Stochastic Systems

Reflecting Brownian motion in two dimensions: Exact asymptotics for the stationary distribution

J. G. Dai and Masakiyo Miyazawa

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We consider a two-dimensional semimartingale reflecting Brownian motion (SRBM) in the nonnegative quadrant. The data of the SRBM consists of a two-dimensional drift vector, a $2\times 2$ positive definite covariance matrix, and a $2\times 2$ reflection matrix. Assuming the SRBM is positive recurrent, we are interested in tail asymptotic of its marginal stationary distribution along each direction in the quadrant. For a given direction, the marginal tail distribution has the exact asymptotic of the form $bx^{\kappa}\exp(-\alpha x)$ as $x$ goes to infinity, where $\alpha$ and $b$ are positive constants and $\kappa$ takes one of the values $-3/2$, $-1/2$, $0$, or $1$; both the decay rate $\alpha$ and the power $\kappa$ can be computed explicitly from the given direction and the SRBM data.

A key tool in our proof is a relationship governing the moment generating function of the two-dimensional stationary distribution and two moment generating functions of the associated one-dimensional boundary measures. This relationship allows us to characterize the convergence domain of the two-dimensional moment generating function. For a given direction $c$, the line in this direction intersects the boundary of the convergence domain at one point, and that point uniquely determines the decay rate $\alpha$. The one-dimensional moment generating function of the marginal distribution along direction $c$ has a singularity at $\alpha$. Using analytic extension in complex analysis, we characterize the precise nature of the singularity there. Using that characterization and complex inversion techniques, we obtain the exact asymptotic of the marginal tail distribution.

Article information

Stoch. Syst., Volume 1, Number 1 (2011), 146-208.

First available in Project Euclid: 24 February 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65] 60K65
Secondary: 60K25: Queueing theory [See also 68M20, 90B22] 60G42: Martingales with discrete parameter 90C33: Complementarity and equilibrium problems and variational inequalities (finite dimensions)

Diffusion process heavy traffic queueing network tail behavior large deviations


Dai, J. G.; Miyazawa, Masakiyo. Reflecting Brownian motion in two dimensions: Exact asymptotics for the stationary distribution. Stoch. Syst. 1 (2011), no. 1, 146--208. doi:10.1214/10-SSY022.

