1 Performance Evaluation of Computer Networks: Part II Objectives r Simulation Modeling r Classification of Simulation Modeling r Discrete-Event Simulation Modeling  Open Jackson networks

2 Simulation Modeling  A simulation model is a computer program that  Replicates the operations of a business process  Estimates rates at which outputs are produced and resources are consumed  Simulation models test the consistency of the facts, logic, and assumptions  Simulation models test the sensitivity of a process to changes in selected assumptions  Simulation models help decision makers to assess the potential benefits, costs, and risks of alternative processes and strategies

3 Classification of Simulation Models  Static vs. Dynamic Simulation Model  Static Simulation Model is a representation of a system at a particular point in time (i.e., time plays no role oEstimating the probability of winning a soccer game oEstimating the value of π  Dynamic Simulation Model is a representation of a system as it evolves over time oExamples include models of a bank, a processor

4 Classification of Simulation Models  Deterministic vs. Stochastic Simulation Models  Deterministic Simulation Model does not contain any probabilistic components Example: a system of differential equations representing a chemical reaction Output are also deterministic r Stochastic Simulation models are those having at least some random input components m Examples include Queuing models (Interarrival times between two consecutive customers and service times are usually random) m They produce output that are also random

5 Classification of Simulation Models  Continuous vs. Discrete Simulation Models  Discrete Simulation models those representing systems whose state changes at discrete points of time  Continuous Simulation models are those that changes of the system occur continuously as the time evolves

6 Discrete-event Simulation Model  Simulation models we consider here are discrete, dynamic, and stochastic. Such models are called Discrete-Event Simulation Models  Changes occur at a separate points of time r What does it change the system state? Events m Event: is an instantaneous occurrence that changes the state of the system m Examples: Arrival of a new customer, a Departure of a customer from a queuing model

7 Discrete-event Simulation Model: Time- Advance Mechanism r Simulation Clock: is a variable in the simulation model that keeps track of the current simulation time (does not depend on the computer time) Simulation clock 0  There are 2 approaches for advancing the simulation clock : 1.Next-event time advance 2.Fixed-increment time advance 0 tt2  t3  t4  t

8 Discrete-event Simulation Model: Next- Event Time Advance r The most common used approach : 1.The simulation clock is initialized to zero 2.Time of occurrence of future events are determined 3.The simulation clock is then advanced to the time of the occurrence of the next event (the event that is scheduled to occur first) 4.The system is updated taking in account that the event has occurred 5.Update the time of the occurrence of the next events 6.Go to step 3 7.Repeat until a stopping criterion is satisfied.

9 Development and Validation of Simulation Models r Building a conceptual model r Verification Phase r Validation Phase

10 Simulation: Verification Phase r The result of the verification phase answers the following question: m Does the simulation code implement the conceptual model correctly? r Sometimes this check can be done formally m A requirement for this is that the system specification (in our case: the conceptual model of the system) is written in a formal language The Z Formal Specification Language r In many cases only certain aspects of a software system can be verified formally, other aspects have to be tested or checked by other means m code inspection, careful inspection of the generated results / event sequences, etc

11 Simulation: Validation Phase r The result of the validation phase answers the following question: m Is the conceptual model correct or at least adequate? r The reason to ask for adequacy is as follows: r In many real-world applications the requirements are vague, unspecific, random and changing over time. m For example, call arrivals to a central office: do they form a Poisson process? Clearly, the correct answer is: no, m However, it is often observed that the Poisson process is a reasonable or adequate m Therefore, it is not too dangerous to use Poisson arrivals as a model

12 Background: The Poisson Distribution r For our purposes, customers arrive at a location in a truly random fashion. That is, there is no way to predict exactly when someone will arrive r The Poisson distribution describes the number of such arrivals during a given period

13 The Exponential Distribution r If the number of arrivals at a location occurs according to the Poisson distribution then, automatically, the distribution of the intervals between successive arrivals must follow the exponential distribution

14 Probability of t  A r If t is an exponentially distributed random variable then the probability that t takes on a value less than a constant, A, is given by the expression:

15 Example r The average interarrival time for customers at a system is 2 minutes and we want to determine the probability that the time of the next arrival occurs: m Within one minute m Within 2 minutes m Between 1 and 2 minutes hence

16 Example, continued

17 Application to Queuing Theory r Packet–switched networks get congested! r Congestion occurs when the number of packets transmitted approaches network capacity r Objective of congestion control: m keep the number of packets entering the network below the level at which performance drops off dramatically r Can we use our queuing theory knowledge to tackle the congestion control problem?

18 At Saturation Point Two Possible Strategies at Node: 1. Discard any incoming packet if no buffer space is available. Therefore look at the Queue Size 2. Exercise flow control over neighbors m May cause congestion to propagate throughout network m Try to control other entity’s queue size

19 Components of A Simple System r Arrivals: Poisson, renewal, general ergodic processes. r Service: exponential service times, general distributions. r Capacity: finite buffer (size=N) or infinite buffer. r Control: scheduling policy (e.g. FCFS, LCFS, etc) Departure Arrival Queue/BufferServer job/task/customer

20 Two Types of Network Topology r Open Networks: all customers can leave the network r Closed Networks: No customers can leave the network

21 (Open) Jackson Networks r There are J queues r Customers arrive at queue l according to independent Poisson processes with rate r The service times in queue l are exponential with rates r Upon leaving queue l, each customer is sent to queue m with probability and leaves the network with probability r The routing decision is independent of the past evolution of the network

22 An Open Jackson Network i j k m