35. Satellites are launched according to a Poisson process with rate . Each satellite will, independently, orbit

35. Satellites are launched according to a Poisson process with rate λ. 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.

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

Hint: Relate this to the M/G/∞ queue.

(b) 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

(a) when k = 0.