On a class of reflected AR(1) processes

O.J. Boxma, M. Mandjes, J. Reed

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2 Citations (Scopus)


In this paper we study a reflectedAR(1) process, i.e. a process (Zn)n obeying the recursion Zn+1 = max{aZn + Xn, 0}, with (Xn)n a sequence of independent and identically distributed (i.i.d.) random variables. We find explicit results for the distribution of Zn (in terms of transforms) in case Xn can be written as Yn - Bn, with (Bn)n being a sequence of independent random variables which are all Exp(λ) distributed, and (Yn)n i.i.d.; when |a| <1 we can also perform the corresponding stationary analysis. Extensions are possible to the case that (Bn)n are of phase-type. Under a heavy-traffic scaling, it is shown that the process converges to a reflected Ornstein-Uhlenbeck process; the corresponding steady-state distribution converges to the distribution of a normal random variable conditioned on being positive.

Original languageEnglish
Pages (from-to)818-832
Number of pages15
JournalJournal of Applied Probability
Issue number3
Publication statusPublished - 1 Sep 2016


  • Queueing
  • Reflected process
  • Scaling limit


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