Solution
= Solution
Reading the $j$th component gives
$$
\boxed{p_{ij}(t)=\frac{r_j}{R}(1-e^{-Rt})+e^{-Rt}\mathbf1_{\{i=j\}}.}
$$
The chain has the interpretation of a rate-$R$ Poisson clock which independently redraws a colour from $\pi$, allowing a redraw of the same colour. No redraw by time $t$ has probability $e^{-Rt}$ and retains the initial state; at least one redraw gives distribution $\pi$.