Stability and Renormalization of Jackson Networks with Non-idling Mobile Servers

A tandem of two queues sharing a pool of servers, where users take the time to switch to the second queue, is used to model a typical pathway through an emergency department (ED), where patients undergo two consultations separated by diagnostic tests. In this article, explicit conditions for ergodicity, transience and null-recurrence are given and proven using linear Lyapunov functions. This result is extended to a Jackson network, with the key feature that the nodes share a pool of servers, with a non-idling service policy. Further, the delay times for customers to move from one node to another must be taken into account. This covers the main characteristics of models for emergency departments, namely priorities (triage) between patients. As for the tandem queue, by scaling the arrival rate and the number of servers by N, we obtain a renormalized process converging to the solution of an ordinary differential equation (ODE) subject to boundary conditions. When the system is ergodic, we discuss the solution of this ODE as t → ∞.