Tail asymptotics for waiting time distribution of an M/M/s queue with general impatient time

Yutaka Sakuma; Atsushi Inoie; Ken’ichi Kawanishi; Masakiyo Miyazawa

doi:10.3934/jimo.2011.7.593

Article Contents

2011, Volume 7, Issue 3: 593-606. Doi: 10.3934/jimo.2011.7.593

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Tail asymptotics for waiting time distribution of an M/M/s queue with general impatient time

1.
Department of Distribution and Information Engineering, Hiroshima National College of Maritime Technology, Osakikamijima-Town, 725-0231
2.
Department of Information Network and Communication, Kanagawa Institute of Technology, Atsugi-City, 243-0292
3.
Department of Computer Science, Gunma University, Kiryu-City, 376-8515
4.
Department of Information Sciences, Tokyo University of Science, Noda-City, 278-8510

Received: September 2010

Revised: May 2011

Published: July 2011

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Abstract

In this paper, we consider an $M/M/s$ queueing model where customers may abandon waiting for service and leave the system without receiving their services. We assume that impatient time on waiting for each customer is an independent and identically distributed nonnegative random variable with a general distribution where the probability distribution is light-tailed and unbounded. The main objective of this paper is to provide an approximation for the waiting time distribution in an analytically tractable form. To this end, we obtain the tail asymptotics of the waiting time distributions of served and impatient customers. By using the tail asymptotics, we show that the fairly good approximations of the waiting time distributions can be obtained in asymptotic region with low numerical complexity.

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Mathematics Subject Classification: Primary: 60K25, 60K25; Secondary: 60K25.

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