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.

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

{\displaystyle {\mathcal {S}}}

, transition rate from state

i

{\displaystyle i}

to

j

{\displaystyle j}

given by

q

i

j

{\displaystyle q_{ij}}

and equilibrium distribution given by

π

{\displaystyle \pi }

, the global balance equations are given by

π

i

∑

j

∈

S

∖

{

i

}

q

i

j

=

∑

j

∈

S

∖

{

i

}

π

j

q

j

i

.

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

for all

i

∈

S

{\displaystyle i\in S}

. Here

π

i

q

i

j

{\displaystyle \pi _{i}q_{ij}}

represents the probability flux from state

i

{\displaystyle i}

to state

j

{\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

j

≠

i

{\displaystyle j\neq i}

into state

i

{\displaystyle i}

. In general it is computationally intractable to solve this system of equations for most queueing models.

Detailed balance

For a continuous time Markov chain (CTMC) with transition rate matrix

Q

{\displaystyle Q}

, if

π

i

{\displaystyle \pi _{i}}

can be found such that for every pair of states

i

{\displaystyle i}

and

j

{\displaystyle j}

π

i

q

i

j

=

π

j

q

j

i

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

holds, then by summing over

j

{\displaystyle j}

, the global balance equations are satisfied and

π

{\displaystyle \pi }

is the stationary distribution of the process. If such a solution can be found the resulting equations are usually much easier than directly solving the global balance equations.

A CTMC is reversible if and only if the detailed balance conditions are satisfied for every pair of states

i

{\displaystyle i}

and

j

{\displaystyle j}

.