Consider a system with two components. We observe the state of the system every hour. A given component operating at time n has probability p of failing before the next observation at time n + 1. A component that was in a failed condition at time n has a probability r of being repaired by time n + 1, independent of how long the component has been in a failed state. The component failures and repairs are mutually independent events. Let Xn be the number of components in operation at time n. The process {Xn, n = 0, 1, . . .} is a discrete time homogeneous Markov chain with state space I = 0, 1, 2.
a) Determine its transition probability matrix, and draw the state diagram.
b) Obtain the steady state probability vector, if it exists.