On the distribution of queueing times for queues with two servers

Abstract

Initially we consider first come first served queues with two or more servers wherein the intervals between the successive arrivals are independently and identically distributed. The customers1 service times are similarly distributed. The method is to define an embedded Markov chain on the moments just before each arrival and thence recurrences ^ which prelate the state of the system just before the (n+1; arrival to that which existed just before the nth. The problem is then specialized to that for two servers and these probability recurrences are used to develop a relationship between the bivariate Laplace transformations of the distribution functions which arise. The problem is reduced to the solution of a single integral equation for the Laplace transformation of the ergodic limiting distribution function by the definition of two compensation functions. These steps are analogous to the veil known probability "sweeping up" operations for one server queues. This integral equation is reduced to a functional equation for the case where the interarrival and service tine distributions are both composed of any integral numbers of exponential stages. This equation is solved in principle for all finite numbers of such stages, and in detail when the service time distribution has one or two stages and the interarrival time distribution any number. The results are checked against a known result for one case and a set of simulation results for another. The agreement is satisfactory. Of particular note are the curious loci of certain singularities. The Thesis also discusses a number of important intermediate results which suggest that the classical Miener-Kopf method for a unidimensional integral equation may generalize to a useful multidimensional result. We conclude with a section which outlines the proofs of a more general Theorem which explores this possibility. Show more