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The kernel method tail asymptotics analytic approach for stationary probabilities of two-dimensional queueing systems

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Abstract

Many queueing systems can be modelled as two-dimensional random walks with reflective boundaries, discrete, continuous or mixed. Stationary probabilities are one of the most sought after statistical quantities in queueing analysis. However, explicit expressions are only available for a very limited number of models. Therefore, tail asymptotic properties become more important, since they provide insightful information on the structure of the tail probabilities, and often lead to approximations, performance bounds, algorithms, among possible other applications. In this survey, we provide key ideas of a kernel method, developed from the classical kernel method in analytic combinatorics, for studying so-called exact tail asymptotic properties in stationary probabilities for this type of random walk.

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References

  1. Adan, I., Foley, R.D., McDonald, D.R.: Exact asymptotics for the stationary distribution of a Markov chain: a production model. Queueing Syst. 62, 311–344 (2009)

    Article  Google Scholar 

  2. Adan, I.J.B.F., Wessels, J., Zijm, W.H.M.: A compensation approach for two-dimensional Markov processes. Adv. Appl. Probab. 25, 783–817 (1993)

    Article  Google Scholar 

  3. Avrachenkov, K., Nain, P., Yechiali, U.: A retrial system with two input streams and two orbit queues. Queueing Syst. 77(1), 1–31 (2014)

    Article  Google Scholar 

  4. Banderier, C., Bousquet-Mélou, M., Denise, A., Flajolet, P., Gardy, D., Gouyou-Beauchamps, D.: Generating functions of generating trees. Discrete Math. 246, 29–55 (2002)

    Article  Google Scholar 

  5. Borovkov, A.A., Mogul’skii, A.A.: Large deviations for Markov chains in the positive quadrant. Russ. Math. Surv. 56, 803–916 (2001)

  6. Bousquet-Mélou, M.: Walks in the quarter plane: Kreweras’ algebraic model. Ann. Appl. Probab. 15, 1451–1491 (2005)

  7. Cohen, J.W., Boxma, O.J.: Boundary Value Problems in Queueing System Analysis. North-Holland Amsterdam (1983)

  8. Dai, H., Dawson, D.A., Zhao, Y.Q.: Kernel method for staionary tails: from discrete to continuous. In: Dawson, D., Kulik, R., Ould Haye, M., Szyszkowicz, B., Zhao, Y. (eds.) Asymptotic Laws and Methods in Stochastics, pp. 297–328. Springer, New York (2015)

    Chapter  Google Scholar 

  9. Dai, H., Dawson, D., Zhao, Y.Q.: Exact tail asymptotics for a three dimensional Brownian tandem queue with intermediate inputs, under revisions. arXiv:org/abs/1807.08425 (2018)

  10. Dai, H., Kong, L., Song, Y.: Exact tail asymptotics for a two-stage queue: complete solution via kernel method. RAIRO-Oper. Res. 51, 1211–1250 (2017)

    Article  Google Scholar 

  11. Dai, H., Zhao, Y.Q.: Wireless 3-hop networks with stealing revisited: a kernel approach. INFOR 51(4), 192–205 (2013)

    Google Scholar 

  12. 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)

    Article  Google Scholar 

  13. Dai, J.G., Miyazawa, M.: Reflecting Brownian motion in two dimensions: exact asymptotics for the stationary distribution. Stoch. Syst. 1, 146–208 (2011)

    Article  Google Scholar 

  14. Dai, J.G., Miyazawa, M.: Stationary distribution of a two-dimensional SRBM: geometric views and boundary measures. Queueng Syst. 74, 181–217 (2013)

    Article  Google Scholar 

  15. Fayolle, G., Iasnogorodski, R.: Two coupled processors: the reduction to a Riemann–Hilbert problem. Z. Wahrscheinlichkeitsth 47, 325–351 (1979)

    Article  Google Scholar 

  16. Fayolle, G., Iasnogorodski, R., Malyshev, V.: Random Walks in the Quarter-Plane, 2nd edn. Springer, New York (2017)

    Book  Google Scholar 

  17. Fayolle, G., King, P.J.B., Mitrani, I.: The solution of certain two-dimensional Markov models. Adv. Appl. Probab. 14, 295–308 (1982)

