Question: A small machine shop consists of three sensitive machines that break down frequently but independently from each other. The company has two service technicians on standby to repair the machines as soon as they break down. For each fully functional machine, the breakdown rate is 0.1 times/h and the time between breakdowns is exponentially distributed. The service technicians work independently of one another, and due to restricted space around the machines, only one technician at a time can work on a machine. For each technician the average service rate is 0.2 machines/h, and the service times follow an exponential distribution.

The operations manager has realized that this system can be modeled as a birth-and-death process with state-dependent arrival and service rates. He has also figured out what these arrival and service rates are, expressed in λ = the machine breakdown rate (the same for all machines) and µ = the service rate for each of the technicians. Table 6.7 shows a summary of what he has discovered.

a. Unfortunately, the operations manager lacks training in analyzing queuing systems, and his analysis of the arrival and service rates is not entirely correct. Show the correct service and arrival rates for the system by drawing a state/rate diagram. The operations manager is very stubborn and does not admit that he is wrong, so he orders you to continue the analysis based on the arrival rates he has specified.

b. Construct the state/rate diagram corresponding to the given arrival and service rates. Develop the balance equations and solve them to determine the steady-state probability distribution. What is the probability that all the machines are functioning?

c. Calculate the expected number of broken machines and the expected time for a machine to be operating again after it has broken down.