In a manufacturing setting, Markov chains can be useful for describing throughput for certain processes. One example of this is a repair process at an electronics manufacturer. For this process, that states of the chain are called "stages". A device enters through "receive" stage and then is moved to the "debug" stage. Depending on the test results from "debug", it is moves to "scrap" or "repair" stages. From "repair", it moves to "test". From the "test" stage, the device will move to either "ship" or back to "debug". About 40% of the time, the debugging stage fails and the device is scrapped. About 80% of the time, the testing phase succeeds and the device is shipped. The average processing times for each transient stage are 10 minutes in receiving, 1 hour in debugging, 30 minutes in repair, and 15 minutes in testing. (The "scrap" and "ship" stages are absorbing stages, and formally have infinite processing times.)

(a)What is the probability that a device in the debug stage eventually returns to the debug stage?

(b)What is the expected number of times a device will enter the debug stage? (Hint: the number of returns is geometrically distributed)

(c)The total load of a device is the expected number of hours needed to handle each device before it is scrapped or shipped. Calculate the total load under the above model description.