In probability theory, a balance equation is an equation that describes the probability flux associated with a Markov chain in and out of states or set of states.

The global balance equations (also known as full balance equations) are a set of equations that characterize the equilibrium distribution (or any stationary distribution) of a Markov chain, when such a distribution exists.

For a continuous time Markov chain with state space ${\displaystyle {\mathcal {S}}}$, transition rate from state ${\displaystyle i}$ to ${\displaystyle j}$ given by ${\displaystyle q_{ij}}$ and equilibrium distribution given by ${\displaystyle \pi }$, the global balance equations are given by

or equivalently

for all ${\displaystyle i\in S}$. Here ${\displaystyle \pi _{i}q_{ij}}$ represents the probability flux from state ${\displaystyle i}$ to state ${\displaystyle j}$. So the left-hand side represents the total flow from out of state i into states other than i, while the right-hand side represents the total flow out of all states ${\displaystyle j\neq i}$ into state ${\displaystyle i}$. In general it is computationally intractable to solve this system of equations for most queueing models.

For a continuous time Markov chain (CTMC) with transition rate matrix ${\displaystyle Q}$, if ${\displaystyle \pi _{i}}$ can be found such that for every pair of states ${\displaystyle i}$ and ${\displaystyle j}$

holds, then by summing over ${\displaystyle j}$, the global balance equations are satisfied and ${\displaystyle \pi }$ is the stationary distribution of the process. If such a solution can be found the resulting equations are usually much easier than directly solving the global balance equations.

A CTMC is reversible if and only if the detailed balance conditions are satisfied for every pair of states ${\displaystyle i}$ and ${\displaystyle j}$.