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A Pollaczek–Khintchine formula for M/G/1 queues with disasters

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Gautam Jain  and
Karl Sigman
Gautam Jain*
Affiliation:
Columbia University
Karl Sigman*
Affiliation:
Columbia University
*
∗Postal address: Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027–6699, USA.
∗Postal address: Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027–6699, USA.
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Abstract

A disaster occurs in a queue when a negative arrival causes all the work (and therefore customers) to leave the system instantaneously. Recent papers have addressed several issues pertaining to queueing networks with negative arrivals under the i.i.d. exponential service times assumption. Here we relax this assumption and derive a Pollaczek–Khintchine-like formula for M/G/1 queues with disasters by making use of the preemptive LIFO discipline. As a byproduct, the stationary distribution of the remaining service time process is obtained for queues operating under this discipline. Finally, as an application, we obtain the Laplace transform of the stationary remaining service time of the customer in service for unstable preemptive LIFO M/G/1 queues.

Keywords

NEGATIVE ARRIVALSRCLWORKLOAD PROCESSPREEMPTIVE LIFOM/G/1 QUEUEREMAINING SERVICE TIME

MSC classification

Primary: 60K25: Queueing theory 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Secondary: 60G10: Stationary processes 60G17: Sample path properties 60G55: Point processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 4 , December 1996 , pp. 1191 - 1200
Copyright
Copyright © Applied Probability Trust 1996 

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