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A positive balance function $\Phi$ expresses each active route's aggregate service as this ratio. With independent Poisson arrivals and exponential document sizes, detailed balance yields stationary weights $\Phi (n){\prod }_r{\rho }_r^{n_r}$, provided their sum is finite.

A stable equilibrium M-M-s queue has a Poisson process of departures with the same rate as its input. Its present population is also independent of past departures, a component of quasireversibility. The birth-death process satisfies detailed balance; reversed upward jumps have a constant arrival rate, so they are generated by an independent Poisson process.

Carrier-sense multiple access lets a station attempt transmission only when interfering neighbors are inactive. An ideal continuous-time model gives inactive station $r$ an exponential attempt rate $e^{{\theta }_r}$ and unit-rate exponential transmission completions. Detailed balance then gives activation weights $e^{\theta \cdot n}$ on the independent sets of the interference graph.

A single-coordinate Gibbs sampler update preserves the remaining coordinates and independently redraws the selected coordinate from its full conditional distribution. The joint measure of old and new states factors into a marginal and two identical conditional factors, making it symmetric. A fixed-probability mixture of such kernels therefore satisfies detailed balance.

The Ehrenfest urn model moves one uniformly chosen ball between two boxes at each step. For $k\ge 1$ balls, the count in one box is a Markov chain on $\{0,\dots ,k\}$. Its stationary distribution is $2^{-k}\left(\genfrac{}{}{0ex}{}{k}{i}\right)$, and detailed balance makes it a reversible Markov chain. The chain has period two, but its mean recurrence time to the all-in-one-box state is still $2^k$.

An involutive Metropolis proposal uses a deterministic bijection $S$ satisfying $S^2=I$. For a volume-preserving $S$ and target probability density function $\rho$, accept $S(z)$ with probability $\min (1,\rho (S(z))/\rho (z))$. The identity $\rho (z)\alpha (z)=\min (\rho (z),\rho (S(z)))$ and a change of variables under $S$ prove detailed balance. A non-unit Jacobian determinant must be included when volume is not preserved.

A last-exit decomposition partitions a transient path event according to the final visit to a finite set. Reversing the finite path before that visit and using detailed balance converts last-exit probabilities into hitting probabilities weighted by an equilibrium measure of a finite set.

The Metropolis–Hastings algorithm proposes $y$ from $q(x,\cdot )$ at state $x$ and accepts it with probabilityThis acceptance rule enforces detailed balance with the target density $\pi$.

For an equilibrium process, a path and its physically time-reversed path have equal probability after including their equilibrium starting weights. Conditional on their endpoints, this meansThis is the path version of detailed balance. A time-even order parameter is read backward without changing sign, while a transport flux changes sign. The Onsager--Machlup path probability implements this reversal in a coarse-grained stochastic description.

For nonconserved order-parameter dynamics, microscopic reversibility with equilibrium weight proportional to $e^{-F/(k_{B}T)}$ requiresIndeed the Onsager--Machlup path probability gives $\log (P_F/P_B)=-2\Gamma \int \dot{p}\delta F/\delta pdt/{\sigma }^2=-2\Gamma (F_2-F_1)/{\sigma }^2$. Equating this to the detailed balance value proves the relation. The functional chain rule is understood with the same midpoint convention as the path action.