Abstract
In this paper, we consider a generalized two-demand queueing model, the same model studied in Wright (Adv. Appl. Prob., 24, 986–1007, 1992). Using this model, we show how the kernel method can be applied to a two-dimensional queueing system for exact tail asymptotics in the stationary joint distribution and also in the two marginal distributions. We demonstrate in detail how to locate the dominant singularity and how to determine the detailed behavior of the unknown generating function around the dominant singularity for a bivariate kernel, which is much more challenging than the analysis for a one-dimensional kernel. This information is the key for characterizing exact tail asymptotics in terms of asymptotic analysis theory. This approach does not require a determination or presentation of the unknown generating function(s).
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Abate, J., Whitt, W.: Asymptotics for M/G/1 low-priority waiting-time tail probabilities. Queueing Syst. 25, 173–233 (1997)
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)
Bender, E.: Asymptotic methods in enumeration. SIAM Rev. 16, 485–513 (1974)
Bousquet-Mélou, M.: Walks in the quarter plane: Kreweras’ algebraic model. Ann. Appl. Probab. 15, 1451–1491 (2005)
Cohen, J.W., Boxma, O.J.: Boundary Value Problems in Queueing System Analysis. North-Holland, Amsterdam (1983)
Fayolle, G., Iasnogorodski, R.: Two coupled processors: the reduction to a Riemann-Hilbert problem. Z. Wahrscheinlichkeitstheor. Verw. Geb. 47, 325–351 (1979)
Fayolle, G., King, P.J.B., Mitrani, I.: The solution of certain two-dimensional Markov models. Adv. Appl. Probab. 14, 295–308 (1982)
Fayolle, G., Iasnogorodski, R., Malyshev, V.: Random Walks in the Quarter-Plane. Springer, New York (1991)
Flajolet, P., Odlyzko, A.: Singularity analysis of generating functions. SIAM J. Discrete Math. 3, 216–240 (1990)
Flajolet, F., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)
Flatto, L., McKean, H.P.: Two queues in parallel. Commun. Pure Appl. Math. 30, 255–263 (1977)
Flatto, L., Hahn, S.: Two parallel queues created by arrivals with two demands I. SIAM J. Appl. Math. 44, 1041–1053 (1984)
Flatto, L.: Two parallel queues created by arrivals with two demands II. SIAM J. Appl. Math. 45, 861–878 (1985)
Guillemin, F., van Leeuwaarden, J.: Rare event asymptotics for a random walk in the quarter plane. Queueing Syst. 67, 1–32 (2011)
Knuth, D.E.: The Art of Computer Programming, Fundamental Algorithms, vol. 1, 2nd edn.. Addison–Wesley, Reading (1969)
Kurkova, I.A., Suhov, Y.M.: Malyshev’s theory and JS-queues. Asymptotics of stationary probabilities. Ann. Appl. Probab. 13, 1313–1354 (2003)
Li, H., Zhao, Y.Q.: Exact tail asymptotics in a priority queue—characterizations of the preemptive model. Queueing Syst. 63, 355–381 (2009)
Li, H., Zhao, Y.Q.: Exact tail asymptotics in a priority queue—characterizations of the non-preemptive model, accepted by Queueing Systems. (2010)
Li, H., Zhao, Y.Q.: A kernel method for exact tail asymptotics—random walks in the quarter plane, submitted (2010)
Lieshout, P., Mandjes, M.: Asymptotic analysis of Lévy-driven tandem queues. Queueing Syst. 60, 203–226 (2008)
Malyshev, V.A.: An analytical method in the theory of two-dimensional positive random walks. Sib. Math. J. 13, 1314–1329 (1972)
Malyshev, V.A.: Asymptotic behaviour of stationary probabilities for two dimensional positive random walks. Sib. Math. J. 14, 156–169 (1973)
Miyazawa, M.: Tail decay rates in double QBD processes and related reflected random walks. Math. Methods Oper. Res. 34, 547–575 (2009)
Miyazawa, M., Rolski, T.: Tail asymptotics for a Lévey-driven tandem queue with an intermediate input. Queueing Syst. 63, 323–353 (2009)
Morrison, J.A.: Processor sharing for two queues with vastly different rates. Queueing Syst. 57, 19–28 (2007)
Wright, P.: Two parallel processors with coupled inputs. Adv. Appl. Probab. 24, 986–1007 (1992)
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Li, H., Zhao, Y.Q. Tail asymptotics for a generalized two-demand queueing model—a kernel method. Queueing Syst 69, 77–100 (2011). https://doi.org/10.1007/s11134-011-9227-0
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DOI: https://doi.org/10.1007/s11134-011-9227-0
Keywords
- Generalized two-demand queueing model
- Generating functions
- Stationary probabilities
- Kernel method
- Asymptotic analysis
- Dominant singularity
- Exact tail asymptotics
- Random walks in the quarter plane