6- In a wireless system with Slotted Aloha access method, the transmission inter

Question

6- In a wireless system with Slotted Aloha access method, the transmission interval is T-1ms and the probability of transmitting a packet by a user in a time slot of length T: [t, 1 + T] is P.-596. Calculate the following quantities in a system with n-10 users. [20 points] The probability of success for one use (for instance for user #7) The probability of successful transmission in a time slot by any of the users: p(user#1 success)+ success) Find the optimal value of pe that maximizes the success probability? (provide details of derivations) a) b) p(user#10 c)

Explanation / Answer

Given,

Pt=5%= 0.05

n=10 user

a) The probability of success for one user(say, user #7)= ((1-Pt)n-1) Pt

= (1-0.05)7-1 * 0.05 = 0.03675

b) The probability of successfull transmission for any user = Pt +(1-Pt) Pt? +....................+ ((1-Pt)n-1) Pt

= Pt +(1-Pt) Pt? +....................+ (1-Pt)9) Pt

=Pt [ 1+(1-Pt) +....................+ (1-Pt)9]

let,

x=(1-Pt)= 0.95

Formula 1+ x+ x2 +......... +xn = 1-Xn +1 / 1- X

= Pt [1-Xn +1 / 1- X]

=0.05 *[ 1-0.5987 / 1- 0.95]

=0.05*8.026

=0.4013

C) optimal value of probability is P= 1 i.e., Smax = 0.36

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