Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-23T05:07:47.460Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on uniformization for dynamic non-negative systems

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Nico M. van Dijk  and
Karel Sladký
Nico M. van Dijk*
Affiliation:
University of Amsterdam
Karel Sladký*
Affiliation:
Institute of Information Theory and Automation, Prague
*
∗Postal address: Department of Economic Sciences and Econometrics, University of Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands. Email address: [email protected]
∗∗Postal address: Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, PO Box 18, 182 08 Prague 8, Czech Republic. Email address: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

The classical technique of uniformization (or randomization) for bounded continuous-time Markov chains and Markov reward structures is extended to dynamic systems generated by arbitrary non-negative generators. Most notably, these include so-called input-output models in economic analysis. The results are of practical interest for both computational and theoretical purposes. Particularly, the recursive computation and the limiting behaviour of cumulative reward structures for non-negative dynamic systems is concluded as a special application. Two numerical examples are included to illustrate the conditions and the results.

Keywords

Uniformizationstochastic generator matrixnon-negative generator matrixinput-output models

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 2 , June 2000 , pp. 329 - 341
Copyright
Copyright © by the Applied Probability Trust 2000 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anderson, W. J. (1991). Continuous-Time Markov Chains: An Applications-Oriented Approach. Springer, New York.Google Scholar
Berman, A., and Plemmons, R. J. (1979). Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York.Google Scholar
Bochner, S. (1955). Harmonic Analysis and the Theory of Probability. University of California Press, Berkeley.Google Scholar
de Souza e Silva, E., and Gail, H. R. (1986). Calculating cumulative operational time distributions of repairable computer systems. IEEE Trans. Computers 35, 322–332.Google Scholar
de Souza e Silva, E., and Gail, H. R. (1986). Calculating availability and performability measures of repairable computer systems using randomization. J. Assoc. Comput. Machinery 36, 171–193.CrossRefGoogle Scholar
van Dijk, N. M. (1991). Transient error bound analysis for continuous-time Markov reward structures. Perf. Evaluat. 13, 147–158.Google Scholar
van Dijk, N. M. (1992). Approximate uniformization for continuous-time Markov chains with an application to performability analysis. Stoch. Proc. Appl. 40, 339–357.CrossRefGoogle Scholar
van Dijk, N. M. and Sladký, K. (1999). Error bounds for nonnegative dynamic models. J. Optimizat. Th. Appl. 101, 449–474.Google Scholar
Dynkin, E. B. (1965). Markov Processes. Translation from the Russian original published by Fizmatgiz, Moscow, 1963.Google Scholar
Grassman, W. (1990). Finding transient solutions in Markovian event systems through randomization. In Numerical Solution of Markov Chains, ed. Steward, W. J. Marcel Dekker, New York, pp. 375–385.Google Scholar
Gross, D., and Miller, D. R. (1984). The randomization technique as a modelling tool and solution procedure for transient Markov processes. Operat. Res. 32, 343–361.Google Scholar
Jensen, A. (1953). Markovian chains as an aid in the study of Markov processes. Scand. Actuarial J. 36, 87–91.Google Scholar
Lewis, P. A. W., and Shedler, G. S. (1979). Simulation of nonhomogeneous Poisson processes by thinning. Naval Res. Logist. Quart. 26, 403–413.Google Scholar
Meiss, T., and Marcowitz, U. (1981). Numerical Solution of Partial Differential Equations. Springer, Berlin.Google Scholar
Melamed, B., and Yadin, M. (1984). Randomization procedures in the computation of cumulative-time distributions over discrete state Markov processes. Operat. Res. 32, 926–943.CrossRefGoogle Scholar
van Moorsel, A. P. A. (1993). Performability evaluation concepts and techniques. Ph.D. Thesis, University of Twente, Enschede, The Netherlands.Google Scholar
van Moorsel, A. P. A., and Sanders, W. H. (1992). Adaptive uniformization. Memoranda Informatica 92–88, Computer Science Department, University of Twente, Enschede, The Netherlands.Google Scholar
van Moorsel, A. P. A., and Haverkort, B. R. (1993). A unified performability evaluation framework for computer and communication systems. Proc. 2nd Int. Workshop on Performability Modelling of Computer and Communication Systems, Le Mont Saint-Michel, France, 1993.Google Scholar
Reibman, A. L., and Trivedi, K. S. (1988). Numerical transient analysis of Markov modes. Comp. Operat. Res. 15, 19–36.Google Scholar
Reibman, A. L., and Trivedi, K. S. (1989). Transient analysis of cumulative measures of Markov model behavior. Commun. Statist. Stoch. Models 5, 683–710.CrossRefGoogle Scholar
Reibman, A. L., Smith, R. M., and Trivedi, K. S. (1989). Markov and Markov reward model transient analysis: an overview of numerical approaches. Eur. J. Operat. Res. 40, 257–267.CrossRefGoogle Scholar
Ross, S. M. (1986). Approximating transition probabilities and mean occupation times in continuous-time Markov chains. Prob. Eng. Inf. Sci. 1, 251–264.Google Scholar
Trivedi, K. S., Reibman, A. L., and Smith, R. M. (1988). Transient analysis of Markov and Markov reward models. In Computer Performance and Reliability, eds Iazeolla, G., Courtois, P. J. and Boxma, O. J. Elsevier, New York, pp. 535–545.Google Scholar