Differential equations important in the M|G|∞ queue system transient behavior and busy period study

Abstract

This paper main subject is the M|G|∞ queue system populational process 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 can be applied in the modelation of many social problems: sickness, unemployment, emigration, ...(see, for instance, Ferreira (1995, 1996, 2003a and 2003b)), 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 and 2003) as a consequence of the transient behaviour study. We put also a special incidence in the study of the mean and the variance of the transient probabilities when the time origin is a busy period beginning instant, as time function. It allows the consideration of very interesting linear differential equations.