We are given a Markov chain that has a single recurrent class which is aperiodic. Suppose that we have found a solution m to the following system of local balance and normalization equations: TiPij = jPji m ; = 1, = i=1 i, j = 1,..., m, i = 1,..., m. (a) Show that the are the steady-state probabilities. = (b) What is the interpretation of the equations iPjPji in terms of expected long- term frequencies of transitions between i and j? (c) Construct an example where the local balance equations are not satisfied by the steady-state probabilities.