Sensitivity Analysis of Joint Characteristics of (Max,+)-Linear Queueing Networks

Abstract

We introduce the concept of D-differentiability of matrices over the (max,+) algebra. Specifically, we view the stochastic (max,+)-linear system x(k + 1) = A(k)⊗ x(k), for k ≥ 0 with x(0) = x0, as a Markov chain the transition dynamic of which is given through the matrices A(k). Elaborating on the product rule of D-differentiability for Markov kernels, we obtain results on differentiability of (max,+)-linear systems and unbiased gradient estimators as well. Moreover, we establish sufficient conditions for deducing the differentiability of a (max,+)-linear system from that of the firing time distributions of the corresponding stochastic event graph. The results hold uniformly on a predefined class of performance functions. We illustrate our approach with an analysis of joint characteristics of waiting times in a (max,+)-linear queueing network.