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

  • Xiuli Chao (a1)

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.

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Postal address: Department of Industrial and Manufacturing Engineering, New Jersey Institute of Technology, Newark, NJ 07102, USA.

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

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References

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Boucherie, R. and Van Duk, N. M. (1994) Local balances in queueing networks with positive and negative customers. Ann. Operat. Res. 48, 463492.
Chao, X., Pinedo, M. and Shaw, D. (1996) Networks of queues with batch services and customer coalescence. J. Appl. Prob. 33, 858869.
Henderson, W. (1993) Queueing networks with negative customers and negative queue lengths. J. Appl. Prob. 30, 931942.
Miyazawa, M. and Wolff, R. (1996) Symmetric queues with batch departures and their networks. Adv. Appl. Prob. 28, 308326.
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, 523544.
Van Duk, N. M. (1993) Queueing Networks and Product Forms. Wiley, New York.
Whittle, P. (1985) Partial balance and insensitivity. J. Appl. Prob. 22, 168176.
Whittle, P. (1986) Systems in Stochastic Equilibrium. Wiley, New York.

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