The time-stationary and Palm inter-arrival times of a Markov-modulated Poisson process need not be ordered

For an irreducible Markov-modulated Poisson process, let $T^\pi$ be the time from a stationary instant to the next arrival and $T^\alpha$ the time from an arrival to the next one in the event-stationary version. Asanjarani and Nazarathy conjectured that $T^\pi$ is stochastically larger than $T^\alpha$, and He listed the same property among open questions on the MMPP class of phase-type distributions. The conjecture is false. The order is equivalent to the hazard rate of $T^\pi$ never exceeding the arrival rate. For a Cox process driven by a non-constant periodic intensity with uniform phase this fails on a set of positive measure. For the MMPP on the directed $n$-cycle with arrival rates $\varepsilon(1+b\cos(2\pi j/n))$ it fails for all small $\varepsilon>0$ whenever $n\ge 12$, and an explicit twelve-state MMPP with positive rational rates has $\mathbb P(T^\pi>19/2)-\mathbb P(T^\alpha>19/2)<-1/300000$, proved in exact rational arithmetic. For reversible modulators the order holds. We also show that the mean of He's MMPP-type distribution need not be convex in the speed of modulation.