Asymptotic Analysis of a Closed Exponential Queueing Network with Unreliable Nodes

A closed exponential queueing network with unreliable nodes was studied. The process of changing the number of serviceable servers in network nodes was considered as the birth–death process. The process of changing the number of customers at the nodes was studied under the asymptotic assumption of a large number of customers. The last-mentioned process divided by the large number of customers is a continuous stochastic process with the Markov property. It was proved that its probability density function satisfies the Fokker–Planck–Kolmogorov equation. The system of differential equations for the first-order and second-order moments of this process was derived.