Two-State Markov Chain. Consider a machine (or a production system) that alternates, in discrete time, between being in operation (state 1) or being down for repair or reloading (state 2). The...

Two-State Markov Chain. Consider a machine (or a production system) that alternates, in discrete time, between being in operation (state 1) or being down for repair or reloading (state 2). The successive durations of time during which the machine is in operation are i.i.d. and the successive time durations the machine is down are i.i.d. and independent of the operation times. Let
_Xn
denote the state of the machine at time n. Determine the type of distributions for the operation times and down times in order for
_Xn
to be a Markov chain on S = {1, 2} with transition matrix