A small company has two servers: A and B. The time to failure for server A is exponentially distributed with a mean of 1/λA hours, and the time to failure for server B is exponentially distributed with a mean of 1/λB hours. The servers also have different repair times. The time to repair server A when it fails is exponentially distributed with a mean if 1/μA hours and the time to repair server B when it fails is exponentially distributed with a mean of 1/μB hours. There is only one repair person available to work on both servers when failure occurs, and each server is considered as good as new after it has been repaired.
a. Give the state-transition-rate diagram of the process.
b. What is the probability that both servers are down?
c. What is the probability that server A is the first to fail given that both servers have failed? d. What is the probability that both servers are up?