A new formula for the transient behaviour of a non-empty M/M/1/∞ queue

This paper studies the most general model of the Markovian queues namely M/M/1/~ queue with any arbitrary number ''i'' of customers being present in the system initially. For this model, a new and simple series form is obtained for the transient state probabilities whence all particular cases concerning the system being empty and steady-state situations can be derived straightaway. The coefficients in this series satisfy iterative recurrence relations which would allow for the rapid and efficient numerical evaluations of the state probabilities. Moreover, a simple algebraic proof to show the equivalence between different formulae of a non-empty M/M/1/~ and the new formula is established.