A Simple Formula for the Transient State Probabilities of an M/M/1/∞Queue

In this paper a series formula is developed for the time-dependent state probabilities for the M/M/1 queue and it is shown that the coefficients in this series satisfy simple recurrence relations which would allow for the rapid and efficient evaluation of the state probabilities. Furthermore, a closed form solution to these recurrence equations is obtained which leads to a simple formula for the transient state probabilities from which the steady state solution is readily isolated