Time inhomogeneity in longest gap and longest run problems

TL;DR

This paper analyzes the timing of gaps and runs in inhomogeneous Poisson processes and Bernoulli trials, providing criteria for finiteness and tail behavior, with applications to computer reliability and failure modeling.

Contribution

It introduces new criteria for the finiteness and tail properties of the first epoch followed by a gap in inhomogeneous Poisson processes and extends these results to non-stationary Bernoulli trials.

Findings

Established criteria for almost sure finiteness of D

Derived logarithmic tail asymptotics in various cases

Applied results to computer reliability models

Abstract

Consider an inhomogeneous Poisson process and let $D$ be the first of its epochs which is followed by a gap of size $\ell>0$. We establish a criterion for $D<\infty$ a.s., as well as for $D$ being long-tailed and short-tailed, and obtain logarithmic tail asymptotics in various cases. These results are translated into the discrete time framework of independent non-stationary Bernoulli trials where the analogue of $D$ is the waiting time for the first run of ones of length $\ell$. A main motivation comes from computer reliability, where $D+\ell$ represents the actual execution time of a program or transfer of a file of size $\ell$ in presence of failures (epochs of the process) which necessitate restart.