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lecturenotes11 - MS&E 223 Simulation Brad Null Lecture...

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MS&E 223 Lecture Notes #11 Simulation Efficiency-Improvement Techniques Brad Null Spring Quarter 2008-09 Page 1 of 9 Efficiency-Improvement Techniques Ref : Law, Chapter 11, Handbook of Simulation, Ch. 10 1. Variance Reduction and Efficiency Improvement The techniques that we will look at typically are designed to reduce the variance of the estimator for the performance measure that we are trying to estimate via simulation. (Reduction of the variance leads to narrower confidence intervals, and hence less computational effort is required to achieve a given level of precision.) The reduction is measured relative to the variance obtained when using “straightforward simulation.” For this reason these techniques are sometimes called variance-reduction methods. One must be careful when evaluating these techniques: they are only worthwhile if the reduction in computer effort due to the variance reduction outweighs the increase in computational effort needed to execute the technique during the simulation, as well as the additional programming complexity. For many of the methods that we will look at, it is obvious that the additional effort to implement the variance reduction is small, so the technique is a clear win. How do we deal with more complicated situations? One fair way to compare two estimation methods is to assume a fixed computer budget of c time units, and compare the confidence-interval width for the two methods at the end of the simulation. Example : Suppose α = E[X] = E[Y]. Should we generate i.i.d. replicates of X or Y in order to estimate α ? Let c be the computer budget. Then, the number of X-observations generated within budget c is N X (c) = max {n 0 : τ X (1) + ... + τ X (n) c} , where τ X (i) is the (random) computer time required to generate X i . Assume (reasonably) that (X 1, τ X (1)), (X 2, τ X (2)), ... is a sequence of i.i.d. pairs. It can be shown that (c) N c 1 X λ X with probability 1 as c , where λ X = ] E[ 1 X τ . Now, note that α X (c) = (c) N 1 = i i X X X (c) N 1 (set α X (c) = 0 if N X (c) = 0)
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MS&E 223 Lecture Notes #11 Simulation Efficiency-Improvement Techniques Brad Null Spring Quarter 2008-09 Page 2 of 9 is the estimator for α (based on X) obtained after expending budget c. Since (c) N X c X λ , it follows that α X (c) - α = α - X (c) N 1 (c) N 1 = i i X X λ = α - λ c 1 i i X X X c 1 D N(0,1) c ] X [ Var X λ = N(0,1) Var[X] ] E[ c 1 X τ , Similarly, if α Y (c) is the estimator based on i.i.d. replications of Y, α Y (c) - α D N(0,1) Var[Y] ] E[ c 1 Y τ . Clearly, we should choose the estimator that minimizes the product of the mean computer time per observation and the variance per observation . This conclusion holds more generally, so the inverse of this product is a reasonable measure of efficiency. 2. Common Random Numbers (CRN) This method applies when comparing two or more alternate system configurations.
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  • Spring '09
  • Probability theory, Randomness, Brad Null, Techniques Spring Quarter, Efficiency-Improvement Techniques Spring

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