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Unformatted text preview: CS4 Modelling and Simulation LN-13 13 Random Variables and Simulation In this lecture note we consider the relationship between random variables and simulation models. Random variables play two important roles in simulation models. We assume that within our models some delays ( hold() in SimJava) will not have deterministic values, but instead will be represented by random variables; similarly when a choice must be made within the behaviour of an entity we will sometimes want the decision to be made probabilistically. Both cases will involve sampling a probability distribution to extract a value each time this part of the entity’s behaviour is reached. For delays we will use continuous random variables (e.g. Sim normal obj in SimJava) and for a choice we will use a boolean random variable (this is not explicitly supported in SimJava). As we will see below, both cases rely on the random number generator. As with the models we have considered earlier in the course, we assume that the vari- ables characterising the behaviour of the system/model, the performance measures or output parameters, are also random variables. For example, a measure might be the number of packets lost per million in a communication network, or the average waiting time for an access to a disk. In general, each run of the simulation model provides a single estimate for these random variables. If we are interested in steady state values the longer we run a simulation the better our estimate will be. However, it still remains a single observation in the sample space. We need more than a single estimate in order to draw conclusions about the system. Thus we use output analysis techniques to improve the quality of an estimate from a single model run, and to develop ways of gaining more observations without excessive computational cost. Realistic simulation models take a long time to run—there is always a trade-off between accuracy of estimates and execution time. 13.1 Random Number Generation Generating random values for variables with a specified random distribution, such as an exponential or normal distribution, involves two steps. First, a sequence of random numbers distributed uniformly between 0 and 1 is obtained. Then the sequence is trans- formed to produce a sequence of random values which satisfy the desired distribution. This second step is sometimes called random variate generation . To obtain a sequence of uniform random numbers, it is suﬃcient to be able to generate a sequence X k of integers in the range [0 , M − 1] since the sequence X k / ( M − 1) will then be approximately uniformly distributed over (0 , 1). In 1951, D.H. Lehmer discovered that the residues of successive powers of a number have good randomness properties. He obtained the k th number in the sequence by dividing the k th power of an integer a by another integer M and taking the remainder....
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- Spring '10
- Probability distribution, Probability theory, probability density function, Cumulative distribution function, random number