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

There is now no separate rate-$\nu$ clockwise transition. The only transitions areBoth preserve $m+n$ modulo $M$. The initial state therefore selects the closed communicating class $m+n\equiv 0(\mathrm{mod}M)$, on which $m\equiv -n(\mathrm{mod}M)$. The population is a birth-death process with birth rate ${\lambda }_{-n}$ and death rate $\mu$.

Define $a_0=1$ and $a_r={\mu }^{-r}{\prod }_{j=0}^{r-1}{\lambda }_{-j}$ for $1\le r<M$. The detailed balance for a birth-death process weights areWriting $n=kM+r$ with $0\le r<M$, the product condition gives $w_n=a_r{\rho }^{kM}$. Summing the resulting geometric series yieldsThis is also the law in part (b) conditioned on $m+n\equiv 0(\mathrm{mod}M)$: the ratio of its weights at consecutive accessible populations is ${\lambda }_{-n}/\mu$. The stability condition is $\mu >\nu$, which uses the product of the periodic birth rates; it does not require $\mu >{\lambda }_m$ for every $m$.