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Unformatted text preview: CS4 Modelling and Simulation LN11 11 Simulation Models—Introduction and Motivation So far in the course the stochastic models which we have considered have been solved ana lytically . What this means is that by carrying out analysis of the system we have been able to deduce the steady state behaviour of the corresponding stochastic process, expressed as a probability distribution over the possible states. Such models can be regarded as a mathematical abstraction of the system. The analytic model is a representation which can be analysed mathematically to deduce the behaviour of the system. In contrast, a stochastic simulation model can be regarded as an algorithmic abstraction of the system—a simulation model gives a representation which when executed reproduces the behaviour of the system. We still assume that the system is characterised by a family of random variables { X ( t ) , t ∈ T } . As the value of time increases, and in response to the “environment” (represented by random variables within the model) the stochastic process progresses from state to state. Any set of instances of { X ( t ) , t ∈ T } can be regarded as a path of a particle moving randomly in a state space, S , its position at time t being X ( t ). These paths are called sample paths . Using the analytic approach of Markov processes we char acterised all possible sample paths by the global balance equations. Using simulation we investigate the sample paths directly. In other words we allow the model to trace out a sample path over the state space. Each run of the simulation model will generate another, usually distinct, sample path. There are many reasons why simulation may be preferable to analytic modelling. Level of Abstraction As we have seen, Markovian modelling relies on many assump tions and abstractions—these may not be appropriate for the system being studied. For example, it may be unrealistic to assume that only one event can happen at any time, or that the interevent times are all exponentially distributed. Simulation models allow us to represent a system at arbitrary levels of detail, although you should remember that there is always a tradeoff between how elaborate the model is and how long it takes to produce a statistically significant run. Transient Analysis In some cases we are not interested in the steady state behaviour of a system, but in its transient behaviour. Some systems never reach a steady state. Even those that do will usually have a “warmup” period while the behaviour settles into the regular pattern which characterises steady state. The analytic solutions we have considered earlier in the course ignore this period since the global balance equations only capture the behaviour after steady state has been reached. However it is not always possible to extract all the information we want about a system from its steady state behaviour. A sample path derived from a simulation model will clearly represent transient behaviour in addition to steady state...
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This note was uploaded on 04/08/2010 for the course COMPUTER E 409232 taught by Professor Mohammadabdolahiazgomiph.d during the Spring '10 term at Islamic University.
 Spring '10
 MohammadAbdolahiAzgomiPh.D

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