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Let $X_n$ be a birth-death process, with birth rates $\lambda_n$ and death rates $\mu_n$ (with $\mu_o=0$ and $\lambda_{-1}=0$). How do you show that the invariant distribution $\pi_i$ is:

$\pi_0=\Big[ 1+ \sum_{k=0}^\infty \frac{\lambda_k\lambda_{k-1}\dots\lambda_0}{\mu_{k+1}\mu_k\dots \mu_1}\Big]^{-1}$

and

$\pi_{n+1}=\frac{\lambda_n}{\mu_{n+1}}\pi_n$?

I used the definition of invariant distribution, and arrived at the formula

$\pi_i=\frac{\pi_{i-1}\lambda_{i-1}+\pi_{i+1}\mu_{i+1}}{\lambda_i+\mu_i}$,

but I have no idea how to use this to prove what I'm being asked to prove.

1 Answers 1

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With $\dfrac{\pi_{i-1}\lambda_{i-1}+\pi_{i+1}\mu_{i+1}}{\lambda_i+\mu_i}$, replace $\pi_{i-1}$ by $ \dfrac{\mu_i}{\lambda_{i-1}} \pi_i$ (just a restatement of $\pi_{n+1}=\frac{\lambda_n}{\mu_{n+1}}\pi_n$) and replace $\pi_{i+1}$ by $ \dfrac{\lambda_i}{\mu_{i+1}}\pi_i$. So long as you are not dividing by $0$, this clearly gives $\pi_i$.

Then note that if $\pi_{n+1}=\frac{\lambda_n}{\mu_{n+1}}\pi_n$ then you need to set $\pi_0=\Big[ 1+ \sum_{k=0}^\infty \frac{\lambda_k\lambda_{k-1}\dots\lambda_0}{\mu_{k+1}\mu_k\dots \mu_1}\Big]^{-1}$ to get $\sum_{n=0}^\infty \pi_n = 1$.

So the distribution you are asked to show is the invariant distribution is indeed an invariant distribution: it is invariant as it satisfies the formula you found, and it is a probability distribution as it is non-negative and sums to $1$. Since there is only one invariant distribution in this case,, this must be it.