Study of the $\mathrm{MMPP/GI/}\infty$ queueing system with random customers' capacities

A queueing system with an infinite number of servers is considered. Customers arrive in the system according to a Markov Modulated Poisson Process (MMPP). Each customer carries a random quantity of work (capacity of the customer). In this study, service time does not depend on the customers' capacities; the latter are used just to fix some additional features of the system's evolution. It is shown that the joint probability distribution of the customers' number and total capacities in the system is two-dimensional Gaussian under the asymptotic condition of an infinitely growing service time. Simulation results allow determining the applicability area of the asymptotic result.