From traditional to lattice paths approach: A review on performance of M/G/1 queueing systems

Abstract

In this paper we review the performance of M/G/1 queueing systems. These systems refer to those systems in which the arrivals are generated by a Poisson process but the service time is permitted to be non-exponential distribution. Note that M/G/1 assumes no explicit distribution for the service time but assume that the service times of all customers be independently and identically distributed (i.i.d). Feature of actual situation in many practical scenarios often follows M/G/1 model. For example when a service station fails and needs repairs, then the amount of time needed to finish the job has a general distribution. Through traditional approach, the steady state or transient solutions were either in terms of Laplace-Stieltjes transforms (LSTs) or other integrals, so that inversion was required to get the probabilities of interest. For general service distributions and complex queuing scenarios, even with the current computing power the inversions are complicated and intractable. Avoiding such problems the discrete time queueing has been applied to derive the solutions. Furthermore, one can use lattice path combinatorics to study queuing processes using limiting process. Lattice paths approach was modified on deriving the busy period density function without using limiting process. The system probabilities obtained in this case are in explicit closed forms that are straight forward to compute.