Risk measures for infinite-server queues driven by a Cox-BESQ process

Abstract

This paper introduces a model for infinite-server queues driven by a Cox process with stochastic intensity given by a squared Bessel (BESQ) process. Exploiting the affine structure of the underlying dynamics, we derive a closed-form expression for the joint moment-generating function (MGF) of the arrival intensity, the number of customers in the system, and the integral of the number of customers in the system over a time interval. This MGF can also be obtained at two (or more) dates, thereby allowing an analysis of the time evolution of infinite-server queues driven by a Cox--BESQ process. The availability of these explicit MGFs enables the development of risk measures for these variables, such as the joint moments of the arrival intensity, the number of customers in the system, and the integral of the number of customers in the system over a time interval, conditional on one of these variables exceeding a critical threshold. These risk measures can be determined at one or two dates. They rely essentially on the explicit form of the MGF of the state variables and on Fourier representations of the risk measures. Numerical experiments demonstrate the practical implementation of the proposed methodology and highlight the relevance of the theoretical results.