Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-24T07:19:52.336Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 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.

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Asmussen, S. (2003).Applied Probability and Queues, 2nd edn.Springer,New York.Google Scholar
[2] Badila, E. S.,Boxma, O. J. and Resing, J. A. C. (2014).Queues and risk processes with dependencies.Stoch. Models 30,390–419.Google Scholar
[3] Billingsley, P. (1968).Convergence of Probability Measures.John Wiley,New York.Google Scholar
[4] Bladt, M. and Nielsen, B. F. (2010).Multivariate matrix-exponential distributions.Stoch. Models 26,1–26.Google Scholar
[5] Brandt, A. (1986).The stochastic equationY n+1=A n Y n +B n with stationary coefficients.Adv. Appl. Prob. 18,211–220.Google Scholar
[6] Brockwell, P. J. and Davis, R. A. (2002).Introduction to Time Series and Forecasting, 2nd edn.Springer,New York.Google Scholar
[7] Cohen, J. H. (1975).The Wiener–Hopf technique in applied probability,In Perspectives in Probability and Statistics, ed. J.Gani,Applied Probability Trust,Sheffield, pp.145–156.Google Scholar
[8] Cohen, J. W. (1982).The Single Server Queue, 2nd edn.North-Holland,Amsterdam.Google Scholar
[9] Diaconis, P. and Freedman, . (1999).Iterated random functions.SIAM Rev. 41,45–76.CrossRefGoogle Scholar
[10] Goldie, C. M. (1991).Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Prob. 1,126–166.Google Scholar
[11] Mills, T. C. (1990).Time Series Techniques for Economists.Cambridge University Press.Google Scholar
[12] Reed, J.,Ward, A. and Zhan, D. (2013).On the generalized Skorokhod problem in one dimension.J. Appl. Prob. 50,16–28.CrossRefGoogle Scholar
[13] Titchmarsh, E. C. (1939).The Theory of Functions, 2nd edn.Oxford University Press..Google Scholar
[14] Vlasiou, M.,Adan, I. J. B. F. and Wessels, J. (2004).A Lindley-type equation arising from a carousel problem.J. Appl. Prob. 41,1171–1181.Google Scholar
[15] Ward, A. R. and Glynn, P. W. (2003).Properties of the reflected Ornstein–Uhlenbeck process.Queueing Systems 44,109–123.CrossRefGoogle Scholar
[16] Whitt, W. (1990).Queues with service times and interarrival times depending linearly and randomly upon waiting times.Queueing Systems Theory Appl. 6,335–351.Google Scholar
[17] Williams, D. (1991).Probability with Martingales.CambridgeUniversity Press.Google Scholar