Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-03T08:34:31.182Z Has data issue: false hasContentIssue false
  • English
  • Français

The Workload in the M/G/1 Queue with Work Removal

Published online by Cambridge University Press:  27 July 2009

Richard J. Boucherie  and
Onno J. Boxma
Richard J. Boucherie
Affiliation:
Department of Econometrics, University of Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands
Onno J. Boxma
Affiliation:
CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands and Tilburg University, Faculty of Economics, P.O. Box 90153, 5000 LE Tilburg, The Netherlands
Get access Rights & Permissions [Opens in a new window]

Extract

We consider an M/G/1 queue with the special feature of additional negative customers, who arrive according to a Poisson process. Negative customers require no service, but at their arrival a stochastic amount of work is instantaneously removed from the system. We show that the workload distribution in this M/G/1 queue with negative customers equals the waiting time distribution in a GI/G/1 queue with ordinary customers only; the effect of the negative customers is incorporated in the new arrival process.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 10 , Issue 2 , April 1996 , pp. 261 - 277
Copyright
Copyright © Cambridge University Press 1996

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.Asmussen, S. (1987). Applied probability and queues. New York: Wiley.Google Scholar
2.Boucherie, R.J. & van Dijk, N.M. (1994). Local balance in queueing networks with positive and negative customers. Annals of Operations Research 48: 463–492.CrossRefGoogle Scholar
3.Boxma, O.J. (1975). The single-server queue with random service output. Journal of Applied Probability 12: 763–778.CrossRefGoogle Scholar
4.Brill, P.H. & Posner, M.J.M. (1977). Level crossings in point processes applied to queues: Singleserver case. Operations Research 25: 662–674.CrossRefGoogle Scholar
5.Chao, X. (1995). Networks of queues with customers, signals and arbitrary service. Operations Research 43: 537–544.CrossRefGoogle Scholar
6.Chao, X. & Pinedo, M. (1993). On generalized networks of queues with positive and negative arrivals. Probability in the Engineering and Informational Sciences 7: 301–334.CrossRefGoogle Scholar
7.Cohen, J.W. (1977). On up- and downcrossings. Journal of Applied Probability 14: 405–410.CrossRefGoogle Scholar
8.Cohen, J.W. (1982). The single server queue. Amsterdam: North-Holland.Google Scholar
9.Cramèr, H. (1955). Collective risk theory. Reprinted from the Jubilee Volume of Skandia Insurance Company. Stockholm: Esselte.Google Scholar
10.Gani, J. & Pyke, R. (1960). The content of a dam as the supremum of an infinitely divisible process. Journal of Mathematics and Mechanics 9: 639–651.Google Scholar
11.Gelenbe, E. (1991). Product-form queueing networks with negative and positive customers. Journal of Applied Probability 28: 656–663.CrossRefGoogle Scholar
12.Gelenbe, E., Glynn, P., & Sigman, K. (1991). Queues with negative arrivals. Journal of Applied Probability 28: 245–250.CrossRefGoogle Scholar
13.Gelenbe, E. & Schassberger, R. (1992). Stability of product form G-networks. Probability in the Engineering and Informational Sciences 6: 271–276.CrossRefGoogle Scholar
14.Grinstein, J. & Rubinovitch, M. (1974). Queues with random service output: The case of Poisson arrivals. Journal of Applied Probability 11: 771–784.CrossRefGoogle Scholar
15.Harrison, P.G. & Pitel, E. (1993). Sojourn times in single-server queues with negative customers. Journal of Applied Probability 30: 943–963.CrossRefGoogle Scholar
16.Harrison, P.G. & Pitel, E. (1994). The M/G/1 queue with negative customers. Research Report, Imperial College, London (to appear in Journal of Applied Probability).Google Scholar
17.Henderson, W. (1993). Queueing networks with negative customers and negative queue lengths. Journal of Applied Probability 30: 931–942.CrossRefGoogle Scholar
18.Jain, G. & Sigman, K. (1994). A Pollaczek-Khintchine formulation for M/G/1 queues with disasters. Research Report, Columbia University, New York, July (to appear in Journal of Applied Probability).Google Scholar
19.Jain, G. & Sigman, K. (1995). A generalization of Pollaczek-Khintchine formula to account for negative arrivals. Research Report, Columbia University, New York, 04.Google Scholar
20.Prabhu, N.U. (1980). Stochastic storage processes. New York: Springer-Verlag.CrossRefGoogle Scholar
21.Rogozin, B.A. (1966). On the distribution of functional related to boundary problems for processes with independent increments. Theory of Probability and Its Applications 11: 580–591.CrossRefGoogle Scholar
22.Wolff, R.W. (1989). Stochastic modeling and the theory of queues. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar