Question

In a multiuser wireless network the data frames are being transmitted when the probability of collision during the transmission is 0.75. The transmitters try to send the collision-free frames using up to 4 independent trials (transmission) as needed. The transmitters indefinitely discard the data frames after 4 failures. Calculate the probability that a data frame can be successfully transmitted (not be discarded). Let ? denote the random number of trails until the frame is transmitted without collision, find ?(? ≤ 5) =?

Answer 1

BIONOMIAL DISTRIBUTION
pmf of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial
I.
mean = np
where
n = total number of repetitions experiment is executed
p = success probability
mean = 8 * 0.75
= 6
II.
variance = npq
where
n = total number of repetitions experiment is executed
p = success probability
q = failure probability
variance = 8 * 0.75 * 0.25
= 1.5
III.
standard deviation = sqrt( variance ) = sqrt(1.5)
=1.2247

Let ? denote the random number of trails until the frame is transmitted without collision,
the probability that a data frame can be successfully transmitted ?(? ≤ 5)
P( X < = 5) = P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= ( 8 5 ) * 0.75^5 * ( 1- 0.75 ) ^3 + ( 8 4 ) * 0.75^4 * ( 1- 0.75 ) ^4 + ( 8 3 ) * 0.75^3 * ( 1- 0.75 ) ^5 + ( 8 2 ) * 0.75^2 * ( 1- 0.75 ) ^6 + ( 8 1 ) * 0.75^1 * ( 1- 0.75 ) ^7 + ( 8 0 ) * 0.75^0 * ( 1- 0.75 ) ^8   
= 0.3215