Question 7.EX.35: Satellites are launched according to a Poisson process with ......
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.
(a) We can view this as an M / G / \infty system where a satellite launching corresponds to an arrival and F is the service distribution. Hence,
P\{X(t)=k\}=e^{-\lambda(t)}[\lambda(t)]^k / k !
where \lambda(t)=\lambda \int_0^t(1-F(s)) d s.
(b) By viewing the system as an alternating renewal process that is on when there is at least one satellite orbiting, we obtain
\lim P\{X(t)=0\}=\frac{1 / \lambda}{1 / \lambda+E[T]}
where T, the on time in a cycle, is the quantity of interest. From part (a)
\lim P\{X(t)=0\}=e^{-\lambda \mu}
where \mu=\int_0^{\infty}(1-F(s)) d s is the mean time that a satellite orbits. Hence,
e^{-\lambda \mu}=\frac{1 / \lambda}{1 / \lambda+E[T]}
so
E[T]=\frac{1-e^{-\lambda \mu}}{\lambda e^{-\lambda \mu}}