Let A be a specified set of states of a continuous-time Markov chain and let T,(t) denote the amount of time spent in A during the time interval [0, 1] given that the chain begins in state i Let Y, ..., Y, be independent exponential random variables with mean A. Suppose the Y, are independent of the Markov chain, and set t,(n) = E[T,(Y, + .. + Y)]

(a) Derive a set of linear equations for 1,(1), i 0

(b) Derive a set of linear equations for 1,(n) in terms of the other 1,(n) and t,(n - 1)

(c) When n is large, for what value of A is t,(n) a good approximation of E[T,(t)]? =