(a) Define a renewal process $\left(X_{t}\right)$ with independent, identically-distributed holding times $S_{1}, S_{2}, \ldots .$ State without proof the strong law of large numbers for $\left(X_{t}\right)$. State without proof the elementary renewal theorem for the mean value $m(t)=\mathbb{E} X_{t}$.

(b) A circular bus route consists of ten bus stops. At exactly 5am, the bus starts letting passengers in at the main bus station (stop 1). It then proceeds to stop 2 where it stops to let passengers in and out. It continues in this fashion, stopping at stops 3 to 10 in sequence. After leaving stop 10, the bus heads to stop 1 and the cycle repeats. The travel times between stops are exponentially distributed with mean 4 minutes, and the time required to let passengers in and out at each stop are exponentially distributed with mean 1 minute. Calculate approximately the average number of times the bus has gone round its route by $1 \mathrm{pm}$.

When the driver's shift finishes, at exactly $1 \mathrm{pm}$, he immediately throws all the passengers off the bus if the bus is already stopped, or otherwise, he drives to the next stop and then throws the passengers off. He then drives as fast as he can round the rest of the route to the main bus station. Giving reasons but not proofs, calculate approximately the average number of stops he will drive past at the end of his shift while on his way back to the main bus station, not including either the stop at which he throws off the passengers or the station itself.