Spontaneous Symmetry-Breaking in Deterministic Queueing Models with Delayed Information

Abstract

The dynamics of a system involving two queues which incorporates customer choice behavior based on delayed queue length information was studied recently. Waiting times in emergency rooms of hospitals, telephone call centers, and various rides in theme parks are some examples where delayed information is provided to the customers. This time delay has an impact on the dynamics of the queues and therefore has the capacity to affect the decision of a customer to choose which queue to wait in. We generalize this queueing model to a finite arbitrary number of queues. The system of delay differential equations for this generalized model is equivariant under a symmetry group. Spontaneous symmetry-breaking occurs in an equivariant dynamical system when the symmetry group of a solution of the equations is lesser than the symmetry group of the equations themselves. In this work, we show that the generalized model exhibits spontaneous symmetry-breaking. In particular, we show that varying the time delay parameter can make a stable equilibrium become unstable, and this switch in stability occurs only at a symmetry-breaking Hopf bifurcation. However, if the number of queues is chosen to be large enough, then the equilibrium is absolutely stable.