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 thesecond queue, is used to model a typical pathway through an emergency department (ED), wherepatients undergo two consultations separated by diagnostic tests. In this article, explicit conditionsfor ergodicity and transience are given and proven via Foster’s criterion, using a linear Lyapunovfunction. This result is extended to a Jackson network, with the key difference that the nodes sharea pool of servers, with a non-idling service policy. Further, the delay times for customers to movefrom one node to another must be taken into account. This covers some of the main features ofmodels for emergency departments, namely priorities (triage) between patients. In the case of thetandem queue, by scaling the arrival rate and the number of servers by N, we obtain a renormalizedprocess converging to the solution of an ordinary differential equation (ODE) subject to boundaryconditions. When the system is ergodic, we discuss the solution of this ODE as t → ∞.