Tail Decay Rates in Double QBD Processes and Related Reflected Random Walks

Published Online:https://doi.org/10.1287/moor.1090.0375

References

  • Adan I. J. B. F., Wessels J., Zijm W. H. M. A compensation approach for two-dimensional Markov processes. Adv. Appl. Probab. (1993) 25(4):783–817CrossrefGoogle Scholar
  • Borovkov A. A. The second rate function and the asymptotic problems of renewal and hitting the boundary for multidimensional random walk. Siberian Math. J. (1996) 37:647–682CrossrefGoogle Scholar
  • Borovkov A. A.Ergodicity and Stability of Stochastic Processes (1998) (John Wiley & Sons, Chichester, UK) . Translated by V. YurinskyGoogle Scholar
  • Borovkov A. A., Mogul'skii A. A. The second rate function and the asymptotic problems of renewal and hitting the boundary for multidimensional random walks. Siberian Math. J. (1996) 37:647–682CrossrefGoogle Scholar
  • Borovkov A. A., Mogul'skii A. A. Large deviations for Markov chains in the positive quadrant. Russian Math. Surveys (2001) 56:803–916CrossrefGoogle Scholar
  • Fayolle G., Malyshev V. A., Menshikov M. V.Topics in the Constructive Theory of Countable Markov Chains (1995) (Cambridge University Press, Cambridge, UK) CrossrefGoogle Scholar
  • Foley R. D., McDonald D. R. Join the shortest queue: Stability and exact asymptotics. Ann. Appl. Probab. (2001) 11:569–607CrossrefGoogle Scholar
  • Foley R. D., McDonald D. R. Large deviations of a modified Jackson network: Stability and rough asymptotics. Ann. Appl. Probab. (2005) 15:519–541CrossrefGoogle Scholar
  • Foley R. D., McDonald D. R. Bridges and networks: Exact asymptotics. Ann. Appl. Probab. (2005) 15:542–586CrossrefGoogle Scholar
  • Foley R. D., McDonald D. R. Personal communication. (2008) February 20Google Scholar
  • Fujimoto K., Takahashi Y., Makimoto N. Asymptotic properties of stationary distributions in two-stage tandem queueing systems. J. Oper. Res. Soc. Japan (1998) 41:118–141Google Scholar
  • Haque L., Liu L., Zhao Y. Q. Sufficient conditions for a geometric tail in a QBD process with countably many levels and phases. Stochastic Models (2005) 21:77–99CrossrefGoogle Scholar
  • He Q., Li H., Zhao Y. Q. Light-tailed behavior in QBD process with countably many phases. Stochastic Models (2009) 25(1):50–75CrossrefGoogle Scholar
  • Ignatyuk I. A., Malyshev V. A., Scherbakov V. V. Boundary effects in a large deviation problems. Russian Math. Survey (1994) 49:41–99CrossrefGoogle Scholar
  • Ignatiouk-Robert I. Sample path large deviations and convergence parameters. Ann. Appl. Probab. (2001) 11:1292–1329CrossrefGoogle Scholar
  • Katou K., Makimoto N., Takahashi Y. Upper bound for the decay rate of the marginal queue-length distribution in a two-node Markovian queueing system. J. Oper. Res. Soc. Japan (2004) 47:314–338Google Scholar
  • Kroese D. P., Scheinhardt W. C. W., Taylor P. G. Spectral properties of the Tandem Jackson network, seen as a quasi-birth-and-death process. Ann. Appl. Probab. (2004) 14:2057–2089CrossrefGoogle Scholar
  • Latouche G., Ramaswami V.Introduction to Matrix Analytic Methods in Stochastic Modeling (1999) (American Statistical Association and the Society for Industrial and Applied Mathematics, Philadelphia) CrossrefGoogle Scholar
  • Leemans H. Probable bounds for the mean queue lengths in a heterogeneous priority queue. Queueing Systems (2000) 36:269–286CrossrefGoogle Scholar
  • Li H., Miyazawa M., Zhao Y. Q. Geometric decay in a QBD process with countable background states with applications to shortest queues. Stochastic Models (2007) 23:413–438CrossrefGoogle Scholar
  • Malyshev V. A. Asymptotic behavior of the stationary probabilities for two-dimensional positive random walks. Siberian Math. J. (1973) 14:109–118CrossrefGoogle Scholar
  • Majewski K. Large deviations of the steady-state distribution of reflected processes with applications to queueing systems. Queueing Systems (1998) 29:351–381CrossrefGoogle Scholar
  • Miyazawa M. Conjectures on decay rates of tail probabilities in generalized Jackson and batch movement networks. J. Oper. Res. Soc. Japan (2003) 46:74–98Google Scholar
  • Miyazawa M. Doubly QBD process and a solution to the tail decay rate problem. Proc. Second Asia-Pacific Sympos. Queueing Theory and Network Appl. (2007) Kobe, Japan:33–42Google Scholar
  • Miyazawa M., Takahashi Y., Takagi H. Two sided DQBD process and solutions to the tail decay rate problem and their applications to the generalized join shortest queue. Advances in Queueing Theory and Network Applications (2009) (Springer, New York) 3–33CrossrefGoogle Scholar
  • Miyazawa M., Zhao Y. Q. The stationary tail asymptotics in the GI/G/1 type queue with countably many background states. Adv. Appl. Probab. (2004) 36:1231–1251CrossrefGoogle Scholar
  • Motyer A. J., Taylor P. G. Decay rates for quasi-birth-and-death processes with countably many phases and tridiagonal block generators. Adv. Appl. Probab. (2006) 38:522–544CrossrefGoogle Scholar
  • Neuts M. F.Matrix-Geometric Solutions in Stochastic Models (1981) (Johns Hopkins University Press, Baltimore) Google Scholar
  • Ramaswami V., Taylor P. G. Some properties of the rate operators in level dependent quasi-birth-and-death processes with a countable number of phases. Stochastic Models (1996) 12:143–164CrossrefGoogle Scholar
  • Sakuma Y., Miyazawa M. On the effect of finite buffer truncation in a two-node Jackson network. J. Appl. Probab. (2005) 42(1):199–222CrossrefGoogle Scholar
  • Seneta E.Non-Negative Matrices and Markov Chains (1981) 2nd ed.(Springer-Verlag, New York) CrossrefGoogle Scholar
  • Shwartz A., Weiss A.Large Deviations for Performance Analysis, Queues, Communications, and Computing (1995) (Chapman & Hall, London) Google Scholar
  • 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. Stochastic Models (2001) 17:1–24CrossrefGoogle Scholar
  • Tweedie R. L. Operator-geometric stationary distributions for Markov chains with applications to queueing models. Adv. Appl. Probab. (1982) 14:368–391CrossrefGoogle Scholar
INFORMS site uses cookies to store information on your computer. Some are essential to make our site work; Others help us improve the user experience. By using this site, you consent to the placement of these cookies. Please read our Privacy Statement to learn more.