Stability of generalized Jackson networks with permanent customers

Jackson queueing networks are typically open or closed: Either all customers join the network and eventually leave it, or no customers ever enter or exit. Here we focus on mixed Jackson networks, with both types of customers, general arrival streams and general service time distributions. We examine the stability of the model in terms of the positive Harris recurrence or transience of a Markov process which describes the state of the system. We show that this stability study reduces to that of an associated macroscopic deterministic model called the fluid model, obtained by an appropriate time-space scaling. This fluid model is shown to coincide with that associated with the equivalent open network, obtained by removing the closed component. As a result, the stability condition for the mixed Jackson network is the same as that for the equivalent open Jackson network.