Queues with Choice via Delay Differential Equations

Abstract

Delay or queue length information has the potential to influence the decision of a customer to join a queue. Thus, it is imperative for managers of queueing systems to understand how the information that they provide will affect the performance of the system. To this end, we construct and analyze two two-dimensional deterministic fluid models that incorporate customer choice behavior based on delayed queue length information. In the first fluid model, customers join each queue according to a Multinomial Logit Model, however, the queue length information the customer receives is delayed by a constant [Formula: see text]. We show that the delay can cause oscillations or asynchronous behavior in the model based on the value of [Formula: see text]. In the second model, customers receive information about the queue length through a moving average of the queue length. Although it has been shown empirically that giving patients moving average information causes oscillations and asynchronous behavior to occur in U.S. hospitals, we analytically and mathematically show for the first time that the moving average fluid model can exhibit oscillations and determine their dependence on the moving average window. Thus, our analysis provides new insight on how operators of service systems should report queue length information to customers and how delayed information can produce unwanted system dynamics.

Numerical Continuation and Bifurcation Analysis in a Harvested Predator-Prey Model with Time Delay using DDE-Biftool

“…Initially, E * is locally asymptotically stable, i.e. all roots of the characteristic equation (10) with τ = 0 lie in the open left-half plane. If one or more roots of equation (10) cross the imaginary axis and move towards the open right-half plane as τ is increased, then E * will switch stability and becomes unstable.…”

“…Time delay has been incorporated in models to reflect certain physical or biological meaning. Examples include optical feedback in laser systems [1,2,3], maturation age in stage structured population models [4], and delayed information in queueing models [10] just to name a few. The theory of delay differential equations (DDEs) [9,12], which has seen extensive growth in the last seventy years or so, can be used to examine the effects of time delay in the dynamical behavior of systems being considered.…”

Queues Driven by Hawkes Processes

“…Not only do these approximations have the potential to describe the moment dynamics, but they can be used to stabilize performance measures like in Pender and Massey [31]. A detailed analysis of these extensions will provide a better understanding how the information that operations managers provide to their customers will affect the dynamics of these real world systems like in Pender et al [32,34,33]. We plan to explore these extensions in subsequent work.…”

Nonlinear Dynamics in Queueing Theory: Determining the Size of Oscillations in Queues with Delay

“…The authors in [24] use DDEs and FDEs to develop two new two-dimensional fluid models of queues that incorporate customer choice based on delayed queue length information, and show that oscillations in queue lengths occur for certain lengths of delay. By comparison, in this paper we prove that the observed behavior is due to a supercritical Hopf bifurcation, and we use two techniques to approximate the size of the amplitude of oscillations.…”

“…Paper outline. This paper considers two models of queues that were originally presented in [24] and [25] as fluid limits of stochastic queueing models. In each model, there are two queues and customers decide which queue to join based on information about the queue length that is delayed.…”