Impact of customers’ impatience on an M/M/1 queueing system subject to differentiated vacations with a waiting server

Abstract

Impatience behavior of an M/M/1 queueing system, which has differentiated server vacations, is considered. Arrivals follow a Poisson process, service is exponentially distributed, as are both vacation types. When the system is empty after waiting for a random period of time, the server takes a vacation and returns after a random duration. If there are still no customers in the system, server can go for a vacation of shorter duration. If the server is in a vacation state, arriving customers become impatient with an individual timer which is exponentially distributed. Explicit expressions for time-dependent probabilities of the system size are obtained in terms of the modified Bessel function of first kind by making use of Laplace transforms and probability generating function techniques along with continued fractions and the confluent hypergeometric function. Explicit expressions for the time-dependent mean and variance of the system size are also derived. Moreover, steady-state probabilities for the system size are derived from the transient state probabilities. Finally, a numerical example is presented to study the behavior of the system.

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