On the time-dependent occupancy and backlog distributions for the GI/G/∞ queue

We consider an infinite server queueing system. An examination of sample path dynamics allows a straightforward development of integral equations having solutions that give time-dependent occupancy (number of customers) and backlog (unfinished work) distributions (conditioned on the time of the first arrival) for the GI/G/∞ queue. These integral equations are amenable to numerical evaluation and can be generalized to characterize GI^X/G/∞ queue. Two examples are given to illustrate the results.