Solving logistics problems using M|G|∞ queue systems busy period

Abstract

In the M|G|∞ queuing systems customers arrive according to a Poisson process at rate λ . Each of them receives immediately after its arrival a service whose length is a positive random variable with distribution function G(.) and mean value α . An important parameter of the system is the traffic intensity ρ = λα . The service of a customer is independent of the services of the other customers and of the arrival process. The busy period of a queuing system begins when a customer arrives there, finding it empty, and ends when a customer leaves the system letting it empty. During the busy period there is always at least one customer in the system. Therefore in a queuing system there is a sequence of idle and busy periods. For these systems with infinite servers the busy period length distribution is difficult to derive, except for a few exceptions. But formulae that allow the calculation of some of the busy period length parameters for the M|G|∞ queuing system are presented. These results can be applied in logistics (see, for instance, Ferreira [4,5] and Ferreira, Andrade and Filipe [9]). For instance, they can be applied to the failures which occur in the operation of an aircraft, shipping or trucking fleet. The customers are the failures. And their service time is the time that goes from the instant at which they occur till the one at which they are completely repaired. Here a busy period is a period in which there is at least one failure waiting for reparation or being repaired. The formulae referred allow the determination of measures of the system performance.