Mean sojourn time in state k, k=0,1,..., for the M|G|∞ queueing system (Exact and approximated expressions)

Abstract

In a M|G|∞ queue system ? is the Poisson process arrival rate, ? is the mean service time, G(.) is the service time distribution function and there are infinite servers. When we consider practical situations to apply this model we do not want necessarily the physical presence of infinite servers. But we only guarantee that when a customer arrives at the system it always finds immediately a server available. Other situations occur when there is no distinction between a customer and its server. So, often, it is very important to manage a group of servers, in order to guarantee that the system works as an infinite server queueing system as it was designed. For this purpose it is important to know the mean sojourn time in state k , k=0,1,.... Here, by state k , we mean that there are k customers in the system, or, that is the same, k servers occupied. Unhappily, only for M|M|∞ (exponential service time) queueing systems we know that mean. But as it was proposed in (Ramalhoto and Girmes, 1977) we will consider that M|G|∞ systems are well approximated by a Markov Renewal Process. And so we will consider the mean sojourn time in state k , k=0,1,... for that process as a good approximation to the ones of the M|G|∞ queueing systems. Then we are going to show some results to the mean sojourn time in state k , k=0,1,... distribution function for the Markov Renewal Process considered.