= Destination-dependent reset chain
{title2=$Q=\mathbf1r^T-RI$}
For jump rates $q_{ij}=r_j$ when $i\ne j$, let $R=\sum_jr_j>0$. The positive-rate states form the unique closed communicating class; every zero-rate state is an open singleton. The stationary law is $\pi_i=r_i/R$, with detailed balance $\pi_iq_{ij}=r_ir_j/R$. When at least two rates are positive, the embedded jump-chain law is proportional to $r_i(R-r_i)$. With only one positive rate, that state is absorbing and the genuine sequence of jump times stops there; an extended discrete chain can use an absorbing self-loop.
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