On a new formula for the transient state probabilities for M/M/1 queues and computational implications

Abstract

Past work relating to the computation of time-dependent state probabilities in M/M/1 queueing systems is reviewed, with emphasis on methods that avoid Bessel functions. A new series formula of Sharma [13] is discussed and its connection with traditional Bessel function series is established. An alternative new series is developed which isolates the steady-state component for all values of traffic intensity and which turns out to be computationally superior. A brief comparison of our formula, Sharma's formula, and a classical Bessel function formula is given for the computation time of the probability that an initially empty system is empty at time t later.