Simulation Steady - State Simulation

We will discuss several techniques for obtaining point estimates and confidence intervals for 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, for s∈S { } } s X { P s) t (X P lim t = = = μ ∞ → For a sequence (Z n : n ≥ 0) of real-valued random variables, we say that Z n ⇒ Z if ()) 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)|] < ∞.