Estimating the average delay of the first N customers in an M/Erlang/1 queue

Queueing problems where the queue starts from empty each day are quite common, yet it is not easy to estimate the average waiting time over all customers for the class of M/G/1 queues. The paper provides an easily programmed algorithm that calculates it exactly for M/Ek/1 queues. The convergence of the average waiting time to the steady-state value turns out to be surprisingly independent of the number of stages. The motivation for this study was a forestry weighbridge problem where the service time distribution depended on whether a truck arrived to an empty, or to a busy, weighbridge. The algorithm is extended to this case.