Networks of ⋅/G/∞\cdot/G/\infty Server Queues with Shot-Noise-Driven Arrival Intensities

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 shot noise. A shot-noise rate emerges as a natural model, if the arrival rate tends to display 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 for analysis. In particular, we perform transient analysis on the number of customers in the queue jointly with the value of the driving shot-noise process. Additionally, we derive heavy-traffic asymptotics for the number of customers 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

Steady-State Analysis and Online Learning for Queues with Hawkes Arrivals

We investigate the long-run behavior of single-server queues with Hawkes arrivals and general service distributions and related optimization problems. In detail, utilizing novel coupling techniques, we establish finite moment bounds for the stationary distribution of the workload and busy period processes. In addition, we are able to show that, those queueing processes converge exponentially fast to their stationary distribution. Based on these theoretic results, we develop an efficient numerical algorithm to solve the optimal staffing problem for the Hawkes queues in a data-driven manner. Numerical results indicate a sharp difference in staffing for Hawkes queues, compared to the classic GI/GI/1 model, especially in the heavy-traffic regime

How to Staff when Customers Arrive in Batches

In settings as diverse as autonomous vehicles, cloud computing, and pandemic quarantines, requests for service can arrive in near or true simultaneity with one another. This creates batches of arrivals to the underlying queueing system. In this paper, we study the staffing problem for the batch arrival queue. We show that batches place a significant stress on services, and thus require a high amount of resources and preparation. In fact, we find that there is no economy of scale as the number of customers in each batch increases, creating a stark contrast with the square root safety staffing rules enjoyed by systems with solitary arrivals of customers. Furthermore, when customers arrive both quickly and in batches, an economy of scale can exist, but it is weaker than what is typically expected. Methodologically, these staffing results follow from novel large batch and hybrid large-batch-and-large-rate limits of the general multi-server queueing model. In the pure large batch limit, we establish the first formal connection between multi-server queues and storage processes, another family of stochastic processes. By consequence, we show that the limit of the batch scaled queue length process is not asymptotically normal, and that, in fact, the fluid and diffusion-type limits coincide. This is what drives our staffing analysis of the batch arrival queue, and what implies that the (safety) staffing of this system must be directly proportional to the batch size just to achieve a non-degenerate probability of customers waiting

Shot-noise queueing models

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