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.

Global balance

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, the global balance equations are given by

. So the left-hand side represents the total flow from out of state i into states other than i. In general it is computationally intractable to solve this system of equations for most queueing models.

Detailed balance

For a continuous time Markov chain (CTMC) with transition rate matrix

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 discrete time Markov chain,

When a solution can be found, as in the case of a CTMC, the computation is usually much quicker than directly solving the global balance equations.

Local balance

In some situations, terms on either side of the global balance equations cancel. The global balance equations can then be partitioned to give a set of local balance equations (also known as partial balance equations, independent balance equations or individual balance equations). These balance equations were first considered by Peter Whittle. to the local balance equations is always a solution to the global balance equations (we can recover the global balance equations by summing the relevant local balance equations), but the converse is not always true. Often, constructing local balance equations is equivalent to removing the outer summations in the global balance equations for certain terms.

During the 1980s it was thought local balance was a requirement for a product-form equilibrium distribution, but Gelenbe's G-network model showed this not to be the case.

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