(a) Define a continuous time Markov chain $X$ with infinitesimal generator $Q$ and jump chain $Y$.

(b) Let $i$ be a transient state of a continuous-time Markov chain $X$ with $X(0)=i$. Show that the time spent in state $i$ has an exponential distribution and explicitly state its parameter.

[You may use the fact that if $S \sim \operatorname{Exp}(\lambda)$, then $\mathbb{E}\left[e^{\theta S}\right]=\lambda /(\lambda-\theta)$ for $\theta<\lambda$.]

(c) Let $X$ be an asymmetric random walk in continuous time on the non-negative integers with reflection at 0 , so that

$$q_{i, j}= \begin{cases}\lambda & \text { if } \quad j=i+1, i \geqslant 0 \\ \mu & \text { if } \quad j=i-1, i \geqslant 1\end{cases}$$

Suppose that $X(0)=0$ and $\lambda>\mu$. Show that for all $r \geqslant 1$, the total time $T_{r}$ spent in state $r$ is exponentially distributed with parameter $\lambda-\mu$.

Assume now that $X(0)$ has some general distribution with probability generating function $G$. Find the expected amount of time spent at 0 in terms of $G$.