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.

Question

(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.

Accepted Answer

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

Question 7.EX.35: Satellites are launched according to a Poisson process with ......

This Question has been Answered.

Already have an account?

Login

Related Answered Questions

For an interarrival distribution F having mean μ, we defined the equilibrium distribution of F, denoted Fe, by Fe(x) =1/μ∫0^x[1-F(y)]dy (a) Show that if F is an exponential distribution, then F = Fe. (b) If for some constant c, F(x)={0, x<c 1, x ≥ c show that Fe is the uniform distribution ...

Verified Answer:

Verified Answer:

Compute the renewal function when the interarrival distribution F is such that 1−F(t) = pe^−μ1t + (1 −p)e^−μ2t ...

Verified Answer:

A machine in use is replaced by a new machine either when it fails or when it reaches the age of T years. If the lifetimes of successive machines are independent with a common distribution F having density f , show that (a) the long-run rate at which machines are replaced equals ...

Verified Answer:

If the mean-value function of the renewal process {N(t), t ≥ 0} is given by m(t) = t/2, t ≥ 0, what is P{N(5) = 0}? ...

Verified Answer:

Compute E[T], the expected time until the pattern h, h, h, t , h, h, h appears, when a coin that comes up heads with probability p and tails with probability q = 1− p is continually flipped. ...

Verified Answer:

Consider a machine that can be in one of three states: good condition, fair condition, or broken down. Suppose that a machine in good condition will remain this way for a mean time μ1 and then will go to either the fair condition or the broken condition with respective probabilities 3/4 and 1/4 ...

Verified Answer:

The rate a certain insurance company charges its policyholders alternates between r1 and r0. A new policyholder is initially charged at a rate of r1 per unit time. When a policyholder paying at rate r1 has made no claims for the most recent s time units, then the rate charged becomes r0 per unit ...

Verified Answer:

Verified Answer:

Although a system needs only a single machine to function, it maintains an additional machine as a backup. A machine in use functions for a random time with density function f and then fails. If a machine fails while the other one is in working condition, then the latter is put in use and, ...

Verified Answer: