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crazy_craig
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Homework Statement
Logistics model: Consider the birth and death chain with birth rates π(n)=a + Bn and death rates μ(n) = (S + yn)n where all four constants (a, B, S, y) are positive. In words, there is immigration at rate a, each particle gives birth at rate B and dies at rate S+yn, i.e., at a rate which starts at rate S and increases linearly due to crowding. Show that the system has a stationary distribution.
Homework Equations
Since this is a birth and death process, we know;
π(n) = λ(n-1)*λ(n-2)* ...*λ(0) * π(0) / μ(n) * μ(n-1) * ... * μ(1)
where the rate from n to n+1 = λ(n) and the rate from n to n-1= μ(n)
The Attempt at a Solution
Using this, we have:
π(n) = π(0) * ∏i=0n-1 λ(i)/μ(i+1)
π(n) = π(0) * ∏i=0n-1 [ a + Bi] / [ (S+y(i + 1)) * (i+1) ]
And this is where I'm stuck. If I can show that the Ʃn π(n) < ∞ , then I could pick π(0) to make the sum 1. I guess I need to show that the above product is a finite value or at least less than some finite value, and then Ʃn π(n) would also be finite...