Unified approach for solving exit problems for additive-increase and multiplicative-decrease processes

TL;DR

This paper develops a unified mathematical framework to solve exit problems for a growth-collapse process, with explicit formulas for Laplace transforms, applicable to models like TCP and earthquake simulations.

Contribution

It introduces a unified first-step analysis method that yields explicit formulas for exit time distributions in growth-collapse processes, including their reflected variants.

Findings

Explicit formulas for Laplace transforms of exit times.

Introduction of scale functions $Z_{\uparrow}$ and $L_{\uparrow}$.

Unified approach applicable to various growth-collapse models.

Abstract

We analyse an additive-increase and multiplicative-decrease (aka growth-collapse) process that grows linearly in time and that experiences downward jumps at Poisson epochs that are (deterministically) proportional to its present position. This process is used for example in modelling of Transmission Control Protocol (TCP) and can be viewed as a particular example of the so-called shot noise model, a basic tool in modeling earthquakes, avalanches and neuron firings. For this process, and also for its reflected versions, we consider one- and two-sided exit problems that concern the identification of the laws of exit times from fixed intervals and half-lines. All proofs are based on a unified first-step analysis approach at the first jump epoch, which allows us to give explicit, yet involved, formulas for their Laplace transforms. All the eight Laplace transforms can be described in terms of two so-called scale functions $Z_{\uparrow}$ and $L_{\uparrow}$. Here $Z_{\uparrow}$ is described in terms of multiple explicit sums, and $L_{\uparrow}$ in terms of an explicit recursion formula. All other Laplace transforms can be obtained from $Z_{\uparrow}$ and $L_{\uparrow}$ by taking limits, derivatives, integrals and combinations of these.