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A Geometric Product-Form Distribution for a Queueing Network by Non-Standard Batch Arrivals and Batch Transfers

Part of: Markov processes Special processes

Published online by Cambridge University Press:  01 July 2016

Masakiyo Miyazawa  and
Peter G. Taylor
Masakiyo Miyazawa*
Affiliation:
Science University of Tokyo
Peter G. Taylor*
Affiliation:
The University of Adelaide
*
∗ Postal address: Department of Information Science, Science University of Tokyo, Noda, Chiba 278, Japan.
∗∗ Postal address: Department of Applied Mathematics, The University of Adelaide, SA 5005, Australia.
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Abstract

We introduce a batch service discipline, called assemble-transfer batch service, for continuous-time open queueing networks with batch movements. Under this service discipline a requested number of customers is simultaneously served at a node, and transferred to another node as, possibly, a batch of different size, if there are sufficient customers there; the node is emptied otherwise. We assume a Markovian setting for the arrival process, service times and routing, where batch sizes are generally distributed.

Under the assumption that extra batches arrive while nodes are empty, and under a stability condition, it is shown that the stationary distribution of the queue length has a geometric product form over the nodes if and only if certain conditions are satisfied for the extra arrivals. This gives a new class of queueing networks which have tractable stationary distributions, and simultaneously shows that the product form provides a stochastic upper bound for the stationary distribution of the corresponding queueing network without the extra arrivals.

Keywords

MARKOVIAN QUEUEING NETWORKBATCH ARRIVALBATCH SERVICESTOCHASTIC ORDERGEOMETRIC DISTRIBUTIONPRODUCT-FORMSTATIONARY DISTRIBUTIONNEGATIVE CUSTOMER

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 29 , Issue 2 , June 1997 , pp. 523 - 544
Copyright
Copyright © Applied Probability Trust 1997 

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