M|G|∞ queue system transient behaviour and busy period

Abstract

This paper main subject is the M|G|∞ queue system transient probabilities study as time functions. We achieve it completely when the time origin is an unoccupied system instant. But we do not get such a goal when the time origin is a busy period beginning instant. We shall see that, in this last situation, the service time length distribution hazard rate function plays a very important role. And so the results got may be useful in the survival analysis field. As the M G queue system can be applied in the modelation of many social problems: sickness, unemployment, emigration, ...(see, for instance, Ferreira (1995 and 1996)), in these situations it is very important to study the busy period length distribution of that system. We show, in this work, that the solution of the problem may be in the resolution of a Ricatti equation generalizing the work of Ferreira (1998) as a consequence of the transient behaviour study.