The use of derivatives for optimizing steady state queues

To design queueing systems in an optimal way, one needs derivatives of the main queueing measures, such as the average number in the system and the throughput. This paper shows how such measures can be obtained in a Markovian environment. For simplicity, attention is restricted to queues with Poisson arrivals, having either a finite buffer capacity or a finite calling population. For these queues, the paper first determines the derivatives of their steady state probabilities, which allows the finding of the derivatives of throughput and average number in the system. A number of examples show how these derivatives can be used for the purpose of optimization.