    Article  Google Scholar 

  18. Feller, W.: An Introduction to Probability Theory and its Applications, vol. II, 2nd edn. Wiley, New York (1971)

    Google Scholar 

  19. Flatto, L., McKean, H.P.: Two queues in parallel. Commun. Pure Appl. Math. 30, 255–263 (1977)

    Article  Google Scholar 

  20. Flatto, L., Hahn, S.: Two parallel queues created by arrivals with two demands I. SIAM J. Appl. Math. 44, 1041–1053 (1984)

    Article  Google Scholar 

  21. Flajolet, F., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press (2009)

  22. Foley, R.D., McDonald, D.R.: Join the shortest queue: stability and exact asymptotics. Ann. Appl. Probab. 11, 569–607 (2001)

    Article  Google Scholar 

  23. Foley, R.D., McDonald, R.D.: Large deviations of a modified Jackson network: stability and rough asymptotics. Ann. Appl. Probab. 15, 519–541 (2005)

    Article  Google Scholar 

  24. Foley, R.D., McDonald, R.D.: Bridges and networks: exact asymptotics. Ann. Appl. Probab. 15, 542–586 (2005)

    Article  Google Scholar 

  25. Guillemin, F., Knessl, C., Van Leeuwaarden, J.S.H.: Wireless multi-hop networks with stealing: large buffer asymptotics via the Ray method. SIAM J. Appl. Math. 71, 1220–1240 (2011)

    Article  Google Scholar 

  26. Guillemin, F., van Leeuwarden, J.S.H.: Rare event asymptotics for a random walk in the quarter plane. Queueing Syst. 67(1), 1–32 (2011)

    Article  Google Scholar 

  27. Guillemin, F., Pinchon, D.: Analysis of generalized processor-sharing systems with two classes of customers and exponential services. J. Appl. Probab. 41(03), 832–858 (2004)

    Article  Google Scholar 

  28. Haque, L.: Tail Behaviour for Stationary Distributions for Two-Dimensional Stochastic Models, Ph.D. Thesis, Carleton University, Ottawa, ON, Canada (2003)

  29. Haque, L., Liu, L., Zhao, Y.Q.: Sufficient conditions for a geometric tail in a QBD process with countably many levels and phases. Stoch. Models 21(1), 77–99 (2005)

    Article  Google Scholar 

  30. He, Q., Li, H., Zhao, Y.Q.: Light-tailed behaviour in QBD process with countably many phases. Stoch. Models 25, 50–75 (2009)

    Article  Google Scholar 

  31. Khanchi, A.: State of a network when one node overloads. Ph.D. Thesis, University of Ottawa (2008)

  32. Khanchi, Aziz: Asymptotic hitting distribution for a reflected random walk in the positive quardrant. Stoch. Models 27, 169–201 (2009)

    Article  Google Scholar 

  33. Kingman, J.F.C.: Two similar queues in parallel. Ann. Math. Statist. 32, 1314–1323 (1961)

    Article  Google Scholar 

  34. Knuth, D.E.: The Art of Computer Programming Fundamental Algorithms, vol. 1, 2nd edn. Addison-Wesley (1969)

  35. Kobayashi, M., Miyazawa, M.: Revisiting the tail asymptotics of the double QBD process: refinement and complete solutions for the coordinate and diagonal directions. In: Latouche, G., Ramaswami, V., Sethuraman, J., Sigman, K., Squillante, M.S., Yao, D. (eds.) Matrix-Analytic Methods in Stochastic Models, pp. 145–185. Springer, Berlin (2013)

    Chapter  Google Scholar 

  36. Kobayashi, M., Miyazawa, M.: Tail asymptotics of the stationary distribution of a two dimensional reflecting random walk with unbounded upward jumps. Adv. Appl. Probab. 46(2), 365–399 (2014)

    Article  Google Scholar 

  37. Kobayashi, M., Miyazawa, M., Zhao, Y.Q.: Tail asymptotics of the occupation measure for a Markov additive process with an M/G/1-type background process. Stoch. Models 26, 463–486 (2010)

    Article  Google Scholar 

  38. Kroese, D.P., Scheinhardt, W.R.W., Taylor, P.G.: Spectral properties of the tandem Jackson network, seen as a quasi-birth-and-death process. Ann. Appl. Probab. 14(4), 2057–2089 (2004)

    Article  Google Scholar 

  39. Kurkova, I.A., Raschel, K.: Explicit expression for the generating function counting Gessel’s walks. Adv. Appl. Math. 47, 414–433 (2011)

