Time Inhomogeneity in Longest Gap and Longest Run Problems

Søren Asmussen*, Jevgenijs Ivanovs, Anders Rønn Nielsen

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Publikation: Bidrag til tidsskriftTidsskriftartikel


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.
TidsskriftStochastic Processes and Their Applications
Udgave nummer2
Sider (fra-til)574-589
StatusUdgivet - 2017
Udgivet eksterntJa


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