Tail behaviour of the area under a random process, with applications to queueing systems, insurance and percolations

TL;DR

This paper investigates the tail behavior of the distribution of the area under workload and queuing processes in single server queues, with implications for risk and percolation theories, highlighting open problems and partial results.

Contribution

It introduces new insights into the tail distributions of workload and queuing process integrals, connecting queueing theory with risk and percolation applications, and presents open problems and conjectures.

Findings

Partial results for special cases of tail behavior

Identification of open problems and conjectures

Connections established between queueing, risk, and percolation theories

Abstract

The areas under workload process and under queuing process in a single server queue over the busy period have many applications not only in queuing theory but also in risk theory or percolation theory. We focus here on the tail behaviour of distribution of these two integrals. We present various open problems and conjectures, which are supported by partial results for some special cases.