A Sample Performance Function of Closed Jackson Queueing Networks

Abstract

A stochastic system such as a queueing network can be specified by system parameters and a set of sequences of random variables that represents the randomness in the system. A “sample performance function” is a measure of system performance as a function of system parameters, for each realization of the random sequences. Although the average of N sample performance functions converges to the expected value of the performance with probability one when N goes to infinity, the average of the derivatives of these N sample performance functions with respect to a parameter may not converge to the derivative of the expected value. In this paper, we study a sample performance function of a closed Jackson queueing network; specifically, the time required by a server to serve a finite number of customers. We show that this sample performance function is a continuous, piecewise linear function of the mean service time. We prove that the average of derivatives of this sample performance function with a given initial state does converge, with probability one, to the derivative of the conditional mean value, given the same initial state. The result shows that the estimate of the derivative of a server's mean throughput in a finite time period with respect to the mean service time obtained by infinitesimal perturbation analysis is strongly consistent.

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