Asymptotical analysis of a non-Markovian queueing system with renewal input process and random capacity of customers

In this paper, the GI/GI/да queueing system (QS) with random capacity customers is studied. The arrival process is a Renewal process. The system has an unlimited number of servers and service times on each server are i.i.d. with distribution function B(x). All customers have a random capacity v > 0 with probability distribution G(y) = P{v < y} and the customers capacities are independent. Moreover, we assume that service time and customers capacity are mutually independent. After the service, customers leave the system and "take away" the capacity. We considered two-dimensional stochastic process {i(t), V(t)}, where i(t) and V(t) denote the number of customers in the system and the total customers capacity at time t, respectively. We proposed the dynamic screening method for its investigation. Note that this method exactly determines the characteristics of the process V(t) since the screened process contains only those customers, which do not finish the service at the moment T. We obtained the system of Kolmogorov differential equations and by using the partial characteristic function, we wrote the main equation: ⁸h(z, ^u1,^u 2,^t) = ⁸h(z, ^u1, ^u2,^t) + ⁸h(0, ^u1,^u 2,^t) [a(z ) -1 + S (t )A(z )(e- G * (u₂) -1)] 8t 8z 8z with the initial condition h(z, u₁, u ₂,1₀ ) = R(z).

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