Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 30 / 1 / a

Write the clock states modulo $M$. The detailed balance equations are $p_m{\lambda }_m=\nu p_{m-1}$. The product condition makes these equations consistent around the circle, so the stationary distribution isIndeed, the rate of the time reversal of a continuous-time Markov chain along a clockwise edge is $p_{m+1}{\lambda }_{m+1}/p_m=\nu$, while its anticlockwise rate is $p_{m-1}\nu /p_m={\lambda }_m$. Thus the clock is a reversible Markov chain.

The reversed clockwise transitions can be driven by a single independent rate-$\nu$ Poisson process, since their rate does not depend on the clock state. They correspond to the original anticlockwise transitions. Consequently the anticlockwise event times form a stationary Poisson process of rate $\nu$. Its events before a given time are independent of the clock state at that time, which is the relevant quasireversibility property. If clockwise and anticlockwise transitions have the same destination, their direction is retained as an event mark when reversing the process.