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The non-homogeneous Markov system in a stochastic environment

Part of: Operations research and management science

Published online by Cambridge University Press:  14 July 2016

N. Tsantas  and
P.-C. G. Vassiliou
N. Tsantas
Affiliation:
Aristotle University of Thessaloniki
P.-C. G. Vassiliou*
Affiliation:
Aristotle University of Thessaloniki
*
∗ Postal address: Statistics and Operations Research Section, Department of Mathematics, Aristotle University of Thessaloniki, 54006 Thessaloniki, Greece.
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Abstract

We introduce and define for the first time the concept of a non-homogeneous Markov system in a stochastic environment (S-NHMS). The problem of finding the expected population structure in an S-NHMS is studied, and important properties among the basic parameters of the S-NHMS are established. Moreover, we study the problem of maintaining the relative sizes of the states in a stochastic environment applying control in the input process. Among other things, we provide the probability of maintaining any vector of relative state sizes. Also strategies for attaining in an optimal way a desired relative structure are designed, with the use of a given algorithm. Finally, an illustration is provided of the present results in a manpower system.

Keywords

STOCHASTIC POPULATION SYSTEMSMANPOWER SYSTEMSNON-HOMOGENEOUS MARKOV CHAINS

MSC classification

Primary: 90B70: Theory of organizations, manpower planning

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 2 , June 1993 , pp. 285 - 301
Copyright
Copyright © Applied Probability Trust 1993 

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References

Bartholomew, D. J. (1973) Stochastic Models for Social Process, 2nd edn. Wiley, Chichester.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. (1977) Maintaining a grade or age structure in a stochastic environment. Adv. Appl. Prob. 11, 603–615.Google Scholar
Bartholomew, D. J. (1982) Stochastic Models for Social Process, 3rd edn. Wiley, Chichester.Google Scholar
Bartholomew, D. J. and Forbes, A. F. (1979) Statistical Techniques for Manpower Planning. Wiley, Chichester.Google Scholar
Borgwardt, K. H. (1987) The Simplex Method. A Probabilistic Analysis. Springer-Verlag, Berlin.Google Scholar
Cohen, J. E. (1976) Ergodicity of age structure in populations with Markovian vital rates, I: countable states. J. Amer. Statist. Assoc. 71, 335–339.CrossRefGoogle Scholar
Cohen, J. E. (1977) Ergodicity of age structure in populations with Markovian vital rates, II: general states. Adv. Appl. Prob. 9, 18–37.Google Scholar
Conlisk, J. (1976) Interactive Markov chains. J. Math. Sociol. 4, 157–185.Google Scholar
Davies, G. S. (1973) Structural control in a graded manpower system. Management Sci. 20, 76–84.CrossRefGoogle Scholar
Dreyfus, S. E. and Law, A. M. (1977) The Art and Theory of Dynamic Programming. Academic Press, New York.Google Scholar
Forbes, A. F. (1971) Markov chain models for manpower systems. In Manpower and Management Science, ed. Bartholomew, D. J. and Smith, A. R., English University Press, London. pp. 93–113.Google Scholar
Grinold, R. D. and Stanford, R. E. (1974) Optimal control of a graded manpower system. Management Sci. 20, 1201–1216.Google Scholar
Johnson, N. I. and Kotz, S. (1969) Distributions in Statistics. Discrete Distributions. Wiley, New York.Google Scholar
Larson, R. E. and Casti, J. L. (1982) Principles of Dynamic Programming, Part II. Marcel Dekker, New York.Google Scholar
Mcclean, S. I. (1976) A continuous time population model with Poisson recruitment. J. Appl. Prob. 13, 348–354.CrossRefGoogle Scholar
Mcclean, S. I. (1978) Continuous-time stochastic models of a multigrade population. J. Appl. Prob. 15, 26–37.Google Scholar
Powell, M. J. D. (1988) A tolerant algorithm for linearly constrained optimization calculations. Report DAMTP/1988/NA17. University of Cambridge.Google Scholar
Schervish, M. J. (1984) Algorithm AS 195. Multivariate normal probabilities with error bound. Appl. Statist. 33, 81–94.Google Scholar
Vajda, S. (1978) Mathematics of Manpower Planning. Wiley, Chichester.Google Scholar
Vassiliou, P.-C. G. (1976) A Markov chain model for prediction of wastage in manpower systems. Operat. Res. Quart. 27, 57–76.CrossRefGoogle Scholar
Vassiliou, P.-C. G. (1978) A high order Markovian model for promotion in manpower systems. J. R. Statist. Soc. A 141, 86–94.Google Scholar
Vassiliou, P.-C. G. (1982) Asymptotic behavior of Markov systems. J. Appl. Prob. 19, 815–857.Google Scholar
Vassiliou, P.-C. G. (1984) Cyclic behavior and asymptotic stability of non-homogeneous Markov systems. J. Appl. Prob. 21, 315–325.CrossRefGoogle Scholar
Vassiliou, P.-C. G. (1986) Asymptotic variability of non-homogeneous Markov systems under cyclic behaviour. Eur. J. Operat. Res. 27, 215–228.Google Scholar
Vassiliou, P.-C. G. and Georgiou, A. C. (1990) Asymptotically attainable structures in nonhomogeneous Markov systems. Operat. Res. 38, 537–545.Google Scholar
Vassiliou, P.-C. G. and Tsantas, N. (1984a) Stochastic control in non-homogeneous Markov systems. Internat. J. Computer Math. 16, 139–155.Google Scholar
Vassiliou, P.-C. G. and Tsantas, N. (1984b) Maintainability of structures in non-homogeneous Markov systems under cyclic behaviour and input control. SIAM J. Appl. Math. 44, 1014–1022.CrossRefGoogle 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