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Partial balances in batch arrival batch service and assemble-transfer queueing networks

Part of: Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Xiuli Chao
Xiuli Chao*
Affiliation:
New Jersey Institute of Technology
*
∗Postal address: Department of Industrial and Manufacturing Engineering, New Jersey Institute of Technology, Newark, NJ 07102, USA.
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Abstract

Recently Miyazawa and Taylor (1997) proposed a new class of queueing networks with batch arrival batch service and assemble-transfer features. In such networks customers arrive and are served in batches, and may change size when a batch transfers from one node to another. With the assumption of an additional arrival process at each node when it is empty, they obtain a simple product-form steady-state probability distribution, which is a (stochastic) upper bound for the original network. This paper shows that this class of network possesses a set of non-standard partial balance equations, and it is demonstrated that the condition of the additional arrival process introduced by Miyazawa and Taylor is there precisely to satisfy the partial balance equations, i.e. it is necessary and sufficient not only for having a product form solution, but also for the partial balance equations to hold.

Keywords

QUEUEING NETWORKSPARTIAL BALANCEPRODUCT FORM SOLUTION

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 34 , Issue 3 , September 1997 , pp. 745 - 752
Copyright
Copyright © Applied Probability Trust 1997 

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Footnotes

This research partially supported by the NSF under DDM-9209526.

References

Boucherie, R. and Van Duk, N. M. (1994) Local balances in queueing networks with positive and negative customers. Ann. Operat. Res. 48, 463–492.CrossRefGoogle Scholar
Chao, X., Pinedo, M. and Shaw, D. (1996) Networks of queues with batch services and customer coalescence. J. Appl. Prob. 33, 858–869.CrossRefGoogle Scholar
Henderson, W. (1993) Queueing networks with negative customers and negative queue lengths. J. Appl. Prob. 30, 931–942.CrossRefGoogle Scholar
Miyazawa, M. and Wolff, R. (1996) Symmetric queues with batch departures and their networks. Adv. Appl. Prob. 28, 308–326.CrossRefGoogle Scholar
Miyazawa, M. and Taylor, P. (1997) A geometric product-form distribution for queueing networks with non-standard batch arrivals and batch transfers. Adv. Appl. Prob. 29, 523–544.CrossRefGoogle Scholar
Van Duk, N. M. (1993) Queueing Networks and Product Forms. Wiley, New York.Google Scholar
Whittle, P. (1985) Partial balance and insensitivity. J. Appl. Prob. 22, 168–176.CrossRefGoogle Scholar
Whittle, P. (1986) Systems in Stochastic Equilibrium. Wiley, New York.Google Scholar