Simple Approximations for the Batch-Arrival Mx/G/1 Queue

Batch-arrival queueing models can be used in 1) many practical situations, such as the analysis of message packetization in data communication systems. In general it is difficult, if not impossible, to find tractable expressions for the waiting-time probabilities of individual customers. It is, therefore, useful to have easily computable approximations for these probabilities. This paper gives such approximations for the single server MX/G/1 model. Exact methods for the computation of the waitingtime distribution in the MX/G/1 queue are discussed in Eikeboom and Tijms (1987), cf. also Chaudhry and Templeton (1983), Neuts (1981) and Tijms (1986). However, these methods apply only for special servicetime distributions and are, in general, not suited for routine calculations in practice. A simple approximation for the tail probabilities of the waiting time was given in Eikeboom and Tijms by using interpolation of the asymptotic expansions for the particular cases of deterministic and exponential services. This approximate approach uses only the first two moments of the service time. This paper presents an alternative approach that uses the actual service-time distribution rather than just its first two moments. This alternative approach starts with the asymptotic expansion of the waitingtime distribution. In Van Ommeren (1988) it is shown that the complementary waiting-time probability of an arbitrary customer in the MX/G/1 queue has an exponentially fast decreasing tail under some mild assumptions. By calculating the decay parameter and the amplitude factor, we get the asymptotic expansion of the waiting-time distribution. Such asymptotic expansions already provide a very powerful tool in practical queueing analysis, cf. Cromie, Chaudhry and Grassman (1979) and Tijms. It turns out that for nonlight traffic this asymptotic expansion can be used as a good first-order approximation for the waitingtime probabilities. Next, by incorporating exact results for other quantities, such as the delay probability and the first two moments of the waiting time, we are able to give an improved second-order approximation. This approximate method performs very well for a wide range of values of the traffic intensity and the coefficients of variation of the service-time distribution and the batch-size distribution. The organization of this paper is as follows. In Section 1, the model is defined and some preliminaries, including the asymptotic expansion of the waiting-time distribution, are given. The second-order approximation is given in Section 2. In Section 3 we give numerical results and discuss the performance of the approximations. The Appendix deals with the motivation of the second-order approximation.