Asymptotic Expansion of Stationary Distribution for Reflected Brownian Motion in the Quarter Plane via Analytic Approach
Published Online:8 May 2017https://doi.org/10.1287/16-SSY218
References
- (2001). Explicit solutions for variational problems in the quadrant. Queueing Systems. Theory and Applications, 37(1–3):259–289. MR1833666Google Scholar
- (1987). Analysis of models reducible to a class of diffusion processes in the positive quarter plane. SIAM Journal on Applied Mathematics, 47(6):1367–1385. MR0916246Google Scholar
- (1992). Multidimensional Integral Transformations. CRC Press. MR1177594Google Scholar
- (1990). Steady-state analysis of reflected Brownian motions: Characterization, numerical methods and queueing applications. ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)–Stanford University. MR2685786Google Scholar
- (1992). Reflected Brownian motion in an orthant: Numerical methods for steady-state analysis. The Annals of Applied Probability, 2(1):65–86. MR1143393Google Scholar
- (2011). Reflecting Brownian motion in two dimensions: Exact asymptotics for the stationary distribution. Stochastic Systems, 1(1):146–208. MR2948920Link, Google Scholar
- (2013). Stationary distribution of a two-dimensional SRBM: geometric views and boundary measures. Queueing Systems, 74(2–3):181–217. MR3054655Google Scholar
- (2009). Reflected Brownian motion in a wedge: Sum-of-exponential stationary densities. Electronic Communications in Probability, 14:1–16. MR2472171Google Scholar
- (1974). Introduction to the Theory and Application of the Laplace Transformation. Springer Berlin Heidelberg, Berlin, Heidelberg. MR0344810Google Scholar
- (1994). Lyapunov functions for semimartingale reflecting Brownian motions. The Annals of Probability, 22(2):680–702. MR1288127Google Scholar
- (1979). Two coupled processors: The reduction to a Riemann-Hilbert problem. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 47(3):325–351. MR0525314Google Scholar
- (1999). Random Walks in the Quarter-Plane. Springer Berlin Heidelberg, Berlin, Heidelberg. MR1691900Google Scholar
- (2015). About a possible analytic approach for walks in the quarter plane with arbitrary big jumps. Comptes Rendus Mathematique, 353(2):89–94. MR3300936Google Scholar
- (1977). The saddle-point method. Izdat. “Nauka”, Moscow. MR0507923Google Scholar
- (1989). Asymptotic methods in analysis. In Analysis I, pages 83–191. Springer. MR0899753Google Scholar
- (1984). Analysis of Brownian motion with drift, confined to a quadrant by oblique reflection (diffusions, Riemann-Hilbert problem). ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)–Stanford University. MR2633392Google Scholar
- (2016). Tutte’s invariant approach for Brownian motion reflected in the quadrant. ESAIM Probab. Stat. to appear.Google Scholar
- (2017). Explicit expression for the stationary distribution of reflected Brownian motion in a wedge. Preprint arXiv:1703.09433.Google Scholar
- (1978). The diffusion approximation for tandem queues in heavy traffic. Advances in Applied Probability, 10(4):886–905. MR0509222Google Scholar
- (2009). Reflected Brownian motion in the quadrant: tail behavior of the stationary distribution. Queueing Systems, 61(2–3):113–138. MR2485885Google Scholar
- (1993). Brownian models of multiclass queueing networks: Current status and open problems. Queueing Systems. Theory and Applications, 13(1–3):5–40. MR1218842Google Scholar
- (1987a). Brownian models of open queueing networks with homogeneous customer populations. Stochastics, 22(2):77–115. MR0912049Google Scholar
- (1987b). Multidimensional reflected Brownian motions having exponential stationary distributions. The Annals of Probability, 15(1):115–137. MR0877593Google Scholar
- (1993). Recurrence and transience of reflecting Brownian motion in the quadrant. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 113, pages 387–399. Cambridge Univ Press. MR1198420Google Scholar
- (1994). The influence of boundaries in problems on large deviations. Rossiĭskaya Akademiya Nauk. Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk, 49(2(296)):43–102. MR1283135Google Scholar
- (2011). Random walks in Z+2 with non-zero drift absorbed at the axes. Bulletin de la Société Mathématique de France, 139:341–387. MR2869310Google Scholar
- (2012). On the functions counting walks with small steps in the quarter plane. Publications mathématiques de l’IHES, 116(1):69–114. MR3090255Google Scholar
- (1998). Martin boundary and elliptic curves. Markov Processes and Related Fields, 4(2):203–272. MR1641546Google Scholar
- (2003). Malyshev’s Theory and JS-Queues. Asymptotics of Stationary Probabilities. The Annals of Applied Probability, 13(4):1313–1354. MR2023879Google Scholar
- (2013). Product-form characterization for a two-dimensional reflecting random walk. Queueing Systems, 77(4):373–391. MR3225816Google Scholar
- (2007). Tandem Brownian queues. Mathematical Methods of Operations Research, 66(2):275–298. MR2342215Google Scholar
- (2008). Asymptotic analysis of Lévy-driven tandem queues. Queueing Systems. Theory and Applications, 60(3–4):203–226. MR2461616Google Scholar
- (1996). Large Deviations of Stationary Reflected Brownian Motions. Stochastic Networks: Theory and Applications.Google Scholar
- (1998). Large deviations of the steady-state distribution of reflected processes with applications to queueing systems. Queueing Systems, 29(2–4):351–381. MR1654452Google Scholar
- (1970). Sluchainye bluzhdaniya. Uravneniya. Vinera-Khopfa v chetverti ploskosti. Avtomorfizmy Galua. Izdat. Moskov. Univ., Moscow.Google Scholar
- (1973). Asymptotic behavior of the stationary probabilities for two-dimensional positive random walks. Siberian Mathematical Journal, 14(1):109–118. MR0433604Google Scholar
- (2013). Analytic combinatorics in several variables, volume 140 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge. MR3088495Google Scholar
- (2012). Counting walks in a quadrant: A unified approach via boundary value problems. Journal of the European Mathematical Society, pages 749–777. MR2911883Google Scholar
- (1988). A boundary property of semimartingale reflecting Brownian motions. Probability Theory and Related Fields, 77(1):87–97. MR0921820Google Scholar
- (1993). Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant. Probability Theory and Related Fields, 96(3):283–317. MR1231926Google Scholar
- (1985). Recurrence classification and invariant measure for reflected Brownian motion in a wedge. Annals of Probabability, 13:758–778. MR0799421Google Scholar
- (1995). Semimartingale reflecting Brownian motions in the orthant. Stochastic Networks, 13. MR1381009Google Scholar
- (1996). On the Approximation of Queuing Networks in Heavy Traffic. In Stochastic Networks: Theory and Applications, Royal Statistical Society Series. F. P. Kelly, S. Zachary, and I. Ziedins, oxford university press edition.Google Scholar

