An M/M/1 Queueing Model Subject to Differentiated Working Vacation and Customer Impatience

Abstract

This paper deals with the stationary and transient analysis of a single server queueing model subject to differentiated working vacation and customer impatience. Customers are assumed to arrive according to a Poisson process and the service times are assumed to be exponentially distributed. When the system empties, the single server takes a vacation of some random duration (Type I) and upon his return if the system is still empty, he takes another vacation of shorter duration (Type II). Both the vacation duration are assumed to follow exponential distribution. Further, the impatient behaviour of the waiting customer due to slow service during the period of vacation is also considered. Explicit expressions for the time dependent system size probabilities are obtained in terms of confluent hyper geometric series and modified Bessel’s function of first kind using Laplace transform, continued fractions and generating function methodologies. Numerical illustrations are added to depict the effect of variations in different parameter values on the time dependent probabilities.KeywordsM/M/1 queueDifferentiated working vacationCustomer impatienceContinued fractionsGenerating functionsConfluent hypergeometric functions