Breaking the Symmetry in Queues with Delayed Information

Abstract

Giving customers queue length information about a service system 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 a two-dimensional deterministic fluid model that incorporates customer choice behavior based on delayed queue length information. Reports in the existing literature always assume that all queues have identical parameters and the underlying dynamical system is symmetric. However, in this paper, we relax this symmetry assumption by allowing the arrival rates, service rates, and the choice model parameters to be different for each queue. Our methodology exploits the method of multiple scales and asymptotic analysis to understand how to break the symmetry. We find that the asymmetry can have a large impact on the underlying dynamics of the queueing system.

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

“…The characteristic equation corresponding to the linear system in Equation ( 9) is det ∆(λ) = 0 (11) where ∆(λ) = λI 2 − M 0 − M 1 e −λσ − M 2 e −λτ and I 2 denotes the 2 × 2 identity matrix, and M 0 , M 1 and M 2 are given in Equation (10). When simplified, the characteristic Equation ( 11) takes the following form…”

“…The queues-with-choice model can also be obtained as a functional law of large numbers' limit of a stochastic queueing process [10]. A version where the corresponding coefficients in the equations are different but with a single time delay has likewise been investigated [11]. Extensions into the more general model involving an arbitrary but finitely many queues were also considered.…”

Synchronisation in TCP networks with Drop-Tail Queues

“…In a broader context, understanding the impact of delayed information on the dynamics of queuing systems is beginning to generate significant research interest in the literature [19,21,26,27,29]. Some common scenarios where delayed information can potentially hurt customer experience are the Internet, mobile services, healthcare services and public transportation, where it is important to communicate the waiting times or the queue length information with the customers [6,30,31]. In this paper, we study queuing systems which arise in the Internet, wherein majority of the traffic is regulated by Transmission Control Protocol (TCP), and congestion notifications are delayed.…”

Queues with Delayed Information: Analyzing the Impact of the Choice Model Function

“…This delay in information can be caused by factors such as the fact that it can take time to process and send information about queue lengths to the customers and thus the information that customers actually receive describes the queue lengths from some amount of time in the past. Queueing systems where customers are provided with delayed information have been studied extensively, see for example Novitzky et al [16], Doldo and Pender [3,2], Doldo et al [6], Novitzky and Pender [13], Novitzky et al [15], Atar and Lipshutz [1], Whitt [23].…”

“…One of the most common choices used in the literature is where the customer is provided with the lengths of the queues from some amount of time in the past. This delay in information is often modeled with a constant delay, as is the case in Mitzenmacher [11], Raina and Wischik [20], Kamath et al [8], Raina et al [21], Lipshutz and Williams [10], Lipshutz [9], Atar and Lipshutz [1], Nirenberg et al [12], Novitzky et al [16], Doldo et al [6], Novitzky et al [15], Pender et al [17,18], Whitt [23].…”