A machine shop operates two identical machines,supervised by a single operator. Each machine requires a service, independent of the other, at random points in time. Theprobabilitythatoneofthemachinesrequiresaserviceinanyveminute periodisp = 0.4. The operator is able to service one machine in 5minutes. Letus approximate the system by dividing the whole period of operation into 5-minute blocks and assume that any machine failure occurs at the start of a block.

(a) Construct a transition diagram for this problem and write down the associated transition matrix. Hint: consider the number of machines that need xing in any given 5 minute period.

(b) If both machines are operating at 11 AM, nd the probabilities that they are operating at 11:05 AM and at 11:10 AM.

(c) Find the steady state probabilities for this process.