The Annals of Applied Probability

A generalized backward scheme for solving nonmonotonic stochastic recursions

P. Moyal

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We propose an explicit construction of a stationary solution for a stochastic recursion of the form $X\circ\theta=\varphi(X)$ on a partially-ordered Polish space, when the monotonicity of $\varphi$ is not assumed. Under certain conditions, we show that an extension of the original probability space exists, on which a solution is well defined, and construct explicitly this extension using a randomized contraction technique. We then provide conditions for the existence of a solution on the original space. We finally apply these results to the stability study of two nonmonotonic queuing systems.

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Ann. Appl. Probab. Volume 25, Number 2 (2015), 582-599.

First available in Project Euclid: 19 February 2015

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Zentralblatt MATH identifier

Primary: 60G10: Stationary processes 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60K25: Queueing theory [See also 68M20, 90B22] 37H99: None of the above, but in this section

Stochastic recursions stationary solutions enriched probability space ergodic theory queuing theory


Moyal, P. A generalized backward scheme for solving nonmonotonic stochastic recursions. Ann. Appl. Probab. 25 (2015), no. 2, 582--599. doi:10.1214/14-AAP1004.

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  • [1] Anantharam, V. and Konstantopoulos, T. (1997). Stationary solutions of stochastic recursions describing discrete event systems. Stochastic Process. Appl. 68 181–194.
  • [2] Anantharam, V. and Konstantopoulos, T. (1999). Corrigendum: “Stationary solutions of stochastic recursions describing discrete event systems” [Stochastic Process. Appl. 68 (1997) 181–194; MR1454831]. Stochastic Process. Appl. 80 271–278.
  • [3] Baccelli, F. and Brémaud, P. (2003). Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences, 2nd ed. Applications of Mathematics (New York) 26. Springer, Berlin.
  • [4] Baccelli, F. and Hebuterne, G. (1981). On queues with impatient customers. In Performance’81 (Amsterdam, 1981) 159–179. North-Holland, Amsterdam.
  • [5] Bacelli, F., Boyer, P. and Hébuterne, G. (1984). Single-server queues with impatient customers. Adv. in Appl. Probab. 16 887–905.
  • [6] Bhattacharya, R. N. and Lee, O. (1988). Ergodicity and central limit theorems for a class of Markov processes. J. Multivariate Anal. 27 80–90.
  • [7] Borovkov, A. and Foss, S. G. (1994). Two ergodicity criteria for stochastically recursive sequences. Acta Appl. Math. 34 125–134.
  • [8] Borovkov, A. A. (1984). Asymptotic Methods in Queuing Theory. Wiley, Chichester.
  • [9] Borovkov, A. A. (1998). Ergodicity and Stability of Stochastic Processes. Wiley, Chichester.
  • [10] Borovkov, A. A. and Foss, S. G. (1992). Stochastically recursive sequences and their generalizations. Siberian Adv. Math. 2 16–81.
  • [11] Brandt, A., Franken, P. and Lisek, B. (1990). Stationary Stochastic Models. Mathematische Lehrbücher und Monographien, II. Abteilung: Mathematische Monographien 78. Akademie-Verlag, Berlin.
  • [12] Flipo, D. (1983). Steady state of loss systems (in French). Comptes Rendus de L’Académie des Sciences de Paris, Ser. I 297 6.
  • [13] Flipo, D. (1988). Charge stationnaire d’une file d’attente à rejet. Application au cas indépendant. Ann. Sci. Univ. Clermont-Ferrand II Probab. Appl. 7 47–74.
  • [14] Foss, S. and Konstantopoulos, T. (2003). Extended renovation theory and limit theorems for stochastic ordered graphs. Markov Process. Related Fields 9 413–468.
  • [15] Lisek, B. (1982). A method for solving a class of recursive stochastic equations. Z. Wahrsch. Verw. Gebiete 60 151–161.
  • [16] Loynes, R. M. (1962). The stability of a queue with non-independent interarrival and service times. Math. Proc. Cambridge Philos. Soc. 58 497–520.
  • [17] Moyal, P. (2010). The queue with impatience: Construction of the stationary workload under FIFO. J. Appl. Probab. 47 498–512.
  • [18] Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. Wiley Series in Probability and Statistics. Wiley, Chichester.
  • [19] Neveu, J. (1983). Construction de files d’attente stationnaires. In Modelling and Performance Evaluation Methodology (Paris, 1983). Lecture Notes in Control and Inform. Sci. 60 31–41. Springer, Berlin.
  • [20] Stoyan, D. (1977). Bounds and approximations in queueing through monotonicity and continuity. Oper. Res. 25 851–863.
  • [21] Stoyan, D. (1983). Comparison Methods for Queues and Other Stochastic Models. Wiley, Chichester.
  • [22] Vlasiou, M. (2007). A non-increasing Lindley-type equation. Queueing Syst. 56 41–52.