Jackson networks with single-line nodes and limited sojourn or waiting times

We consider an exponential queueing network with single-line nodes that differs from a Jackson network only in that the time claims sojourn in the system (waiting time for the servicing) at the network's nodes is limited by an exponentially distributed random value. Claims serviced at nodes and claims that have not finished servicing (that exceeded maximal waiting time) move along the network according to different routing matrices. We prove a theorem that generalizes Jackson's theorem for open and closed networks.