a birth-death process
A birth-death process is a continuous-time Markov chain on the integers (or {0,1,2,...}) that can only jump to a neighbouring state: up by one ('birth') or down by one ('death'). It is the most important explicitly solvable family of CTMCs, the model for population sizes, queue lengths, and any quantity that changes by single units, and the testbed where every general CTMC concept (holding times, balance, explosion, reversibility) becomes a concrete formula.
On state space {0,1,2,...} the generator has q_{n,n+1} = lambda_n (birth rate), q_{n,n-1} = mu_n (death rate, with mu_0 = 0), and q_n = lambda_n + mu_n. The holding time in n is exponential(lambda_n + mu_n); the jump chain is a nearest-neighbour random walk going up with probability lambda_n/(lambda_n+mu_n). Because jumps are nearest-neighbour, the chain is automatically reversible, so detailed balance gives the stationary distribution in closed product form pi(n) proportional to prod_{k=1}^n (lambda_{k-1}/mu_k), which is a probability distribution exactly when this series is summable (the positive-recurrence condition). Several classical processes are special cases: the Poisson process (mu_n = 0, lambda_n = lambda constant), the Yule pure birth process (lambda_n = n*lambda), the linear birth-death (lambda_n = n*lambda, mu_n = n*mu) modeling a branching population, and the M/M/1 and M/M/c queues (constant or capped rates).
Birth-death processes are central because their tridiagonal structure makes almost everything computable: stationary distributions, expected hitting and extinction times (by solving a second-order recurrence), and explosion (the pure-birth criterion sum 1/lambda_n = infinity for non-explosion). The honest cautions are the general CTMC ones made concrete here: a linear birth-death population can go extinct (absorption at 0) with positive probability even when births outnumber deaths on average, and a pure-birth process with rapidly growing lambda_n explodes in finite time. Reversibility is special to the nearest-neighbour structure — add a jump that skips a state and you generally lose detailed balance.
An M/M/1 queue is a birth-death chain with lambda_n = lambda, mu_n = mu (n >= 1). Detailed balance gives pi(n) = (1-rho) rho^n with rho = lambda/mu, a geometric stationary distribution — provided rho < 1 (positive recurrence). If rho >= 1 the series diverges: the queue is unstable, with no stationary distribution.
The M/M/1 queue: stable with a geometric stationary law iff traffic intensity rho = lambda/mu < 1.
Nearest-neighbour structure makes a birth-death process automatically reversible (detailed balance, product-form pi). This fails the moment jumps can skip states. And a supercritical linear birth-death population still hits extinction at 0 with positive probability — drift up on average does not preclude absorption.