An exact FCFS waiting time analysis for a general class of G/G/s queueing systems

Abstract

A closed form expression for the waiting time distribution under FCFS is derived for the queueing system MGE_k/MGE_m/s, where MGE_n is the class of mixed generalized Erlang probability density functions (pdfs) of ordern, which is a subset of the Coxian pdfs that have rational Laplace transform. Using the calculus of difference equations and based on previous results of the author, it is proved that the waiting time distribution is of the form 1-$\sum\nolimits_{j = l}^{(\begin{array}{*{20}c} {s + m - l} \\ s \\ \end{array} )} {L_j e} ^{ - u_j t}$, under the assumption that the rootsU _j are distinct, i.e. belongs to the Coxian class of distributions of order$(\begin{array}{*{20}c} {s + m - l} \\ s \\ \end{array} )$. The present approach offers qualitative insight by providing exact and asymptotic expressions, generalizes and unifies the well known theories developed for the G/G/1,G/M/s systems and leads to an$O(k^3 (\begin{array}{*{20}c} {s + m - l} \\ s \\ \end{array} )^3 )$ algorithm, which is polynomial if only one of the parameterss orm varies, and is exponential if both parameters vary. As an example, numerical results for the waiting time distribution of the MGE₂/MGE₂/s queueing system are presented.