On the Transient Behavior of the M / M /1 and M / M /1/ M Queues

This paper gives a transient analysis of the classic M/M/ 1 and M/M/ 1/ K queues. Our results are asymptotic as time and queue length become simultaneously large for the infinite capacity queue, and as the system’s storage capacity K becomes large for the finite capacity queue. We give asymptotic expansions for p n ( t ), which is the probability that the system contains n customers at time t . We treat several cases of initial conditions and different traffic intensities. The results are based on (i) asymptotic expansion of an exact integral representation for p n ( t ) and (ii) applying the ray method to a scaled form of the forward Kolmogorov equation which describes the time evolution of p n ( t ).