Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-24T13:11:34.951Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

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

Chao, X. and Pinedo, M. (1993) On generalized networks of queues with positive and negative arrivals. Prob. Eng. Info. Sci. 7, 301–334.Google Scholar
Fakinos, D. (1981) The G/G/1 queueing system with a particular queue discipline. J. R. Statist. Soc. B 43, 190–196.Google Scholar
Gelenbe, E. (1991) Product form networks with negative and positive customers. J. Appl. Prob. 28, 656–663.CrossRefGoogle Scholar
Gelenbe, E. (1993) G-networks with triggered customer movement. J. Appl. Prob. 30, 742–748.Google Scholar
Gelenbe, E., Glynn, P. and Sigman, K. (1991) Queues with negative arrivals. J. Appl. Prob. 28, 245–250.CrossRefGoogle Scholar
Harrison, P. G. and Pitel, E. (1993) Sojourn times in single-server queues with negative customers. J. Appl. Prob. 30. 943–963.CrossRefGoogle Scholar
Henderson, W. (1993) Queueing networks with negative customers and negative queue lengths. J. Appl. Prob. 30. 931–942.CrossRefGoogle Scholar
Jain, G. (1996) A rate conservation analysis of queues and networks with work removal. PhD dissertation. Industrial Engineering and Operations Research, Columbia University.Google Scholar
Miyazawa, M. (1983) The derivation of invariance relations in complex queueing systems with stationary inputs. Adv. Appl. Prob. 15, 874–885.Google Scholar
Sigman, K. (1994) Stationary Marked Point Processes: An Intuitive Approach. Chapman and Hall, New York.Google Scholar
Wolff, R. W. (1989) Stochastic Modeling and the Theory of Queues. Prentice Hall, Englewood Cliffs, NJ.Google Scholar