The importance of modeling machine downtime correctly in simulation studies was discussed in Industrial Engineering (Aug. 1990). The paper presented simulation results for a single-machine tool system with the following properties:

    • The interarrival times of jobs are exponentially distributed with a mean of 1.25 minutes.

    • The amount of time the machine operates before breaking down is exponentially distributed with a mean of 540 minutes.

    • The repair time (in minutes) for the machine has a gamma distribution with parameters α = 2 and β = 30.

a. Find the probability that two jobs arrive for processing at most 1 minute apart.

b. Find the probability that the machine operates for at least 720 minutes (12 hours) before breaking down.

c. Find the mean and variance of the repair time for the machine. Interpret the results.

d. Find the probability that the repair time for the machine exceeds 120 minutes.