36. Each of n skiers continually, and independently, climbs up and then skis down a particular slope.
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36. 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}?
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