A part processed in a flexible manufacturing system

(FMS) is routed through a set of operations, some of which are sequential and some of which are parallel. In addition, an FMS operation can be processed by alternative machines. An article in IEEE Transactions (Mar.

1990) gave an example of an FMS with four machines operating independently. The repair rates for the machines (i.e., the time, in hours, it takes to repair a failed machine) are exponentially distributed with means p, = 1, p2 = 2, p, = .5, and p, = .5, respectively.

a. Find the probability that the repair time for machine 1 exceeds one hour.

b. Repeat part a for machine 2.

c. Repeat part a for machines 3 and 4.

d. If all four machines fail simultaneously, find the probability that the repair time for the entire system exceeds one hour.