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Queueing networks by negative customers and negative queue lengths

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

W. Henderson
W. Henderson*
Affiliation:
The University of Adelaide
*
∗ Postal address: Teletraffic Research Centre, Department of Applied Mathematics, The University of Adelaide, Box 498, G.P.O., Adelaide SA 5001, Australia.
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Abstract

A number of papers have recently appeared in the literature in which customers, in moving from node to node in the network arrive as either a positive customer or as a batch of negative customers. A positive customer joining its queue increases the number of customers at the queue by 1 and each negative customer decreases the queue length by 1, if possible. It has been shown that the equilibrium distribution for these networks assumes a geometric product form, that certain partial balance equations prevail and that the parameters of the geometric distributions are, as in Jackson networks, the service facility throughputs of customers. In this paper the previous work is generalised by allowing state dependence into both the service and routing intensities and by allowing the possibility, although not the necessity, for negative customers to build up at the nodes.

Keywords

STATE DEPENDENCEPRODUCT FORM

MSC classification

Primary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 4 , December 1993 , pp. 931 - 942
Copyright
Copyright © Applied Probability Trust 1993 

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References

[1] Baskett, F., Chandy, M., Muntz, R. and Palacios, J. (1975) Open, closed and mixed networks of queues with different classes of customers. J. Assoc. Comput. Mach. 22, 248–260.Google Scholar
[2] Boucherie, R. J. and Van Dijk, N. M. (1992) Local balance in queueing networks with positive and negative customers, Research Report 1992-1, Free University, Amsterdam.Google Scholar
[3] Gelenbe, E. (1991) Product-form queueing networks with negative and positive customers, J. Appl. Prob. 28, 656–663.Google Scholar
[4] Gelenbe, E. and Schassberger, R. (1990) Note on the stability of G-networks. Research Report 90-8, Université René Descartes, Paris.Google Scholar
[5] Henderson, W., Northcote, B. S. and Taylor, P. G. (1992) Geometric equilibrium distributions for queues with interactive batch departures. Ann. Operat. Res. Google Scholar
[6] Jackson, J. (1963) Jobshop-like queueing systems. Management Sci. 10, 131–142.CrossRefGoogle Scholar
[7] Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
[8] Whittle, P. (1968) Equilibrium distributions for an open migration process. J. Appl. Prob. 5, 567–571.Google Scholar