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.

Do you mean long run proportion of time?
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:

so for

calculating the brackets yields:

so my final solution would be: 1-1=0 which doesn't seem right.