for y:

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29 Trace-Driven Simulation Uses time-ordered record of events on real system as input Example: to compare memory management, use trace of page reference patterns as input, and can model and simulate page replacement algorithms Note, need trace to be independent of system Example: if had trace of disk events, could not be used to study page replacement since events are dependent upon current algorithm

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30 Advantages of Trace-Driven Simulations 1. Credibility 2. Easy Validation: Compare simulation with measured 3. Accurate Workload: Models correlation and interference 4. Detailed Trade-Offs: Detailed workload Can study small changes in algorithms 5. Less Randomness: Trace deterministic input Fewer repetitions 6. Fair Comparison: Better than random input 7. Similarity to the Actual Implementation: Trace-driven model is similar to the system Can understand complexity of implementation

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31 Disadvantages of Trace-Driven Simulations 1. Complexity: More detailed 2. Representativeness: Workload changes with time, equipment 3. Finiteness: Few minutes fill up a disk 4. Single Point of Validation: One trace = one point 5. Detail 6. Trade-Off: Difficult to change workload

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32 Discrete Event Simulations A simulation using a discrete state model of the system is DISCRETE EVENT SIMULATION Continuous-event simulations – the state of the system takes continuous values Typical components: Event scheduler Simulation Clock and a Time Advancing Mechanism System State Variables Event Routines Input Routines Report Generator Initialization Routines Trace Routines Dynamic Memory Management Main Program

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33 Components of Discrete Event Simulations Event scheduler – linked list of events waiting Schedule event X at time T Hold event X for interval dt Cancel previously scheduled event X Hold event X indefinitely until scheduled by other event Schedule an indefinitely scheduled event Note, event scheduler executed often, so has significant impact on performance Simulation Clock and a Time Advancing Mechanism Global variable representing simulated time (maintained by the scheduler) Two approaches Unit-time approach: increment time and check for events Event-driven approach: move to the next event in queue

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34 Components of Discrete Events Sims (cont ’ d) System State Variable Global variables describing the state of the systems (e.g., the umber of jobs in CPU scheduling simulation) Local variables (e.g., CPU time required for a job is placed in the data structure for that particular job) Event Routines -- one per event; update state variables and schedule other events E.g., job arrivals, job scheduling, and job departure Input Routines Get model parameters (e.g., means CPU time per job) from the user Very parameters in a range

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35 Components of Discrete Events Sims (cont ’ d) Report Generator Output routines run at the end of the simulation Initialization Routines Set the initial state of the system state variables. Initialize seeds. Trace Routines Print out intermediate variables as the simulation proceeds On/off feature Dynamic Memory Management New entities are created and old ones are destroyed Periodic garbage collection Main Program Tie everything together

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36 Event-Set Algorithms Event Set = Ordered linked list of future event notices Insert vs. Execute next 1. Ordered Linked List: SIMULA, GPSS, and GASP IV Search from left or from right HeadTail Next Previous Event n Next Previous Event 1 Next Previous Event 2 Code for event 1 Code for event 2 Code for event n

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37 Event-Set Algorithms (cont ’ d) 2. Indexed Linear List Array of indexes No search to find the sub-list Fixed or variable t. Only the first list is kept sorted Head 1Tail 1 Head 3Tail 3 Head 2Tail 2 t t+ tt+ t t+n tt+n t

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38 Event-Set Algorithms (Cont) 3. Tree Structures: Binary tree log2 n Special case: Heap: Event is a node in binary tree (a) Tree representation of a heap

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39 Summary Common Mistakes: Detail, Invalid, Short Discrete Event, Continuous time, nonlinear models Monte Carlo Simulation: Static models Trace driven simulation: Credibility, difficult trade-offs Even Set Algorithms: Linked list, indexed linear list, heaps

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Analysis of Simulation Results

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41 Overview Analysis of Simulation Results Model Verification Techniques Model Validation Techniques Transient Removal Terminating Simulations Stopping Criteria: Variance Estimation Variance Reduction

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42 Model Verification vs. Validation The model output should be close to that of real system Make assumptions about behavior of real systems 1 st step, test if assumptions are reasonable Validation, or representativeness of assumptions 2 nd step, test whether model implements assumptions Verification, or correctness Four Possibilities 1.Unverified, Invalid 2.Unverified, Valid 3.Verified, Invalid 4.Verified, Valid

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43 Model Verification Techniques 1.Top Down Modular Design 2.Anti-bugging 3.Structured Walk-Through 4.Deterministic Models 5.Run Simplified Cases 6.Trace 7.On-Line Graphic Displays 8.Continuity Test 9.Degeneracy Tests 10.Consistency Tests 11.Seed Independence

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44 Top Down Modular Design Divide and Conquer Modules = Subroutines, Subprograms, Procedures Modules have well defined interfaces Can be independently developed, debugged, and maintained Top-down design Hierarchical structure Modules and sub-modules

