Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-24T05:21:00.306Z Has data issue: false hasContentIssue false
  • English
  • Français

State-dependent signalling in queueing networks

Part of: Operations research and management science Special processes

Published online by Cambridge University Press:  01 July 2016

W. Henderson ,
B. S. Northcote  and
P. G. Taylor
W. Henderson
Affiliation:
University of Adelaide
B. S. Northcote
Affiliation:
University of Adelaide
P. G. Taylor*
Affiliation:
University of Adelaide
*
* Postal address: Teletraffic Research Centre, Department of Applied Mathematics, GPO Box 498, Adelaide, SA 5005, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

It has recently been shown that networks of queues with state-dependent movement of negative customers, and with state-independent triggering of customer movement have product-form equilibrium distributions. Triggers and negative customers are entities which, when arriving to a queue, force a single customer to be routed through the network or leave the network respectively. They are ‘signals' which affect/control network behaviour. The provision of state-dependent intensities introduces queues other than single-server queues into the network.

This paper considers networks with state-dependent intensities in which signals can be either a trigger or a batch of negative customers (the batch size being determined by an arbitrary probability distribution). It is shown that such networks still have a product-form equilibrium distribution. Natural methods for state space truncation and for the inclusion of multiple customer types in the network can be viewed as special cases of this state dependence. A further generalisation allows for the possibility of signals building up at nodes.

Keywords

PRODUCT FORMTRIGGERED MOTIONBATCH DEPARTURESSTATE DEPENDENCE

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) 90B22: Queues and service
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 26 , Issue 2 , June 1994 , pp. 436 - 455
Copyright
Copyright © Applied Probability Trust 1994 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

References

[1] Boucherie, R., (1992) Product form in queueing networks. PhD thesis, Free University, Amsterdam.Google Scholar
[2] Boucherie, R. and Van Dijk, N. (1992) Local balance in queueing networks with negative customers. Research memorandum 1992-1, Free University, Amsterdam.Google Scholar
[3] Chandy, K. M. and Martin, A. J. (1983) A characterization of product form queueing networks. J. Assoc. Comput. Mech. 30, 286–299.Google Scholar
[4] Dunford, N. and Schwartz, N. J. (1958) Linear Operators, Part 1. Interscience, New York.Google Scholar
[5] Gelenbe, E. (1991) Product form networks with negative and positive customers. J. Appl. Prob. 28, 656–663.CrossRefGoogle Scholar
[6] Gelenbe, E., Glynn, P., and Sigman, K. (1991) Queues with negative arrivals. J. Appl. Prob. 28, 245–250.Google Scholar
[7] Gelenbe, E. (1993) G-networks with triggered customer movement. J. Appl. Prob. 30, 742–748.Google Scholar
[8] Gelenbe, E. (1992) Negative customers with batch removal. Personal communication.Google Scholar
[9] Henderson, W. (1993) Queuing networks with negative customers and negative queue lengths. J. Appl. Prob. 30, 931–942.Google Scholar
[10] Henderson, W. and Taylor, P. (1989) Insensitivity of processes with interruptions. J. Appl. Prob. 26, 242–258.Google Scholar
[11] Henderson, W., Northcote, B., and Taylor, P. (1993) Geometric equilibrium distributions for queues with interactive batch departures. Ann. Operat. Res. Special Issue on Queueing Networks. Google Scholar
[12] Jackson, J. (1957) Networks of waiting lines. Operat. Res. 5, 518–521.Google Scholar
[13] Kelly, F. P. (1985) Reversibility and Stochastic Networks Wiley, Chichester.Google Scholar
[14] Miller, , Stationary equations in continuous time Markov chains. Trans. Amer. Math. Soc. 109, 35–44.Google Scholar
[15] Ross, S. M. (1989) Introduction to Probability Models, 4th edn. Academic Press, New York.Google Scholar

Reference added in proof

[16] Northcote, B. S. (1993) Signalling in Product Form Queueing Networks. PhD thesis, University of Adelaide.Google Scholar