Transient Analysis of an M/M/1 Queueing Model with Two Kinds of Differentiated Working Vacation

Abstract

This paper deals with the time-dependent analysis of an M/M/1 queueing model subject to differentiated working vacation. When the system becomes empty, the server takes a vacation of a particular duration (Type I). When the server finds an empty system upon his return, the server might take another vacation of a shorter duration (Type II). Both type I and type II vacation are exponentially distributed. Further, it is assumed that the service continues at a slower rate during the vacation period, rather than completely stopping the service. Explicit expressions for the time-dependent system size probabilities are obtained in terms of modified Bessel function of first kind using generating functions and continued fraction methodology. Specific performance measures like the time-dependent mean and variance are also obtained. Numerical illustrations are added to depict the effect of variation in different parameter values on the time-dependent probabilities, mean and variance. Also, as tends to infinity, the time dependent probabilities are deduced to the corresponding steady-state probabilities.