Q1. Consider a slotted ALOHA type network in which each node attempts to transmi

Question

Q1. Consider a slotted ALOHA type network in which each node attempts to transmit a frame in each slot with probability p. It is assumed here that each node always has a frame to send and that the node transmits with probability p for a fresh frame as well as for a frame that has already suffered a collision. Suppose there are N nodes. Then the probability that a given slot is a successful slot is the probability that one of the nodes transmits and that the remaining N-1 nodes do not transmit. The probability that a given node transmits is p; the probability that the remaining nodes do not transmit is (1-p)N-1. Therefore the probability a given node has a success is p(1-p) Because there are N nodes, the probability that any one of the N nodes has a success is Mp(1-p)N-1. This is also the efficiency of this slotted ALOHA protocol. Given that there are N participants, determine p that maximizes this efficiency. Also determine the maximum efficiency for th Hint: To determine p that maximizes this efficiency, take a derivative of Mp(1-p)N-1 with respect to p and then find the value of p for which this derivative is zero. For the second part of the question, substitute this determined value ofp in Np(1-py-and then determine what will you get as N approaches infinity

Explanation / Answer

Given

E'(p) = N(1-p) N-1 - N (N-1) (1-p)N-2

E'(p) = N(1-p) N-2 ( 1 - p - p(N-1))

E'(p)= 0 **** to find maximum value of efficiency *****

so ,

N (1-p) N-2 ( 1 - p - p(N-1)) =0

(1-p) N-2 =0 => p=1.

( 1 - p - p(N-1)) =0 => 1-p(1+n-1)=0 => 1-p(n)=0 => p=1/n

E(p)= N (1/N) (1- (1/N) )N-1

E(p) = (1 - (1/N)) N-1 => (1- (1/N))N / (1-(1/N)) -----------1)

lim N->infinite (1-(1/N))N = 1/e

lim N->infinite (1-(1/N)) = 1

so put them both in eq 1

E(p) = 1/e

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