Chapter 6, Problem P11P

a.

Explanation of Solution

Given:

Consider that there are four nodes “A”, “B”, “C”, and “D”. These four nodes are active and competing for access to a channel using slotted ALOHA. Each nodes have infinite number of packets.

To find: Probability (p) that the node “A” succeeds for the first time in slot number 5.

Solution:

The probability (p) that the node “A” succeeds for the first time in slot number 5 is ${\text{(1-p(A))}}^{\text{4}}\text{p(A) --- (1)}$.

Here,

p (A) is the probability that the node “A” succeeds in any slot. That is: p (A transmits, B, C, and D does not transmit).

$\text{p(A)=p}\left(\text{A transmit}\right)\text{ * p}\left(\text{B does not transmit}\right)\text{ * p}\left(\text{C does not transmit}\right)\text{ * p(D does not transmit)}$

b.

Explanation of Solution

Given:

Consider that there are four nodes “A”, “B”, “C”, and “D”. These four nodes are active and competing for access to a channel using slotted ALOHA. Each nodes have infinite number of packets.

To find: Probability (p) that any of the nodes “A”, “B”, “C”, or “D” succeeds in slot number 4.

Solution:

From (a), it is clear that if any node succeeds in any slot, the probability is ${\text{p(node)=p(1-p)}}^{\text{3}}$.

So,

• The probability that the node “A” succeeds in slot number 4 is: ${\text{p(A)=p(1-p)}}^{\text{3}}$
• The probability that the node “B” succeeds in slot number 4

c.

Explanation of Solution

Given:

Consider that there are four nodes “A”, “B”, “C”, and “D”. These four nodes are active and competing for access to a channel using slotted ALOHA. Each nodes have infinite number of packets.

To find: Probability (p) that 1^st success occurs on slot number 3.

Solution:

Probability of some node occurs in a slot = ${\text{4p(1-p)}}^{\text{3}}$

Probability of no node succeeds in a slot = ${\text{1-4p(1-p)}}^{\text{3}}$

Probability of the first success occurs in slot number 3 is determined by multiplying the probability of some node succeeds in a slot and probability of no node succeeds in a slot.

Let p(S) = probability of the 1^st success occurs in slot number 3

d.

Explanation of Solution

Given:

Consider that there are four nodes “A”, “B”, “C”, and “D”. These four nodes are active and competing for access to a channel using slotted ALOHA. Each nodes have infinite number of packets.

To find: Efficiency of the four node system.

Solution:

The efficiency of the four node system is determined by the success of any nodes succeeds in a slot...