On the Transient Behaviour of Fractional $$m/m/\infty $$ Queues

We study some features of the transient probability distribution of a fractional M/ M/infinity queueing system. Such model is constructed as a suitable timechanged birth-death process. The fractional differential-difference problem is studied for the corresponding probability distribution and a fractional partial differential equation is obtained for the generating function. Finally, the interpretation of the system as an actual M/M/8 queue and as a M/ M/1 queue with responsive server is given and some conditioned virtual waiting times are studied.