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Time inhomogeneity in longest gap and longest run problems

Author

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  • Asmussen, Søren
  • Ivanovs, Jevgenijs
  • Nielsen, Anders Rønn

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.

Suggested Citation

  • Asmussen, Søren & Ivanovs, Jevgenijs & Nielsen, Anders Rønn, 2017. "Time inhomogeneity in longest gap and longest run problems," Stochastic Processes and their Applications, Elsevier, vol. 127(2), pages 574-589.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:2:p:574-589
    DOI: 10.1016/j.spa.2016.06.018
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    References listed on IDEAS

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    1. Demetrios Antzoulakos, 1999. "On Waiting Time Problems Associated with Runs in Markov Dependent Trials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 51(2), pages 323-330, June.
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