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45 Top Down Modular Design (cont ’ d) Computer Network Simulator for Congestion Control studies

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46 Top Down Modular Design (cont ’ d)

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47 Verification Techniques Anti-bugging: Include self-checks Probabilities = 1 Jobs left = Generated - Serviced Structured Walk-Through Explain the code another person or group Works even if the person is sleeping Deterministic Models: Use constant values Run Simplified Cases Only one packet Only one source Only one intermediate node

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48 Verification Techniques (cont ’ d) Trace = Time-ordered list of events and variables Several levels of detail Events trace Procedure trace Variables trace User selects the detail Include on and off

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49 Verification Techniques (cont ’ d) On-Line Graphic Displays Make simulation interesting Help selling the results More comprehensive than trace

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50 Verification Techniques (cont ’ d) Continuity Test Run for different values of input parameters Slight change in input slight change in output If not, investigate Before After

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51 Verification Techniques (cont ’ d) Degeneracy Tests: Try extreme configuration and workloads One CPU, Zero disk Consistency Tests Similar result for inputs that have same effect Four users at 100 Mbps vs. Two at 200 Mbps Build a test library of continuity, degeneracy and consistency tests Seed Independence: Similar results for different seeds

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52 Model Validation Techniques Ensure assumptions used are reasonable Final simulated system should be like the real system Unlike verification, techniques to validate one simulation may be different from one model to another Three key aspects to validate Assumptions Input parameter values and distributions Output values and conclusions Compare validity of each to one or more of Expert intuition Real system measurements Theoretical results 9 combinations - Not all are always possible, however

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53 Expert Intuition Most practical and common way Experts = Involved in design, architecture, implementation, analysis, marketing, or maintenance of the system Present assumption, input, output Better to validate one at a time See if the experts can distinguish simulation vs. measurement Throughput Which alternative looks invalid? Why?

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54 Real System Measurements Most reliable and preferred May be unfeasible because system does not exist or too expensive to measure That could be why simulating in the first place! But even one or two measurements add an enormous amount to the validity of the simulation Should compare input values, output values, workload characterization Use multiple traces for trace-driven simulations Can use statistical techniques (confidence intervals) to determine if simulated values different than measured values

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55 Theoretical Results Can be used to compare a simplified system with simulated results May not be useful for sole validation but can be used to complement measurements or expert intuition E.g.: measurement validates for one processor, while analytic model validates for many processors Note, there is no such thing as a “fully validated” model Would require too many resources and may be impossible Can only show is invalid Instead, show validation in a few select cases, to lend confidence to the overall model results

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56 Transient Removal Most simulations only want steady state Remove initial transient state Trouble is, not possible to define exactly what constitutes end of transient state Use heuristics: Long runs Proper initialization Truncation Initial data deletion Moving average of replications Batch means

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57 Long Runs Use very long runs Effects of transient state will be amortized But … wastes resources And tough to choose how long is “enough” Recommendation … don’t use long runs alone

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58 Proper Initialization Start simulation in state close to expected state Ex: CPU scheduler may start with some jobs in the queue Determine starting conditions by previous simulations or simple analysis May result in decreased run length, but still may not provide confidence that are in stable condition

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59 Truncation Assume variability during steady state is less than during transient state Variability measured in terms of range (min, max) If a trajectory of range stabilizes, then assume that in stable state Method: Given n observations {x 1, x 2, …, x n } Ignore first l observations Calculate (min,max) of remaining n-l Repeat for l = 1…n Stop when l+1th observation is neither min nor max

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60 Truncation Example Sequence: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 10, 9, 10, 11, 10, 9… Ignore first (l=1), range is (2, 11) and 2 nd observation (l+1) is the min Ignore second (l=2), range is (3,11) and 3 rd observation (l+1) is min Finally, l=9 and range is (9,11) and 10 th observation is neither min nor max So, discard first 9 observations Transient Interval

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61 Truncation Example 2 (1 of 2) Find duration of transient interval for: 11, 4, 2, 6, 5, 7, 10, 9, 10, 9, 10, 9, 10

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62 Truncation Example 2 (2 of 2) Find duration of transient interval for: 11, 4, 2, 6, 5, 7, 10, 9, 10, 9, 10, 9, 10 When l=3, range is (5,10) and 4 th (6) is not min or max So, discard only 3 instead of 6 “Real” transient Assumed transient

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63 Initial Data Deletion (1 of 3) Study average after some initial observations are deleted from sample If average does not change much, must be deleting from steady state However, since randomness can cause some fluctuations during steady state, need multiple runs (w/different seeds) Given m replications size n each with x ij – jth observation of ith replication Note j varies along time axis and i varies across replications

