Optimal Service Time Distribution for an M/G/1 Waiting Queue

Suppose that in an M/G/1 waiting queue, the server can choose between two service time distributions. We look for the choice that enables us to minimize the expected value of a cost criterion that takes into account the cost incurred by working faster and the time needed to empty the waiting line. The random final time is the first time there is no customer waiting for service. When the service times are exponential random variables, we can appeal to dynamic programming to obtain the optimal solution. In the general case, conditional probability is used. Particular problems in which the capacity of the system is finite are solved explicitly.