SHARE
SPREAD
HELP

The Tradition of Sharing

Help your friends and juniors by posting answers to the questions that you know. Also post questions that are not available.

To start with, Sr2Jr’s first step is to reduce the expenses related to education. To achieve this goal Sr2Jr organized the textbook’s question and answers. Sr2Jr is community based and need your support to fill the question and answers. The question and answers posted will be available free of cost to all.

The pleasure of all reading is doubled when one lives with another who shares the same. -Katherine Mansfield

 

#
Authors:
James F. Kurose, Keith W. Ross
Chapter:
The Link Layer: Links,access Networks, And Lans
Exercise:
Problems
Question:11 | ISBN:9780132856201 | Edition: 6

Question

Suppose four active nodes—nodes A, B, C and D—are competing for access to a channel using slotted ALOHA. Assume each node has an infinite number of packets to send. Each node attempts to transmit in each slot with probability p. The first slot is numbered slot 1, the second slot is numbered slot 2, and so on. a. What is the probability that node A succeeds for the first time in slot 5?

b. What is the probability that some node (either A, B, C or D) succeeds in slot 4?

c. What is the probability that the first success occurs in slot 3?

d. What is the efficiency of this four-node system?

TextbookTextbookTextbookTextbookTextbookTextbookTextbookTextbookTextbookTextbookTextbookTextbookTextbookTextbookTextbookTextbookTextbook

Answer

a.

To calculate the probability that node A succeeds for the first time in slot 5, we need to consider the probability that A does not succeed in the first four slots and succeeds in the fifth slot.

The probability that A does not succeed in a slot is (1 - p), and the probability that it succeeds is p.

Therefore, the probability that A does not succeed in the first four slots is (1 - p)^4.

The probability that A succeeds in the fifth slot is p.So, the probability that node A succeeds for the first time in slot 5 is:\small P(A succeeds in slot 5) = (1 - p)^4 \times p

 

b.

To calculate the probability that some node (either A, B, C, or D) succeeds in slot 4, we can consider the complementary probability of no node succeeding in slot 4.

The probability that a specific node does not succeed in a slot is (1 - p).

Since there are four nodes, the probability that none of them succeeds in a slot is \small (1 - p)^4.

Therefore, the probability that at least one node succeeds in slot 4 is \small 1 - (1 - p)^4.

 

 

c.

To calculate the probability that the first success occurs in slot 3, we need to consider the probability that no node succeeds in the first two slots and at least one node succeeds in slot 3.

The probability that a specific node does not succeed in a slot is (1 - p).

The probability that no node succeeds in the first two slots is\small (1 - p)^2.

The probability that at least one node succeeds in slot 3 is \small 1 - (1 - p)^4 (similar to the previous scenario).

So, the probability that the first success occurs in slot 3 is:

\small P(first success in slot 3) = (1 - p)^2 [1 - (1 - p)^4]

 

d.

The efficiency of the four-node system in slotted ALOHA can be defined as the average channel utilization. Since each node has an infinite number of packets to send, the system is continuously transmitting.

In slotted ALOHA, the probability of a successful transmission in a slot is p.

Therefore, the expected number of successful transmissions in a slot is p.

In this four-node system, there are four slots per unit time.

So, the expected number of successful transmissions in four slots is 4p.

Since there are four nodes, the total number of attempted transmissions in four slots is 4.

Therefore, the efficiency of the four-node system is given by:

Efficiency = (Expected number of successful transmissions) / (Total number of attempted transmissions) = 4p / 4 = p

Thus, the efficiency of the four-node system is equal to the transmission probability, p.

0 0

Discussions

Post the discussion to improve the above solution.