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

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

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Article ID: iaor20002515
Country: United States
Volume: 26
Issue: 3/4
Start Page Number: 301
End Page Number: 342
Publication Date: Nov 1997
Journal: Queueing Systems
Authors: , ,
Keywords: petri nets, queueing networks
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 queueing networks where we assume that the arrival process of customers (tokens, jobs, etc.) is Poisson. Under natural Cramér 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 λ. 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.

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