Tail asymptotics for a generalized two-demand queueing model—a kernel method

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).

[1]  V. A. Malyshev.  An analytical method in the theory of two-dimensional positive random walks , 1972 .

[2]  Paul E. Wright,et al.  Two parallel processors with coupled inputs , 1992, Advances in Applied Probability.

[3]  Philippe Flajolet,et al.  Singularity Analysis of Generating Functions , 1990, SIAM J. Discret. Math..

[4]  Ward Whitt,et al.  Asymptotics for M/G/1 low-priority waiting-time tail probabilities , 1997, Queueing Syst. Theory Appl..

[5]  Masakiyo Miyazawa,et al.  Tail asymptotics for a Lévy-driven tandem queue with an intermediate input , 2009, Queueing Syst. Theory Appl..

[6]  E. Bender.  Asymptotic Methods in Enumeration , 1974 .

[7]  Donald E. Knuth,et al.  The Art of Computer Programming, Volume I: Fundamental Algorithms, 2nd Edition , 1997 .

[8]  David Thomas,et al.  The Art in Computer Programming , 2001 .

[9]  V. A. Malyshev.  Asymptotic behavior of the stationary probabilities for two-dimensional positive random walks , 1973 .

[10]  G. Fayolle,et al.  The solution of certain two-dimensional markov models , 1980, PERFORMANCE '80.

[11]  L. Flatto,et al.  Two parallel queues created by arrivals with two demands. II , 1984 .

[12]  Michel Mandjes,et al.  Asymptotic analysis of Lévy-driven tandem queues , 2008, Queueing Syst. Theory Appl..

[13]  Kilian Raschel,et al.  Random walks in the quarter plane absorbed at the boundary: exact and asymptotic , 2009, 0902.2785.

[14]  J. Hunter.  Two Queues in Parallel , 1969 .

[15]  Mireille Bousquet-Mélou,et al.  Generating functions for generating trees , 2002, Discret. Math..

[16]  Mireille Bousquet-M'elou,et al.  Walks in the quarter plane: Kreweras’ algebraic model , 2004, math/0401067.

[17]  G. Fayolle,et al.  Random Walks in the Quarter-Plane: Algebraic Methods, Boundary Value Problems and Applications , 1999 .

[18]  P. Flajolet,et al.  Analytic Combinatorics: RANDOM STRUCTURES , 2009 .

[19]  L. Flatto,et al.  Erratum: Two Parallel Queues Created by Arrivals with Two Demands I , 1985 .

[20]  Fabrice Guillemin,et al.  Rare event asymptotics for a random walk in the quarter plane , 2011, Queueing Syst. Theory Appl..

[21]  Hui Li,et al.  Exact tail asymptotics in a priority queue—characterizations of the preemptive model , 2009, Queueing Syst. Theory Appl..

[22]  John A. Morrison.  Processor sharing for two queues with vastly different rates , 2007, Queueing Syst. Theory Appl..

[23]  Hui Li,et al.  Exact tail asymptotics in a priority queue—characterizations of the non-preemptive model , 2011, Queueing Syst. Theory Appl..

[24]  G. Fayolle,et al.  Two coupled processors: The reduction to a Riemann-Hilbert problem , 1979 .

[25]  Masakiyo Miyazawa,et al.  Tail Decay Rates in Double QBD Processes and Related Reflected Random Walks , 2009, Math. Oper. Res..

[26]  Onno Boxma,et al.  Boundary value problems in queueing system analysis , 1983 .

[27]  I. Kurkova,et al.  Malyshev's theory and JS-queues. Asymptotics of stationary probabilities , 2003 .

[28]  Philippe Flajolet,et al.  Analytic Combinatorics , 2009 .