Balance Equation - Global Balance

Global Balance

The global balance equations (also known as full balance equations) 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_ij and equilibrium distribution given by, the global balance equations are given for every state i in S by

Here 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.

For a discrete time Markov chain with transition matrix P and equilibrium distribution the global balance equation is