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TRANSIENT ANALYSIS OF LINEAR BIRTH–DEATH PROCESSES WITH IMMIGRATION AND EMIGRATION

Published online by Cambridge University Press:  16 April 2004

Yuxi Zheng* ,
Xiuli Chao[dagger]  and
Xiaomei Ji
Yuxi Zheng*
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
Xiuli Chao[dagger]
Affiliation:
Department of Industrial Engineering, North Carolina State University, Raleigh, North Carolina 27695-7906, E-mail: [email protected]
Xiaomei Ji
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
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Abstract

Linear birth–death processes with immigration and emigration are major models in the study of population processes of biological and ecological systems, and their transient analysis is important in the understanding of the structural behavior of such systems. The spectral method has been widely used for solving these processes; see, for example, Karlin and McGregor [11]. In this article, we provide an alternative approach: the method of characteristics. This method yields a Volterra-type integral equation for the chance of extinction and an explicit formula for the z-transform of the transient distribution. These results allow us to obtain closed-form solutions for the transient behavior of several cases that have not been previously explicitly presented in the literature.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 18 , Issue 2 , April 2004 , pp. 141 - 159
Copyright
© 2004 Cambridge University Press

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