Question

A data packet is transmitted over a lossy wireless channel with packet error rate 0.3. Most communication systems adopt the Automatic Repeat reQuest (ARQ) protocol. That is, if a packet is successfully (with probability 0.7) or unsuccessfully (with probability 0.3) received, the receiver respectively sends a one-bit ‘ACK’ or ‘NAK’ message (over a separate feedback channel) to the transmitter. After receiving an ‘ACK’, the transmitter will move on to the next packet transmission; otherwise, the transmitter will re-transmit the same packet, and this process continues until the transmitter receives an ‘ACK’ message.

(a) [2 points] Assume the feedback channel is reliable, let X denote the total number of transmissions (including re-transmission) for one packet, calculate P(X=3) and the expectation of X. (b) [1 points] Assume the feedback channel is reliable, given the first transmission fails, re-calculate the expectation of X. (c) [2 points] Assume the feedback channel is unreliable so that the NAK/ACK feedback bit flips with probability 0.2 (i.e., with 20% of chance the transmitter receives incorrect feedback). In this case, derive the Probability Mass Function (PMF) of X.

Answer 1

in each trial ca ut p be the probability of correct transmission ane X be the number of transmissions for each packet. :XN geometric (p) with p.m.b PIX=x) = pq x=1,2,... Dow. where p=0.7. ... P(x=3 = p q 3-1 = pq² = 0.7 (0.3) = 0.063 & f(x) = a 1 = I = 1.4286 0.7

( frons = pqH! 71* >> 1) = f(x, >1) f(x>1) pan- - P(X=1 = pqxl 1-Þ final) = pqx-2 ,x=2,3,.... ELXIX>1) - & fem /*>1) - * * p xq* 3 (-1 +) que (31) q** + 3.242] 4:2 22 X