# Balance equation

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]}

## Global balance

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 *S*, transition rate from state *i* to *j* given by *q*_{ij} and equilibrium distribution given by , the global balance equations are given by^{[3]}

or equivalently

for all *i* in *S*. Here 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 into state i. In general it is computationally intractable to solve this system of equations for most queueing models.^{[4]}

For a discrete time Markov chain with transition matrix *Q* and equilibrium distribution , the global balance equation has the same form as above, except that should be interpreted as the transition probability and not the transition rate.

## Detailed balance

For a continuous time Markov chain (CTMC) with transition rate matrix *Q*, if can be found such that for every pair of states *i* and *j*

holds, then by summing over *j*, the global balance equations are satisfied and is the stationary distribution of the process.^{[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 *i* and *j*.

A discrete time Markov chains (DTMC) with transition matrix *P* and equilibrium distribution is said to be in detailed balance if for all pairs *i* and *j*,^{[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.

## 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**,^{[2]} **independent balance equations**^{[7]} or **individual balance equations**^{[8]}).^{[1]} These balance equations were first considered by Peter Whittle.^{[8]}^{[9]} The resulting equations are somewhere between detailed balance and global balance equations. Any solution 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.^{[12]}

## Notes

- 1 2 3 Harrison, Peter G.; Patel, Naresh M. (1992).
*Performance Modelling of Communication Networks and Computer Architectures*. Addison-Wesley. ISBN 0-201-54419-9. - 1 2 3 Kelly, F. P. (1979).
*Reversibility and stochastic networks*. J. Wiley. ISBN 0-471-27601-4. - ↑ Chandy, K.M. (March 1972). "The analysis and solutions for general queueing networks".
*Proc. Sixth Annual Princeton Conference on Information Sciences and Systems, Princeton U*. Princeton, N.J. pp. 224–228. - 1 2 Grassman, Winfried K. (2000).
*Computational probability*. Springer. ISBN 0-7923-8617-5. - ↑ Bocharov, Pavel Petrovich; D'Apice, C.; Pechinkin, A.V.; Salerno, S. (2004).
*Queueing theory*. Walter de Gruyter. p. 37. ISBN 90-6764-398-X. - ↑ Norris, James R. (1998).
*Markov Chains*. Cambridge University Press. ISBN 0-521-63396-6. Retrieved 2010-09-11. - ↑ Baskett, F.; Chandy, K. Mani; Muntz, R.R.; Palacios, F.G. (1975). "Open, closed and mixed networks of queues with different classes of customers".
*Journal of the ACM*.**22**: 248–260. doi:10.1145/321879.321887. - 1 2 Whittle, P. (1968). "Equilibrium Distributions for an Open Migration Process".
*Journal of Applied Probability*.**5**(3): 567–571. doi:10.2307/3211921. JSTOR 3211921. - ↑ Chao, X.; Miyazawa, M. (1998). "On Quasi-Reversibility and Local Balance: An Alternative Derivation of the Product-Form Results".
*Operations Research*.**46**(6): 927–933. doi:10.1287/opre.46.6.927. JSTOR 222945. - ↑ Boucherie, Richard J.; van Dijk, N.M. (1994). "Local balance in queueing networks with positive and negative customers".
*Annals of Operations Research*.**48**: 463–492. doi:10.1007/bf02033315. - ↑ Chandy, K. Mani; Howard, J.H., Jr; Towsley, D.F. (1977). "Product form and local balance in queueing networks".
*Journal of the ACM*.**24**: 250–263. doi:10.1145/322003.322009. - ↑ Gelenbe, Erol (Sep 1993). "G-Networks with Triggered Customer Movement".
*Journal of Applied Probability*.**30**(3): 742–748. doi:10.2307/3214781. JSTOR 3214781.