Q # 4 Suppose three active nodes (nodes A, B and C) are competing for access to
ID: 3872496 • Letter: Q
Question
Q # 4 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 a node A succeeds for the first time in slot 4? b. What is the probability that some node (either A,B, or C) succeeds in slot 2? c. What is the probability that the first success occurs in slot 4? d. What is the efficiency of this three-node system?Explanation / Answer
a)
Ans: (1– p(A))3 p(A)
Here, p(A) = probability that A succeeds wich means p(A transmits but B&C does not)
= p(A transmits) p(B not transmit) p(C not transmit)
= p(1–p) (1–p) = p(1– p)^2
But given that p(A) succeeds first time in 4th slot.
= (1– p(A))3 p(A)
b)
p(A succeeds in 2nd slot) = p(1-p)
p(B succeeds in 2nd slot) = p(1-p)
p(C succeeds in 2nd slot) = p(1-p)
d)efficiency = p(success in a slot)
= 3p(1-p)2
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