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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
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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References

[1] Boucherie, R. J. and Van Dijk, N. M. (1991) Product forms for queueing networks with state-dependent multiple job transitions. Adv. Appl. Prob. 23, 152–187.Google Scholar
[2] Chao, X. and Pinedo, M. (1993) On generalized networks of queues with positive and negative arrivals. Prob. Eng. Inf. Sci. 7, 301–304.Google Scholar
[3] Chao, X., Pinedo, M. and Shaw, D. (1996) A network of assembly queues with product-form solution. J. Appl. Prob. 33, 858–869.CrossRefGoogle Scholar
[4] Dunford, N. and Schwartz, N. J. (1958) Linear Operators. Part 1. Interscience, New York.Google Scholar
[5] Gelenbe, E. (1991) Product-form queueing networks with negative and positive customers. J. Appl. Prob. 28, 656–663.CrossRefGoogle Scholar
[6] Gelenbe, E. (1993) G-networks with signals and batch removal. Prob. Eng. Inf. Sci. 7, 335–342.Google Scholar
[7] Gelenbe, E. (1993) G-networks with triggered customer movement. J. Appl. Prob. 30, 742–748.CrossRefGoogle Scholar
[8] Gross, D. and Harris, C. M. (1985) Fundamentals of Queueing Theory. 2nd edn. Wiley, New York.Google Scholar
[9] Henderson, W., Pearce, C. E. M., Pollett, P. K. and Taylor, P. G. (1992) Connecting internally balanced quasireversible Markov processes. Adv. Appl. Prob. 24, 934–959.CrossRefGoogle Scholar
[10] Henderson, W. and Taylor, P. G. (1990) Product form in networks of queues with batch arrivals and batch services. Queueing Systems 6, 71–88.Google Scholar
[11] Henderson, W. and Taylor, P. G. (1991) Some new results of queueing networks with batch movement. J. Appl. Prob. 28, 409–421.CrossRefGoogle Scholar
[12] Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
[13] Kelly, F. P. (1982) Networks of quasi-reversible nodes. In Applied Probability Computer Science: The Interface. Vol. I. ed. Disney, R. L. and Ott, T. J. pp. 3–26.Google Scholar
[14] Kingman, J. F. C. (1970) Inequalities in the theory of queues. J. R. Statist. Soc. B. 32, 102–110.Google Scholar
[15] Kleinrock, L. (1975) Queueing Systems. Volume 1: Theory. Wiley, New York.Google Scholar
[16] Massey, W. A. (1987) Stochastic orderings for Markov processes on partially ordered spaces. Math. Operat. Res. 12, 350–367.CrossRefGoogle Scholar
[17] Miyazawa, M. (1994) On the characterization of departure rules for discrete-time queueing networks with batch movements and its applications. Queueing Systems 18, 149–166.Google Scholar
[18] Miyazawa, M. (1995) A note on my paper: On the characterization of departure rules for discrete-time queueing networks with batch movements and its applications. Queueing Systems 19, 455–448.CrossRefGoogle Scholar
[19] Miyazawa, M. and Wolff, W. R. (1996) Symmetric queues with batch departures and their networks. Adv. Appl. Prob. 28, 308–326.CrossRefGoogle Scholar
[20] Serfozo, R. F. (1989) Poisson functionals of Markov processes and queueing networks. Adv. Appl. Prob. 21, 595–616.Google Scholar
[21] Serfozo, R. F. (1993) Queueing networks with dependent nodes and concurrent movements. Queueing Systems 13, 143–182.CrossRefGoogle Scholar
[22] Stoyan, D. (1983) Comparison Methods for Queues and other Stochastic Models. Wiley, New York.Google Scholar