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64 Initial Data Deletion (2 of 3) Get mean trajectory: x j = (1/m) x ij j=1,2,…,n Get overall mean x = (1/n) x j j=1,2,…,n Set l=1. Assume transient state l long, delete first l and repeat for remaining n-l x l = (1/(n-l)) x j j=l+1,…,n Compute relative change (x l – x) / x Repeat with l from 1 to n-1. Plot. Relative change graph will stabilize at knee. Choose l there and delete 1 through l

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65 Initial Data Deletion (3 of 3) x ij j xjxj j xlxl l (x l – x) / x l transient interval knee

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66 Moving Average of Independent Replications Compute mean over moving time window Get mean trajectory x j = (1/m) x ij j=1,2,…,n Set k=1. Plot moving average of 2k+1 values: Mean x j = 1/(2k+1) (x j+l ) With j=k+1, k+2,…,n-k With l=-k to k Repeat for k=2,3… and plot until smooth Find knee. Value at j is length of transient phase. xjxj j Mean x j j transient interval knee

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67 Batch Means Run for long time N observations Divide up into batches m batches size n each so m = N/n Compute batch mean (x i ) Compute var of batch means as function of batch size (X is overall mean) Var(x) = (1/(m-1)) (x i -X) 2 Plot variance versus size n When n starts decreasing, have transient Responses Observation number n2n3n4n5n Variance of batch means transient interval Batch size n (Ignore)

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68 Terminating Simulations For some simulations, transition state is of interest no transient removals required Sometimes upon termination you also get final conditions that do not reflect steady state Can apply transition removal conditions to end of simulation Take care when gathering at end of simulation E.g.: mean service time should include only those that finish Also, take care of values at event times E.g.: queue length needs to consider area under curve Say t=0 two jobs arrive, t=1 one leaves, t=4 2 nd leaves qlengths q 0 =2, q 1 =1 q 4 =0 but q average not (2+1+0)/3=1 Instead, area is so q average 5/4=1.25

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69 Stopping Criteria: Variance Estimation Run until confidence interval is narrow enough For Independent observations: Independence not applicable to most simulations Large waiting time for ith job Large waiting time for (i+1)th job For correlated observations:

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70 Variance Estimation Methods 1. Independent Replications 2. Batch Means 3. Method of Regeneration

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71 Independent Replications Assumes that means of independent replications are independent Conduct m replications of size n+n 0 each 1. Compute a mean for each replication: 2. Compute an overall mean for all replications:

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72 Independent Replications (cont ’ d) 3. Calculate the variance of replicate means: 4. Confidence interval for the mean response is: Keep replications large to avoid waste Ten replications generally sufficient

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73 Batch Means Also called method of sub-samples Run a long simulation run Discard initial transient interval, and Divide the remaining observations run into several batches or sub-samples. 1. Compute means for each batch: 2. Compute an overall mean:

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74 Batch Means (cont ’ d) 3. Calculate the variance of batch means: 4. Confidence interval for the mean response is: Less waste than independent replications Keep batches long to avoid correlation Check: Compute the auto-covariance of successive batch means: Double n until autocovariance is small

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75 Case Study 25.1: Interconnection Networks Indirect binary n-cube networks: Used for processor-memory interconnection Two stage network with full fan out. At 64, autocovariance < 1% of sample variance

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76 Method of Regeneration Behavior after idle period does not depend upon the past history System takes a new birth Regeneration point Note: The regeneration point are the beginning of the idle interval. (not at the ends as shown in the book). Regeneration Points Queue Length

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77 Method of Regeneration (cont ’ d) Regeneration cycle: Between two successive regeneration points Use means of regeneration cycles Problems: Not all systems are regenerative Different lengths Computation complex Overall mean Average of cycle means Cycle means are given by:

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78 Method of Regeneration (cont ’ d) Overall mean: 1. Compute cycle sums: 2. Compute overall mean: 3. Calculate the difference between expected and observed cycle sums:

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79 Method of Regeneration (cont ’ d) 4. Calculate the variance of the differences: 5. Compute mean cycle length: 6. Confidence interval for the mean response is given by: 7. No need to remove transient observations

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80 Method of Regeneration: Problems 1. The cycle lengths are unpredictable. Can't plan the simulation time beforehand. 2. Finding the regeneration point may require a lot of checking after every event. 3. Many of the variance reduction techniques can not be used due to variable length of the cycles. 4. The mean and variance estimators are biased

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81 Variance Reduction Reduce variance by controlling random number streams Introduce correlation in successive observations Problem: Careless use may backfire and lead to increased variance. For statistically sophisticated analysts only Not recommended for beginners

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82 Summary Verification = Debugging Software development techniques Validation Simulation = Real Experts involvement Transient Removal: Initial data deletion, batch means Terminating Simulations = Transients are of interest Stopping Criteria: Independent replications, batch means, method of regeneration Variance reduction is not for novice

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