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April  2016, 12(2): 653-666. doi: 10.3934/jimo.2016.12.653

## Analysis of an M/M/1 queue with vacations and impatience timers which depend on the server's states

 1 Department of Statistics, College of Sciences, Yanshan University, Qinhuangdao 066004 2 Department of Intelligence and Informatics, Konan University, Kobe 658-8501 3 School of Economics and Management, Yanshan University, Qinhuangdao 066004, China

Received  September 2014 Revised  March 2015 Published  June 2015

We consider an M/M/1 queueing system with vacations and impatient customers. Whenever a customer arrives at the system, it activates an random impatience timer". If the customer's service has not been completed before the customer's impatience timer expires, the customer abandons the queue, and never returns. It is assumed that the impatience timer depends on the server's states. We analyze both multiple and single vacation scenarios and derive the probability generating functions of the number of customers in the system when the server is in vacation period and busy period. Then, we obtain explicit expressions for various performance measures such as the mean system sizes when the server is either on vacation or busy, the proportion of customers served, and the average rate of abandonments due to impatience. We present some numerical results for multiple vacation scenario to show the effects of the parameters of impatience timers on some performance measures. Finally, we show some inequalities on some performances under the single vacation policy and under multiple vacation policy.
Citation: Dequan Yue, Wuyi Yue, Guoxi Zhao. Analysis of an M/M/1 queue with vacations and impatience timers which depend on the server's states. Journal of Industrial & Management Optimization, 2016, 12 (2) : 653-666. doi: 10.3934/jimo.2016.12.653
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