Birth and death process - example transition probabilities

Question

Suppose we have a birth-death process with state space S = $\{1,2\}$, $\lambda_1 = 2$ and $\mu_2 = 1$. Find transition probabilities.

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I have drawn how my process looks like:

     --- 2 -->
  (1)         (2)
     <-- 1 ---

I also know that r.v. being waiting time between state 1 and 2 denoted $T_1$ is $$\text{Exp}(\lambda_1 + \mu_1) = \text{Exp}(2)$$. I am to calculate how $$\mathbb{P}_{i,n}(t) = \mathbb{P}(X(t)=n|X(0) = i)$$ looks like.

No further progress has been made. I appreciate your help.

Answer 1

The infinitesimal generator for this continuous-time Markov chain is $$ Q = \begin{pmatrix} -2 &2\\ 1 & -1 \end{pmatrix}. $$ We can compute the transition probability matrix $P(t)$ by matrix exponentiation: \begin{align} P(t) &= e^{tQ}\\ &= \sum_{n=0}^\infty \frac{(tQ)^n}{n!}\\ &= \large\left( \begin{array}{cc} \frac{1}{3}+ \frac{2 e^{-3 t}}{3} & \frac{2}{3}-\frac{2 e^{-3 t}}{3} \\ \frac{1}{3}-\frac{e^{-3 t}}{3} & \frac{2}{3}+\frac{e^{-3 t}}{3}. \end{array} \right) \end{align}