Time inhomogeneity in longest gap and longest run problems

TY - JOUR

T1 - Time inhomogeneity in longest gap and longest run problems

AU - Asmussen, Søren

AU - Ivanovs, Jevgenijs

AU - Rønn-Nielsen, Anders

PY - 2017/2/1

Y1 - 2017/2/1

N2 - 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.

AB - 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.

U2 - 10.1016/j.spa.2016.06.018

DO - 10.1016/j.spa.2016.06.018

M3 - Journal article

SN - 0304-4149

VL - 127

SP - 574

EP - 589

JO - Stochastic Processes and Their Applications

JF - Stochastic Processes and Their Applications

IS - 2

ER -