Analysis of customers' impatience in an M/M/1 queue with working vacations

Dequan Yue; Wuyi Yue; Gang Xu

doi:10.3934/jimo.2012.8.895

Article Contents

2012, Volume 8, Issue 4: 895-908. Doi: 10.3934/jimo.2012.8.895

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Analysis of customers' impatience in an M/M/1 queue with working vacations

1.
Department of Statistics, College of Sciences, Yanshan University, Qinhuangdao 066004
2.
Department of Intelligence and Informatics, Konan University, 8-9-1 Okamoto, Kobe 658-8501

Received: August 31, 2011

Revised: June 30, 2012

Published: September 2012

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Abstract

In this paper, we analyze an M/M/1 queueing system with working vacations and impatient customers. We examine the case that the customers' impatience is due to a working vacation. During a working vacation, customers are served at a slower than usual service rate and are likely to become impatient. Whenever a customer arrives in the system and realizes that the server is on vacation, the customer activates an ``impatience timer" which is exponentially distributed. If a customer's service has not been completed before the customer's timer expires, the customer leaves the queue, never to return. By analyzing this model, we derive the probability generating functions of the number of customers in the system when the server is in a service period and a working vacation, respectively. We further obtain the closed-form expressions for various performance measures, including the mean system size, the mean sojourn time of a customer served, the proportion of customers served and the rate of abandonment due to impatience. Finally, we present some numerical results to demonstrate effects of some parameters on these performance measures of the system.

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Mathematics Subject Classification: Primary: 60K25; Secondary: 90B22.

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