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{{about|balance equations in probability theory|the concept of balanced equations in chemistry|chemical 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.<ref name="harr">{{Cite book | last = Harrison | first = Peter G. | authorlink = Peter G. Harrison | last2 = Patel | first2 = Naresh M. | title = Performance Modelling of Communication Networks and Computer Architectures | publisher = Addison-Wesley | year = 1992 | isbn = 0-201-54419-9}}</ref>

==Global balance==

The '''global balance equations''' (also known as '''full balance equations'''<ref name="kelly">{{cite book |title=Reversibility and stochastic networks |last=Kelly |first=F. P. |authorlink=Frank Kelly (mathematician) |year=1979 |publisher=J. Wiley |isbn=0-471-27601-4 |url=http://www.statslab.cam.ac.uk/~frank/BOOKS/kelly_book.html }}</ref>) are a set of equations that in principle can always be solved to give the [[equilibrium distribution]] of a Markov chain (when such a distribution exists). For a Markov chain with state space ''S'', transition rate from state ''i'' to ''j'' given by ''q''<sub>''ij''</sub> and equilibrium distribution given by <math>\scriptstyle{\pi}</math>, the global balance equations are given for every state ''i'' in ''S'' by<ref>{{cite conference | first = K.M. | last = Chandy | authorlink = K.M. Chandy | title = The analysis and solutions for general queueing networks  | booktitle = Proc. Sixth Annual Princeton Conference on Information Sciences and Systems, Princeton U. | pages = 224&ndash;228 | date = March 1972 | location = Princeton, N.J. }}</ref>

::<math>\sum_{j \in S\setminus \{i\}} \pi_i q_{ij} = \sum_{j \in S\setminus \{i\}} \pi_j q_{ji}.</math>

Here <math>\pi_i q_{ij}</math> represents the probability flux from state ''i'' to state ''j''. In general it is computationally intractable to solve this system of equations for most queueing models.<ref name="grass">{{cite book |title=Computational probability |last=Grassman |first=Winfried K. |year=2000 |publisher=Springer |isbn=0-7923-8617-5 }}</ref>

For a [[discrete time Markov chain]] with transition matrix ''P'' and equilibrium distribution <math>\pi</math> the global balance equation is

::<math>\sum_{j \in S\setminus \{i\}} \pi_i p_{ij} = \sum_{j \in S\setminus \{i\}} \pi_j p_{ji}.</math>

==Detailed balance==

{{See also|Detailed balance}}

For a [[continuous time Markov chain]] (CTMC) with [[transition rate matrix]] ''Q'', if <math>\pi_i</math> can be found such that for every pair of states ''i'' and ''j''

::<math>\pi_i q_{ij} = \pi_j q_{ji}</math>

holds, then the global balance equations are satisfied and <math>\pi</math> is the stationary distribution of the process.<ref name="boch">{{Cite book | last = Bocharov| first = Pavel Petrovich| last2 = D'Apice| first2 = C.| last3 = Pechinkin| first3 = A.V.| last4 = Salerno| first4 = S.| title = Queueing theory| publisher = Walter de Gruyter| year = 2004| pages = 37| isbn = 90-6764-398-X}}</ref> If such a solution can be found the resulting equations are usually much easier than directly solving the global balance equations.<ref name="grass" />

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]] with transition matrix ''P'' and equilibrium distribution <math>\pi</math> is said to be in detailed balance if for all pairs ''i'' and ''j'',<ref>{{cite book |title=Markov Chains |last=Norris |first=James R. |authorlink=James R. Norris |year=1998 |publisher=[[Cambridge University Press]] |isbn=0-521-63396-6 |url=http://www.statslab.cam.ac.uk/~james/Markov/ |accessdate=2010-09-11}}</ref>

::<math>\pi_i p_{ij} = \pi_j p_{ji}.</math>

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

==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''',<ref name="kelly" /> '''independent balance equations'''<ref>{{Cite journal | last = Baskett | first = F. | last2 = Chandy | first2 = K. Mani | author2-link = K.M. Chandy | last3 = Muntz | first3 = R.R. | last4 = Palacios | first4 = F.G. | title = Open, closed and mixed networks of queues with different classes of customers | journal = Journal of the ACM | volume = 22 | pages = 248&ndash;260 | year = 1975 | doi = 10.1145/321879.321887 | postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->}}</ref> or '''individual balance equations'''<ref name="whittle">{{cite jstor|3211921}}</ref>).<ref name="harr" /> These balance equations were first considered by [[Peter Whittle]].<ref name="whittle" /><ref>{{cite jstor|222945}}</ref> The resulting equations are somewhere between detailed balance and global balance equations. Any solution <math>\pi</math> 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.<ref name="kelly" /> Often, constructing local balance equations is equivalent to removing the outer summations in the global balance equations for certain terms.<ref name="harr" />

During the 1980s it was thought local balance was a requirement for a [[product-form solution|product-form equilibrium distribution]],<ref>{{Cite journal | first = Richard J. | last = Boucherie | first2 = N.M. | last2 = van Dijk | contribution = Local balance in queueing networks with positive and negative customers | contribution-url = http://www.springerlink.com/content/g013851405418642/ | title = Annals of Operations Research | year = 1994 | pages = 463&ndash;492 | volume = 48 | postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->}}</ref><ref>{{Cite journal | first = K. Mani | last = Chandy | author-link = K.M. Chandy | first2 = J.H., Jr | last2 = Howard | first3 = D.F. | last3 = Towsley | contribution = Product form and local balance in queueing networks | contribution-url = http://portal.acm.org/citation.cfm?id=322009 | title = Journal of the ACM | year = 1977 | pages = 250&ndash;263 | volume = 24 | postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->}}</ref> but [[Erol Gelenbe|Gelenbe]]'s [[G-network]] model showed this not to be the case.<ref>{{Cite journal | doi = 10.2307/3214781 | title = G-Networks with Triggered Customer Movement | first = Erol | last = Gelenbe | authorlink = Erol Gelenbe | journal = Journal of Applied Probability | volume = 30 | issue = 3 | date = Sep., 1993 | pages = 742–748 | postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. --> | jstor = 3214781}}</ref>

==Notes==

{{Reflist}}

{{Queueing theory}}

[[Category:Articles with inconsistent citation formats]]
[[Category:Queueing theory]]
[[Category:Stochastic processes]]