A computer sends a "connection request" message to the server and should receive $"$ac$-$

knowledgment" signal from the server for keeping the Internet connection of $10$ minutes. How$-$

ever, the computer could receive the "FAIL" message with the probability $\frac{2}{3},$ and if so$,$ the

computer should resend the "connection request" message to the server. Let's say it takes $10$

seconds to send the "connection request" message and to receive the "acknowledgment" or

"FAIL" message. Then, answer the following questions.

$($a$)$ When should the computer start sending "connection request" message to have a $99\%$

chance of successfully keeping the Internet for $10$ minutes.

$($b$)$ Find the mean and variance of the time that it takes to keep the Internet connection.