Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-24T22:54:40.074Z Has data issue: false hasContentIssue false
  • English
  • Français

Periodic strong ergodicity in non-homogeneous Markov systems

Published online by Cambridge University Press:  14 July 2016

Ioannis I. Gerontidis
Ioannis I. Gerontidis*
Affiliation:
University of Thessaloniki
*
∗Postal address: Mathematics Department, University of Thessaloniki, 54006 Thessaloniki, Greece.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper presents a unified treatment of the convergence properties of nonhomogeneous Markov systems under different sets of assumptions. First the periodic case is studied and the limiting evolution of the individual cyclically moving subclasses of the state space of the associated Markov replacement chain is completely determined. A special case of the above result is the aperiodic or strongly ergodic convergence. Two numerical examples from the literature on manpower planning highlight the practical aspect of the theoretical results.

Keywords

MANPOWER PLANNINGNON-STATIONARY MARKOV CHAINSREPLACEMENT PROCESSSTOCHASTIC POPULATION MODELS
Type
Research Papers
Information
Journal of Applied Probability , Volume 28 , Issue 1 , March 1991 , pp. 58 - 73
Copyright
Copyright © Applied Probability Trust 1991 

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

Bartholomew, D. J. (1971) The statistical approach to manpower planning. Statistician 20, 3–26.Google Scholar
Bartholomew, D. J. (1973) Stochastic Models for Social Processes, 2nd edn. Wiley, New York.Google Scholar
Bartholomew, D. J. (1975) A stochastic control problem in the social sciences. Bull. Int. Statist. Inst. 46, 670–680.Google Scholar
Bartholomew, D. J. (1982) Stochastic Models for Social Processes, 3rd edn. Wiley, New York.Google Scholar
Bartholomew, D. J. and Forbes, A. F. (1979) Statistical Techniques for Manpower Planning. Wiley, New York.Google Scholar
Bowerman, B., David, H. T. and Isaacson, D. (1977) The convergence of Cesaro averages for certain non-stationary Markov chains. Stoch. Proc. Appl. 5, 221–230.Google Scholar
Conlisk, J. (1976) Interactive Markov chains. J. Math. Sociol 4, 157–185.Google Scholar
Darroch, J. N. and Seneta, E. (1965) On quasi-stationary distributions in absorbing discrete time finite Markov chains. J. Appl. Prob. 2, 88–100.Google Scholar
Feichtinger, G. and Mehlmann, A. (1976) The recruitment trajectory corresponding to particular stock sequences in Markovian person-flow models. Math. Operat. Res. 1, 175–184.Google Scholar
Gantmacher, F. R. (1959) Applications of the Theory of Matrices. Interscience Publishers, New York.Google Scholar
Hodge, R. W. (1966) Occupational mobility as a probability process. Demography 3, 19–34.Google Scholar
Iosifescu, M. (1980) Finite Markov Processes and their Applications. Wiley, New York.Google Scholar
Isaacson, D. and Luecke, G. R. (1978) Strongly ergodic Markov chains and rates of convergence using spectral conditions. Stoch. Proc. Appl. 7, 113–121.Google Scholar
Isaacson, D. L. and Madsen, W. R. (1976) Markov Chains. Wiley, New York.Google Scholar
Keilson, J. (1979) Markov Chain Models – Rarity and Exponentiality. Springer-Verlag, Berlin.Google Scholar
Keilson, J. and Wishart, D. M. G. (1964) A central limit theorem for processes defined on a finite Markov chain. Proc. Camb. Phil. Soc. 60, 547–567.Google Scholar
Pollard, J. H. (1966) On the use of direct product matrix in analysing certain stochastic population models. Biometrika 53, 397–415.Google Scholar
Pollard, J. H. (1973) Mathematical Models for the Growth of Human Populations. Cambridge University Press.Google Scholar
Rogoff, N. (1953) Recent Trends in Occupational Mobility. Free Press, Glencoe, Illinois.Google Scholar
Tsaklidis, G. and Vassiliou, P.-C. G. (1988) Asymptotic periodicity of the vector of variances and covariances in non-homogeneous Markov systems. J. Appl. Prob. 25, 21–33.Google Scholar
Vassiliou, P.-C. G. (1982) Asymptotic behaviour of Markov systems. J. Appl. Prob. 19, 851–857.Google Scholar
Vassiliou, P.-C. G. (1984) Cyclic behaviour and asymptotic stability of non-homogeneous Markov systems. J. Appl. Prob. 21, 315–325.Google Scholar
Vassiliou, P.-C. G. (1986) Asymptotic variability of non-homogeneous Markov systems under cyclic behaviour. Europ. J. Oper. Res. 27, 215–228.Google Scholar
Vassiliou, P.-C. G. and Gerontidis, I. (1985) Variances and covariances of the grade sizes in manpower systems. J. Appl. Prob. 22, 583–597.Google Scholar
Vassiliou, P.-C. G. and Tsaklidis, G. (1989) The rate of convergence of the vector of variances and covariances in non-homogeneous Markov systems. J. Appl. Prob. 26, 776–783.Google Scholar
Young, A. and Vassiliou, P.-C. G. (1974) A non-linear model on the promotion of staff. J. R. Statist. Soc. A 137, 584–595.Google Scholar