Transient analysis of M/M/1 fractional queue

Queuing models consisting of servers which may not work with full efficiency are modelled with the help of fractional differential equations. Such problems with partial activity of the server are designed with the help of differential-difference equations involving fractional derivatives whose solutions are expressed in terms of Mittag-Leffler function. In this paper, we propose an alternative approach which gives a transient solution of fractional M/M/1 queue in matrix form. The results obtained by this new approach are justified by comparing them with solutions of classical queue which are available in the literature. Efficacy of the model can be assessed by computing its state probabilities and also measures such as expected number of customers in the system etc. Also, the variations in these measures with respect to partial activity of the server have been presented graphically and numerically. Further, a mathematical procedure to find optimal efficiency of the server has been discussed.