The queueing system MX/G/1 and its ramifications

This paper deals with the bulk arrival queueing system M X /G/1 and its ramifications. In the system M X /G/1, customers arrive in groups of size X (a random variable) by a Poisson process, the service times distribution is general, and there is a single server. Although some results for this queueing system have appeared in various books, no unified account of these, as is being presented here, appears to have been reported so far. The chief objectives of the paper are (i) to unify by an elegant procedure the relationships between the p.g.f.'s magnified image where P n and P n + are the limiting probabilities of queue lengths being n at random and departure epochs respectively, (ii) to correct an error in the paper by Krakowski and generalize his results and (iii) to discuss some other interesting cases of the system M X /G/1 and its special cases.