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Discrete-time singularly perturbed Markov chains: aggregation, occupation measures, and switching diffusion limit

Part of: Probability theory on algebraic and topological structures Limit theorems Asymptotic theory Markov processes

Published online by Cambridge University Press:  22 February 2016

G. Yin ,
Q. Zhang  and
G. Badowski
G. Yin*
Affiliation:
Wayne State University
Q. Zhang*
Affiliation:
University of Georgia
G. Badowski*
Affiliation:
Wayne State University
*
∗ Postal address: Department of Mathematics, Wayne State University, Detroit, MI 48202, USA. Email address: gyin@math.wayne.edu
∗∗ Postal address: Department of Mathematics, University of Georgia, Athens, GA 30602, USA.
∗∗∗ Current address: Institute for Systems Research, University of Maryland, College Park, MD 20742, USA.
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Abstract

This work is devoted to asymptotic properties of singularly perturbed Markov chains in discrete time. The motivation stems from applications in discrete-time control and optimization problems, manufacturing and production planning, stochastic networks, and communication systems, in which finite-state Markov chains are used to model large-scale and complex systems. To reduce the complexity of the underlying system, the states in each recurrent class are aggregated into a single state. Although the aggregated process may not be Markovian, its continuous-time interpolation converges to a continuous-time Markov chain whose generator is a function determined by the invariant measures of the recurrent states. Sequences of occupation measures are defined. A mean square estimate on a sequence of unscaled occupation measures is obtained. Furthermore, it is proved that a suitably scaled sequence of occupation measures converges to a switching diffusion.

Keywords

Markov chaindiscrete timesingular perturbationoccupation measureswitching diffusionaggregationweak convergence

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60B10: Convergence of probability measures 60F17: Functional limit theorems; invariance principles 34E05: Asymptotic expansions
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 2 , June 2003 , pp. 449 - 476
Copyright
Copyright © Applied Probability Trust 2003 

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Footnotes

*

Supported in part by the National Science Foundation under grants DMS-9877090.

**

Supported in part by the USAF Grant F30602-99-2-0548 and ONR Grant N00014-96-1-0263.

***

Supported in part by Wayne State University.

References

[1] Abbad, M., Filar, J. A. and Bielecki, T. R. (1992). Algorithms for singularly perturbed limiting average Markov control problems. IEEE Trans. Automatic Control 37, 1421–1425.Google Scholar
[2] Bielecki, T. R. and Stettner, L. (1998). Ergodic control of a singularly perturbed Markov process in discrete time with general state and compact action spaces. Appl. Math. Optimization 38, 261–281.Google Scholar
[3] Blankenship, G. (1981). Singularly perturbed difference equations in optimal control problems. IEEE Trans. Automatic Control 26, 911–917.Google Scholar
[4] Courtois, P. J. (1977). Decomposability: Queueing and Computer System Applications. Academic Press, New York.Google Scholar
[5] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. John Wiley, New York.Google Scholar
[6] Fleming, W. H. and Rishel, R. W. (1975). Deterministic and Stochastic Optimal Control. Springer, New York.Google Scholar
[7] Hoppensteadt, F. C. and Miranker, W. L. (1977). Multitime methods for systems of difference equations. Studies Appl. Math. 56, 273–289.Google Scholar
[8] Iosifescu, M. (1980). Finite Markov Processes and Their Applications. John Wiley, Chichester.Google Scholar
[9] Khasminskii, R. Z. (1966). On stochastic processes defined by differential equations with a small parameter. Theory Prob. Appl. 11, 211–228.Google Scholar
[10] Khasminskii, R. Z., Yin, G. and Zhang, Q. (1996). Asymptotic expansions of singularly perturbed systems involving rapidly fluctuating Markov chains. SIAM J. Appl. Math. 56, 277–293.Google Scholar
[11] Khasminskii, R. Z., Yin, G. and Zhang, Q. (1997). Constructing asymptotic series for probability distributions of Markov chains with weak and strong interactions. Quart. Appl. Math. 55, 177–200.Google Scholar
[12] Kushner, H. J. (1984). Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory. MIT Press, Cambridge, MA.Google Scholar
[13] Kushner, H. J. and Huang, H. (1980). Averaging methods for the asymptotic analysis of learning and adaptive systems with small adjustment rate. SIAM J. Control Optimization 19, 635–650.Google Scholar
[14] Liu, R. H., Zhang, Q. and Yin, G. (2002). Asymptotically optimal controls of hybrid linear quadratic regulators in discrete time. Automatica 38, 409–419.Google Scholar
[15] Naidu, D. S. (1988). Singular Perturbation Methodology in Control Systems. Peter Peregrinus, Stevenage.CrossRefGoogle Scholar
[16] Pan, Z. G. and Başar, T. (1995). H ∞-control of Markovian jump linear systems and solutions to associated piecewise-deterministic differential games. In New Trends in Dynamic Games and Applications, ed. Olsder, G. J., Birkhäuser, Boston, MA, pp. 61–94.Google Scholar
[17] Pervozvanskii, A. A. and Gaitsgori, V. G. (1988). Theory of Suboptimal Decisions: Decomposition and Aggregation. Kluwer, Dordrecht.Google Scholar
[18] Phillips, R. G. and Kokotovic, P. V. (1981). A singular perturbation approach to modelling and control of Markov chains. IEEE Trans. Automatic Control 26, 1087–1094.Google Scholar
[19] Sethi, S. P. and Zhang, Q. (1994). Hierarchical Decision Making in Stochastic Manufacturing Systems. Birkhäuser, Boston.Google Scholar
[20] Simon, H. A. and Ando, A. (1961). Aggregation of variables in dynamic systems. Econometrica 29, 111–138.Google Scholar
[21] Tse, D. N. C., Gallager, R. G. and Tsitsiklis, J. N. (1995). Statistical multiplexing of multiple time-scale Markov streams. IEEE J. Selected Areas Commun. 13, 1028–1038.Google Scholar
[22] Yin, G. and Zhang, Q. (1998). Continuous-time Markov Chains and Applications: A Singular Perturbation Approach. Springer, New York.Google Scholar
[23] Yin, G. and Zhang, Q. (2000). Singularly perturbed discrete-time Markov chains. SIAM J. Appl. Math. 61, 834–854.Google Scholar
[24] Yin, G., Zhang, Q. and Badowski, G. (2000). Asymptotic properties of a singularly perturbed Markov chain with inclusion of transient states. Ann. Appl. Prob. 10, 549–572.Google Scholar
[25] Zhang, Q. and Yin, G. (1999). On nearly optimal controls of hybrid LQG problems. IEEE Trans. Automatic Control 44, 2271–2282.Google Scholar