The queuing system M/ GB/ l and its ramifications

Abstract

This paper deals with the bulk service queuing system M/ G B /1 and its ramifications. In the system M/ G B /1, customers arrive singly by a Poisson process and are served by a single server in order of arrival in batches of fixed maximum size B. The distribution of service time is general and an idle server begins service immediately if one or more customers await service. The main aims of the paper are: 1. (i) to generalize Jaiswal's queuing system M/ E R B /1; 2. (ii) to unify Bailey's and Jaiswal's generalized results and obtain relationships between the pgf's (probability generating functions) P q(z)= ∑ n=0 ∞ P n,1 z n and P +(z)= ∑ n=0 ∞ P n z n + where P n,1 and P n + are the limiting probabilities of n customers in the queue (at random epoch) with server being busy and n in the queue immediately after a departure epoch respectively; 3. (iii) to obtain a simple derivation of the transform of the distribution of waiting time first obtained by Downton and later by others, and to suggest ways of getting computational results in the case when expected values are considered at random epochs rather than at departure epochs as discussed by Bailey and Downton. The supplementary variable technique is used in discussing the above results. It is also demonstrated that Little's formula is true for M/ G B /1 as it should, since it is independent of the distributions involved. Some other operational parameters of this are also discussed. Examples of extensive numerical results for P 0 + and L + are presented in tabular form.