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]

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 ${\displaystyle {\mathcal {S}}}$, transition rate from state ${\displaystyle i}$ to ${\displaystyle j}$ given by ${\displaystyle q_{ij}}$ and equilibrium distribution given by ${\displaystyle \pi }$, the global balance equations are given by^[3]

${\displaystyle \pi _{i}\sum _{j\in S\setminus \{i\}}q_{ij}=\sum _{j\in S\setminus \{i\}}\pi _{j}q_{ji}.}$

for all ${\displaystyle i\in S}$. Here ${\displaystyle \pi _ {i}q_ {ij}}$ represents the probability flux from state ${\displaystyle i}$ to state ${\displaystyle j}$. 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 ${\displaystyle j\neq i}$ into state ${\displaystyle i}$. 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${\displaystyle Q}$, if ${\displaystyle \pi _{i}}$ can be found such that for every pair of states ${\displaystyle i}$ and ${\displaystyle j}$

${\displaystyle \pi _{i}q_{ij}=\pi _{j}q_{ji}}$

holds, then by summing over ${\displaystyle j}$, the global balance equations are satisfied and ${\displaystyle \pi }$ 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 ${\displaystyle i}$ and ${\displaystyle j}$.

A discrete time Markov chain (DTMC) with transition matrix ${\displaystyle P}$ and equilibrium distribution ${\displaystyle \pi }$ is said to be in detailed balance if for all pairs ${\displaystyle i}$ and ${\displaystyle j}$,^[6]

${\displaystyle \pi _{i}p_{ij}=\pi _{j}p_{ji}.}$

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.

En algunas situaciones, los términos a ambos lados de las ecuaciones de balance global se cancelan. Las ecuaciones de balance global se pueden entonces particionar para dar un conjunto de ecuaciones de balance local (también conocidas como ecuaciones de balance parcial , ^{[ 2 ]} ecuaciones de balance independiente ^{[ 7 ]} o ecuaciones de balance individual ^{[ 8 ]} ). ^{[ 1 ]} Estas ecuaciones de balance fueron consideradas por primera vez por Peter Whittle . ^{[ 8 ] [ 9 ]} Las ecuaciones resultantes se encuentran en algún punto entre las ecuaciones de balance detallado y las ecuaciones de balance global. Cualquier solución${\displaystyle \pi }$La solución de las ecuaciones de balance locales siempre es una solución de las ecuaciones de balance globales (podemos recuperar las ecuaciones de balance globales sumando las ecuaciones de balance locales correspondientes), pero lo contrario no siempre es cierto. ^{[ 2 ]} A menudo, construir ecuaciones de balance locales equivale a eliminar las sumas externas en las ecuaciones de balance globales para ciertos términos. ^{[ 1 ]}

Durante la década de 1980 se pensó que el equilibrio local era un requisito para una distribución de equilibrio en forma de producto , ^{[ 10 ] [ 11 ]} pero el modelo de red G de Gelenbe demostró que este no era el caso. ^{[ 12 ]}