Wiener-Hopf Factorizations for a Multidimensional Markov Additive Process and their Applications to Reflected Processes

Published Online:https://doi.org/10.1287/12-SSY069

References

  • Alsmeyer, G. (1994). On the Markov renewal theorem. Stochastic Processes and their Applications 50 37–56. MR1262329Google Scholar
  • Arjas, E. and Speed, T. P. (1973). Symmetric Wiener-Hopf factorisations in Markov additive processes. Probability Theory and Related Fields 26 105–118. MR0331515Google Scholar
  • Asmussen, S. (2003). Applied probability and queues, second ed. Applications of Mathematics (New York) 51. Springer-Verlag, New York. Stochastic Modelling and Applied Probability. MR1978607 (2004f:60001)Google Scholar
  • Billingsley, P. (2000). Probability and Measure, 2nd ed. Wiley.Google Scholar
  • Borovkov, A. A. and Mogul’skiĭ, A. A. (2001). Large deviations for Markov chains in the positive quadrant. Russian Mathematical Surveys 56 803–916. MR1892559 (2002m:60045)Google Scholar
  • Collamore, J. (1996). Hitting probabilities and large deviations. The Annals of Probability 24 2065–2078. MR1415241Google Scholar
  • Dai, J. G. and Miyazawa, M. (2011). Reflecting Brownian motion in two dimensions: Exact asymptotics for the stationary distribution. Stochastic Systems 1 146–208.AbstractGoogle Scholar
  • Dinges, H. (1969). Wiener-Hopf Faktorisierung für substochastische Übergangs-funktionen in angeordneten Räumen. Z. Wahrsch. Verw. Gebiete 11 152–164. MR0248907Google Scholar
  • Dupuis, P. and Ellis, R. S. (1997). A weak convergence approach to the theory of large deviations. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons Inc., New York. A Wiley-Interscience Publication. MR1431744 (99f:60057)Google Scholar
  • Fayolle, G., Malyshev, V. A. and Men’shikov, M. V. (1995). Topics in the constructive theory of countable Markov chains. Cambridge University Press, Cambridge. MR1331145 (96k:60174)Google Scholar
  • Foley, R. D. and McDonald, D. R. (2001). Join the shortest queue: stability and exact asymptotics. Ann. Appl. Probab. 11 569–607. MR1865017Google Scholar
  • Glynn, P. and Whitt, W. (1994). Logarithmic Asymptotics for Steady-State Tail Probabilities in a Single-Server Queue. J. Appl. Probability 31 131–156. MR1274722Google Scholar
  • Grassmann, W. K. and Heyman, D. P. (1990). Equilibrium distribution of blocked-structured Markov chains with repeating rows. J. Appl. Probability 27 557–576. MR1067022Google Scholar
  • Kaspi, H. (1982). On the symmetric Wiener-Hopf factorization for Markov additive processe. Z. Wahrsch. Verw. Gebiete 59 179–196. MR0650610Google Scholar
  • Kim, B. and Sohraby, K. (2006). Tail behavior of the queue size and waiting time in a queue with discrete autoregressive arrivals. Advances in Applied Probability 38 1116–1131. MR2285696Google Scholar
  • Kobayashi, M. and Miyazawa, M. (2011). Tail asymptotics of the stationary distribution of a two dimensional reflecting random walk with unbounded upward jumps. Preprint.Google Scholar
  • Loynes, R. M. (1962). The stability of a queue with non-independent inter-arrival and service times. Proceedings of Cambridge Philosophical Society 58 497–520. MR0141170Google Scholar
  • Majewski, K. (2004). Large Deviation Bounds for Single Class Queueing Networks and Their Calculation. Queueing Syst. Theory Appl. 48 103–134. MR2097523Google Scholar
  • Markushevich, A. I. (1977). Theory of functions of a complex variable. Vol. I, II, III, English ed. Chelsea Publishing Co., New York. Translated and edited by Richard A. Silverman. MR0444912 (56 ##3258)Google Scholar
  • Miyazawa, M. (2004). A Markov Renewal Approach to M/G/1 Type Queues with Countably Many Background States. Queueing Systems 46 177–196. MR2072282Google Scholar
  • Miyazawa, M. (2009). Tail decay rates in double QBD processes and related reflected random walks. Math. Oper. Res. 34 547–575. MR2555336LinkGoogle Scholar
  • Miyazawa, M. (2011). Light tail asymptotics in multidimensional reflecting processes for queueing networks. TOP, an official journal of the Spanish Society of Statistics and Operations Research 19 233–299. MR2859501Google Scholar
  • Miyazawa, M. and Rolski, T. (2009). Tail asymptotics for a Lévy-driven tandem queue with an intermediate input. Queueing Syst. 63 323–353. MR2576017Google Scholar
  • Miyazawa, M. and Zhao, Y. Q. (2004). The stationary tail asymptotics in the GI/G/1-type queue with countably many background states. Adv. in Appl. Probab. 36 1231–1251. MR2119862 (2005h:60280)Google Scholar
  • Neuts, M. F. (1981). Matrix-geometric solutions in stochastic models: an algorithm approach. The John Hopkins University Press, Baltimore, MD. MR0618123Google Scholar
  • Ney, P. and Nummelin, E. (1987). Markov additive processes I. Eigenvalue properties and limit theorems. Annals of Probability 15 561–592. MR0885131Google Scholar
  • Ney, P. and Nummelin, E. (1987). Markov additive processes II. Large deviations. Annals of Probability 15 593–609. MR0885132Google Scholar
  • Nummelin, E. (1984). General irreducible Markov chains and non-negative operators. Cambridge University Press. MR0776608Google Scholar
  • Pitman, J. W. (1974). An identity for stopping times of a Markov process. In Studies in Probability and Statistics (E. J. Williams, ed.) 41–57. Jerusalem Academic Press. MR0431384Google Scholar
  • Rockafellar, R. T. (1970). Convex analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton, N.J. MR0274683 (43 ##445)Google Scholar
  • Shwartz, A. and Weiss, A. (1995). Large deviations for performance analysis. Chapman & Hall, New York. MR1335456Google Scholar
  • Tweedie, R. L. (1982). Operator-geometric stationary distributions for Markov chains, with application to queueing models. Advances in Applied Probability 14 368–391. MR0650129Google Scholar
  • Zhao, Y. Q., Li, W. and Braun, W. J. (2003). Censoring, Factorizations, and Spectral Analysis for Transition Matrices with Block-Repeating Entries. Methodology and Computing in Applied Probability 5 35–58. MR1997776Google Scholar
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