Perturbation Analysis Gives Strongly Consistent Sensitivity Estimates for the M/G/1 Queue

The technique of perturbation analysis has recently been introduced as an efficient way to compute parameter sensitivities for discrete event systems. Thus far, the statistical properties of perturbation analysis have been validated mainly through experiments. This paper considers, for an M/G/1 queueing system, the sensitivity of mean system time of a customer to a parameter of the arrival or service distribution. It shows analytically that (i) the steady state value of the perturbation analysis estimate of this sensitivity is unbiased, and (ii) a perturbation analysis algorithm implemented on a single sample path of the system gives asymptotically unbiased and strongly consistent estimates of this sensitivity. (No previous knowledge of perturbation analysis is assumed, so the paper also serves to introduce this technique to the unfamiliar reader.) Numerical extensions to GI/G/1 queues, and applications to optimization problems, are also illustrated.