Time inhomogeneity in longest gap and longest run problems

Søren Asmussen, Jevgenijs Ivanovs and Anders Rønn Nielsen

Stochastic Processes and their Applications, 2017, vol. 127, issue 2, 574-589

Abstract: Consider an inhomogeneous Poisson process and let D be the first of its epochs which is followed by a gap of size ℓ>0. We establish a criterion for D<∞ 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 ℓ. A main motivation comes from computer reliability, where D+ℓ represents the actual execution time of a program or transfer of a file of size ℓ in presence of failures (epochs of the process) which necessitate restart.

Keywords: Bernoulli trials; Heads runs; Tail asymptotics; Poisson point process; Delay differential equation; Computer reliability (search for similar items in EconPapers)
Date: 2017
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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:127:y:2017:i:2:p:574-589

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DOI: 10.1016/j.spa.2016.06.018

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