New performance sensitivity formulae for a class of product-form queueing networks

Perturbation analysis (PA) applies a dynamic point of view to the sample paths of stochastic systems; the realization factor, one of the main concepts of PA, measures the final effect of a perturbation on system performance and provides a novel approach in obtaining performance sensitivities. In this paper, we solve analytically the set of equations for realization factors of a two-server cyclic network. We prove an invariance property of the performance sensitivity for Norton's aggregation. Using the results, we derive closed-form formulae for the derivatives of performance measures in a closed queueing network with load-dependent exponential servers. The performance measures have two general forms: customer average and time average. In contrast with the usual approach based on product-form solutions, our results provide additional insights into the performance sensitivity of closed queueing networks and have immediate applications to problems of optimal control. The general formulae are expressed in terms of Buzen's algorithm with a computational complexity comparable to that of the formulae obtained by directly taking the derivatives of the product-form solutions.