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A generalized backward scheme for solving nonmonotonic stochastic recursions. (English) Zbl 1318.60038
Summary: 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 queueing systems.

##### MSC:
 60G10 Stationary stochastic processes 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60K25 Queueing theory (aspects of probability theory) 37H99 Random dynamical systems
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##### References:
 [1] Anantharam, V. and Konstantopoulos, T. (1997). Stationary solutions of stochastic recursions describing discrete event systems. Stochastic Process. Appl. 68 181-194. · Zbl 0911.60076 [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. · Zbl 0911.60076 [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. · Zbl 1021.60001 [4] Baccelli, F. and Hebuterne, G. (1981). On queues with impatient customers. In Performance’ 81 ( Amsterdam , 1981) 159-179. North-Holland, Amsterdam. · Zbl 0539.90030 [5] Bacelli, F., Boyer, P. and Hébuterne, G. (1984). Single-server queues with impatient customers. Adv. in Appl. Probab. 16 887-905. · Zbl 0549.60091 [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. · Zbl 0658.60094 [7] Borovkov, A. and Foss, S. G. (1994). Two ergodicity criteria for stochastically recursive sequences. Acta Appl. Math. 34 125-134. · Zbl 0810.60029 [8] Borovkov, A. A. (1984). Asymptotic Methods in Queuing Theory . Wiley, Chichester. · Zbl 0544.60085 [9] Borovkov, A. A. (1998). Ergodicity and Stability of Stochastic Processes . Wiley, Chichester. · Zbl 0917.60005 [10] Borovkov, A. A. and Foss, S. G. (1992). Stochastically recursive sequences and their generalizations. Siberian Adv. Math. 2 16-81. · Zbl 0848.60025 [11] Brandt, A., Franken, P. and Lisek, B. (1990). Stationary Stochastic Models. Mathematische Lehrbücher und Monographien , II. Abteilung : Mathematische Monographien 78 . Akademie-Verlag, Berlin. · Zbl 0707.60084 [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. · Zbl 0659.60125 [14] Foss, S. and Konstantopoulos, T. (2003). Extended renovation theory and limit theorems for stochastic ordered graphs. Markov Process. Related Fields 9 413-468. · Zbl 1043.60025 [15] Lisek, B. (1982). A method for solving a class of recursive stochastic equations. Z. Wahrsch. Verw. Gebiete 60 151-161. · Zbl 0465.60060 [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. · Zbl 0203.22303 [17] Moyal, P. (2010). The queue with impatience: Construction of the stationary workload under FIFO. J. Appl. Probab. 47 498-512. · Zbl 1208.60099 [18] Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. Wiley Series in Probability and Statistics . Wiley, Chichester. · Zbl 0999.60002 [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. · Zbl 0548.60092 [20] Stoyan, D. (1977). Bounds and approximations in queueing through monotonicity and continuity. Oper. Res. 25 851-863. · Zbl 0391.60086 [21] Stoyan, D. (1983). Comparison Methods for Queues and Other Stochastic Models . Wiley, Chichester. · Zbl 0536.60085 [22] Vlasiou, M. (2007). A non-increasing Lindley-type equation. Queueing Syst. 56 41-52. · Zbl 1165.90411
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