A data transmission system uses messages of duration T seconds. After each messa
ID: 3133451 • Letter: A
Question
A data transmission system uses messages of duration T seconds. After each message transmission, the transmitter stops and waits T seconds for a reply from the receiver. The receiver immediately replies with a message indicating that a message was received correctly. The transmitter proceeds to send a new message if it receives a reply within T seconds; otherwise, it re-transmits the previous message. Suppose that messages can be completely garbled while in transit and that this occurs with probability p. Find the maximum possible rate at which messages can be successfully transmitted from the transmitter to the receiver.Explanation / Answer
use Poisson random variable with a parameter which is usually used to count the number of eventswith a certain time interval. is the average numberof event occurrences in a specified time interval, which is There. Here will refer to the average number offailures occurred during T.
Based on geometric distribution, the failure probability would be:1- (1-p)2 since (1-p)2 is successprobability when there is no garbled messages for the originalmessage transmitted and ACK. Then, the average number offailure before the first success is
[1-{1- (1-p)2}]/1-(1-p)2=(1-p)2/(2p-p2)
Then in a Poisson distribution,
= (1-p)2/(2p-p2)/T --> actually Iam not sure this part to represent average # of failures in aspecific time interval.
When k =0, meaning the probability with no failure occurred:
e-=exp{(-(1-p)2/(2p-p2)/T}
When there is no failure, it means a message can bedelivered successfully.
This probability will follow a geometric distribution again duringT seconds. Then the average of this maximum is for k = 1, 2,3, 4....
1/e- =exp((1-p)2/(2p-p2)/T) -->maximum possiblle