Question

Assume, as in Exercise 35(c), that $m$ terminals at Transend Realty are attached to a communication line linked to a computer. Suppose also that $Y$ terminals are ready to transmit, where $Y \geq 1$. Let $X$ be the number of polls required to find the first terminal in the ready state. Prove the following results (due to Russell Ham): (a) $E[X \mid Y=n]=\left(\frac{m+1}{n+1}\right)$. (b) $E\left[X^2 \mid Y=n\right]=\left[1+2\left(\frac{m-n}{n+1}\right)\right]\left(\frac{m+1}{n+1}\right)$.

   Assume, as in Exercise 35(c), that $m$ terminals at Transend Realty are attached to a communication line linked to a computer. Suppose also that $Y$ terminals are ready to transmit, where $Y \geq 1$. Let $X$ be the number of polls required to find the first terminal in the ready state. Prove the following results (due to Russell Ham):
(a) $E[X \mid Y=n]=\left(\frac{m+1}{n+1}\right)$.
(b) $E\left[X^2 \mid Y=n\right]=\left[1+2\left(\frac{m-n}{n+1}\right)\right]\left(\frac{m+1}{n+1}\right)$.

Probability, Statistics, and Queuing Theory with Computer Science Applications, Second Edition (Computer Science and Scientific Computing)

Arnold O. Allen 2nd Edition

Chapter 2, Problem 36

Instant Answer

Step 1

We have $m$ terminals, and $Y$ of these terminals are ready to transmit, where $Y$ is a random variable with $Y \geq 1$. The variable $X$ represents the number of polls required to find the first terminal that is ready to transmit. We are given that $Y = n$ for Show more…

Show all steps

Thanks for your feedback!

Assume, as in Exercise 35(c), that $m$ terminals at Transend Realty are attached to a communication line linked to a computer. Suppose also that $Y$ terminals are ready to transmit, where $Y \geq 1$. Let $X$ be the number of polls required to find the first terminal in the ready state. Prove the following results (due to Russell Ham): (a) $E[X \mid Y=n]=\left(\frac{m+1}{n+1}\right)$. (b) $E\left[X^2 \mid Y=n\right]=\left[1+2\left(\frac{m-n}{n+1}\right)\right]\left(\frac{m+1}{n+1}\right)$.

Play audio

Feedback

Ace is your personal tutor. It breaks down any question with clear steps so you can learn.

Start Using Ace

Continue solving this problem

Create a free account to:

View full step-by-step solution
Ask follow-up questions with Ace AI
Save progress and study later

Please give Ace some feedback