In: Computer Science
(a) Give the probability that A wins this second back-off race
immediately after his first collision.
Answer:------------
For the second back-off race,
A picks kA(2) to be either 0 or 1 with equal
probability,so 1/2 for each.
B picks kB(2) from (0, 1, 2, 3) with probability 1/4 for
each choice.
A wins the second backoff race if kA(2) <
kB(2).
P[A wins] = P[kA(2) < kB(2)]
= P[kA(2) = 0] × P[kB(2) > 0] +
P[kA(2) = 1] × P[kB(2) > 1]
= 1/2 × 3/4 + 1/2 × 2/4
= 3/8 + 2/8
= 5/8
(b) Suppose A wins this second back-off race. A transmits ?2 and
when it is finished, A and B collide again as A tries to transmit
?3 and B tries once more to transmit ?1. Give the probability that
A wins this third back-off race immediately after the first
collision.
Answer:------------
In this case, again A picks kA(3) to be either 0 or 1
with probability 1/2 each,
while B picks kB(3) from (0, 1, 2, 3, 4, 5, 6, 7), each
with probability 1/8:
P[A wins] = P[kA(3) < kB(3)]
= P[kA(3) = 0] × P[kB(3) > 0] +
P[kA(3) = 1] × P[kB(3) > 1]
= 1/2 × 7/8 + 1/2 × 6/8
= 7/16 + 6/16
= 13/16