Finite Perturbation Analysis

One can view the basic goal of PA for DEDS as the reconstruction of an arbitrarily perturbed sample path from a nominal path. Under deterministic similarity, the simple infinitesimal perturbation analysis (IPA) rules described in Chapters 3 and 4 and extended in Chapter 5 compute a perturbed path infinitesimally different from the nominal for the purpose of gradient calculation. The computation of the perturbation propagations can be efficiently done since the “critical timing path” or the “future event schedule” between the nominal and perturbed paths remains the same. However, as pointed out before in the limit of a path of a very long duration or an experiment with very large ensembles of runs, deterministic similarity will always be violated1. In such cases, the IPA rules which ignore the order changes of events have been proved to give unbiased and consistent estimates for performance gradients of only certain classes of DEDS (see Chapters 4 and 5). Although the domain of application of IPA is constantly being expanded, it is nevertheless important to devise methodologies which can overcome the basic problem of “event sequence order change,” or “discontinuous performance measure.” In other words, the key question is “how can one reconstruct the perturbed path from the nominal path short of essentially running a separate new simulation I experiment?” In this and the next chapter we address this question directly and suggest an alternative which we believe is more efficient than brute force reconstruction. To initiate the discussion, let us first dispel the seemingly intuitive notion that one cannot generate a sample path x(t;θ+Δθ,ξ) from an x(t;θ,ξ) when Δθ≠0. A simple view of this general problem of perturbation analysis of DEDS and the efficient construction of multiple sample paths of a DEDS under different values of a can be obtained by appealing to some fundamental procedures in discrete event simulation.