On the waiting time distribution in a GI/G/1 queue with a Coxian–2 service time distribution

The purpose of this paper is to derive in an alternative way the result that the complementary waiting–time distribution function in the Gl/G/I queue is the sum of two exponential functions when the service time has a Coxian–2 distribution. The idea is to interpret this type of service–time distribution as the sum of a stochastic number of exponentially distributed phases. In this way the model can be seen as a special G/^x/W/1 batch–arrival queue where the batch–size distribution is deduced from the Coxian service–time distribution. For the latter model we give an embedded Markov–chain approach. Because of the special form of the batch–size distribution the steady–state distribution of this Markov chain can be represented as the sum of two geometric terms of which the coefficients can be explicitly given. From this result the waiting–time distribution can be deduced immediately. Apart from its didactic interest the result can be useful to obtain simple approximations for more general GI/G/1 models.