Time inhomogeneity in longest gap and longest run problems

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.