Question

6. (20pts) Consider two nodes, A and B, that use the slotted ALOHA protocol to contend fora shared channel. Suppose they both have unlimited frames to send. For each slot, A has probability p_A to send and B has probability p_B to send For each slot, what is the probability that A sends its frame successfully? what is the probability that B sends its frame successfully? what is the probability that either Aor B sends a frame successfully? what is the probability that both A and B send and thus collide? What is the probability that neither A or B sends a frame? What are the average throughputs of A and B, respectively? a. b.

Clarification for Problem 6b: The average throughput of A (or B) means the average number of frames of A (or B) that can be transmitted successfully during every slot.

Answer 1

In slotted aloha we divide the time into slots such that at the starting of the slot the frame will send the data on the link.

a)

If two or more hosts transmit the data at the same time then the collisions will happen.

Here we have two hosts A and B.

The probability of A to send a packet is 0.5 and the probability of A not to send packet is 0.5.

similarly, the probability of B to send a packet is 0.5 and the probability of B not to send packet is 0.5.

The probability that A sends the data without colliding = probability of A sending data * probability of B not sending

= (0.5) * (0.5) = 0.25

The probability that B sends the data without colliding = probability of B sending data * probability of A not sending

= (0.5) * (0.5) = 0.25

The probability that either A or B sends data without colliding = sum of above 2 results = 0.25 + 0.25 = 0.5

The probability that both A and B transmit data and it will collide = probability of A sending * probability of B sending

= 0.5 * 0.5 = 0.25

The probability that neither A nor B transmits data = probability of A not sending * probability of B not sending

= 0.5 * 0.5 = 0.25

b)

Throughput, S = G e-G, where G= Number of frames per T, Here G = 1

then s = 1/e = 0.368

So, the average throughput of A and B is 0.368