Suppose four active nodes—nodes A, B, C and D—are competing for access to a channel using slottedALOHA. Assume each node has an infinite number of packets to send. Each node attempts to transmitin 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 normalized throughput of this four-node system?
Question
Suppose four active nodes—nodes A, B, C and D—are competing for access to a channel using slottedALOHA. Assume each node has an infinite number of packets to send. Each node attempts to transmitin 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 normalized throughput of this four-node system?
Solution
a. The probability that node A succeeds for the first time in slot 5 is the probability that node A transmits in slot 5 and no other node transmits in slot 5, and that node A does not transmit in slots 1 through 4. This is given by:
P(A succeeds in slot 5) = P(A transmits in slot 5) * P(no other node transmits in slot 5) * P(A does not transmit in slots 1 through 4)
= p * (1-p)^3 * (1-p)^4
b. The probability that some node (either A, B, C or D) succeeds in slot 4 is the probability that exactly one node transmits in slot 4. This is given by:
P(some node succeeds in slot 4) = P(exactly one node transmits in slot 4)
= 4 * p * (1-p)^3
c. The probability that the first success occurs in slot 3 is the probability that exactly one node transmits in slot 3 and no node transmits in slots 1 and 2. This is given by:
P(first success in slot 3) = P(exactly one node transmits in slot 3) * P(no node transmits in slots 1 and 2)
= 4 * p * (1-p)^3 * (1-p)^8
d. The normalized throughput of this four-node system is the expected number of successful transmissions per slot. In slotted ALOHA, the throughput is maximized when p = 1/N, where N is the number of nodes. So the normalized throughput is:
Throughput = N * p * (1-p)^(N-1)
= 4 * (1/4) * (1-(1/4))^(4-1)
= 1/2
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