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 journalJournal articleResearchpeer-review


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 languageEnglish
JournalStochastic Processes and Their Applications
Issue number2
Pages (from-to)574-589
Publication statusPublished - Feb 2017
Externally publishedYes


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

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