Assume that in a city the population regenerates itself at an exponential rate , and dies at an exponential rate . Furthermore assume that immigrants arrive according to an exponential rate ; when the population in the given city reaches size no further immigrants are permitted. If and for how long will the city stop accepting new immigrants.
I figure that this is a birth and death process with different birth rates at different population sizes, thus we would have a birth rate of:
and the death rate would simply be
if I wanted to find the expected time the city would reject immigrants I would , which would give me:
calculating the brackets yields:
so my final solution would be: 1-1=0 which doesn't seem right.