Transient exponential-Erlang queues and steady-state simulation

The transient probabilistic structure of M/E m /1 and E m /M/1 queues initialized in an arbitrary deterministic state is derived in discrete time. Computational algorithms for obtaining the required probabilities are provided, and their application in calculating a variety of system performance measures is illustrated. The results are used to investigate the question of initializing simulations of systems such as these to promote rapid convergence to steady state, if that is the object of the simulation. These results are consistent with earlier studies for transient queueing systems, such as the M/M/s, but allow greater flexibility in specification of interarrival or service-time models inherent in the Erlang distributions.