  40. Kurkova, I., Raschel, K.: Passage time from four to two blocks of opinions in the voter model and walks in the quarter plane. Queueing Syst. 72, 219–234 (2013)

    Article  Google Scholar 

  41. Kurkova, I.A., Suhov, Y.M.: Malyshev’s theory and JS-queues. Asymptotics of stationary probabilities. Ann. Appl. Probab. 13, 1313–1354 (2003)

  42. Li, L., Miyazawa, M., Zhao, Y.: Geometric decay in a QBD process with countable background states with applications to a join-the-shortest-queue model. Stoch. Models 23, 413–438 (2007)

    Article  Google Scholar 

  43. Li, H., Song, Y., Zhao, Y.Q.: Exact tail asymptotics for the absorbing time—Revisit of the voter model, preprint (2014)

  44. Li, H., Tavakoli, J., Zhao, Y.Q.: Analysis of exact tail asymptotics for singular random walks in the quarter plane. Queueing Syst. 74, 151–179 (2013)

    Article  Google Scholar 

  45. Li, H., Zhao, Y.Q.: A retrial queue with a constant retrial rate, server break downs and impatient customers. Stoch. Models 21, 531–550 (2005)

    Article  Google Scholar 

  46. Li, H., Zhao, Y.Q.: Tail asymptotics for a generalized two demand queueing model—a kernel method. Queueing Syst. 69, 77–100 (2011)

    Article  Google Scholar 

  47. Li, H., Zhao, Y.Q.: A kernel method for exact tail asymptotics—random walks in the quarter plane. Queueing Models Serv. Manag. 1(1), 95–129 (2018)

    Google Scholar 

  48. Li, W., Liu, Y., Zhao, Y.Q.: Exact tail asymptotics for fluid models driven by an \(M/M/c\) queue. Queueing Syst. 91(3–4), 319–346 (2019)

    Article  Google Scholar 

  49. Lieshout, P., Mandjes, M.: Asymptotic analysis of Lévy-driven tandem queues. Queueing Syst. 60, 203–226 (2008)

    Article  Google Scholar 

  50. Liu, L., Miyazawa, M., Zhao, Y.Q.: Geometric decay in level-expanding QBD models. Ann. Oper. Res. 160, 83–98 (2008)

    Article  Google Scholar 

  51. Liu, Y., Wang, P., Zhao, Y.Q.: Exact stationary tail asymptotics for a Markov modulated two-demand model—in terms of a kernel method. In: Proceedings of the Ninth International Conference on Matrix-Analytic Methods in Stochastic Models, June 28–30, Budapest, Hungary, pp. 51–58 (2016)

  52. Malyshev, V.A.: An analytical method in the theory of two-dimensional positive random walks. Sib. Math. J. 13, 1314–1329 (1972)

    Google Scholar 

  53. Malyshev, V.A.: Asymptotic behaviour of stationary probabilities for two dimensional positive random walks. Sib. Math. J. 14, 156–169 (1973)

    Article  Google Scholar 

  54. McDonald, D.R.: Asymptotics of first passage times for random walk in an orthant. Ann. Appl. Probab. 9, 110–145 (1999)

    Article  Google Scholar 

  55. Miyazawa, M.: The Markov renewal approach to \(M/G/1\) type queues with countably many background states. Queueing Syst. 46, 177–196 (2004)

    Article  Google Scholar 

  56. Miyazawa, M.: Doubly QBD process and a solution to the tail decay rate problem. In: Proceedings of the Second Asia-Pacific Symposium on Queueing Theory and Network Applications, Kobe, Japan, pp. 33–42 (2007)

  57. Miyazawa, M.: Tail decay rates in double QBD processes and related reflected random walks. Math. OR 34, 547–575 (2009)

    Article  Google Scholar 

  58. Miyazawa, M.: Two sided DQBD process and solutions to the tail decay rate problem and their applications to the generalized join shortest queue. In: Yue, W., Takahashi, Y., Takaki, H. (eds.) Advances in Queueing Theory and Network Applications, pp. 3–33. Springer, New York (2009)

    Chapter  Google Scholar 

  59. Miyazawa, M.: Light tail asymptotics in multidimensional reflecting processes for queueing networks. TOP 19(2), 233–299 (2011)

    Article  Google Scholar 

  60. Miyazawa, M., Rolski, T.: Tail asymptotics for a Lévey-driven tandem queue with an intermediate input. Queueing Syst. 63, 323–353 (2009)

