Consider a flexible manufacturing system in which 10 parts are always in process. Each part requires two operations. Each part begins by having operation 1 done at machine 1.

Then, with probability .75 the part has operation 2 processed on machine 2, and with probability .25 the part has operation 2 processed on machine 3. Once a part completes operation 2, the part leaves the system and is immediately replaced by another part. We are given the following machine rates (the time for each operation is exponentially distributed):

m1  .25 minute, m2  .48 minute, and m3  .08 minute.

a Find the probability distribution of the number of parts at each machine.

b Find the expected number of parts present at each machine.

c What fraction of the time is each machine busy?

d How many parts per minute are completed by each machine?