Heterogeneous system GI/GI(n)/∞ with random customers capacities

In the paper, we consider a queuing system with n types of customers. We assume that each customer arrives at the queue according to a renewal process and takes a random resource amount, independent of their service time. We write Kolmogorov integro-differential equation, which, in general, cannot be analytically solved. Hence, we look for the solution under the condition of infinitely growing a service time, and we obtain multi-dimensional asymptotic approximations. We show that the n-dimensional probability distribution of the total resource amounts is asymptotically Gaussian, and we look at its accuracy via Kolmogorov distance.