Modern Stochastics: Theory and Applications

We study a time-changed variant of the Erlang queue by taking the first hitting time of a mixed stable subordinator as the time-changing component. We call it the mixed time-changed Erlang queue. We derive the system of fractional differential equations that governs its state probabilities. The explicit expressions for the state probabilities of mixed time-changed Erlang queue and their Laplace transform are derived. Also, an equivalent representation of this time-changed queue in terms of phases is provided, and its mean queue length is obtained. Some of its distributional properties such as the distribution of its inter-arrival times, inter-phase times, service times and busy period are derived. Later, its conditional waiting time is discussed and some plots of sample paths simulation are presented.