Time Inhomogeneity in Longest Gap and Longest Run Problems

Søren Asmussen; Jevgenijs Ivanovs; Anders Rønn Nielsen

doi:10.1016/j.spa.2016.06.018

Time Inhomogeneity in Longest Gap and Longest Run Problems

Søren Asmussen^*, Jevgenijs Ivanovs, Anders Rønn Nielsen

^*Corresponding author for this work

Research output: Contribution to journal › Journal article › Research › peer-review

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.

Original language	English
Journal	Stochastic Processes and Their Applications
Volume	127
Issue number	2
Pages (from-to)	574-589
ISSN	0304-4149
DOIs	https://doi.org/10.1016/j.spa.2016.06.018
Publication status	Published - 2017
Externally published	Yes

Keywords

Bernoulli trials
Heads runs
Tail asymptotics
Poisson point process
Delay differential equation
Computer reliability

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title = "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 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, Bernoulli trials, Heads runs, Tail asymptotics, Poisson point process, Delay differential equation, Computer reliability",

author = "S{\o}ren Asmussen and Jevgenijs Ivanovs and Nielsen, {Anders R{\o}nn}",

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journal = "Stochastic Processes and Their Applications",

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TY - JOUR

T1 - Time Inhomogeneity in Longest Gap and Longest Run Problems

AU - Asmussen, Søren

AU - Ivanovs, Jevgenijs

AU - Nielsen, Anders Rønn

PY - 2017

Y1 - 2017

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.

KW - Bernoulli trials

KW - Heads runs

KW - Tail asymptotics

KW - Poisson point process

KW - Delay differential equation

KW - Computer reliability

KW - Bernoulli trials

KW - Heads runs

KW - Tail asymptotics

KW - Poisson point process

KW - Delay differential equation

KW - Computer reliability

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DO - 10.1016/j.spa.2016.06.018

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EP - 589

JO - Stochastic Processes and Their Applications

JF - Stochastic Processes and Their Applications

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