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  • [1] Avram, F., Dai, J. G., and Hasenbein, J. J. (2001). Explicit solutions for variational problems in the quadrant. Queueing Syst. 37, 1-3, 259–289.
  • [2] Berman, A. and Plemmons, R. J. (1979). Nonnegative matrices in the mathematical sciences. Academic Press [Harcourt Brace Jovanovich Publishers], New York. Computer Science and Applied Mathematics.
  • [3] Dai, J. G. and Harrison, J. M. (1992). Reflected Brownian motion in an orthant: numerical methods for steady-state analysis. Ann. Appl. Probab. 2, 1, 65–86.
  • [4] Dieker, A. B. and Moriarty, J. (2009). Reflected Brownian motion in a wedge: sum-of-exponential stationary densities. Electron. Commun. Probab. 14, 1–16.
  • [5] Doetsch, G. (1974). Introduction to the theory and application of the Laplace transformation. Springer-Verlag, New York. Translated from the second German edition by Walter Nader.
  • [6] Dupuis, P. and Ramanan, K. (2002). A time-reversed representation for the tail probabilities of stationary reflected Brownian motion. Stochastic Process. Appl. 98, 2, 253–287.
  • [7] Dupuis, P. and Williams, R. J. (1994). Lyapunov functions for semimartingale reflecting Brownian motions. Ann. Probab. 22, 2, 680–702.
  • [8] Fayolle, G., Iasnogorodski, R., and Malyshev, V. (1999). Random walks in the quarter-plane. Applications of Mathematics (New York), Vol. 40. Springer-Verlag, Berlin. Algebraic methods, boundary value problems and applications.
  • [9] Foley, R. D. and McDonald, D. R. (2005). Bridges and networks: exact asymptotics. Ann. Appl. Probab. 15, 1B, 542–586.
  • [10] Harrison, J. M. (1978). The diffusion approximation for tandem queues in heavy traffic. Adv. in Appl. Probab. 10, 4, 886–905.
  • [11] Harrison, J. M. and Hasenbein, J. J. (2009). Reflected Brownian motion in the quadrant: tail behavior of the stationary distribution. Queueing Syst. 61, 2-3, 113–138.
  • [12] Harrison, J. M. and Nguyen, V. (1993). Brownian models of multiclass queueing networks: current status and open problems. Queueing Systems Theory Appl. 13, 1-3, 5–40.
  • [13] Harrison, J. M. and Williams, R. J. (1987). Brownian models of open queueing networks with homogeneous customer populations. Stochastics 22, 2, 77–115.
  • [14] Harrison, J. M. and Williams, R. J. (1987). Multidimensional reflected Brownian motions having exponential stationary distributions. Ann. Probab. 15, 1, 115–137.
  • [15] Hobson, D. G. and Rogers, L. C. G. (1993). Recurrence and transience of reflecting Brownian motion in the quadrant. Math. Proc. Cambridge Philos. Soc. 113, 2, 387–399.
  • [16] Ignatyuk, I. A., Malyshev, V. A., and Shcherbakov, V. V. (1994). The influence of boundaries in problems on large deviations. Uspekhi Mat. Nauk 49, 2(296), 43–102.
  • [17] Kobayashi, M. and Miyazawa, M. (2011). Tail asymptotics of the stationary distribution of a two dimensional reflecting random walk with unbounded upward jumps. Preprint.
  • [18] Kobayashi, M., Miyazawa, M., and Zhao, Y. Q. (2010). Tail asymptotics of the occupation measure for a Markov additive process with an $M/G/1$-type background process. Stoch. Models 26, 3, 463–486.
  • [19] Lieshout, P. and Mandjes, M. (2007). Tandem Brownian queues. Math. Methods Oper. Res. 66, 2, 275–298.
  • [20] Lieshout, P. and Mandjes, M. (2008). Asymptotic analysis of Lévy-driven tandem queues. Queueing Syst. 60, 3-4, 203–226.
  • [21] Majewski, K. (1996). Large Deviations of Stationary Reflected Brownian Motions. In Stochastic Networks: Theory and Applications ( F. P. Kelly, S. Zachary and I. Ziedins, eds.) Oxford University Press.
  • [22] Majewski, K. (1998). Large deviations of the steady-state distribution of reflected processes with applications to queueing systems. Queueing Systems Theory Appl. 29, 2-4, 351–381.
  • [23] 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.
  • [24] Miyazawa, M. (2009). Tail decay rates in double QBD processes and related reflected random walks. Math. Oper. Res. 34, 3, 547–575.
  • [25] Miyazawa, M. and Kobayashi, M. (2011). Conjectures on tail asymptotics of the stationary distribution for a multidimensional SRBM. Queueing Systems. to appear.
  • [26] Miyazawa, M. and Rolski, T. (2009). Tail asymptotics for a Lévy-driven tandem queue with an intermediate input. Queueing Syst. 63, 1-4, 323–353.
  • [27] Reiman, M. I. and Williams, R. J. (1988). A boundary property of semimartingale reflecting Brownian motions. Probab. Theory Related Fields 77, 1, 87–97.
  • [28] Rockafellar, R. T. (1970). Convex analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton, N.J.
  • [29] Taylor, L. M. and Williams, R. J. (1993). Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant. Probab. Theory Related Fields 96, 3, 283–317.
  • [30] Williams, R. J. (1995). Semimartingale reflecting Brownian motions in the orthant. In Stochastic networks. IMA Vol. Math. Appl., Vol. 71. Springer, New York, 125–137.
  • [31] Williams, R. J. (1996). On the approximation of queueing networks in heavy traffic. In Stochastic Networks: Theory and Applications ( F. P. Kelly, S. Zachary and I. Ziedins, eds.). Royal Statistical Society. Oxford University Press.