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On a class of reflected AR(1) processes

Part of: Markov processes Special processes

Published online by Cambridge University Press:  24 October 2016

Onno Boxma ,
Michel Mandjes  and
Josh Reed
Onno Boxma*
Affiliation:
Eindhoven University of Technology and EURANDOM
Michel Mandjes*
Affiliation:
University of Amsterdam and CWI
Josh Reed*
Affiliation:
NYU Stern School of Business
*
* Postal address: Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: o.j.boxma@tue.nl
** Research is partly funded by the NWO Gravitation project NETWORKS, grant number 024.002.003.
**** Postal address: NYU Stern School of Business, 44 West 4th Street, New York, NY 10012, USA. Email address: jreed@stern.nyu.edu
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Abstract

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In this paper we study a reflected AR(1) process, i.e. a process (Z n )n obeying the recursion Z n +1= max{aZ n +X n ,0}, with (X n )n a sequence of independent and identically distributed (i.i.d.) random variables. We find explicit results for the distribution of Z n (in terms of transforms) in case X n can be written as Y n −B n , with (B n )n being a sequence of independent random variables which are all Exp(λ) distributed, and (Y n )n i.i.d.; when |a|<1 we can also perform the corresponding stationary analysis. Extensions are possible to the case that (B n )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.

Keywords

Reflected processqueueingscaling limit

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60J05: Discrete-time Markov processes on general state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 3 , September 2016 , pp. 818 - 832
Copyright
Copyright © Applied Probability Trust 2016 

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