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A linear birth and death process under the influence of another process

Published online by Cambridge University Press:  14 July 2016

Prem S. Puri
Prem S. Puri*
Affiliation:
Purdue University
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Abstract

Let {X 1 (t), X 2 (t), t ≧ 0} be a bivariate birth and death (Markov) process taking non-negative integer values, such that the process {X 2(t), t ≧ 0} may influence the growth of the process {X 1(t), t ≧ 0}, while the process X 2 (·) itself grows without any influence whatsoever of the first process. The process X 2 (·) is taken to be a simple linear birth and death process with λ2 and µ 2 as its birth and death rates respectively. The process X 1 (·) is also assumed to be a linear birth and death process but with its birth and death rates depending on X 2 (·) in the following manner: λ (t) = λ1 (θ + X 2 (t)); µ(t) = µ 1 (θ + X 2 (t)). Here λ i, µi and θ are all non-negative constants. By studying the process X 1 (·), first conditionally given a realization of the process {X 2 (t), t ≧ 0} and then by unconditioning it later on by taking expectation over the process {X 2 (t), t ≧ 0} we obtain explicit solution for G in closed form. Again, it is shown that a proper limit distribution of X 1 (t) always exists as t→∞, except only when both λ1 > µ 1 and λ2 > µ 2. Also, certain problems concerning moments of the process, regression of X 1 (t) on X 2 (t); time to extinction, and the duration of the interaction between the two processes, etc., are studied in some detail.

Keywords

BIRTH AND DEATH PROCESSESINTERACTING GROWTH PROCESSESMARKOV PROCESSESDURATION OF INTERACTION

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 12 , Issue 1 , March 1975, pp. 1 - 17
Copyright
Copyright © Applied Probability Trust 1975 

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