Question

Suppose three active nodes–nodes a, b, and c–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? (Hint: Node a succeeds for the first time in slot 5 means: (1) a doesn't succeed for the first 4 slots; and (2) a succeeds in slot 5.)
b. What is the probability that the first success occurs in slot 3? (Hint: The first success occurs in slot 3 means no nodes succeed in the first two slots and one node succeeds in slot 3.)

Answer 1

To calculate the probability of node a succeeding for the first time in slot 5, multiply the probability of node a not succeeding in the first 4 slots with the probability of succeeding in slot 5. The probability that the first success occurs in slot 3 is found by multiplying the probability of no nodes succeeding in the first two slots with the probability of one node succeeding in slot 3.

Step-by-step explanation:

a. To calculate the probability that node a succeeds for the first time in slot 5, we need to find the probability that node a doesn't succeed in the first 4 slots and succeeds in slot 5. The probability that node a doesn't succeed in a slot is (1-p), where p is the probability of a successful transmission. Therefore, the probability of node a not succeeding in the first 4 slots is (1-p)^4. The probability that node a succeeds in slot 5 is p. So, the overall probability is (1-p)^4 * p.

b. Similarly, to find the probability that the first success occurs in slot 3, we need to find the probability that no nodes succeed in the first two slots and one node succeeds in slot 3. The probability of no nodes succeeding in a slot is (1-p), and the probability of one node succeeding is p. Therefore, the probability that no nodes succeed in the first two slots is (1-p)^2, and the probability of one node succeeding in slot 3 is p. So, the overall probability is (1-p)^2 * p.