On the waiting time distribution in a generalized GI/G/1 queueing system

Abstract

The model considered in this paper describes a queueing system in which the station is dismantled at the end of a busy period and re-established on arrival of a new customer, in such a way that the closing-down process consists of N 1 phases of random duration and that a customer 𝒞 n who arrives while the station is being closed down must wait a random time idn (i = 1, ···, N 1) if the ith phase is going on at the arrival instant. (For each fixed index i, the random variables idn are identically distributed.) A customer 𝒞 n arriving when the closing-down of the station is already accomplished has to wait a random time (N 1 + 1) dn corresponding to the set up time of the station. Besides, a customer 𝒞 n who arrives when the station is busy has to wait an additional random time 0 dn. We thus have (N 1 + 2) types of “delay” (additional waiting time). Similarly, we consider (N 2 + 2) types of service time and (N 3 + 2) probabilities of joining the queue. This may be formulated as a model with (N + 2) types of triplets (delay, service time, probability of joining the queue). We consider the general case where the random variables defining the model all have an arbitrary distribution. The process {wn }, where wn denotes the waiting time of customer 𝒞 n if he joins the queue at all, is not necessarily Markovian, so that we first study (by algebraic considerations) the transient behaviour of a Markovian process {vn } related to {wn }, and then derive the distribution of the variables wn.