In this paper we study the number of customers in infinite-server queues with a self-exciting (Hawkes) arrival process. Initially we assume that service requirements are exponentially distributed and that the Hawkes arrival process is of a Markovian nature. We obtain a system of differential equations that characterizes the joint distribution of the arrival intensity and the number of customers. Moreover, we provide a recursive procedure that explicitly identifies (transient and stationary) moments. Subsequently, we allow for non-Markovian Hawkes arrival processes and nonexponential service times. By viewing the Hawkes process as a branching process, we find that the probability generating function of the number of customers in the system can be expressed in terms of the solution of a fixed-point equation. We also include various asymptotic results: we derive the tail of the distribution of the number of customers for the case that the intensity jumps of the Hawkes process are heavy tailed, and we consider a heavy-traffic regime. We conclude by discussing how our results can be used computationally and by verifying the numerical results via simulations.

Infinite-server queues with Hawkes input

By adopting and extending lessons from the air traffic control system, we argue that a nationwide remote monitoring system for driverless vehicles could increase safety dramatically, speed these vehicles' deployment, and provide employment. It is becoming clear that fully driverless vehicles will not be able to handle "edge" cases in the near future, suggesting that new methods are needed to monitor remotely driverless vehicles' safe deployment. While the remote operations concept is not new, a super-human driver is needed to handle sudden, critical events. We envision that the remote operators do not directly drive the vehicles, but provide input on high level tasks such as path-planning, object detection and classification. This can be achieved via input from multiple individuals, coordinated around a task at a moment's notice. Assuming a 10% penetration rate of driverless vehicles, we show that one remote driver can replace 14,840 human drivers. A comprehensive nationwide interoperability standard and procedure should be established for the remote monitoring and operation of driverless vehicles. The resulting system has potential to be an order of magnitude safer than today's ground transportation system. We articulate a research and policy roadmap to launch this nationwide system. Additionally, this hybrid human-AI system introduces a new job category, likely a source of employment nationwide.

/pdf/beyond-safety-drivers-applying-air-traffic-control-33unlmp06i.pdf

Beyond safety drivers: Applying air traffic control principles to support the deployment of driverless vehicles.

Diffusion limits for networks of Markov-modulated infinite-server queues

We provide a survey of so-called shot-noise queues: queueing models with the special feature that the server speed is proportional to the amount of work it faces. Several results are derived for the workload in an M/G/1 shot-noise queue and some of its variants. Furthermore, we give some attention to queues with general workload-dependent service speed. We also discuss linear stochastic fluid networks, and queues in which the input process is a shot-noise process.

/pdf/shot-noise-queueing-models-dogapzdj9d.pdf

Shot-noise queueing models

This paper considers a queuing system with a finite buffer, a constant main MAP flow, and an additional periodic (non-periodic) Poisson piecewise-constant flow of customers. The eigenvalues of the probability translation matrix of a Kolmogorov system with constant input intensities is analyzed. The definition of the transition mode time based on the analysis of the probability translation matrix determinant is introduced for the first time. An analytical solution to the Kolmogorov equation system for a queuing system with piecewise constant arrival and service intensities is found, the solutions for a queuing system with periodic arrival and service intensities are analyzed, and numerical calculations illustrating this approach are presented.

/pdf/analysis-of-a-map-m-1-n-queue-with-periodic-and-non-periodic-2uyb7dny.pdf

Analysis of a MAP/M/1/N Queue with Periodic and Non-Periodic Piecewise Constant Input Rate

We study infinite-server queues in which the arrival process is a Cox process (or doubly stochastic Poisson process), of which the arrival rate is given by a shot-noise process. A shot-noise rate emerges naturally in cases where the arrival rate tends to exhibit sudden increases (or shots) at random epochs, after which the rate is inclined to revert to lower values. Exponential decay of the shot noise is assumed, so that the queueing systems are amenable to analysis. In particular, we perform transient analysis on the number of jobs in the queue jointly with the value of the driving shot-noise process. Additionally, we derive heavy-traffic asymptotics for the number of jobs in the system by using a linear scaling of the shot intensity. First we focus on a one-dimensional setting in which there is a single infinite-server queue, which we then extend to a network setting.

/pdf/networks-of-cdot-g-infty-queues-with-shot-noise-driven-4x6xk08lt8.pdf

Networks of \cdot /G/\infty queues with shot-noise-driven arrival intensities

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Networks of \cdot /G/\infty queues with shot-noise-driven arrival intensities