Sensitivity analysis of joint characteristics of (max, plus )-linear queueing networks

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) circle times x(k), for k greater than or equal to 0 with x(0) = x(0), as a Maxkov chain the transition dynamic of which is given through the matrices A(k). Elaborating on the product rule of D-differentiability for Maxkov 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. Copyright (C) 2001 IFAC.