Heterogeneous system MMPP/GI(2)/$\infty$ with random customers capacities

A heterogeneous queuing system with an infinite number of servers is considered in this paper. Customers arrive in the system according to a Markov Modulated Poisson Process. The type of incoming customer is defined as $i$-type with probability $p_i \, (i = 1,2)$. Each customer carries a random quantity of work (capacity of the customer). In this study service time does not depend on the customers capacities. It is shown that the joint probability distribution of the customers number and total capacities in the system is multidimensional Gaussian distribution under the asymptotic condition of an infinitely growing service time. Simulation results allow us to determine an applicability area of the asymptotic result.