For each of the following write T (True) or F (False) :

(a) All finite state space irreducible continuous-time Markov chains have a unique limiting probability distribution.

(b) The counting process of a Poisson process, {N(t) : t 0}, is a Birth and Death process.

(c) Little's Law (l = w) requires that the arrival times of customers forms a renewal process.

(d) A Birth and Death process is a special case of a continuous-time Markov chain.

(e) Let {X(t)} be an irreducible positive recurrent continuous-time Markov chain. Fix a state i, let X(0) = i and let tn(i), n 1, denote the consecutive times that X(t) re-visits state i. Then the point process {tn(i) : n 1} defines a renewal process.

(f) A state i for a continuous-time Markov chain {X(t) : t 0} is recurrent if and only if it is recurrent for its embedded discrete time Markov chain {Xn :n0}.

(g) It is possible for a continuous-time Markov chain to be null recurrent while its embedded discrete-time Markov chain is positive recurrent.

(h) If Tj,j denotes the return time to state j given that X(0) = j for a positive recurrent continuous-time Markov chain {X(t)}, then 1/E(Tj,j) is the long-run rate that this Markov chain visits state j.

(i) If an irreducible continuous-time Markov chain does not have a probability solution to the Global Balance Equations, then the Markov chain must be transient.

(j) The Birth and Death Balance equations always have a probability solution.