Approximation to the Transient Solution of the M/E_k/1 Queue

This paper considers the Erlang queueing system M/E_k/1, where customers arrive at random at mean rate λ and the service times have an Erlang distribution with parameter k and mean service rate μ. It is difficult to obtain a transient solution in explicit form to the queue equations because of their complex structure. We propose a simple method of computing W_q(t)—the mean waiting time of a customer arriving in the queue at time t, based on an exponential function approximation.