Each of n skiers continually, and independently, climbs up and then skis down a particular slope. The time it takes skier i to climb up has distribution Fi , and it is independent of her time to ski down, which has distribution Hi, i = 1, . . . , n.

Let N(t) denote the total number of times members of this group have skied down the slope by time t . Also, let U(t) denote the number of skiers climbing up the hill at time t .

(a) What is limt→∞N(t)/t?

(b) Find limt→∞E[U(t)].

(c) If all Fi are exponential with rate λ and all Gi are exponential with rate μ, what is P{U(t) = k}?