Results are obtained concerning the distribution of ranked relative lengths of excursions of a recurrent Markov process from a point in its state space whose inverse local time process is a stable subordinator. It is shown that for a large class of random times $T$ the distribution of relative excursion lengths prior to $T$ is the same as if $T$ were a fixed time. It follows that the generalized arc-sine laws of Lamperti extend to such random times $T$. For some other random times $T$, absolute continuity relations are obtained which relate the law of the relative lengths at time $T$ to the law at a fixed time.