Derivative Calculation through the Matrix-Geometric Solution Method

In this paper we show that through using the matrix-geometric solution method we can express the derivatives of the stationary queue-length probabilities of the M/G/1/K queue or M/G/1/(¿, K) multiclasses queue in terms of the M/G/1 queues with infinite buffers. We first establish these expressions for phase type service time distributions. Then we extend the result to general service time distributions The results can be used in optimization of networks consisting of such nodes with finite buffers.