(i) Explain briefly what is meant by saying that a continuous-time Markov chain $X(t)$ is a birth-and-death process with birth rates $\lambda_{i}>0, i \geqslant 0$, and death rates $\mu_{i}>0$, $i \geqslant 1$.

(ii) In the case where $X(t)$ is recurrent, find a sufficient condition on the birth and death parameters to ensure that

$$\lim _{t \rightarrow \infty} \mathbb{P}(X(t)=j)=\pi_{j}>0, \quad j \geqslant 0$$

and express $\pi_{j}$ in terms of these parameters. State the reversibility property of $X(t)$.

Jobs arrive according to a Poisson process of rate $\lambda>0$. They are processed individually, by a single server, the processing times being independent random variables, each with the exponential distribution of rate $\nu>0$. After processing, the job either leaves the system, with probability $p, 0<p<1$, or, with probability $1-p$, it splits into two separate jobs which are both sent to join the queue for processing again. Let $X(t)$ denote the number of jobs in the system at time $t$.

(iii) In the case $1+\lambda / \nu<2 p$, evaluate $\lim _{t \rightarrow \infty} \mathbb{P}(X(t)=j), j=0,1, \ldots$, and find the expected time that the processor is busy between two successive idle periods.

(iv) What happens if $1+\lambda / \nu \geqslant 2 p$ ?