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Markov Population Processes

J. F. C. Kingman
Journal of Applied Probability
Vol. 6, No. 1 (Apr., 1969), pp. 1-18
DOI: 10.2307/3212273
Stable URL: http://www.jstor.org/stable/3212273
Page Count: 18
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Markov Population Processes
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Abstract

The processes of the title have frequently been used to represent situations involving numbers of individuals in different categories or colonies. In such processes the state at any time is represented by the vector n = (n1, n2, ⋯, nk), where ni is the number of individuals in the ith colony, and the random evolution of n is supposed to be that of a continuous-time Markov chain. The jumps of the chain may be of three types, corresponding to the arrival of a new individual, the departure of an existing one, or the transfer of an individual from one colony to another. Models with this structure have been proposed for the evolution of populations, for instance by Bartlett [2]. They are implicit in the theory of epidemics [1], [3], and in the closely related study of rumours by Daley and Kendall [6]. The competition processes considered by Reuter [16] are of this kind, as are the models for networks of queues studied by Jackson [8]. Jackson's results have recently been the subject of an elegant generalisation by Whittle [20], [21], the potential applications of which (to the theory of traffic flow [18] for instance) remain as yet unexplored. The purpose of this paper is to present a systematic account of the methods available to analyse these processes, with particular reference to the calculation of stationary distributions (when they exist). When the process is reversible, such calculations are very easy, and conditions for this to occur are examined in Section 2, where it is shown that in a certain sense reversibility is a 'local' property. Section 3 is devoted to a discussion, within the present formulation, of Whittle's results, and of their relation to methods based on reversibility. A particular case to which Whittle's method can be applied is the linear system studied in Section 4, but here a more direct technique, generalising the classical work of Bartlett [2] is available. Finally, Sections 5 and 6 contain an analysis of 'flow models', population processes in which at most one individual can occupy any colony, for which a quite different method is appropriate.

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