Chapter 6, Problem P20P

a.

Explanation of Solution

Given:

Frames have constant length (L) = kRS

Where;

k – Constant value

R – Transmission rate

S – Length of the slot

Consider that the propagation delay is less than the length of the slot.

To find: Efficiency of the protocol for the fixed “N” (Nodes) and “p” (probability).

Solution:

Consider the random variable “A” that denotes the number of slots is ${\text{P(A=m)=β(1-β)}}^{\text{m-1}}$. Here, $\text{β}$ is the probability of success.

This is considered as the geometric distribution, which has the mean $\frac{\text{1}}{\text{β}}$. The consecutive wasted slots is $\text{W = A - 1}$. That is,

$\text{x = E[W]}$

Substitute the value of “W” in the above equation. We get,

$\begin{array}{l}\text{x = E[A-1]}\\ \text{x = }\frac{\text{1-β}}{\text{β}}\\ {\text{β=Np(1-p)}}^{\text{N-1}}\end{array}$

Formula to find the efficiency is: $\text{x=}\frac{\text{k}}{\text{k+x}}$

b.

Explanation of Solution

Given:

Frames have constant length (L) = kRS

Where;

k – Constant value

R – Transmission rate

S – Length of the slot

Consider that the propagation delay is less than the length of the slot.

To find: To find the probability (p) that maxims the efficiency for the fixed number of nodes (N).

Solution:

By minimizing “x”, the efficiency can be maximized which is equivalent to maximize $\text{β}$

c.

Explanation of Solution

Given:

Frames have constant length (L) = kRS

Where;

k – Constant value

R – Transmission rate

S – Length of the slot

Consider that the propagation delay is less than the length of the slot.

To find: Efficiency that approaches to infinity.

Solution:

We know that: $\text{Efficiency = }\frac{\text{k}}{\text{k+}\frac{{\text{1-Np(1-p)}}^{\text{N-1}}}{{\text{Np(1-p)}}^{\text{N-1}}}}$

Substitute $\frac{\text{1}}{\text{e}}$ for ${\text{Np(1-p)}}^{\text{N-1}}$

d.

Explanation of Solution

Given:

Frames have constant length (L) = kRS

Where;

k – Constant value

R – Transmission rate

S – Length of the slot

Consider that the propagation delay is less than the length of the slot.

To find: Efficiency that approaches to 1...