Instead of writing differential equations, one can obtain the solution in a simpler fashion by directly considering flow balance for each state, i.e. by claiming that, at equilibrium, the total probability flow entering a state must be balanced by the total probability flow leaving the state and that this will hold for each state of the system.

This leads to the following approach.

(a) Draw the state transition diagram as shown earlier (b) Draw closed boundaries and equate flows across this boundary. Any closed boundary may be chosen for this. If the closed boundary encloses state k (as shown in the figure), then we get Flow entering state k = λ k -1Pk -1 + µk+1Pk+1 = ( λk + µk )Pk = Flow leaving state k as the desired flow balance equation for state k .
	Global Balance Equation for state k

(c) Solve the equations in (b) along with the normalization condition to get the equilibrium state distribution.

Equations of this type which balance the flow leaving and entering a state for each state are referred to as Global Balance Equations. These can always be written for a system at equilibrium though they may be harder to solve than the Detailed Balance Equations given in the next slide

 

It would be even simpler in this case to consider a closed boundary which is actually closed at infinity as shown in the figure

This would lead to the following equation Flow from state k-1 to k = Flow from state k to k-1                             λk-1Pk-1 =µkPk

Detailed Balance Equation

Equations of this type are referred to as Detailed Balance Equations since they essentially balance the flows state-by-state between neighbouring states. The solution for this will be the same as that obtained earlier using the Global Blance Equations.

While Detailed Balance Equations are easy to write for the Birth-Death system, one has to be careful as it may not be always possible to write something this simple for a general Markov Chain. (In that case, it would be advisable to verify that the equations being written essentially do balance flows across a closed boundary.)

One can choose other closed boundaries as well and obtain the same solution. For solving a given Markov Chain, it may be a good idea to try writing flow balance equations in a way such that the overall set of equations is easy to solve for the equilibrium state probabilities.