Stationarity equations (${\pi }_j={\sum }_i{\pi }_i{P}_{ij}$)

For a discrete-time Markov chain with transition matrix $P$, these equations, together with ${\pi }_j\ge 0$ and ${\sum }_j{\pi }_j=1$, say that $\pi$ is a stationary distribution. Incoming probability mass equals the current mass at each state. They are global balance equations and do not require detailed balance.