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.
[1] The global balance equations (also known as full balance equations[2]) 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
, the global balance equations are given by[3] or equivalently for all
represents the probability flux from state
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
In general it is computationally intractable to solve this system of equations for most queueing models.
[4] For a continuous time Markov chain (CTMC) with transition rate matrix
, the global balance equations are satisfied and
[5] If such a solution can be found the resulting equations are usually much easier than directly solving the global balance equations.
[4] A CTMC is reversible if and only if the detailed balance conditions are satisfied for every pair of states
A discrete time Markov chain (DTMC) with transition matrix
,[6] 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.
In some situations, terms on either side of the global balance equations cancel.
[1] 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.
[2] Often, constructing local balance equations is equivalent to removing the outer summations in the global balance equations for certain terms.
[1] During the 1980s it was thought local balance was a requirement for a product-form equilibrium distribution,[10][11] but Gelenbe's G-network model showed this not to be the case.