Transient analysis of an M/M/1 queueing system subject to differentiated vacations

This paper presents the transient analysis of an M/M/1 queueing system wherein the server is subject to two types of vacation, namely: vacation after the busy period and vacation taken immediately after the server has just returned from a previous vacation to find that there are no waiting customers. The arrivals are allowed to join the queue according to a Poisson distribution and the service takes place according to an exponential distribution. Whenever the system is empty, the server goes for a vacation and returns to the system after some random duration. If the server finds no waiting customer, he immediately takes another vacation of shorter duration. This paper presents explicit expressions for the time-dependent system size probabilities in terms of modified Bessel function of the first kind using generating function and Laplace transform techniques. Certain interesting performance measures like the mean and variance of the number of customers in the system, probability of the server to be in a busy or vacation state at an arbitrary time are explicitly obtained. Numerical illustrations are added to support the theoretical results.