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
UR - https://sfx-45cbs.hosted.exlibrisgroup.com/45cbs?url_ver=Z39.88-2004&url_ctx_fmt=info:ofi/fmt:kev:mtx:ctx&ctx_enc=info:ofi/enc:UTF-8&ctx_ver=Z39.88-2004&rfr_id=info:sid/sfxit.com:azlist&sfx.ignore_date_threshold=1&rft.object_id=954925512454&rft.object_portfolio_id=&svc.holdings=yes&svc.fulltext=yes
U2 - 10.1016/j.spa.2016.06.018
DO - 10.1016/j.spa.2016.06.018
M3 - Journal article
AN - SCOPUS:84997531444
VL - 127
SP - 574
EP - 589
JO - Stochastic Processes and Their Applications
JF - Stochastic Processes and Their Applications
SN - 0304-4149
IS - 2
ER -