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Consider a birth-death process on $\{0,1,2,\dots \}$ with constant birth rate $\lambda >0$, constant death rate $\delta >0$ at every positive state, and no deaths from zero. The reflecting boundary is essential: the result changes if zero is an absorbing state. Detailed balance would giveWhen $\lambda <\delta$, these weights normalize to a Geometric distribution, so the irreducible chain is positive recurrent. When $\lambda =\delta$, the embedded jump chain is the symmetric reflected random walk: it returns to zero almost surely, but its constant equilibrium weights have infinite sum, so the chain is null recurrent. When $\lambda >\delta$, its upward-biased embedded random walk has probability $\delta /\lambda <1$ of hitting zero from state one, making the chain transient. The holding rates are bounded, so nonexplosion creates no additional qualification here.

A Continuous-time Markov chain on a finite or countable state space is specified by transition rates $q_{ij}\ge 0$ for $i\ne j$, with finite total exit rate $q_i={\sum }_{j\ne i}{q}_{ij}$. At a state with $q_i>0$, it waits an exponential time of rate $q_i$ and then moves to $j$ with probability $q_{ij}/q_i$. A state with $q_i=0$ is absorbing: it has infinite holding time rather than a zero-rate exponential waiting law.

Its transition-rate matrix is $Q$, with off-diagonal entries $q_{ij}$ and diagonal entries $q_{ii}=-q_i$, so each row sums to zero. The rates describe first-order transition probabilities: $\mathbb{P}(X_{t+h}=j\mid X_t=i)=q_{ij}h+o(h)$ for $j\ne i$. On a finite state space, the transition semigroup is $P_t=e^{tQ}$ and explosion is impossible. On a countably infinite state space, finite individual exit rates do not alone exclude infinitely many jumps in finite time; nonexplosion must be checked separately.

For two states with rates $q_{01}=a>0$ and $q_{10}=b>0$, the mean holding times are $1/a$ and $1/b$. Detailed balance gives Stationary distribution $(b/(a+b),a/(a+b))$, andThis starts at one and relaxes to the stationary mass of state zero. Changing holding rates can preserve an embedded jump chain while changing the proportions of elapsed time spent in its states.

An Equilibrium distribution of a continuous-time Markov chain is a probability row vector $\pi$ invariant under its transition semigroup:It satisfies $\pi Q=0$ when the generator sums are defined. An invariant law is a property of the entire evolution, so formal generator equations must be interpreted with the process's nonexplosion and domain conditions.

For a finite irreducible chain, the equilibrium distribution is unique and every transition row converges to it. For a general finite chain, the limiting row from state $i$ is a mixture of the stationary distributions ${\pi }^C$ of its closed irreducible classes:The chain eventually enters one of these classes and then mixes within it. If every closed class is a single absorbing state, this reduces to absorption probabilities at those states. Merely having some absorbing states does not exclude other closed classes.

Holding times explain the link to an embedded jump chain. For a finite irreducible chain with strictly positive exit rates $q_i$ and jump-chain Stationary distribution $\alpha$, the time-stationary distribution isVisit frequency is multiplied by mean time $1/q_i$ per visit. In the countable setting, these time weights need a finite positive normalizer and a nonexplosive process before they define an equilibrium probability. For an alternating two-state jump chain, $\alpha =(1/2,1/2)$ and rates $a,b$ give $\pi =(b/(a+b),a/(a+b))$: slower holding rates increase time occupancy.

A continuous-time Markov chain with finite holding rates is nonexplosive when its jump times have no finite accumulation almost surely. It then remains in its specified state space at every finite time, rather than being lost to a cemetery state after infinitely many jumps. Bounded rates are sufficient for nonexplosion, but finite rates at individual states alone are not.