    Article  Google Scholar 

  61. Miyazawa, M., Zhao, Y.Q.: The stationary tail asymptotics in the \(GI/G/1\) type queue with countably many background states. Adv. Appl. Probab. 36(4), 1231–1251 (2004)

    Article  Google Scholar 

  62. Motyer, Allan J., Taylor, Peter G.: Decay rates for quasi-birth-and-death process with countably many phases and tri-diagonal block generators. Adv. Appl. Probab. 38, 522–544 (2006)

    Article  Google Scholar 

  63. Ozawa, T.: Asymptotics for the stationary distribution in a discrete-time two-dimensional quasi-birth-and-death process. Queueing Syst. 74, 109–149 (2013)

    Article  Google Scholar 

  64. Ozawa, T., Kobayashi, M.: Exact asymptotic formulae of the stationary distribution of a discrete-time two-dimensional QBD process. Queueing Syst. 90, 351–403 (2018)

    Article  Google Scholar 

  65. Raschel, K.: Green functions and Martin compactification for killed random walks related to SU(3). Electron. Commun. Probab. 15, 176–190 (2010)

    Article  Google Scholar 

  66. Song, Y., Liu, Z., Dai, H.: Exact tail asymptotics for a discrete-time preemptive priority queue. Acta Math. Appl. Sin., English Ser. 31(1), 43–58 (2015)

    Article  Google Scholar 

  67. Song, Y., Liu, Z., Zhao, Y.Q.: Exact tail asymptotics—revisit of a retrial queue with two input streams and two orbits. Ann. Oper. Res. 247(1), 97–120 (2016)

    Article  Google Scholar 

  68. Song, Y., Lu, H.: Exact tail asymptotics for the Israeli queue with retrials and non-persistent customers. Under Review (2021)

  69. Takahashi, Y., Fujimoto, K., Makimoto, N.: Geometric decay of the steady-state probabilities in a quasi-birth-and-death process with a countable number of phases. Stoch. Models 17(1), 1–24 (2001)

    Article  Google Scholar 

  70. Tang, J., Zhao, Y.Q.: Stationary tail asymptotics of a tandem queue with feedback. Ann. Oper. Res. 160, 173–189 (2008)

    Article  Google Scholar 

  71. van Houtum, G.J., Adan, I.J.B.F., Wessels, J., Zijm, W.H.M.: Performance analysis of parallel identical machines with a generalized shortest queue arrival mechanism. OR Spektrum 23, 411–428 (2001)

    Article  Google Scholar 

  72. Williams, R.J.: Semimartingale reflecting Brownian motions in the orthant. In: Stochastic Networks. IMA. Vol. Math. Appl., vol. 71, pp. 125–137. Springer, New York (1995)

    Chapter  Google Scholar 

  73. Williams, R.J.: On the approximation of queueing networks in heavy traffic. In: Kelly, F.P., Zachary, S., Ziedins, I. (eds.) Stochastic Networks: Theory and Applications. Oxford University Press (1996)

  74. Wright, P.: Two parallel processors with coupled inputs. Adv. Appl. Probab. 24, 986–1007 (1992)

    Article  Google Scholar 

  75. Ye, W.: Longer-Queue-Serve-First System. M.Sc. Thesis, School of Mathematics and Statistics, Carleton University. https://curve.carleton.ca/system/files/theses/29025.pdf (2012)

  76. Ye, W., Li, H., Zhao, Y.Q.: Tail behaviour for longest-queue-served-first queueing system—a random walk in the half plane. Stoch. Models 31(3), 452–493 (2015)

    Article  Google Scholar 

  77. Zafari, Z.: The Examt Tail Asymptotics Behaviour of the Joint Stationary Distributions of the Generalized-JSQ Model. Master’s Thesis, University of British Columbia (2012)

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Acknowledgements

The author thanks the anonymous reviewer and the guest editor for their constructive valuable comments and suggestions for the improvement of the presentation of this work, and acknowledges that this work was supported in part through a Discovery Grant of NSERC.

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  1. School of Mathematics and Statistics, Carleton University, Ottawa, ON, K1S 5B6, Canada

    Yiqiang Q. Zhao

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Zhao, Y.Q. The kernel method tail asymptotics analytic approach for stationary probabilities of two-dimensional queueing systems. Queueing Syst 100, 95–131 (2022). https://doi.org/10.1007/s11134-021-09727-6

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