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Relaxed Markov processes

Published online by Cambridge University Press:  01 July 2016

P. Whittle
P. Whittle*
Affiliation:
University of Cambridge
*
∗Postal address: Statistical Laboratory, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB, U.K.
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Abstract

The concept of relaxing a Markov process is introduced; this is the creation of additional transitions between ergodic classes of the process in such a way as to conserve the existing equilibrium distribution within ergodic classes. The ‘open' version of a ‘closed' model of migration, polymerisation etc. often has this character. As further examples, generalized versions of Jackson networks and networks with clustering nodes are given.

Keywords

PARTIAL BALANCEQUASI-REVERSIBILITYJACKSON NETWORKSPOLYMERISATION
Type
Research Article
Information
Advances in Applied Probability , Volume 15 , Issue 4 , December 1983 , pp. 769 - 782
Copyright
Copyright © Applied Probability Trust 1983 

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References

Balescu, R. (1975) Equilibrium and Nonequilibrium Statistical Mechanics. Wiley, New York.Google Scholar
Gordon, W. J. and Newell, G. F. (1967) Closed queueing systems with exponential servers. Operat. Res. 15, 254–265.Google Scholar
Jackson, J. R. (1957) Networks of waiting lines. Operat. Res. 5, 518–521.Google Scholar
Jackson, J. R. (1963) Jobshop-like queueing systems. Management Sci. 10, 131–142.CrossRefGoogle Scholar
Kelly, F. P. (1975) Markov processes and Markov random fields. Bull. Internat. Inst. Statist. 46, 397–404.Google Scholar
Kelly, F. P. (1976) Networks of queues. Adv. Appl. Prob. 8, 416–432.Google Scholar
Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
Kelly, F. P. (1982) Networks of quasi-reversible nodes. In Applied Probability-Computer Science, the Interface: Proceedings of the ORSA-TIMS Boca Raton Symposium, ed. Disney, R., Birkhauser Boston, Cambridge, Ma.Google Scholar
Kreuzer, H. J. (1981) Nonequilibrium Thermodynamics and its Statistical Foundations. Oxford University Press, London.Google Scholar
Lavenda, B. H. (1978) Thermodynamics of Irreversible Processes. Macmillan, London.Google Scholar
Muntz, R. R. (1972) Poisson departure processes and queueing networks. IBM Research Report RC 4145, IBM Thomas J. Watson Research Center, Yorktown Heights, NY.Google Scholar
Prigogine, I. (1962) Non-Equilibrium Statistical Mechanics. Wiley Interscience, New York.Google Scholar
Spitzer, F. (1971) Random Fields and Interacting Particle Systems. Mathematical Association of America, Washington, DC.Google Scholar
Whittle, P. (1965a) Statistical processes of aggregation and polymerisation. Proc. Camb. Phil. Soc. 61, 475–495.Google Scholar
Whittle, P. (1965b) The equilibrium statistics of a clustering process in the uncondensed phase. Proc. R. Soc. A 285, 501–519.Google Scholar
Whittle, P. (1967) Nonlinear migration processes. Bull. Internat. Inst. Statist. 42, 642–647.Google Scholar
Whittle, P. (1968) Equilibrium distributions for an open migration process. J. Appl. Prob. 5, 567–571.CrossRefGoogle Scholar
Whittle, P. (1972) Statistics and critical points of polymerisation processes. Proc. Symp. Statistical and Probabilistic Problems in Metallurgy, Suppl. Adv. Appl. Prob., 199–215.Google Scholar
Whittle, P. (1977) Co-operative effects in assemblies of stochastic automata. Proc. Symp. to Honour Jerzy Neyman, Polish Scientific Publishers, Warsaw, 335–343.Google Scholar
Whittle, P. (1980a) Polymerisation processes with intrapolymer bonding. I. One type of unit. Adv. Appl. Prov. 12, 94–115.CrossRefGoogle Scholar
Whittle, P. (1980b) Polymerisation processes with intrapolymer bonding. II. Stratified processes. Adv. Appl. Prob. 12, 116–134.Google Scholar
Whittle, , (1980C) Polymerisation processes with intrapolymer bonding. III. Several types of unit. Adv. Appl. Prob. 12, 135–153.Google Scholar
Whittle, P. (1981) A direct derivation of the equilibrium distribution for a polymerisation process. Theory Prob. Appl. 26, 344–355.CrossRefGoogle Scholar