A group of 10 students share 2 computers. For each student the following is true: events that require a computer occur at the rate of 0.5/hour, the average time per computer use is 20 minutes. Both service and arrival processes are Poisson. When a student tries to use a computer and finds them both busy, the student wastes his or her time watching television.

Answer numerical questions with the Stochastic Analysis Add-in.

a. Construct the CTMC Matrix that describes this situation.

b. At steady-state what proportion of the total computer time are the computers idle?

c. At steady-state what proportion of the time are students wasting time watching the TV?

d. Change the situation so that a computer is allowed to fail. The average in-use time for a computer to fail is 30 hours. The in-use time only includes time when the computer is actually in use, not when it is idle. The time between failures for an individual computer has an exponential distribution. Say failed computers are not repaired. Model this situation as a CTMC. How long should the students expect to have any service from their computers (at least one operating)?

e. Use the assumptions of part d, but add the possibility of repair. Whenever a computer fails, it is immediately taken to a repair shop. The time for repair has an exponential distribution with a mean of 10 hours. Model this situation as a CTMC. Use stead-state results to determine the proportion of the time that one computer is available and the proportion of the time that two computers are available.