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

Abstract

In this paper, we analyze an M/M/1 queueing system with workingvacations and impatient customers. We examine the case that thecustomers' impatience is due to a working vacation. During a workingvacation, customers are served at a slower than usual service rateand are likely to become impatient. Whenever a customer arrives inthe system and realizes that the server is on vacation, the customeractivates an ``impatience timer' which is exponentially distributed.If a customer's service has not been completed before the customer'stimer expires, the customer leaves the queue, never to return. Byanalyzing this model, we derive the probability generating functionsof the number of customers in the system when the server is in aservice period and a working vacation, respectively. We furtherobtain the closed-form expressions for various performance measures,including the mean system size, the mean sojourn time of a customerserved, the proportion of customers served and the rate ofabandonment due to impatience. Finally, we present some numericalresults to demonstrate effects of some parameters on theseperformance measures of the system.