In: Statistics and Probability
A device has two states: ON and OFF. When it works normally, it spends a random time in the ON state, then a random time in the OFF state, then again in the ON state, and so on. However, when switched from OFF to ON, the device fails with probability 0.05 and a replacement process is triggered immediately. Once a new device is in place, it is turned ON. Suppose that the average durations in the ON and OFF states are 40 and 20 minutes, respectively, and that it takes 20 minutes to replace a failed device.
(a) In the long run what is the rate at which devices are replaced?
(b) In the long run what is the fraction of time that the device is ON?
a)
The number of cycles the device undergoes before failing is distributed as Geometric(0.05) as it fails with p = 0.05 after every cycle. Each cycle takes 1 hour to complete. Thus, if N is the number of cycles before failure,
as given for a Geometric distribution with parameter p, taking values in 1, 2,..
Thus , expected number of cycles before device needs to be replaced is 20. Therefore, expected time before device needs to be replaced is 20 hrs or 1200 minutes. Including the 20 minutes of replacement time, a new device starts working, on an average 1220 minutes after the previous one.
b)
The device is either on, off, or being replaced. On an average, the device spends 20 minutes out of 1220 minutes under repair. Thus, the remaining 1200 out of 1220 minutes is divided between On and Off. The time we spend in On, is double the amount of time we spends in Off. Thus, we spend 800 out of 1220 minutes in On, i.e.