Abstract
We present a geometric interpretation of a product form stationary distribution for a \(d\)-dimensional semimartingale reflecting Brownian motion (SRBM) that lives in the nonnegative orthant. The \(d\)-dimensional SRBM data can be equivalently specified by \(d+1\) geometric objects: an ellipse and \(d\) rays. Using these geometric objects, we establish necessary and sufficient conditions for characterizing product form stationary distribution. The key idea in the characterization is that we decompose the \(d\)-dimensional problem to \(\frac{1}{2}d(d-1)\) two-dimensional SRBMs, each of which is determined by an ellipse and two rays. This characterization contrasts with the algebraic condition of Harrison and Williams (Ann Probab 15:115–137, 1987b). A \(d\)-station tandem queue example is presented to illustrate how the product form can be obtained using our characterization. Drawing the two-dimensional results in Avram et al. (Queueing Syst 37:259–289, 2001), Dai and Miyazawa (Queueing Syst 74:181–217, 2013), we discuss potential optimal paths for a variational problem associated with the three-station tandem queue.
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References
Avram, F., Dai, J.G., Hasenbein, J.J.: Explicit solutions for variational problems in the quadrant. Queueing Syst. 37, 259–289 (2001)
Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York (1979)
Bramson, M., Dai, J.G., Harrison, J.M.: Positive recurrence of reflecting Brownian motion in three dimensions. Ann. Appl. Probab 20, 753–783 (2010)
Dai, J.G., Harrison, J.M.: Reflected Brownian motion in an orthant: numerical methods for steady-state analysis. Ann. Appl. Probab. 2, 65–86 (1992)
Dai, J.G., Kurtz, T.G.: Characterization of the stationary distribution for a semimartingale reflecting Brownian motion in a convex polyhedron. Preprint (1994)
Dai, J.G., Miyazawa, M.: Reflecting Brownian motion in two dimensions: exact asymptotics for the stationary distribution. Stoch. Syst. 1, 146–208 (2011)
Dai, J.G., Miyazawa, M.: Stationary distribution of a two-dimensional SRBM: geometric views and boundary measures. Queueing Syst. 74, 181–217 (2013)
Dai, J.G., Miyazawa, M., Wu, J.: Decomposable stationary distribution of a multidimensional SRBM (2013, submitted). arXiv:1312.1378
Dai, J.G., Miyazawa, M., Wu, J.:A multi-dimensional SRBM: Geometric views of its product form stationary distribution. arXiv:1312.1758
Dai, J.G., Williams, R.J.: Existence and uniqueness of Semimartingale reflecting Brownian motions in convex polyhedrons. Theory Probab. Appl. 40, 1–40 (1995). Correctional note: 2006, 59, 346–347
Harrison, J.M., Hasenbein, J.J.: Reflected Brownian motion in the quadrant: tail behavior of the stationary distribution. Queueing Syst. 61, 113–138 (2009)
Harrison, J.M., Nguyen, V.: Brownian models of multiclass queueing networks: current status and open problems. Queueing Syst. 13, 5–40 (1993)
Harrison, J.M., Williams, R.J.: Brownian models of open queueing networks with homogeneous customer populations. Stochastics 22, 77–115 (1987a)
Harrison, J.M., Williams, R.J.: Multidimensional reflected Brownian motions having exponential stationary distributions. Ann. Probab. 15, 115–137 (1987b)
Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, New York (1991)
Kang, W., Kelly, F., Lee, N., Williams, R.: State space collapse and diffusion approximation for a network operating under a fair bandwidth sharing policy. Ann. Appl. Probab. 19, 1719–1780 (2009)
Kharroubi, A.E., Yaacoubi, A., Tahar, A.B., Bichard, K.: Variational problem in the non-negative orthant of \({\cal R}^{3}\): reflective faces and boundary influence cones. Queueing Syst. 70, 299–337 (2012)
Latouche, G., Miyazawa, M.: Product-form characterization for a two-dimensional reflecting random walk. Queueing Syst. To appear (2014)
Liang, Z., Hasenbein, J.J.: Optimal paths in large deviations of symmetric reflected Brownian motion in the octant. Stoch. Syst. 3, 187–229 (2013)
Majewski, K.: Large deviations of stationary reflected Brownian motions. In: Kelly, F.P., Zachary, S., Ziedins, I. (eds.) Stochastic Networks: Theory and Applications. Oxford University Press, Oxford (1996)
Majewski, K.: Heavy traffic approximations of large deviations of feedforward queueing networks. Queueing Syst. 28, 125–155 (1998a)
Majewski, K.: Large deviations of the steady-state distribution of reflected processes with applications to queueing systems. Queueing Syst. 29, 351–381 (1998b)
Reiman, M.I., Williams, R.J.: A boundary property of semimartingale reflecting Brownian motions. Probab. Theory Relat. Fields 77, 87–97 (1988). Correction: 80, 633 (1989)
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Research supported in part by NSF Grants CMMI-1030589, CNS-1248117, CMMI-1335724, and JSPS Grant 24310115.
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J. G. Dai on leave from Georgia Institute of technology.
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Dai, J.G., Miyazawa, M. & Wu, J. A multi-dimensional SRBM: geometric views of its product form stationary distribution. Queueing Syst 78, 313–335 (2014). https://doi.org/10.1007/s11134-014-9411-0
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DOI: https://doi.org/10.1007/s11134-014-9411-0
Keywords
- Semimartingale reflecting Brownian motion
- Variational problem
- Skew symmetry condition
- Queueing networks
- Diffusion approximations