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A characterisation of (max,+)-linear queueing systems

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Abstract

The (max,+)-algebra has been successfully applied to many areas of queueing theory, like stability analysis and ergodic theory. These results are mainly based on two ingredients: (1) a (max,+)-linear model of the time dynamic of the system under consideration, and (2) the time-invariance of the structure of the (max,+)-model. Unfortunately, (max,+)-linearity is a purely algebraic concept and it is by no means immediate if a queueing network admits a (max,+)-linear representation satisfying (1) and (2). In this paper we derive the condition a queueing network must meet if it is to have a (max,+)-linear representation. In particular, we study (max,+)-linear systems with time-invariant transition structures. For this class of systems, we find a surprisingly simple necessary and sufficient condition for (max,+)-linearity, based on the flow of customers through the network.

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  1. EURANDOM, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    Bernd Heidergott

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  1. Bernd Heidergott
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Heidergott, B. A characterisation of (max,+)-linear queueing systems. Queueing Systems 35, 237–262 (2000). https://doi.org/10.1023/A:1019102429650

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