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INFINITE-SERVER QUEUES WITH SYSTEM'S ADDITIONAL TASKS AND IMPATIENT CUSTOMERS

Published online by Cambridge University Press:  25 September 2008

Eitan Altman  and
Uri Yechiali
Eitan Altman
Affiliation:
INRIA, Sophia Antipolis, France E-mail: altman@sophia.inria.fr
Uri Yechiali
Affiliation:
Department of Statistics and Operations Research, School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel E-mail: uriy@post.tau.ac.il
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Abstract

A system is operating as an M/M/∞ queue. However, when it becomes empty, it is assigned to perform another task, the duration U of which is random. Customers arriving while the system is unavailable for service (i.e., occupied with a U-task) become impatient: Each individual activates an “impatience timer” having random duration T such that if the system does not become available by the time the timer expires, the customer leaves the system never to return. When the system completes a U-task and there are waiting customers, each one is taken immediately into service. We analyze both multiple and single U-task scenarios and consider both exponentially and generally distributed task and impatience times. We derive the (partial) probability generating functions of the number of customers present when the system is occupied with a U-task as well as when it acts as an M/M/∞ queue and we obtain explicit expressions for the corresponding mean queue sizes. We further calculate the mean length of a busy period, the mean cycle time, and the quality of service measure: proportion of customers being served.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 22 , Issue 4 , October 2008 , pp. 477 - 493
Copyright
Copyright © Cambridge University Press 2008

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