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Error Bounds for Augmented Truncations of Discrete-Time Block-Monotone Markov Chains under Geometric Drift Conditions

Part of: Markov processes Special processes

Published online by Cambridge University Press:  04 January 2016

Hiroyuki Masuyama
Hiroyuki Masuyama*
Affiliation:
Kyoto University
*
∗ Postal address: Department of Systems Science, Graduate School of Informatics, Kyoto University, Kyoto, 606-8501, Japan. Email address: [email protected]
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Abstract

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In this paper we study the augmented truncation of discrete-time block-monotone Markov chains under geometric drift conditions. We first present a bound for the total variation distance between the stationary distributions of an original Markov chain and its augmented truncation. We also obtain such error bounds for more general cases, where an original Markov chain itself is not necessarily block monotone but is blockwise dominated by a block-monotone Markov chain. Finally, we discuss the application of our results to GI/G/1-type Markov chains.

Keywords

Augmented truncationblock monotonicityblockwise dominancepathwise orderinggeometric drift conditionlevel-dependent QBDM/G/1-type Markov chainGI/M/1-type Markov chainGI/G/1-type Markov chain

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60K25: Queueing theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 1 , March 2015 , pp. 83 - 105
Copyright
© Applied Probability Trust 

References

Andrew, A. L., Chu, K.-W. E. and Lancaster, P. (1993). Derivatives of eigenvalues and eigenvectors of matrix functions. SIAM J. Matrix Anal. Appl. 14, 903–926.Google Scholar
Baumann, H. and Sandmann, W. (2010). Numerical solution of level dependent quasi-birth-and-death processes. Procedia Comput. Sci. 1, 1561–1569.Google Scholar
Brémaud, P. (1999). Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues. Springer, New York.Google Scholar
Bright, L. and Taylor, P. G. (1995). Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes. Commun. Statist. Stoch. Models 11, 497–525.CrossRefGoogle Scholar
Chung, K. L. (1967). Markov Chains with Stationary Transition Probabilities, 2nd edn. Springer, New York.Google Scholar
Daley, D. J. (1968). Stochastically monotone Markov chains. Z. Wahrscheinlichkeitsth. 10, 305–317.CrossRefGoogle Scholar
Gibson, D. and Seneta, E. (1987). Monotone infinite stochastic matrices and their augmented truncations. Stoch. Process. Appl. 24, 287–292.Google Scholar
He, Q.-M. (2014). Fundamentals of Matrix-Analytic Methods. Springer, New York.Google Scholar
Horn, R. A. and Johnson, C. R. (1990). Matrix Analysis. Cambridge University Press.Google Scholar
Keilson, J. and Kester, A. (1977). Monotone matrices and monotone Markov processes. Stoch. Process. Appl. 5, 231–241.CrossRefGoogle Scholar
Kimura, T., Masuyama, H. and Takahashi, Y. (2013). Subexponential asymptotics of the stationary distributions of GI/G/1-type Markov chains. Stoch. Models 29, 190–239.CrossRefGoogle Scholar
Li, H. and Zhao, Y. Q. (2000). Stochastic block-monotonicity in the approximation of the stationary distribution of infinite Markov chains. Commun. Statist. Stoch. Models 16, 313–333.CrossRefGoogle Scholar
Liu, Y. (2010). Augmented truncation approximations of discrete-time Markov chains. Operat. Res. Lett. 38, 218–222.CrossRefGoogle Scholar
Lucantoni, D. M. (1991). New results on the single server queue with a batch Markovian arrival process. Commun. Statist. Stoch. Models. 7, 1–46.Google Scholar
Lund, R. B., Meyn, S. P. and Tweedie, R. L. (1996). Computable exponential convergence rates for stochastically ordered Markov processes. Ann. Appl. Prob. 6, 218–237.Google Scholar
Masuyama, H. and Takine, T. (2005). Algorithmic computation of the time-dependent solution of structured Markov chains and its application to queues. Stoch. Models 21, 885–912.CrossRefGoogle Scholar
Meyn, S. and Tweedie, R. L. (2009). Markov Chains and Stochastic Stability, 2nd edn. Cambridge University Press.Google Scholar
Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester.Google Scholar
Phung-Duc, T., Masuyama, H., Kasahara, S. and Takahashi, Y. (2010). A simple algorithm for the rate matrices of level-dependent QBD processes. In Proc. 5th Internat. Conf. Queueing Theory Network Appl., ACM, New York, pp. 46–52.Google Scholar
Takine, T. (2000). A new recursion for the queue length distribution in the stationary BMAP/G/1 queue. Commun. Statist. Stoch. Models 16, 335–341.Google Scholar
Takine, T., Matsumoto, Y., Suda, T. and Hasegawa, T. (1994). Mean waiting times in nonpreemptive priority queues with Markovian arrival and i.i.d. service processes. Performance Evaluation 20, 131–149.Google Scholar
Tijms, H. C. (2003). A First Course in Stochastic Models. John Wiley, Chichester.CrossRefGoogle Scholar
Tweedie, R. L. (1998). Truncation approximations of invariant measures for Markov chains. J. Appl. Prob. 35, 517–536.CrossRefGoogle Scholar
Zhao, Y. Q., Li, W. and Braun, W. J. (1998). Infinite block-structured transition matrices and their properties. Adv. Appl. Prob. 30, 365–384.Google Scholar