Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-24T19:53:03.825Z Has data issue: false hasContentIssue false
  • English
  • Français

Markov population processes

Published online by Cambridge University Press:  14 July 2016

J.F.C. Kingman
J.F.C. Kingman*
Affiliation:
University of Sussex
Get access Rights & Permissions [Opens in a new window]

Summary

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 nt 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.

Type
Research Papers
Information
Journal of Applied Probability , Volume 6 , Issue 1 , April 1969 , pp. 1 - 18
Copyright
Copyright © Applied Probability Trust 

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

[1] Bailey, N. T. J. (1957) The Mathematical Theory of Epidemics. Griffin, London.Google Scholar
[2] Bartlett, M. S. (1949) Some evolutionary stochastic processes. J. R. Statist. Soc. B 11, 211–229.Google Scholar
[3] Bartlett, M. S. (1956) Deterministic and stochastic models for recurrent epidemics. Proc. Third Berkeley Symp. Math. Statist. and Prob. 4, 81–109.Google Scholar
[4] Beneš, V. E. (1965) Mathematical Theory of Connecting Networks and Telephone Traffic. Academic Press, New York and London.Google Scholar
[5] Breiman, L. (1963) The Poisson tendency in traffic distribution. Ann. Math. Statist. 34, 308–311.CrossRefGoogle Scholar
[6] Daley, D. J. and Kendall, D. G. (1965) Stochastic rumours. J. Inst. Math. Applic. 1, 42–55.Google Scholar
[7] Derman, C. (1955) Some contributions to the theory of denumerable Markov chains. Trans. Amer. Math. Soc. 79, 541–555.Google Scholar
[8] Jackson, J. R. (1957) Networks of waiting lines. Operat. Res. 5, 518–521.Google Scholar
[9] Kendall, D. G. and Reuter, G E. H. (1957) The calculation of the ergodic projection for Markov chains and processes with a countable infinity of states. Acta Math. 97, 103–143.Google Scholar
[10] Kendall, D. G. (1959) Unitary dilations of one-parameter semigroups of Markov transition operators, and the corresponding integral representations for Markov processes with a countable infinity of states. Proc. Lond. Math. Soc. (3) 9, 417–431.Google Scholar
[11] Kendall, D. G. (1964) Some recent work and further problems in the theory of queues. Teor. Veroyat. Primen. 9, 3–15.Google Scholar
[12] Kingman, J. F. C. (1961) The ergodic behaviour of random walks. Biometrika 48, 391–396.Google Scholar
[13] Kingman, J. F. C. (1961) Two similar queues in parallel. Ann. Math. Statist. 32, 1314–1323.Google Scholar
[14] Nash-Williams, C. St. J. A. (1959) Random walk and electric currents int networks. Proc. Camb. Phil. Soc. 55, 181–194.Google Scholar
[15] Reich, E. (1957) Waiting times when queues are in tandem. Ann. Math. Statist. 28, 768–773.Google Scholar
[16] Reuter, G. E. H. (1961) Competition processes. Proc. Fourth Berkeley Symp. Math. Statist. and Prob. 2, 421–430.Google Scholar
[17] Saaty, T. L. (1965) Stochastic network flows: advances in networks of queues. Proc. Symp. Congestion Theory 86–107, University of North Carolina Press, Chapel Hill.Google Scholar
[18] Weiss, G. H. (1965) A survey of some recent research in road traffic. Proc. Symp. Congestion Theory 253–288, University of North Carolina Press, Chapel Hill.Google Scholar
[19] Weiss, G. H. and Herman, R. (1962) Statistical properties of low-density traffic. Quart. Appl. Math. 22, 121–130.Google Scholar
[20] Whittle, P. (1967) Nonlinear migration processes. Proc. 36th Session of the International Statistical Institute.Google Scholar
[21] Whittle, P. (1968) Equilibrium distributions for an open migration process. J. Appl. Prob. 5, 567–571.Google Scholar