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September 2016 On a class of reflected AR(1) processes
Onno Boxma, Michel Mandjes, Josh Reed
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J. Appl. Probab. 53(3): 818-832 (September 2016).

Abstract

In this paper we study a reflected AR(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 YnBn, 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.

Citation

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Onno Boxma. Michel Mandjes. Josh Reed. "On a class of reflected AR(1) processes." J. Appl. Probab. 53 (3) 818 - 832, September 2016.

Information

Published: September 2016
First available in Project Euclid: 13 October 2016

zbMATH: 1351.60121
MathSciNet: MR3570096

Subjects:
Primary: 60K25
Secondary: 60J05

Keywords: Queueing , reflected process , Scaling limit

Rights: Copyright © 2016 Applied Probability Trust

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Vol.53 • No. 3 • September 2016
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