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.

Understanding how delayed information impacts queueing systems is an important area of research. However, much of the current literature neglects one important feature of many queueing systems, namely non-stationary arrivals. Non-stationary arrivals model the fact that customers tend to access services during certain times of the day and not at a constant rate. In this paper, we analyze two two-dimensional deterministic fluid models that incorporate customer choice behavior based on delayed queue length information with time varying arrivals. In the first model, customers receive queue length information that is delayed by a constant ∆. In the second model, customers receive information about the queue length through a moving average of the queue length where the moving average window is ∆. We analyze the impact of the time varying arrival rate and show using asymptotic analysis that the time varying arrival rate does not impact the critical delay unless the frequency of the time varying arrival 1 arXiv:1701.05443v1 [math.DS]

Internet and mobile services often provide waiting time or queue length information to customers. This information allows a customer to determine whether to remain in line or, in the case of multiple lines, better decide which line to join. Unfortunately, there is usually a delay associated with waiting time information. Either the information itself is stale, or it takes time for the customers to travel to the service location after having received the information. Recent empirical and theoretical work uses functional dynamical systems as limiting models for stochastic queueing systems. This work has shown that if information is delayed long enough, a Hopf bifurcation can occur and cause unwanted oscillations in the queues. However, it is not known how large the oscillations are when a Hopf bifurcation occurs. To answer this question, we model queues with functional differential equations and implement two methods for approximating the amplitude of these oscillations. The first approximation is analytic and yields a closed-form approximation in terms of the model parameters. The second approximation uses a statistical technique, and delivers highly accurate approximations over a wider range of parameters.

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Abstract. Motivated by a classical theorem of Schoenberg we prove that an n + 1 point finite metric space has strict 2-negative type if and only if it can be isometrically embedded in the Euclidean space R n of dimension n but it cannot be isometrically embedded in any Euclidean space R r of dimension r < n. We use this result as a technical tool to study "roundness" properties of additive metrics with a particular focus on ultrametrics and leaf metrics. The following conditions are shown to be equivalent for a metric space (X, d):(1) X is ultrametric, (2) X has infinite roundness, (3) X has infinite generalized roundness, (4) X has strict p-negative type for all p ≥ 0, and (5) X admits no p-polygonal equality for any p ≥ 0. As all ultrametric spaces have strict 2-negative by (4) we thus obtain a short new proof of Lemin's theorem: Every n + 1 point ultrametric space is isometrically embeddable into some Euclidean space as an affinely independent set. Motivated by a question of Lemin, Shkarin introduced the class M of all finite metric spaces that may be isometrically embedded into ℓ 2 as an affinely independent set. The results of this paper show that Shkarin's class M consists of all finite metric spaces of strict 2-negative type. We also note that it is possible to construct an additive metric space whose generalized roundness is exactly ℘ for each ℘ ∈ [1, ∞].