A queueing/inventory and an insurance risk model

TL;DR

This paper analyzes a queueing model with inventory and risk features, using Wiener-Hopf techniques to derive steady-state distributions and exploring related insurance risk models.

Contribution

It introduces a novel queueing/inventory model with stochastic inventory removal, applying Wiener-Hopf factorisation to obtain explicit steady-state results.

Findings

Derived steady-state workload distributions for constant removal rate

Extended analysis to linear removal rate with algebraic conditions

Connected queueing model results to insurance risk theory

Abstract

We study an M/G/1-type queueing model with the following additional feature. The server works continuously, at fixed speed, even if there are no service requirements. In the latter case, it is building up inventory, which can be interpreted as negative workload. At random times, with an intensity {\omega}(x) when the inventory is at level x > 0, the present inventory is removed, instantaneously reducing the inventory to zero. We study the steady-state distribution of the (positive and negative) workload levels for the cases {\omega}(x) is constant and {\omega}(x) = ax. The key tool is the Wiener-Hopf factorisation technique. When {\omega}(x) is constant, no specific assumptions will be made on the service requirement distribution. However, in the linear case, we need some algebraic hypotheses concerning the Laplace-Stieltjes transform of the service requirement distribution. Throughout the paper, we also study a closely related model coming from insurance risk theory. Keywords: M/G/1 queue, Cramer-Lundberg insurance risk model, workload, inventory, ruin probability, Wiener-Hopf technique. 2010 Mathematics Subject Classification: 60K25, 90B22, 91B30, 47A68.