Consider a communications path through a packet switching network with bit errors that occur

independently with a probability of$6$

$10$

$$eP $.$ Packets travelling through the network are a fixed

length of $864$ bits consisting of the following fields:

Destination Address $32$ bits

Information Field $800$ bits

CRC $32$ bits

$($a$)$ What is the probability that a packet is lost before arriving at the destination? $($Assume that a

packet is lost if the destination address is corrupted during transit. You may assume that

the probability that a packet is lost due to queue overflow at switches is negligible.$)$ What

is the probability that a packet arriving at the destination is detected as having errors,

assuming that the CRC is capable of detecting all errors?

$($b$)$ How many packets are expected to be sent before a packet needs to be retransmitted?

$($c$)$ Assume that $1000$ packets are to be sent. What is the probability that three or more packets

need to be retransmitted?

$($d$)$ Assuming that the CRC is capable of detecting one, two, or three bit error patterns, what is

the probability that a received packet has errors that are detected? What is the probability

that errors are not detected? Comment.

Question $2$

A communications protocol implements an error control scheme by sending the information and

CRC fields of a packet $3$ times and performing a majority vote on each bit of the fields. For

example, at the receiver, bit i of the information field is determined by a majority vote on the $3$

copies of bit i$.$ For the protocol, each packet is $864$ bits long, consisting of a $32-$bit destination

field $($which is not triplicated, but is only sent once$),$ an $800-$bit information field and a $32-$bit

CRC$.$ At the transmitter, the CRC scheme is applied to the packet information field before the

bits are triplicated..

$($a$)$ Assuming that bit errors occur independently in the communications channel with a

probability of$6$

$10$

$$eP $,$ determine the probability that a bit in the information field is in

error after the majority vote is applied at the receiver.

$($b$)$ Under this scenario, repeat Question $1($a$)$ and $($b$).$