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.
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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 with state space S, transition rate from state i to j given by qij and equilibrium distribution given by
or equivalently
for all i in S. Here
For a discrete time Markov chain with transition matrix Q and equilibrium distribution
Detailed balance
For a continuous time Markov chain (CTMC) with transition rate matrix Q, if
holds, then by summing over j, the global balance equations are satisfied and
A CTMC is reversible if and only if the detailed balance conditions are satisfied for every pair of states i and j.
A discrete time Markov chains (DTMC) with transition matrix P and equilibrium distribution
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. The resulting equations are somewhere between detailed balance and global balance equations. Any solution
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.