The lifetime of a certain machine is a random variable having an exponential distribution with parameter λ. When the machine breaks down, there is a probability equal to p (respectively, 1 - p) that the failure is of type I (resp., II). In the case of a type I failure, the machine is out of use for an exponential time, with mean equal to l/μ time unit(s). To repair a type II failure, two independent operations must be performed. Each operation takes an exponential time with mean equal to l/μ.
(a) Define a state space such that the process {X(t),t ≥ 0}, where X{t) denotes the state of the system at time t, is a continuous-time Markov chain.
(b) Calculate, assuming the existence of the limiting probabilities, the probability that the machine will be functioning at a (large enough) given time instant.