the balance equations
The balance equations are the algebraic conditions a probability distribution pi must satisfy to be a stationary (equilibrium) distribution of a continuous-time Markov chain. They are the continuous-time analogue of pi = pi P for discrete chains, and they are what you solve to find the long-run fractions of time the chain spends in each state. The word 'balance' is literal: in equilibrium the probability flowing into each state equals the probability flowing out.
A distribution pi on S is stationary if pi Q = 0, written out as the global balance equations: for every state j, sum_{i != j} pi(i) q_ij = pi(j) q_j. The left side is the total probability flux into j, the right side the total flux out; equilibrium means they balance. If the chain is irreducible and positive recurrent, pi is unique and is also the limiting distribution: P_t(i,j) -> pi(j) as t -> infinity for every i, and pi(j) is the long-run fraction of time in j. Recall pi here is the CTMC stationary distribution, related to the jump-chain stationary nu by pi(j) proportional to nu(j)/q_j. A much stronger and very useful special case is detailed balance: pi(i) q_ij = pi(j) q_ji for every pair i, j, meaning the flux balances pairwise, edge by edge, not just in total. A chain admitting such a pi is reversible — run backwards in time it is statistically the same chain — and detailed balance lets you read off pi almost by inspection (e.g. by a spanning-tree or product-form argument).
Balance equations are the everyday tool for steady-state analysis of queues, networks and physical systems. Two honest points. Solving the global balance equations on an infinite state space requires also checking that the solution is summable (normalizable to a probability) — a positive solution that sums to infinity signals null recurrence or transience, not equilibrium. And detailed balance is a genuine restriction: most chains are NOT reversible (e.g. a unidirectional cycle has a stationary distribution but no detailed-balance solution), so do not assume detailed balance holds; when it fails you must solve the full global balance system.
For a birth-death chain on {0,1,2,...} with birth rates lambda_n and death rates mu_n, detailed balance pi(n) lambda_n = pi(n+1) mu_{n+1} gives the product form pi(n) = pi(0) * prod_{k=1}^{n} (lambda_{k-1}/mu_k), normalized so the pi(n) sum to 1 (which requires the series to converge).
Birth-death chains are reversible, so detailed balance gives the stationary law in closed product form — provided the normalizing series converges.
Detailed balance (reversibility) is a strict special case: most chains satisfy only global balance pi Q = 0. On infinite state spaces a nonnegative solution of the balance equations is the stationary distribution only if it is summable; an unsummable solution means null recurrence or transience.