Stochastic decompositions in bivariate risk and queueing models with mutual assistance

TL;DR

This paper investigates stochastic decompositions in bivariate risk and queueing models with mutual assistance, revealing that key quantities can be expressed as sums of independent components and proposing a general method for establishing such decompositions.

Contribution

It introduces a novel stochastic decomposition approach for models with two-way interactions and provides a general method to derive these decompositions from kernel equations.

Findings

Decomposition of joint survival probability and workload into independent sums.

Method for deriving decompositions from kernel equations using difference of independent variables.

Decomposition applies to reflected Brownian motion in orthants and related face measures.

Abstract

We consider two bivariate models with two-way interactions in context of risk and queueing theory. The two entities interact with each other by providing assistance but otherwise evolve independently. We focus on certain random quantities underlying the joint survival probability and the joint stationary workload, and show that these admit stochastic decomposition. Each one can be seen as an independent sum of respective quantities for the two models with one-way interaction. Additionally, we discuss a rather general method of establishing decompositions from a given kernel equation by identifying two independent random variables from their difference, which may be useful for other models. Finally, we point out that the same decomposition is true for uncorrelated Brownian motion reflected to stay in an orthant, and it concerns the face measures appearing in the basic adjoint relationship.