Balance equation

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 \mathcal{S}, transition rate from state i to j given by q_{ij} and equilibrium distribution given by \pi, the global balance equations are given by :: \pi_i \sum_{j \in S\setminus \{i\}} q_{ij} = \sum_{j \in S\setminus \{i\}} \pi_j q_{ji}. for all i \in S. Here \pi_i q_{ij} 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 j \neq i into state 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, if \pi_i can be found such that for every pair of states i and j ::\pi_i q_{ij} = \pi_j q_{ji} holds, then by summing over j, the global balance equations are satisfied and \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. ::\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. ==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, or individual balance equations). The resulting equations are somewhere between detailed balance and global balance equations. Any solution \pi 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. but Gelenbe's G-network model showed this not to be the case. ==Notes==