Stationary distribution of Jackson networks with an exponential constraint on the sojourn time of claims

An exponential queuing network containing nodes of two types — single-line and multi-line is considered. In contrast to Jackson networks, the sojourn time at the nodes of the network is a random variable, the conditional distribution of which is exponential for a fixed number of claims in the node. For single-line nodes, the routing matrices of the served and non-served customers are, generally speaking, different, and for multi-line nodes they coincide. A sufficient ergodicity condition is established and a stationary distribution is found.