Let $Q$ be the Q-matrix of a continuous time Markov process ($q_{ii}=-\sum\limits_{j\ne i}q_{ij}$ and $q_{ij}\ge 0\ \forall i\ne j)$
The probability matrix of the jump chain corresponding to the continuous process (the discrete time Markov chain that models where the continuous time process is going to jump next) is given by $(P)_{ij}=\begin{cases}0& \text{if } i=j\\{q_{ij}\over -q_{ii}}&\text{otherwise} \end{cases}$
The invariant distribution $\lambda$ of the continuous time process is given by $\lambda Q=0$ whereas the invariant distribution $\pi$ of the chain must satisfy $\pi P=\pi$
My question is: why do we have $\frac{\pi_i}{\pi_j}=\frac{\lambda_i}{\lambda_j}\frac{-q_{ii}}{-q_{jj}}$?