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Many stochastic systems have arrival processes that exhibit clustering behavior. In these systems, arriving entities influence additional arrivals to occur through selfexcitation of the arrival process. In this paper, we analyze an infinite server queueing system in which the arrivals are driven by the self-exciting Hawkes process and where service follows a phase-type distribution or is deterministic. In the phase-type setting, we derive differential equations for the moments and a partial differential equation for the moment generating function; we also derive exact expressions for the transient and steady-state mean, variance, and covariances. Furthermore, we also derive exact expressions for the auto-covariance of the queue and provide an expression for the cumulant moment generating function in terms of a single ordinary differential equation. In the deterministic service setting, we provide exact expressions for the first and second moments and the queue auto-covariance. As motivation for our Hawkes queueing model, we demonstrate its usefulness through two novel applications. These applications are trending internet traffic and arrivals to nightclubs. In the web traffic setting, we investigate the impact of a click. In the nightclub or Club Queue setting, we design an optimal control problem for the optimal rate to admit club-goers.

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.

In this paper, we introduce a new approximation for estimating the dynamics of multiserver queues with abandonment. The approximation involves a four-dimensional dynamical system that uses the skewness and kurtosis of the queueing distribution via the Gram Charlier expansion. We show that the additional information captured in the skewness and kurtosis allows us to estimate the dynamics of the mean and variance much better than fluid and diffusion limit theorems or other methods that use only mean and variance behavior. Lastly, our approach also yields accurate approximations for the probability of delay, which is an important metric for quality of service. Introduction.Motivated by the need for better approximations for the performance of small and medium sized service systems, such as emergency care centers and small data centers, we introduce a new, four-dimensional dynamical system approximation for queueing systems using the mean, variance, skewness, and kurtosis of these of dynamic rate Markov processes. Better approximations for these queueing models are needed as they help managers to optimally staff and accurately maintain quality of service metrics imposed by service level agreements. Since real service systems like call centers experience time varying behavior and large arrivals of customers and have multiple agents ready to deliver service, Markovian service networks are the class of time inhomogeneous stochastic processes that capture all these dynamics. Our canonical queueing model assumes the customer arrival process is a nonhomogeneous Poisson process. We also have c(t) servers at time t with i.i.d. (independent identically distributed) service times that are exponentially distributed with time dependent rate μ(t). Finally, all the customers have i.i.d. abandonment times that are also exponentially distributed with time varying rate β(t). This model is known as the M t /M t /C t + M t queueing model, where the +M is included for abandonment. Using the functional strong law of large numbers (FSLLN) developed for our family of Markovian service networks in Mandelbaum, Massey, and Reiman [8], one can show that the limiting behavior of Markovian service networks can be described by a deterministic nonlinear ordinary differential equation. A more refined functional central limit theorem (FCLT), also developed in [8], yields that the behavior of the network can be described by a Gaussian diffusion that solves a stochastic differential equation.The Gaussian diffusion from the FCLT relies on the fact that the amount of time that the mean number of customers is equal to the number of servers is of measure zero. However, when the mean number of customers lingers around the number of

Across a wide variety of applications, the self-exciting Hawkes process has been used to model phenomena in which the history of events influences future occurrences. However, there may be many situations in which the past events only influence the future as long as they remain active. For example, a person spreads a contagious disease only as long as they are contagious. In this paper, we define a novel generalization of the Hawkes process that we call the ephemerally self-exciting process. In this new stochastic process, the excitement from one arrival lasts for a randomly drawn activity duration, hence the ephemerality. Our study includes exploration of the process itself as well as connections to well-known stochastic models such as branching processes, random walks, epidemics, preferential attachment, and Bayesian mixture models. Furthermore, we prove a batch scaling construction of general, marked Hawkes processes from a general ephemerally self-exciting model, and this novel limit theorem both provides insight into the Hawkes process and motivates the model contained herein as an attractive self-exciting process in its own right.