Open Access
Volume 26, 2022
Page(s) 26 - 68
Published online 13 January 2022
  1. F. Avram, J.G. Dai and J.J. Hasenbein, Explicit solutions for variational problems in the quadrant. Queueing Syst. 37 (2001) 259–289. [CrossRef] [MathSciNet] [Google Scholar]
  2. C. Banderier, M. Bousquet-Mélou, A. Denise, P. Flajolet, D. Gardy and D. Gouyou-Beauchamps, Generating functions of generating trees. Discrete Math. 246 (2002) 29–55. [CrossRef] [MathSciNet] [Google Scholar]
  3. M. Bousquet-Mélou, Walks in the quarter plane: Kreweras’ algebraic model. Ann. Appl. Probab. 15 (2005) 1451–1491. [MathSciNet] [Google Scholar]
  4. V. Bhaskar and P. Lallement, Modeling a supply chain using a network of queues. Appl. Math. Model. 34 (2010) 2074–2088. [CrossRef] [MathSciNet] [Google Scholar]
  5. H.S. Dai, D.A. Dawson and Y.Q. Zhao, Kernel method for stationary tails: from discrete to continuous, in Asymptotic Laws and Methods in Stochastics, edited by D.A. Dawson, R. Kulik, M. Ould Haye, B. Szyszkowicz, Y.Q. Zhao (2015) 297–327. [CrossRef] [Google Scholar]
  6. J.G. Dai and J.M. Harrison, Reflecting Brownian motion in an orthant: Numerical methods for steady-state analysis. Ann. Appl. Prob. 2 (1992) 65–86. [Google Scholar]
  7. J.G. Dai and M. Miyazawa, Reflecting Brownian motion in two dimensions: exact asymptotics for the stationary distribution. Stoch. Syst. 1 (2011) 146–208. [CrossRef] [MathSciNet] [Google Scholar]
  8. K. Debicki, M. Mandjes and M. van Uitert, A tandem queue with Lévy input: a new representation of the downstream queue length. Prob. Eng. Inform. Sci. 21 (2007) 83–107. [CrossRef] [Google Scholar]
  9. P. Dupuis and K. Ramanan, A time-reversed representation for the tail probabilities of stationary reflected Brownian motion. Stoch. Process. Appl. 98 (2002) 253–287. [CrossRef] [Google Scholar]
  10. G. Fayolle, R. Iasnogorodski and V. Malyshev, Random Walks in the Quarter-Plane, second ed. Springer, New York (2017). [CrossRef] [Google Scholar]
  11. S. Franceschi and I. Kurkova, Asymptotic expansion of stationary distribution for reflected Brownian motion in the quarter plane via analytic approach. Stoch. Syst. 7 (2017) 32–94. [CrossRef] [MathSciNet] [Google Scholar]
  12. G. Giambene, Queuing Theory and Telecommunications. Springer, Boston (2014). [CrossRef] [Google Scholar]
  13. M.K. Govil and M. Fu, Queueing theory in manufacturing: a survey. J. Manufactur. Syst. 18 (1999) 214–240. [CrossRef] [Google Scholar]
  14. J.M. Harrison and J.J. Hasenbein, Reflected Brownian motion in the quadrant: Tail behavior of the stationary distribution. Queu. Syst. 61 (2009) 113–138. [CrossRef] [Google Scholar]
  15. J. Harrison and M. Reiman, On the distribution of multidimensional reflected Brownian motion. SIAM J. Appl. Math. 41 (1981) 345–361. [CrossRef] [MathSciNet] [Google Scholar]
  16. J. Harrison and M. Reiman, Reflected Brownian motion on an orthant. Ann. Probab. 9 (1981) 302–308. [MathSciNet] [Google Scholar]
  17. J.M. Harrison and R.J. Williams, Brownian models of open queueing networks with homogeneous customer populations. Stochastic 22 (1987) 77–115. [CrossRef] [Google Scholar]
  18. T. Konstantopoulous, G. Last and S.J. Lin, On a class of Lévy stochastic networks. Queu. Syst. 46 (2004) 409–437. [CrossRef] [Google Scholar]
  19. G.R. Lawlor, L’Hôspital’s rule for multivariable functions. Am. Math. Monthly 127 (2020) 717–725. [CrossRef] [Google Scholar]
  20. H. Li and Y.Q. Zhao, Tail asymptotics for a generalized two-dimensional queueing model – a kernel method. Queu. Syst. 69 (2011) 77–100. [CrossRef] [Google Scholar]
  21. H. Li and Y.Q. Zhao, A kernel method for exact tail asymptotics-random walks in the quarter plane. Queu. Models Serv. Manag. 1 (2018) 95–129. [Google Scholar]
  22. P. Lieshout and M. Mandjes, Tandem Brownian queues. Math. Methods Oper. Res. 66 (2007) 275–298. [CrossRef] [MathSciNet] [Google Scholar]
  23. P. Lieshout and M. Mandjes, Asymptotic analysis of Lévy-driven tandem queues. Queu. Syst. 60 (2008) 203–226. [CrossRef] [Google Scholar]
  24. M. Mandjes, Packet models revisited: tandem and priority systems. Queu. Syst. 47 (2004) 363–377. [CrossRef] [Google Scholar]
  25. K. Majewski, Large deviations of the steady state distribution of reflected processes with applications to queueing systems. Queu. Syst. 29 (1998) 351–381. [CrossRef] [Google Scholar]
  26. A.I. Markushevich, Theory of Functions of A Complex Variable. Vol. I.II.III, English ed. Chelsea Publishing Co., New York (1977). [Google Scholar]
  27. M. Miyazawa and T. Rolski, Tail asymptotics for a Lévy-driven tandem queue with an intermediate input. Queu. Syst. 63 (2009) 323–353. [CrossRef] [Google Scholar]
  28. R. Narasimhan, Several Complex Variables. The University of Chicago Press, Chicago and London (1964). [Google Scholar]
  29. T. Robertazzi, Computer Networks and Systems — Queueing Theory and Performance Evaluation, third ed. Springer, New York (2000). [Google Scholar]
  30. S.I. Resnick, Extreme Values, Regular Variation, and Point Processes. Springer, New York (1987). [CrossRef] [Google Scholar]
  31. V. Schmitz, Copulas and Stochastic Processes. Ph.D thesis: Achen University, 2003. [Google Scholar]
  32. S. Varadhan and R. Williams, Brownian motion in a wedge with oblique reflection. Commun. Pure Appl. Math. 38 (1985) 405–443. [CrossRef] [Google Scholar]
  33. R. Williams, Recurrence classification and invariant measure for reflected Brownian motion in a wedge. Ann. Probab. 13 (1985) 758–778. [CrossRef] [MathSciNet] [Google Scholar]
  34. R.J. Williams, Semimartingale reflecting Brownian motions in the orthant. In “IMA Volumes in Mathematics and Its Applications, Volume 71”, edited by F.R. Kelly, R.J. Williams (1995) 125–137. [CrossRef] [Google Scholar]
  35. R.J. Williams, On the approximation of queueing networks in heavy traffic. In “ Stochastic Networks: Theory and Applications,” edited by F.P. Kelly, S. Zachary, I. Ziedins (1996) 35–56. [Google Scholar]
  36. W. Whitt, Stochastic-Process Limits. Springer, New York (2002). [CrossRef] [Google Scholar]
  37. Y.Q. Zhao, Kernel method-an analytic approach for tail asymptotics in stationary probabilities of 2-dimensional queueing systems. Preprint arXiv:2101.11661 (2021). [Google Scholar]

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