Sensitivity analysis of discrete event systems by the “push out” method

Abstract

Let$l(v) = E_{v_1 } \{ L(Y,v_2 )\}$ be the expected performance measure of adiscrete event system (DES), whereL is the sample performance depending on the vector of parametersv ₂ and driven by an input vectorY, which has a probability density function (pdf)f(y, v ₁),v=(v ₁,v ₂) is a vector of parameters, and the subscriptv ₁ in E$E_{v_1 } L$ indicates that the expectation is taken with respect to the pdff(y, v ₁). Suppose thatl(v) is not available analytically and we want to evaluate (estimate) it, as well as the associated sensitivities ∇^k l(v),k=1, 2, ...simultaneously for different values ofv=(v ₁,v ₂) via simulation. In this paper, we show that in some cases interesting for applications, we can estimatel(v) and ∇^k l(v),k=1,2, ... by using the so-called “push out” technique. More precisely, we show that it is possible to replace the original sample performance by an auxiliary one while “pushing out” the parameter vectorv ₂ from the original sample performance functionL(Y,v ₂) to a pdf$\tilde f$(x,v ₁,v ₂) associated with the original onef(y,v ₁). We also show how both the auxiliary sample performance and the associated pdf can be obtained from their original counterparts and how to combine them together to perform sensitivity analysis for the original DES. Particular emphasis will be placed on the case where the sample performance functionL(y,v ₂) isneither analytically available nor everywhere differentiable in v ₂. We finally discuss the advantage of the proposed method and present numerical results supporting our theory.