Transient and stationary waiting times in (max,+)-linear systems with Poisson input

We consider a certain class of vectorial evolution equations, which are linear in the (max, +) semi-field. They can be used to model several types of discrete event systems, in particular queueing networks where we assume that the arrival process of customers (tokens, jobs, etc.) is Poisson. Under natural Cramer type conditions on certain variables, we show that the expected waiting time which the nth customer has to spend in a given subarea of such a system can be expanded analytically in an infinite power series with respect to the arrival intensity X. Furthermore, we state an algorithm for computing all coefficients of this series expansion and derive an explicit finite representation formula for the remainder term. We also give an explicit finite expansion for expected stationary waiting times in (max,+)-linear systems with deterministic queueing services.