Queues and risk models with simultaneous arrivals

TL;DR

This paper explores the connection between queueing and risk models with simultaneous arrivals, deriving transforms for workload and survival probabilities, and analyzing models with ordered and unordered claim sizes.

Contribution

It provides new Laplace transform formulas for joint workloads and survival probabilities in multi-dimensional queueing and risk models with simultaneous arrivals, including ordered and unordered claim sizes.

Findings

Laplace-Stieltjes transform of joint stationary workload distribution

Laplace transform of survival probability in multivariate risk models

Stochastic decomposition of workload vector

Abstract

We focus on a particular connection between queueing and risk models in a multi-dimensional setting. We first consider the joint workload process in a queueing model with parallel queues and simultaneous arrivals at the queues. For the case that the service times are ordered (from largest in the first queue to smallest in the last queue) we obtain the Laplace-Stieltjes transform of the joint stationary workload distribution. Using a multivariate duality argument between queueing and risk models, this also gives the Laplace transform of the survival probability of all books in a multivariate risk model with simultaneous claim arrivals and the same ordering between claim sizes. Other features of the paper include a stochastic decomposition result for the workload vector, and an outline how the two-dimensional risk model with a general two-dimensional claim size distribution (hence without ordering of claim sizes) is related to a known Riemann boundary value problem.