Abstract
We are concerned with the stationary distribution of a d-dimensional semimartingale reflecting Brownian motion on a nonnegative orthant, provided it is stable, and conjecture about the tail decay rate of its marginal distribution in an arbitrary direction. Due to recent studies, the conjecture is true for d=2. We show its validity for the skew symmetric case for a general d.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Avram, F., Dai, J.G., Hasenbein, J.J.: Explicit solutions for variational problems in the quadrant. Queueing Syst. 37, 259–289 (2001)
Borovkov, A.A., Mogul’skii, A.A.: Large deviations for Markov chains in the positive quadrant. Russ. Math. Surv. 56, 803–916 (2001)
Chen, H., Yao, D.D.: Fundamentals of Queueing Networks, Performance, Asymptotics, and Optimization. Springer, New York (2001)
Dai, J.G., Miyazawa, M.: Reflecting Brownian motion in two dimensions: Exact asymptotics for the stationary distribution (2010, submitted for publication). http://queue3.is.noda.tus.ac.jp/miyazawa/mm-paper
Foley, R.D., McDonald, D.R.: Large deviations of a modified Jackson network: stability and rough asymptotics. Ann. Appl. Probab. 15, 519–541 (2005)
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., Reiman, M.I.: Reflected Brownian motion in an orthant. Ann. Probab. 9, 302–308 (1981)
Harrison, J.M., Williams, R.J.: Brownian models of open queueing networks with homogeneous customer populations. Stochastics 22, 77–115 (1987)
Kurkova, L.A., Suhov, Y.M.: Malyshev’s theory and JS-queues. Asymptotics of stationary probabilities. Ann. Appl. Probab. 13, 1313–1354 (2003)
Lieshout, P., Mandjes, M.: Brownian tandem queues. Math. Methods Oper. Res. 66, 275–298 (2007)
Lieshout, P., Mandjes, M.: Asymptotic analysis of Lévy-driven tandem queues. CWI report, PNA-R0809 (2008)
Majewski, K.: Large deviations of the steady state distribution of reflected processes with applications to queueing systems. Queueing Syst. 29, 351–381 (1998)
Miyazawa, M.: Conjectures on decay rates of tail probabilities in generalized Jackson and batch movement networks. J. Oper. Res. Soc. Jpn. 46(1), 74–98 (2003)
Miyazawa, M.: Tail decay rates in double QBD processes and related reflected random walks. Math. Oper. Res. 34, 547–575 (2009)
Miyazawa, M.: Light tail asymptotics in multidimensional reflecting processes for queueing networks. Top (2011, to appear). http://queue3.is.noda.tus.ac.jp/miyazawa/mm-paper
Miyazawa, M., Rolski, T.: Exact asymptotics for a Levy-driven tandem queue with an intermediate input. Queueing Syst. 63, 323–353 (2009)
Taylor, L.M., Williams, R.J.: Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant. Probab. Theory Relat. Fields 96, 283–317 (1993)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Miyazawa, M., Kobayashi, M. Conjectures on tail asymptotics of the marginal stationary distribution for a multidimensional SRBM. Queueing Syst 68, 251–260 (2011). https://doi.org/10.1007/s11134-011-9251-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11134-011-9251-0
Keywords
- Queueing network
- Semi-martingale reflecting Brownian motion
- Stationary distribution
- Tail asymptotic
- Tail decay rate
- Stationary inequality
- Multidimensional moment generating function