Definition. Consider a CTMC with states \(i=0,1,2\ldots,n\). Let \(\lambda_{i}\) be the rate that \(i\) transitions to \(i+1\), \(\mu_{i}\) be the rate that \(i\) transitions to \(i-1\), and \(\lambda_{i} + \mu_{i}\) be the rate that \(i\) transitions to a new state. This CTMC is known as a birth-death process.
The limiting probabilities satisfies the balance equations. Then,
\[\begin{align} \lambda_{0} P_{0} &= \mu_{1} P_{1} \\ (\lambda_{1} + \mu_{1}) P_{1} &= \lambda_{0} P_{0} + \mu_{2} P_{2} \Rightarrow \lambda_{1} P_{1} = \mu_{2} P_{2} \end{align}\]
By induction, we find \(\lambda_{k-1} P_{k-1} = \mu_{k} P_{k}\), so
\[P_{k} = \frac{\lambda_{k-1}}{\mu_{k}} P_{k-1} = \frac{\lambda_{k-1}\cdots \lambda_{0}}{\mu_{k}\cdots \mu_{1}}P_{0}.\]
Let \(\lambda_{i} = \lambda\), \(\mu_{i} = \mu\).
\[\begin{align} \sum_{k=0}^{n} P_{k} &= \sum_{k=0}^{n} \left( \frac{\lambda}{\mu} \right)^{k} P_{0} = P_{0} \left( \frac{1 - (\lambda/\mu)^{n+1}}{1 - \lambda/\mu} \right) = 1 \\ P_{0} &= \frac{1 - \lambda/\mu}{1 - (\lambda/\mu)^{n+1}} \\ P_{k} &= \left( \frac{\lambda}{\mu} \right)^{k}\frac{1 - \lambda/\mu}{1 - (\lambda/\mu)^{n+1}} \end{align}\]
If the state space were infinite and \(\lambda/\mu < 1\), then
\[P_{k} = \left( \frac{\lambda}{\mu} \right)^{k}\left(1 - \frac{\lambda}{\mu}\right) = P(\mbox{Geom}(\lambda/\mu) = k + 1),\]
so the distribution of limiting probabilities is \(\mbox{Geom}(\lambda/\mu) - 1\).
Let \(\lambda_{i} = (n-i)\lambda\), \(\mu_{i} = i\mu\) and suppose we have \(n+1\) states.
\[\begin{align} P_{k} &= \frac{(n-k)(n-k+1)\cdots n}{k(k-1)\cdots 1}\left( \frac{\lambda}{\mu} \right)^{k}P_{0} = \binom{n}{k}\left( \frac{\lambda}{\mu} \right)^{n}P_{0} \\ P_{0}\sum_{k=0}^{n} \binom{n}{k} \left( \frac{\lambda}{\mu} \right)^{k} &= P_{0}(1 + \lambda/\mu)^{n} = 1 \\ P_{0} &= \frac{1}{(1 + \lambda/\mu)^{n}} \\ P_{k} &= \binom{n}{k}\left( \frac{\lambda}{\mu} \right)^{k}\frac{1}{(1 + \lambda/\mu)^{n}} \\ &= \binom{n}{k}\left( \frac{\lambda}{\lambda + \mu} \right)^{k}\left( \frac{\mu}{\lambda + \mu} \right)^{n-k} \\ &= P(\mbox{Bin}(n, \lambda/(\lambda + \mu)) = k), \end{align}\]
so the distribution of limiting probabilities is \(\displaystyle \mbox{Bin}\left(n, \frac{\lambda}{\lambda + \mu}\right)\).
\[\begin{align} \sum_{k=1}^{n} \frac{1}{k!}\left( \frac{\lambda}{\mu} \right)^{k}P_{0} &= P_{0}e^{\lambda/\mu} = 1 \\ P_{0} &= e^{-\lambda/\mu} \\ P_{k} &= \frac{1}{k!}\left( \frac{\lambda}{\mu} \right)^{k}e^{-\lambda/\mu} \\ &= P(\mbox{Poisson}(\lambda/\mu) = k) \end{align}\]
so the distribution of limiting probabilities is \(\displaystyle \mbox{Poisson}(\lambda/\mu)\).