(i) Define an $M / M / 1$ queue. Justifying briefly your answer, specify when this queue has a stationary distribution, and identify that distribution. State and prove Burke's theorem for this queue.

(ii) Let $\left(L_{1}(t), \ldots, L_{N}(t), t \geqslant 0\right)$ denote a Jackson network of $N$ queues, where the entrance and service rates for queue $i$ are respectively $\lambda_{i}$ and $\mu_{i}$, and each customer leaving queue $i$ moves to queue $j$ with probability $p_{i j}$ after service. We assume $\sum_{j} p_{i j}<1$ for each $i=1, \ldots, N$; with probability $1-\sum_{j} p_{i j}$ a customer leaving queue $i$ departs from the system. State Jackson's theorem for this network. [You are not required to prove it.] Are the processes $\left(L_{1}(t), \ldots, L_{N}(t), t \geqslant 0\right)$ independent at equilibrium? Justify your answer.

(iii) Let $D_{i}(t)$ be the process of final departures from queue $i$. Show that, at equilibrium, $\left(L_{1}(t), \ldots, L_{N}(t)\right)$ is independent of $\left(D_{i}(s), 1 \leqslant i \leqslant N, 0 \leqslant s \leqslant t\right)$. Show that, for each fixed $i=1, \ldots, N,\left(D_{i}(t), t \geqslant 0\right)$ is a Poisson process, and specify its rate.