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Networks of queues with batch services and customer coalescence

Part of: Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Xiuli Chao ,
Michael Pinedo  and
Dequan Shaw
Xiuli Chao*
Affiliation:
New Jersey Institute of Technology
Michael Pinedo*
Affiliation:
Columbia University
Dequan Shaw*
Affiliation:
GTE Laboratories
*
∗ Postal address: Division of Industrial and Management Engineering, New Jersey Institute of Technology, Newark, NJ 07102, USA.
∗∗ Postal address: Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027, USA.
∗∗∗ Postal address: GTE Laboratories, 40 Sylvan Road, Waltham, MA 02254, USA.
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Abstract

Consider a queueing network with batch services at each node. The service time of a batch is exponential and the batch size at each node is arbitrarily distributed. At a service completion the entire batch coalesces into a single unit, and it either leaves the system or goes to another node according to given routing probabilities. When the batch sizes are identical to one, the network reduces to a classical Jackson network. Our main result is that this network possesses a product form solution with a special type of traffic equations which depend on the batch size distribution at each node. The product form solution satisfies a particular type of partial balance equation. The result is further generalized to the non-ergodic case. For this case the bottleneck nodes and the maximal subnetwork that achieves steady state are determined. The existence of a unique solution is shown and stability conditions are established. Our results can be used, for example, in the analysis of production systems with assembly and subassembly processes.

Keywords

QUEUEING NETWORKSBATCH SERVICESCUSTOMER COALESCENCEPRODUCT FORM SOLUTIONFIXED POINTNON-ERGODIC QUEUESSTABILITY CONDITIONS

MSC classification

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

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 3 , September 1996 , pp. 858 - 869
Copyright
Copyright © Applied Probability Trust 1996 

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Footnotes

Research partially supported by the NSF under grant DDM-9209526.

Research partially supported by the NSF under grant ECS 91–14689.

References

Boucherie, R. J. and Van Dijk, N. M. (1991) Product form for queueing networks with state dependent multiple job transitions. Adv. Appl. Prob. 23, 152–187.Google Scholar
Çinlar, E. (1975) Introduction to Stochastic Processes. Prentice Hall, New Jersey.Google Scholar
Chao, X. and Pinedo, M. (1993) On generalized networks of queues with positive and negative arrivals. Prob. Eng. Inf. Sci. 7, 301–334.Google Scholar
Goodman, J. B. and Massey, W. (1984) The non-ergodic Jackson network. J. Appl. Prob. 21, 860–869.Google Scholar
Gross, D. and Harris, C. (1985) Fundamentals of Queueing Theory. 2nd edn. Wiley, New York.Google Scholar
Henderson, W. (1993) Queueing networks with negative customers and negative queue lengths. J. Appl. Prob. 30, 931–942.CrossRefGoogle Scholar
Henderson, W. and Taylor, P. G. (1990) Product form in networks of queues with batch arrivals and batch services. QUESTA 6, 71–88.Google Scholar
Jackson, J. R. (1963) Job-shop like queueing systems. Management Sci. 10, 131–142.Google Scholar
Kelly, F. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
Miyazawa, M. and Wolff, R. (1996) Symmetric queues with batch departures and their networks. Adv. Appl. Prob. 28, 308–326.Google Scholar
Schauder, J. (1930) Der Fixpunktsatz in Funktionalraumen. Studia Math. 2, 171–180.Google Scholar
Serfozo, R. (1989) Poisson functionals of Markov processes and queueing networks, Adv. Appl. Prob. 21, 595–616.Google Scholar
Stoyan, D. (1983) Comparison Methods for Queues and Other Stochastic Models. Wiley, New York.Google Scholar