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Generalizing the Pollaczek-Khintchine Formula to Account for Arbitrary Work Removal

Published online by Cambridge University Press:  27 July 2009

Gautam Jain  and
Karl Sigman
Gautam Jain
Affiliation:
Department of Industrial Engineering and Operations Research, Columbia University, New York, New York 10027
Karl Sigman
Affiliation:
Department of Industrial Engineering and Operations Research, Columbia University, New York, New York 10027
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Extract

Recently, a Pollaczek-Khintchine-like formulation for M/G/l queues with disasters has been obtained. A disaster is said to occur if a negative arrival causes all the customers (and therefore work) to depart from the system immediately. This study generalizes this result further, as it is shown to hold even when negative arrivals cause only part of the work to be demolished. In other words, an arbitrary amount of work, following a known distribution, is allowed to be removed at a negative event. Under these circumstances, a general approach for obtaining the Pollaczek–Khintchine Formula is proposed, which is then illustrated via several examples. Typically, it is seen that the formulainvolves certain parameters that are not explicitly known. The formula itself is made possible due to the number in system being geometric under preemptive last in-first out discipline.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 10 , Issue 4 , October 1996 , pp. 519 - 531
Copyright
Copyright © Cambridge University Press 1996

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References

1.Bardhan, I. & Sigman, K. (1995). Stationary regimes for inventory processes. Stochastic Processes and Applications 56: 77–86.CrossRefGoogle Scholar
2.Borovkov, A.A. (1976). Stochastic processes in queueing theory. New York: Springer-Verlag.CrossRefGoogle Scholar
3.Boucherie, R.J. & Boxma, O.J. (1996). The workload in the M/G/l queue with work removal. Probability in the Engineering and Informational Sciences 10: 261–277.CrossRefGoogle Scholar
4.Fakinos, D. (1981). The G/G/l queueing system with a particular queue discipline. Journal of the Royal Statistical Society Series B 43: 190–196.Google Scholar
5.Gelenbe, E. (1991). Product form networks with negative and positive customers. Journal of Applied Probability 28: 656–663.CrossRefGoogle Scholar
6.Gelenbe, E. (1993). G-networks with triggered customer movement. Journal of Applied Probability 30: 742–748.CrossRefGoogle Scholar
7.Jain, G. (1996). A rate conservation analysis of queues and networks with work removal. Ph.D. dissertation, Industrial Engineering and Operations Research, Columbia University.Google Scholar
8.Jain, G. & Sigman, K. (1996). A Pollaczek-Khintchine formula for M/G/l queues with disasters. Journal of Applied Probability (to appear).CrossRefGoogle Scholar
9.Prabhu, N.U. (1980). Stochastic storage processes: Queues, insurance risk, and dams. New York: Springer-Verlag.CrossRefGoogle Scholar
10.Sigman, K. (1994). Stationary marked point processes: An intuitive approach. New York: Chapman and Hall.Google Scholar
11.Wolff, R.W. (1989). Stochastic modeling and the theory of queues. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar