Local balance

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 S, transition rate from state i to j given by q_ij and equilibrium distribution given by ${\displaystyle \scriptstyle {\pi }}$, the global balance equations are given by

or equivalently

for all i in S. Here ${\displaystyle \pi _{i}q_{ij}}$ represents the probability flux from state i to state 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 i. In general it is computationally intractable to solve this system of equations for most queueing models.

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