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 af dette arbejde

Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › peer review

Abstract

Consider an inhomogeneous Poisson process and let D be the first of its epochs which is followed bya gap of size ℓ > 0. We establish a criterion for D < ∞ a.s., as well as for D being long-tailed andshort-tailed, and obtain logarithmic tail asymptotics in various cases. These results are translated into thediscrete time framework of independent non-stationary Bernoulli trials where the analogue of D is thewaiting time for the first run of ones of length ℓ. A main motivation comes from computer reliability, whereD +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.

Originalsprog	Engelsk
Tidsskrift	Stochastic Processes and Their Applications
Vol/bind	127
Udgave nummer	2
Sider (fra-til)	574-589
ISSN	0304-4149
DOI	https://doi.org/10.1016/j.spa.2016.06.018
Status	Udgivet - feb. 2017
Udgivet eksternt	Ja

Emneord

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

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10.1016/j.spa.2016.06.018Licens: Ikke-specificeret

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Citationsformater

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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}",

year = "2017",

month = feb,

doi = "10.1016/j.spa.2016.06.018",

language = "English",

volume = "127",

pages = "574--589",

journal = "Stochastic Processes and Their Applications",

issn = "0304-4149",

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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 - 2017/2

Y1 - 2017/2

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

U2 - 10.1016/j.spa.2016.06.018

DO - 10.1016/j.spa.2016.06.018

M3 - Journal article

AN - SCOPUS:84997531444

SN - 0304-4149

VL - 127

SP - 574

EP - 589

JO - Stochastic Processes and Their Applications

JF - Stochastic Processes and Their Applications

IS - 2

ER -