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The efficiency and heavy traffic properties of the score function method in sensitivity analysis of queueing models. (English) Zbl 0745.65088

Many modern complex stochastic systems can be modelled as discrete-event dynamic systems (DEDS). Due to the complex interactions of such discrete events over time, sensitivity and stability analysis of such systems are important tasks.
The authors assume that their DEDS is driven by an i.i.d. input sequence \(Y_ t\), \(t=1,2,\dots\), where \(Y_ t\) is generated from the density \(f(y,v)\), where \(v\) is a governing parameter (typically \(Y_ t\) and \(v\) are vector-valued). Moreover, it is assumed that the output sequence \(L_ t\), \(t=1,2,\dots\), which is called the sample performance process, settles in the steady state and becomes a stationary and ergodic process.
Numerous methods have been suggested for estimating the expected steady state performance \(l(v)=E_ vL=\lim_{t\to\infty}E_ vL_ t\). In this paper the authors are interested in estimating the sensitivities (gradient \(\nabla_{l_ k}(v)\), Hessian \(\nabla^ 2l_ k(v)\) etc.) and in optimization of \(l(v)\), where \(v\in V\subset R^ n\) and \(V\) being a given region. Particular attention is given to the score function (SF) method (the likelihood method) for sensitivity analysis of DEDS.
Following topics are discussed in more details: Basic formulas for the sensitivities are given; a heavy traffic framework for sensitivity analysis is shown; performance evaluation for the SF method is presented; control variate procedures for improving the accuracy of the simulation estimators are discussed.
As a motivating example, the authors consider the estimation of the gradient of the waiting time \(l(v)\) at a particular node of a \(k\)-station queueing network with respect to the service rate vector \(v\) and/or with respect to the routing probability matrix \(P\).
In addition, the paper also contains a brief summary of some fundamental facts about the relevant simulation estimators, central limit theorems and confidence intervals as well as a number of examples and numerical tables illustrating and supporting the theory discussed.
Reviewer: J.Antoch (Praha)

MSC:

65C99 Probabilistic methods, stochastic differential equations
65C20 Probabilistic models, generic numerical methods in probability and statistics
60K25 Queueing theory (aspects of probability theory)
60J65 Brownian motion
60F17 Functional limit theorems; invariance principles
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