On the optimal switching level for an M/G/1 queueing system

A cost function is studied for an M/G/1 queueing model for which the service rate of the virtual waiting time process U_t for U_t<K differs from that for U_t > K. The costs considered are costs for maintaining the service rate, costs for switching the service rate and costs proportional to the inventory U_t. The relevant cost factors for the system operating below level K differ from those when U_t > K. The cost function which is considered only for the stationary situation of the U_t-process expresses the average cost per unit time. The problem is to find that K for which the cost function reaches a minimum. Criteria for the possibly optimal cases are found; they have an interesting intuitive interpretation, and require the knowledge of only the first moment of the service time distribution.