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

TL;DR

This paper analyzes infinite-server queues with shot-noise-driven Cox process arrivals, providing transient and heavy-traffic asymptotic analysis, and extends the model from a single queue to a network setting.

Contribution

It introduces a novel model of queues with shot-noise-driven arrival rates and develops analytical methods for transient and heavy-traffic analysis.

Findings

Derived explicit transient distribution of queue length.

Established heavy-traffic limit theorems for the system.

Extended analysis from single queue to network setting.

Abstract

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.