Consider a communication system that adopts the Automatic Repeat reQuest (ARQ) protocol. That is, when a packet is successfully (with probability 1- α) or unsuccessfully (with probability α) received, the receiver respectively sends the transmitter a one-bit ‘ACK’ or ‘NAK’ message over a separate feedback channel. 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) Assume the data packet of N binary bits contains k data bits and r parity check bits (i.e., N=k+r). During a packet transmission, each bit may flip with probability p (i.e., the bit error rate is p). We also know that the receiver is able to detect up to r bit flips errors (i.e., the packet will be unsuccessfully received if there are more than r bit errors). Please express α (the probability of unsuccessful packet transmission) in terms of N, r and p.(b) Let α=0.2 and assume the feedback channel is also unreliable so that the NAK/ACK feedback bit flips with probability 0.1 (i.e., with 10% of chance the transmitter receives incorrect feedback). Let Y denote the total number of transmissions required for the transmitter to send one packet, derive the Probability Mass Function (PMF) and the Expectation of Y.
Consider a communication system that adopts the Automatic Repeat reQuest (ARQ) protocol. That is, when a packet is successfully (with probability 1- α) or unsuccessfully (with probability α) received, the receiver respectively sends the transmitter a one-bit ‘ACK’ or ‘NAK’ message over a separate feedback channel. 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) Assume the data packet of N binary bits contains k data bits and r parity check bits (i.e., N=k+r). During a packet transmission, each bit may flip with probability p (i.e., the bit error rate is p). We also know that the receiver is able to detect up to r bit flips errors (i.e., the packet will be unsuccessfully received if there are more than r bit errors). Please express α (the probability of unsuccessful packet transmission) in terms of N, r and p.
(b) Let α=0.2 and assume the feedback channel is also unreliable so that the NAK/ACK feedback bit flips with probability 0.1 (i.e., with 10% of chance the transmitter receives incorrect feedback). Let Y denote the total number of transmissions required for the transmitter to send one packet, derive the Probability Mass Function (PMF) and the Expectation of Y.
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