A data transmission system uses messages of duration T seconds. After the messag
ID: 3332387 • Letter: A
Question
A data transmission system uses messages of duration T seconds. After the 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 retransmits 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
There are two components in the system a Transmitter and Receiver.
Suppose that message m is transmitted by transmitter at time instant t. There are two possibilities
The probability that message delivery fails is given by p (garbled message). The probability that message delivery is successful is (1-p).
After sending message the transmitter waits for T seconds and either retransmits the old message. The second case can happen if receiver indicates failure of message or acknowledgement is not received in time period T.
The data transmission rate will decrease either if more messages are garbled or acknowledgement is not received. Each can occur with probability p. We can treat events message garbled and acknowledgement garbled as independent. Then probability that message will have to be retransmitted is given by p(2-p).
This case can be captured by Binomial scenario as follows.
The number of messages which will be retransmitted follows Binomial distribution with n and p(2-p).
The expected number of retransmitted messages is np(2-p).
The rate of retransmission is given by (number of messages transmitted/ time period). In this case rate of retransmission ® will be obtained and will be R = [ np(2-p)/D].
In order to maximize rate of successful data transmission we must minimize R.