# Transient and Stationary Waiting Times in $max,$-Linear Systems with Poisson Input

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1 MISTRAL - Modeling of Computer Systems and Telecommunication Networks : Research and Software Development CRISAM - Inria Sophia Antipolis - Méditerranée

Abstract : 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 stochastic service systems 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 $n$-th 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 $\lambda$. 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 service.

Keywords : WAITING TIMES ANALYTICITY TAYLOR SERIES EXPANSION POISSON PROCESS STOCHASTIC RECURRENCE EQUATION VECTORIAL EVOLUTION EQUATION STATIONARY STATE VARIABLE QUEUEING NETWORK STOCHASTIC PETRI NET

Autor: François Baccelli - Sven Hasenfuss Volker Schmidt

Fuente: https://hal.archives-ouvertes.fr/

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