Bubbling, Bistable Limit Cycles and Quasi-Periodic Oscillations in Queues with Delayed Information

Juancho A Collera

doi:10.3390/sym14020376

Abstract

We consider a model describing the length of two queues that incorporates customer choice behavior based on delayed queue length information. The symmetric case, where the values of the time-delay parameter in each queue are the same, was recently studied. It was shown that under some conditions, the stable equilibrium solution becomes unstable as the common time delay passes a threshold value. This one-time stability switch occurs only at a symmetry-breaking Hopf bifurcation where a family of stable asynchronous limit-cycle solutions arise. In this paper, we examine the non-symmetric case, wherein the values of the time-delay parameter in each queue are different. We show that, in contrast to the symmetric case, the non-symmetric case allows bubbling, multiple stability switches and coexistence of distinct families of stable limit cycles. An investigation of the dynamical behavior of the non-symmetric system in a neighborhood of a double-Hopf bifurcation using numerical continuation explains the occurrence of the bistable limit cycles. Quasi-periodic oscillations were also observed due to the presence of torus bifurcations near the double-Hopf bifurcation. These identifications of the underlying mechanisms that cause unwanted oscillations in the system give a better understanding of the effects of providing delayed information and consequently help in better management of queues.

Highlights

Companies often publish the wait time of the services they provide
We derive rigorously the conditions for absolute stability and conditions that allow switches on the stability of the equilibrium of the system (1). We show that this non-symmetric case, i.e., the system (1) with σ 6= τ, allows bubbling, multiple stability switches and the coexistence of distinct families of stable limit cycles
To further explain the presence of coexisting stable limit cycles in the system (1), as shown in Figure 10, we examine the dynamics of the system in a neighborhood with a double-Hopf bifurcation

Summary

Introduction

Companies often publish the wait time of the services they provide. Frequently, this information is neither up-to-date, nor real-time. Wait times in hospital emergency rooms, call centers, and amusement park rides are some examples of delayed information divulged by companies to their customers. In [1], the impact of delay announcements on the coordination within hospital networks was investigated They showed that patients take delay information into account when choosing a provider of emergency services. This leads to improvements in performance of the network due to the increased coordination. The authors showed that these factors cause oscillations in the queues and such oscillatory behavior mimics the observed data dynamics All these examples point to the hypothesis that providing delayed information impacts the system dynamics of the queues and in turn affects customer choice

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Conclusion