Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-20T01:11:00.611Z Has data issue: false hasContentIssue false
  • English
  • Français

On Generalized Networks of Queues with Positive and Negative Arrivals

Published online by Cambridge University Press:  27 July 2009

Xiuli Chao  and
Michael Pinedo
Xiuli Chao
Affiliation:
Division of Industrial and Management Engineering, Department of Industrial and Mechanical Engineering, New Jersey Institute of Technology, Newark, New Jersey 07102
Michael Pinedo
Affiliation:
Department of Industrial Engineering and Operations Research, Columbia University, New York, New York 10027
Get access Rights & Permissions [Opens in a new window]

Abstract

Consider a generalized queueing network model that is subject to two types of arrivals. The first type represents the regular customers; the second type represents signals. A signal induces a regular customer already present at a node to leave. Gelenbe [5] showed that such a network possesses a product form solution when each node consists of a single exponential server. In this paper we study a number of issues concerning this class of networks. First, we explain why such networks have a product form solution. Second, we generalize existing results to include different service disciplines, state-dependent service rates, multiple job classes, and batch servicing. Finally, we establish the relationship between these networks and networks of quasi-reversible queues. We show that the product form solution of the generalized networks is a consequence of a property of the individual nodes viewed in isolation. This property is similar to the quasi-reversibility property of the nodes of a Jackson network: if the arrivals of the regular customers and of the signals at a node in isolation are independent Poisson, the departure processes of the regular customers and the signals are also independent Poisson, and the current state of the system is independent of the past departure processes.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 7 , Issue 3 , July 1993 , pp. 301 - 334
Copyright
Copyright © Cambridge University Press 1993

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

1.Baskett, F., Chandy, K.M., Munz, R.R., & Palacios, F. (1975). Open, closed, and mixed networks of queues with different classes of customers. Journal of the Association for Computing Machines 22: 248–260.CrossRefGoogle Scholar
2.Breiman, L. (1968). Probability. New York: Wiley.Google Scholar
3.Chao, X. (1992). On generalized networks of queues with customers, signals and arbitrary service time distributions. Operations Research (to appear).Google Scholar
4.Fourneau, J.M. & Gelenbe, E. (1992). Multiple class G-networks. ORSA Technical Committee on Computer Science Conference, Williamsburg, VA, 01 1992.CrossRefGoogle Scholar
5.Gelenbe, E. (1991). Product form queueing networks with positive and negative customers. Journal of Applied Probability 28: 656–663.CrossRefGoogle Scholar
6.Gopal, P.M. & Kadaba, B.K. (1989). Network delay considerations for packetized voice. Performance Evaluation, 9: 167–180.CrossRefGoogle Scholar
7.Gordon, W.J. & Newell, G.F.(1967). Closed queueing systems with exponential servers. Operations Research 16: 254–265.CrossRefGoogle Scholar
8.Jackson, J.R. (1957). Networks of waiting lines. Operations Research 5: 518–521.CrossRefGoogle Scholar
9.Jackson, J.R. (1963). Job-shop like queueing systems. Management Science 10: 131–142.CrossRefGoogle Scholar
10.Kelly, F. (1975). Networks of queues with customers of different types. Journal of Applied Probability 12: 542–554.CrossRefGoogle Scholar
11.Kelly, F. (1976). Networks of queues. Advances in Applied Probability 8: 416–432.CrossRefGoogle Scholar
12.Kelly, F. (1979). Reversibility and stochastic networks. New York: Wiley & Sons.Google Scholar
13.Kelly, F. (1982). Networks of quasi-reversible nodes. In Disney, R. & Ott, T. (eds.), Applied probability and computer science: The interface. Boston: Birkhauser, pp. 3–26.CrossRefGoogle Scholar
14.Serfozo, R. (1989). Poisson functional of Markov processes and queueing networks. Advances in Applied Probability 21: 595–616.CrossRefGoogle Scholar
15.Walrand, J. (1983). A probabilistic look at networks of quasi-reversible queues. IEEE Transactions on Information Theory IT-29(6): 825–831.CrossRefGoogle Scholar
16.Walrand, J. (1988). Introduction to queueing networks. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar