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Evacuation of a Yule process with immigration

Published online by Cambridge University Press:  14 July 2016

Mark Brown ,
Sheldon Ross  and
Richard Shorrock
Mark Brown
Affiliation:
City College, CUNY
Sheldon Ross
Affiliation:
University of California
Richard Shorrock
Affiliation:
University of Montreal
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Abstract

Individuals arrive at a geographical area that is initially empty, in accordance with a pure birth process with birth parameters λj = jλ + θ, j ≧ 0. Due to contamination, this geographical area is unsafe for its population and at some fixed time T in the future everyone in the area will be evacuated and no further immigration will be allowed.

Suppose now than an intermediate evacuation time τ, 0 ≦ τ ≦ T, at which time everyone present in the area would be evacuated, is to be chosen. The area would then again fill up with individuals between times τ and T, and, at T, the final evacuation would be made and the area would be permanently sealed off. The problem is to choose τ so as to minimise the total expected cost incurred by time T, where a cost g(x) is incurred for each individual that spends a time x in the area.

Keywords

YULE PROCESSOPTIMAL EVACUATION TIMEINFINITESIMAL LOOK AHEAD STOPPING RULES
Type
Short Communications
Information
Journal of Applied Probability , Volume 12 , Issue 4 , December 1975 , pp. 807 - 811
Copyright
Copyright © Applied Probability Trust 1975 

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References

[1] Neuts, M. and Resnick, S. (1971) On the times of births in a linear birthprocess. J. Austral. Math. Soc. 12, 473–475.CrossRefGoogle Scholar
[2] Ross, S. (1971) Infinitesimal look ahead stopping rules. Ann. Math. Statist. 42, 297–303.CrossRefGoogle Scholar
[3] Ross, S. (1969) Optimal dispatching of a Poisson process. J. Appl. Prob. 6, 692–699.CrossRefGoogle Scholar