*28. Satellites are launched according to a Poisson process with rate A.

Each satellite will, independently, orbit the earth for a random time having distribution F. Let X{t) denote the number of satellites orbiting at time t.

(i) Determine P{X(t) = k}.

Hint: Relate this to the M/G/oo queue.

(ii) If at least one satellite is orbiting, then messages can be transmitted and we say that the system is functional. If the first satellite is orbited at time t = 0, determine the expected time that the system remains functional.

Hint: Make use of part (i) when k = 0.