Journal of Applied Probability

On a class of reflected AR(1) processes

Onno Boxma, Michel Mandjes, and Josh Reed

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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.

Article information

J. Appl. Probab., Volume 53, Number 3 (2016), 818-832.

First available in Project Euclid: 13 October 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K25: Queueing theory [See also 68M20, 90B22]
Secondary: 60J05: Discrete-time Markov processes on general state spaces

Reflected process queueing scaling limit


Boxma, Onno; Mandjes, Michel; Reed, Josh. On a class of reflected AR(1) processes. J. Appl. Probab. 53 (2016), no. 3, 818--832.

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