Approximate Transient Analysis of Some Queuing Systems

We propose approximations for the time-dependent queue size distribution P_n(t) and the mean queue size E[X(t)] of an M/M/1 queue. With a Laplace transform-generating function technique, the exact analysis involves the complex root u* inside the unit circle of a certain quadratic. The exact value of u* leads to transforms that, when inverted, yield rather complicated expressions for P_n(t) and E[X(t)]. We discuss three iterative procedures that give successive rational approximations for u*. These approximations yield, for the queue size distribution and the mean queue size, approximate transforms that are rational and easier to invert. We obtain sharp bounds for P_n(t) and E[X(t)] for a wide range of parameter values by performing the second iteration and combining the approximate results given by the iterative procedures.

The methods can be applied to approximate the behavior of more complicated queues. Applications to the M/M/2 queue and the M/E_p/1 queue are discussed briefly.