Stochastic decomposition in discrete-time queues with generalized vacations and applications

Sofian De Clercq; Wouter Rogiest; Bart Steyaert; Herwig Bruneel

doi:10.3934/jimo.2012.8.925

Article Contents

2012, Volume 8, Issue 4: 925-938. Doi: 10.3934/jimo.2012.8.925

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Stochastic decomposition in discrete-time queues with generalized vacations and applications

1.
SMACS Research Group, Ghent University, St.-Pietersnieuwstraat 41, 9000 Gent

Received: August 31, 2011

Revised: June 30, 2012

Published: September 2012

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Abstract

For several specific queueing models with a vacation policy, the stationary system occupancy at the beginning of a random slot is distributed as the sum of two independent random variables. One of these variables is the stationary number of customers in an equivalent queueing system with no vacations. For models in continuous time with Poissonian arrivals, this result is well-known, and referred to as stochastic decomposition, with proof provided by Fuhrmann and Cooper. For models in discrete time, this result received less attention, with no proof available to date. In this paper, we first establish a proof of the decomposition result in discrete time. When compared to the proof in continuous time, conditions for the proof in discrete time are somewhat more general. Second, we explore four different examples: non-preemptive priority systems, slot-bound priority systems, polling systems, and fiber delay line (FDL) buffer systems. The first two examples are known results from literature that are given here as an illustration. The third is a new example, and the last one (FDL buffer systems) shows new results. It is shown that in some cases the queueing analysis can be considerably simplified using this decomposition property.

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Mathematics Subject Classification: Primary: 60K25, 68M20; Secondary: 90B22.

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