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Interconnected birth and death processes

Published online by Cambridge University Press:  14 July 2016

Prem S. Puri
Prem S. Puri*
Affiliation:
Purdue University, Lafayette
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Summary

Two cases of multiple linearly interconnected linear birth and death processes are considered. It is found that in general the solution of the Kolmogorov differential equations for the probability generating function (p.g.f) g of the random variables involved is not obtainable by standard methods, although one can obtain moments of the random variables from these equations. A method is considered for obtaining an approximate solution for g. This is based on the introduction of a sequence of stochastic processes such that the sequence {f(n)} of their p.g.f.'s tends to g as n → ∞ in an appropriate manner. The method is applied to the simple case of two birth and death processes with birth and death rates λi and μi, i = 1,2, interconnected linearly with transition rates v and δ (see Figure 2). For this case some limit theorems are established and the probability of ultimate extinction of both the processes is considered. In addition, for the special cases (i) λ1 = δ = 0, with the remaining rates time dependent and (ii) λ2 = δ = 0, with the remaining rates constant, explicit solutions for g are obtained and studied.

Type
Research Papers
Information
Journal of Applied Probability , Volume 5 , Issue 2 , August 1968 , pp. 334 - 349
Copyright
Copyright © Applied Probability Trust 1968 

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References

[1] Apostol, T. M. (1957) Mathematical Analysis. Addison Wesley Co., Reading, U.S.A. Google Scholar
[2] Armitage, P. (1952) The statistical theory of bacterial populations subject to mutation. J.R.Statist. Soc B14, 1–33.Google Scholar
[3] Bartlett, M. S. (1955) An Introduction to Stochastic Processes. Cambridge University Press.Google Scholar
[4] Fowler, J. H. et al. (1957) The cellular composition of hemopoietic spleen colonies. J. Cellular Physiology 69, 65–71.Google Scholar
[5] Gani, J. and Yeo, G. F. (1965) Some birth-death and mutation models for phage reproduction. J. Appl. Prob. 2, 150–161.Google Scholar
[6] Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.Google Scholar
[7] Kamke, E. (1948) Differentialgleichungen, Lösungsmethoden und Lösungen, I. Chelsea Publishing Co., New York.Google Scholar
[8] Mitchison, M. B. (1954) Problems of drug resistance. Brit. Med. Bull. 10, 115–124.Google Scholar
[9] Nissen-Meyer, S. (1966) Analysis of effects of antibiotics on bacteria by means of stochastic models. Biometrics 22, 761–780.CrossRefGoogle Scholar
[10] Siminovitch, L. et al. (1963) The distribution of colony-forming cells among spleen colonies. J.Cellular and Comparative Physiology 62, 327–336.CrossRefGoogle ScholarPubMed
[11] Till, J. E. et al. (1964) A stochastic model of stem cell proliferation, based on the growth of spleen colony-forming cells. Proc. Nat. Acad. Sci. USA 51, 29–36.Google Scholar
[12] Wiggins, A. D. (1957) A Statistical Study of the Mechanism of Bacterial Toxicity. Ph.D. Thesis, University of California, Berkeley.Google Scholar