Is it not true that in order for multiple state model probabilities to be equal to the corresponding multiple decrement rates all non active states have to be absorbing? In this case in order for p^ar = (aq)^r retirement has to be an absorbing state and the transition probability from retirement to death has to be equal to zero. But in this question mu is non zero!?
What you say is true - in order for the probabilities in the differential equation to be dependent probabilities, we must assume that mu = 0. That's what's done in the solution (and in the notes). However, this is just an assumption we make in order to obtain the dependent probabilities. mu isn't actually zero - just having retired doesn't mean that you'll never die.
But the question tells us mu=0.05 so how can we assume it to be zero? I thought the whole theory for this was that the time t decrement rates are just the probabilities of leaving the active state via that decrement by time t, and will therefore include the probabilities of subsequently leaving from that state to another afterwards by time t. The transition probabilities t_p^ ij however are the probabilities of being a state j at time t given that you were in state i at time 0. As I see it therefore in order for t_(aq)^r to be equal to t_p^ar there can be no possible further decrements out of state r by time t .
