Question

Relays are commonly used in communication systems to extend the range of a transmitter (source), but they are also used to increase link reliability. A decode-and-forward (DF) relay receives a message from the source, checks for errors, and then retransmits the same message to the receiver (destination). The destination now has two chances to decode the message without errors: on the direct source-destination (s-d) link and over the source-relaydestination (s-r-d) path. If an error occurs on both the s-d link and the s-r-d path, then the message transmission fails. Suppose that a radio (destination) can successfully receive a message from the radio tower (source) directly on the s-d link with probability p0; the relay can successfully receive the message from the radio tower (source) on the s-r link with the probability p1; and the radio (destination) can successfully receive a message from the relay on the r-d link with probability p2. Assume that the s-d, s-r, and r-d links a independent. (a) What is the probability that at least one copy of a message transmitted from the source (s) reaches the receiver (r).? (b) If p0 = 0.3, p1 = 0.8 and p2 = 0.7, how much more reliable is the system with the relay than without the relay (i.e., what is the improvement in the probability of success)?

Answer 1

Probability that the message is successfully transmitted from the transmitter to the reciever in the system with relay = $p_{1}p_{2}$

(a) Probability that at least one copy of a message transmitted from the source reaches the reciever

= 1 - P(none of the copy of the message transmitted from the source reaches the reciever)

= 1 - ()

po * P1P2

=

1- popipz

(b) Probability that the message is successfully transmitted from the transmitter to the reciever in the system with relay = $p_{1}p_{2}$ = 0.56

   Probability that the message is successfully transmitted from the transmitter to the reciever in the system without relay

= $p_{0}$ = 0.3

Thus, there is an improvement of probability of 0.26 as the probability of success (or 0.56/0,30 = 1.867 times more compared to direct transmission)