Series Expansions of Generalized Matrix Products

We consider generalized products of random matrices. They arise in discrete event systems (DES), such as queueing networks or stochastic Petri nets, where they are used to express the state transition dynamic. Instances of such DES are those whose state dynamic can be modelled through a matrix-vector multiplication in conventional, max-plus and min-plus algebra. We will present a Taylor series approach to numerical evaluation of finite horizon performance characteristics of systems modelled by generalized matrix products. The cornerstone of our analysis is the introduction of a differential calculus, based on the concept of weak derivative of a random matrix. We illustrate our results with a couple of numerical computations performed on a classical DES example.