Markov Birth Death Chain Show Stationary Distribution

[crazy_craig]

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=0^n-1 λ(i)/μ(i+1)

π(n) = π(0) * ∏_i=0^n-1 [ a + Bi] / [ (S+y(i + 1)) * (i+1) ]

And this is where I'm stuck. If I can show that the Ʃ_n 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 Ʃ_n π(n) would also be finite...

 

[Ray Vickson]

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=0^n-1 λ(i)/μ(i+1)

π(n) = π(0) * ∏_i=0^n-1 [ a + Bi] / [ (S+y(i + 1)) * (i+1) ]

And this is where I'm stuck. If I can show that the Ʃ_n 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 Ʃ_n π(n) would also be finite...

Since a, B, S and y are > 0 we have $$0 < f(i) \equiv \frac{a + Bi}{S + y(i+1)} \leq K$$ for some easily-computable constant K = K(a,B,S,y). Thus
$$0 < \prod_{i=0}^{n-1} \frac{f(i)}{i+1} \leq \frac{K^n}{n!}.$$

RGV