Perturbation Analysis

Perturbation analysis (PA) is a sample path technique for analyzing changes in the performance of stochastic systems due to changes in system parameters. In terms of stochastic simulation | the main setting for the application of PA | the objective is to estimate sensitivities of the performance measures of interest with respect to system parameters while obtaining estimates of performance itself, without the need for additional simulation runs. The primary application is gradient estimation during the simulation of discrete-event systems, e.g., queueing and inventory systems. Besides their importance in sensitivity analysis, these gradient estimators are a critical component in gradient-based simulation optimization methods. Let l() be a performance measure of interest with parameter (possibly vector) of interest. We are interested in those systems where l() cannot be easily obtained through analytical means and therefore must be estimated from sample paths, e.g., via stochastic simulation. We denote by L(; !) the sample performance obtained from a sample path realization ! such that l() = EL(; !)]. The goal of PA is to eeciently estimate the eeects on l of a perturbation ?! + , using information from a sample path ! at. We distinguish between two major classes of problems that PA addresses: ! 0: estimating the gradient dl() dd ; when l is diierentiable in ; 6 = 0: estimating changes due to a nite perturbation, i.e., l(+): In the former case, no perturbation is ever actually introduced into the system (or simulation), although the idea of a perturbation may be employed as a heuristic tool in preliminary analysis. To sort out the abundance of acronyms in the PA eld, we provide a brief deenition of each corresponding approach, along with a reference. Among gradient estimation techniques, by far the most well-known is innnitesimal perturbation analysis (IPA), which simply uses the sample derivative dL=dd to estimate dl=dd. It is straightforward to implement and very computationally eecient; however, as we shall discuss shortly in more detail, its applicability is not universal. The books by Ho and Cao (1991), Glasserman (1991), and Cao (1994) cover IPA in detail. A very general and well-developed extension of IPA is smoothed perturbation analysis (SPA), based on the ideas of conditional expectation (Gong and Ho 1987). Although its applicability is quite broad, its implementation is usually very problem dependent. The book by Fu and Hu (1997) covers this method in full generality. Other gradient estimation techniques …