A small specialized workshop is used to perform periodic check-up on some high-tech devices. The check up is made in three stages whose durations are independent random variables with geometric distribution with averages of 2, 3 and 1 days, respectively, for the 1st, 2nd and 3rd stage. The workshop issmall, only one device can be processed at once. In addition, when a step is completed, the next step of the work can not begin until the beginning of the next day business. When step 3 is completed, the development of the device is terminated and the Work resumed on another device at the beginning of the next business day. Let Xn be the step During the day number n.
(A) show that {Xn, n = 0, 1, 2,. . .} Is an irreducible Markov chain and build its matrix of transition probabilities.
(B) Calculate its stationary probabilities. Note: no matrix inversion is required.
(C) Suppose that the first step requires the presence of three (3) technicians while the second stage involves one(1),and third requires two (2). On average how many technicians are working each day?
(D) On average, how many technician-day does it take to make the check-up of a device like this?