(a) Define what it means to say that $\pi$ is an equilibrium distribution for a Markov chain on a countable state space with Q-matrix $Q=\left(q_{i j}\right)$, and give an equation which is satisfied by any equilibrium distribution. Comment on the possible non-uniqueness of equilibrium distributions.

(b) State a theorem on convergence to an equilibrium distribution for a continuoustime Markov chain.

A continuous-time Markov chain $\left(X_{t}, t \geqslant 0\right)$ has three states $1,2,3$ and the Qmatrix $Q=\left(q_{i j}\right)$ is of the form

$$Q=\left(\begin{array}{ccc} -\lambda_{1} & \lambda_{1} / 2 & \lambda_{1} / 2 \\ \lambda_{2} / 2 & -\lambda_{2} & \lambda_{2} / 2 \\ \lambda_{3} / 2 & \lambda_{3} / 2 & -\lambda_{3} \end{array}\right)$$

where the rates $\lambda_{1}, \lambda_{2}, \lambda_{3} \in[0, \infty)$ are not all zero.

[Note that some of the $\lambda_{i}$ may be zero, and those cases may need special treatment.]

(c) Find the equilibrium distributions of the Markov chain in question. Specify the cases of uniqueness and non-uniqueness.

(d) Find the limit of the transition matrix $P(t)=\exp (t Q)$ when $t \rightarrow \infty$.

(e) Describe the jump chain $\left(Y_{n}\right)$ and its equilibrium distributions. If $\widehat{P}$is the jump probability matrix, find the limit of $\widehat{P}^{n}$ as $n \rightarrow \infty$.