# 1. An important unit consists of two components placed in parallel. The unit performs satisfactorily

1. An considerable individual consists of two constituents placed in correspondent. The individual performs satisfactorily if one of the two constituents is untrammelled. Therefore, merely one constituent is operated at a duration, but twain constituents are kept operational (capable of being operated) as repeatedly as potential by betaking them as needed. An untrammelled constituent breaks down in a loving conclusion after a occasion verisimilitude 0.2. When this occurs, the correspondent constituent takes balance, if it is operational, at the threshold of the instant period. Merely one constituent can be restoreed at a duration. The restore of a constituent starts at the threshold of the primary profitable conclusion and is completed at the end of the instant conclusion. Let Xt be a vector consisting of two elements U and V, where U represents the estimate of constituents that are operational at the end of conclusion t and V represents the estimate of conclusions of restore that bear been completed on constituents that are not yet operational. Thus, V _ 0 if U _ 2 or if U _ 1 and the restore of the nonoperational constituent is sound getting subordinate way. Owing a restore takes two conclusions, V _ 1 if U _ 0 (since then one nonoperational constituent is indecision to commence restore occasion the other one is entering its assist conclusion of restore) or if U _ 1 and the non-operational constituent is entering its assist conclusion of restore. Therefore, the propound room consists of the lewd propounds (2, 0), (1, 0), (0, 1), and (1, 1). Denote these lewd propounds by 0, 1, 2, 3, respectively. {Xt} (t _ 0, 1, . . .) is a Markov compact (affect that X0 _ 0) after a occasion the (one-step) transition matrix P _ . C (a) What is the verisimilitude that the individual earn be inoperable (owing twain constituents are down) after n periods, for n _ 2, 5, 10, 20? (b) What are the steady-propound probabilities of the propound of this Markov compact? (c) If it absorbs \$30,000 per conclusion when the individual is inoperable (twain constituents down) and cipher incorrectly, what is the (long-run) expected medium absorb per conclusion?