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.

Distribution of the sum of independent identically distributed symmetric lattice vectors is approximated by the accompanying compound Poisson law and the second-order Hipp-type signed compound Poisson measure. Bergström-type asymptotic expansion is constructed. The accuracy of approximation is estimated in the total variation metric and, in many cases, is of the order $O({n^{-1}})$.