Stochastic Systems

Wiener-Hopf factorizations for a multidimensional Markov additive process and their applications to reflected processes

Masakiyo Miyazawa and Bert Zwart

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We extend the framework of Neuts’ matrix analytic approach to a reflected process generated by a discrete time multidimensional Markov additive process. This Markov additive process has a general background state space and a real vector valued additive component, and generates a multidimensional reflected process. Our major interest is to derive a closed form formula for the stationary distribution of this reflected process. To this end, we introduce a real valued level, and derive new versions of the Wiener-Hopf factorization for the Markov additive process with the multidimensional additive component. In particular, it is represented by moment generating functions, and we consider the domain for it to be valid.

Our framework is general enough to include multi-server queues and/or queueing networks as well as non-linear time series which are currently popular in financial and actuarial mathematics. Our results yield structural results for such models. As an illustration, we apply our results to extend existing results on the tail behavior of reflected processes.

A major theme of this work is to connect recent work on matrix analytic methods to classical probabilistic studies on Markov additive processes. Indeed, using purely probabilistic methods such as censoring, duality, level crossing and time-reversal (which are known in the matrix analytic methods community but date back to Arjas & Speed [2] and Pitman [29]), we extend and unify existing results in both areas.

Article information

Stoch. Syst., Volume 2, Number 1 (2012), 67-114.

First available in Project Euclid: 24 February 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K25: Queueing theory [See also 68M20, 90B22] 60J05: Discrete-time Markov processes on general state spaces
Secondary: 90B22: Queues and service [See also 60K25, 68M20] 60F10: Large deviations

Markov-additive processes Wiener-Hopf factorization R-G decomposition occupation measure multidimensional reflecting process stationary distribution tail asymptotics light tail


Miyazawa, Masakiyo; Zwart, Bert. Wiener-Hopf factorizations for a multidimensional Markov additive process and their applications to reflected processes. Stoch. Syst. 2 (2012), no. 1, 67--114. doi:10.1214/12-SSY069.

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  • [1] Alsmeyer, G. (1994). On the Markov renewal theorem. Stochastic Processes and their Applications 50 37–56.
  • [2] Arjas, E. and Speed, T. P. (1973). Symmetric Wiener-Hopf factorisations in Markov additive processes. Probability Theory and Related Fields 26 105–118.
  • [3] Asmussen, S. (2003). Applied probability and queues, second ed. Applications of Mathematics (New York) 51. Springer-Verlag, New York. Stochastic Modelling and Applied Probability.
  • [4] Billingsley, P. (2000). Probability and Measure, 2nd ed. Wiley.
  • [5] Borovkov, A. A. and Mogul’skiĭ, A. A. (2001). Large deviations for Markov chains in the positive quadrant. Russian Mathematical Surveys 56 803–916.
  • [6] Collamore, J. (1996). Hitting probabilities and large deviations. The Annals of Probability 24 2065–2078.
  • [7] Dai, J. G. and Miyazawa, M. (2011). Reflecting Brownian motion in two dimensions: Exact asymptotics for the stationary distribution. Stochastic Systems 1 146-208.
  • [8] Dinges, H. (1969). Wiener-Hopf Faktorisierung für substochastische Übergangs-funktionen in angeordneten Räumen. Z. Wahrsch. Verw. Gebiete 11 152–164.
  • [9] Dupuis, P. and Ellis, R. S. (1997). A weak convergence approach to the theory of large deviations. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons Inc., New York. A Wiley-Interscience Publication.
  • [10] Fayolle, G., Malyshev, V. A. and Men’shikov, M. V. (1995). Topics in the constructive theory of countable Markov chains. Cambridge University Press, Cambridge.
  • [11] Foley, R. D. and McDonald, D. R. (2001). Join the shortest queue: stability and exact asymptotics. Ann. Appl. Probab. 11 569–607.
  • [12] Glynn, P. and Whitt, W. (1994). Logarithmic Asymptotics for Steady-State Tail Probabilities in a Single-Server Queue. J. Appl. Probability 31 131–156.
  • [13] Grassmann, W. K. and Heyman, D. P. (1990). Equilibrium distribution of blocked-structured Markov chains with repeating rows. J. Appl. Probability 27 557–576.
  • [14] Kaspi, H. (1982). On the symmetric Wiener-Hopf factorization for Markov additive processe. Z. Wahrsch. Verw. Gebiete 59 179–196.
  • [15] Kim, B. and Sohraby, K. (2006). Tail behavior of the queue size and waiting time in a queue with discrete autoregressive arrivals. Advances in Applied Probability 38 1116–1131.
  • [16] Kobayashi, M. and Miyazawa, M. (2011). Tail asymptotics of the stationary distribution of a two dimensional reflecting random walk with unbounded upward jumps. Preprint.
  • [17] Loynes, R. M. (1962). The stability of a queue with non-independent inter-arrival and service times. Proceedings of Cambridge Philosophical Society 58 497–520.
  • [18] Majewski, K. (2004). Large Deviation Bounds for Single Class Queueing Networks and Their Calculation. Queueing Syst. Theory Appl. 48 103–134.
  • [19] Markushevich, A. I. (1977). Theory of functions of a complex variable. Vol. I, II, III, English ed. Chelsea Publishing Co., New York. Translated and edited by Richard A. Silverman.
  • [20] Miyazawa, M. (2004). A Markov Renewal Approach to M/G/1 Type Queues with Countably Many Background States. Queueing Systems 46 177–196.
  • [21] Miyazawa, M. (2009). Tail decay rates in double QBD processes and related reflected random walks. Math. Oper. Res. 34 547–575.
  • [22] Miyazawa, M. (2011). Light tail asymptotics in multidimensional reflecting processes for queueing networks. TOP, an official journal of the Spanish Society of Statistics and Operations Research 19 233–299.
  • [23] Miyazawa, M. and Rolski, T. (2009). Tail asymptotics for a Lévy-driven tandem queue with an intermediate input. Queueing Syst. 63 323–353.
  • [24] Miyazawa, M. and Zhao, Y. Q. (2004). The stationary tail asymptotics in the $GI/G/1$-type queue with countably many background states. Adv. in Appl. Probab. 36 1231–1251.
  • [25] Neuts, M. F. (1981). Matrix-geometric solutions in stochastic models: an algorithm approach. The John Hopkins University Press, Baltimore, MD.
  • [26] Ney, P. and Nummelin, E. (1987). Markov additive processes I. Eigenvalue properties and limit theorems. Annals of Probability 15 561–592.
  • [27] Ney, P. and Nummelin, E. (1987). Markov additive processes II. Large deviations. Annals of Probability 15 593–609.
  • [28] Nummelin, E. (1984). General irreducible Markov chains and non-negative operators. Cambridge University Press.
  • [29] Pitman, J. W. (1974). An identity for stopping times of a Markov process. In Studies in Probability and Statistics ( E. J. Williams, ed.) 41–57. Jerusalem Academic Press.
  • [30] Rockafellar, R. T. (1970). Convex analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton, N.J.
  • [31] Shwartz, A. and Weiss, A. (1995). Large deviations for performance analysis. Chapman & Hall, New York.
  • [32] Tweedie, R. L. (1982). Operator-geometric stationary distributions for Markov chains, with application to queueing models. Advances in Applied Probability 14 368–391.
  • [33] Zhao, Y. Q., Li, W. and Braun, W. J. (2003). Censoring, Factorizations, and Spectral Analysis for Transition Matrices with Block-Repeating Entries. Methodology and Computing in Applied Probability 5 35–58.