A Queueing System with $\chi^2$ Service-Time Distribution

Abstract

A stochastic process associated with a queueing system is specified by knowledge of (i) the input, (ii) the queue discipline, and (iii) the service mechanism. A system in which the input is of the "general independent" type and the service times independent and identically distributed according to an arbitrary, general law is given the label $GI/G/s,$ where $s$ is the number of servers (see Kendall [4]). An appointment system for arrivals (or regular service times) is designated by $D$ (deterministic); $M$ describes random arrivals (or negative-exponential service times); and $E_k$ (Erlangian) indicates that a scale-modified $\chi^2$ distribution with 2$k$ degrees of freedom governs the input (or service mechanism). Note that $M$ is equivalent to $E_1.$ The following study was suggested by Kendall in order to extend his description of the system $GI/M/s$ (see [4]) to the system $GI/E_k/s.$ This service time is thought of as the sum of $k$ independent components, identically distributed with negative-exponential distributions. The general system $GI/E_k/s,$ however, appears currently to be intractable in this form, so that we confine ourselves, in this paper, to the system $GI/E_k/1.$ We analyse this with the aid of an embedded Markov chain deriving the stationary distribution for the number of customers in the system at epochs of arrival (equation 1.16) and the distribution of the waiting time for an arbitrary customer (equation 1.21). Lindley [5] has discussed the problem of the waiting time in the system $D/E_k/1,$ solving for this particular example an integral equation governing all systems of the type $GI/G/1:$ the equivalence of our waiting time distribution is demonstrated in Section 2. Pollaczek ([6] and [7]) and Smith [8] have also considered systems of this kind.