Skip to main content Accessibility help

On Generalized Networks of Queues with Positive and Negative Arrivals

  • Xiuli Chao (a1) and Michael Pinedo (a2)


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.



Hide All
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: 248260.
2.Breiman, L. (1968). Probability. New York: Wiley.
3.Chao, X. (1992). On generalized networks of queues with customers, signals and arbitrary service time distributions. Operations Research (to appear).
4.Fourneau, J.M. & Gelenbe, E. (1992). Multiple class G-networks. ORSA Technical Committee on Computer Science Conference, Williamsburg, VA, 01 1992.
5.Gelenbe, E. (1991). Product form queueing networks with positive and negative customers. Journal of Applied Probability 28: 656663.
6.Gopal, P.M. & Kadaba, B.K. (1989). Network delay considerations for packetized voice. Performance Evaluation, 9: 167180.
7.Gordon, W.J. & Newell, G.F.(1967). Closed queueing systems with exponential servers. Operations Research 16: 254265.
8.Jackson, J.R. (1957). Networks of waiting lines. Operations Research 5: 518521.
9.Jackson, J.R. (1963). Job-shop like queueing systems. Management Science 10: 131142.
10.Kelly, F. (1975). Networks of queues with customers of different types. Journal of Applied Probability 12: 542554.
11.Kelly, F. (1976). Networks of queues. Advances in Applied Probability 8: 416432.
12.Kelly, F. (1979). Reversibility and stochastic networks. New York: Wiley & Sons.
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. 326.
14.Serfozo, R. (1989). Poisson functional of Markov processes and queueing networks. Advances in Applied Probability 21: 595616.
15.Walrand, J. (1983). A probabilistic look at networks of quasi-reversible queues. IEEE Transactions on Information Theory IT-29(6): 825831.
16.Walrand, J. (1988). Introduction to queueing networks. Englewood Cliffs, NJ: Prentice-Hall.


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed