Time-dependent and steady state solution of an M[x]/(G)/1 queueing system with server's long and short vacations

Abstract

We study the time-dependent and the steady state behaviour of an  M[x]/(G)/1 queue with server’s long and short vacations. Customers arrive at the system in a Poisson stream in batches of variable size. The service time of a customer served by the server is assumed to have a general distribution. At each service completion epoch, the server may opt to take a long vacation with probability p1 and a short vacation with probability p2 or else with probability p3 (p1+p2+p3=1) he may opt to continue to be available in the system. The  long and short vacation periods of the server  are assumed to have general distributions with different mean vacation times. We obtain  time-dependent probability generating functions for the queue size. The corresponding steady state results including the stability condition, steady state probabilities of different states of the system, the mean queue size and mean waiting time in the queue as well as in the system have been derived in explicit and closed forms. Several particular cases of interest including some earlier known results  have been derived. Many numerical examples confirming validity of the theoretical results have been peformed.