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lecturenotes10 - MS&E 223 Simulation Brad Null Lecture...

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MS&E 223 Lecture Notes #10 Simulation Steady-State Simulation Brad Null Spring Quarter 2008-09 Page 1 of 12 Steady-State Simulation Reading: Law, Chapter 9, Handbook of Simulation , Chapter 15. We will discuss several techniques for obtaining point estimates and confidence intervals for steady-state performance measures. 1. Steady-State Performance Measures Let (X(t): t 0) be a GSMP with (discrete) state space S that represents the underlying stochastic process of a simulation. Fix a real-valued function f defined on S and set Y(t) = f(X(t)) for t 0. There are several possible ways to define steady-state performance measures. Definition 1 : We say that (Y(t): t 0) has a well-defined time-average limit α if 1 du ) u ( Y lim P t 0 t 1 t = α = μ for any initial distribution μ of the GSMP. Remark : When we say “for any initial distribution μ ”, we mean, strictly speaking, that for a family of GSMP’s indexed by μ (all other building blocks the same) the above limit statement holds for each member of the family. Definition 2 : We say that (Y(t): t 0) has a well-defined steady-state mean α if there exists a random variable X such that X(t) X and α = E[f(X)] exists. Remark : Recall that X(t) X means that (X(t): t 0) converges in distribution to X. That is, for any initial distribution μ , { } } s X { P s ) t ( X P lim t = = = μ for s S For a sequence (Z n : n 0) of real-valued random variables, we say that Z n Z if { } } x Z { P x Z P lim n n = for all points x at which the distribution function of Z is continuous. E.g., ( 29 ) 1 , 0 ( N X n X n μ - σ in the CLT. Remark : A necessary and sufficient condition for E[f(X)] to exist is that E[|f(X)|] < .
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MS&E 223 Lecture Notes #10 Simulation Steady-State Simulation Brad Null Spring Quarter 2008-09 Page 2 of 12 Remark : In the GSMP case, if we fix μ and write p(t,s) = P μ {X(t) = s} and π (s) = P{X = s}, then convergence in distribution means that p(t,s) π (s) as t for all s S. We can then write E[f(X)] = π S s ) s ( ) s ( f Remark : If the function f is bounded and continuous, then X(t) X implies that E[f(X(t))] E[f(X)] as t , so that α (in Definition 2) also can be viewed as a limiting expected value. In general, the existence of one kind of steady-state performance measure does not imply the existence of any other steady-state performance measure. When different measures exist, they will usually coincide. We will focus primarily on time-average limits. 2. Appropriateness of Steady-State Analysis Steady-state analysis can be convenient for several reasons: (1) The time-average limit α provides an approximation for the cumulative cost C(t) = t 0 ds Y(s) , namely C(t) t α . Given that we are often interested in such costs, this can be a very useful approximation, provided t is large. Note that the distribution and moments of C(t) are hard to compute exactly, even when the underlying process is a CTMC.
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