A recursive method to the optimal control of an M/G/1 queueing system with finite capacity and infinite capacity

We study a single removable server in an infinite and a finite queueing systems with Poisson arrivals and general distribution service times. The server may be turned on at arrival epochs or off at service completion epochs. We present a recursive method, using the supplementary variable technique and treating the supplementary variable as the remaining service time, to obtain the steady state probability distribution of the number of customers in a finite system. The method is illustrated analytically for three different service time distributions: exponential, 3-stage Erlang, and deterministic. Cost models for infinite and finite queueing systems are respectively developed to determine the optimal operating policy at minimum cost.