Transient Analysis of an M/M/1 Queue with Bernoulli Vacation and Vacation Interruption

Abstract

This paper presents the transient analysis of an \( M/M/1 \) queueing system subject to Bernoulli vacation and vacation interruption. 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 can take either a working vacation or an ordinary vacation with certain probabilities. During working vacation, the arrivals are allowed to join the queue and the service takes place according to an exponential distribution, but with a slower rate. The vacation times are also assumed to be exponentially distributed. Further, during working vacation, the server has the option to either continue the vacation or interrupt and transit to the regular busy period. During ordinary vacation, the arrivals are allowed to join the queue but no service takes place. The sever returns to the system on completion of the vacation duration. Upon returning from vacation, the server continues to provide service to the waiting customers (if any) or takes another vacation (working or ordinary vacation) if the system is empty. Explicit expressions are obtained for the time dependent system size probabilities in terms of modified Bessel function of the first kind using generating function and Laplace transform techniques. Numerical illustrations are added to support the theoretical results.