Perfect Sampling of Markov Chains with Piecewise Homogeneous Events

Perfect sampling is a technique that uses coupling arguments to provide a sample from the stationary distribution of a Markov chain in a finite time without ever computing the distribution. This technique is very efficient if all the events in the system have monotonicity property. However, in the general (non-monotone) case, this technique needs to consider the whole state space, which limits its application only to chains with a state space of small cardinality. We propose here a new approach for the general case that only needs to consider two trajectories. Instead of the original chain, we use two bounding processes (envelopes) and we show that, whenever they couple, one obtains a sample under the stationary distribution of the original chain. We show that this new approach is particularly effective when the state space can be partitioned into pieces where envelopes can be easily computed. We further show that most Markovian queueing networks have this property and we propose efficient algorithms for some of them.

💡 Research Summary

The paper addresses a fundamental limitation of perfect sampling methods for Markov chains: the reliance on monotonicity. Classical perfect sampling, such as Coupling From The Past (CFTP), works efficiently when every transition preserves a partial order, allowing the algorithm to track only the minimal and maximal states. In non‑monotone settings, however, the state space must be explored exhaustively, which quickly becomes infeasible for any realistic system.

To overcome this obstacle, the authors introduce a novel framework based on piecewise homogeneous events and envelopes (bounding processes). The key idea is to partition the global state space (\mathcal{S}) into a finite collection of pieces ({\mathcal{S}k}{k=1}^K) such that, within each piece, the transition mechanism is homogeneous – i.e., the same transition rule applies to every state in the piece. Typical examples include queueing networks where each node’s queue length evolves by simple (\pm 1) increments, or systems where a subset of variables changes according to identical stochastic rules.

Within this partitioned structure the authors define two stochastic processes, the lower envelope (\underline{X}_t) and the upper envelope (\overline{X}_t). At any time (t) these envelopes bound all possible realizations of the original chain: for every sample path (X_t) we have (\underline{X}_t \le X_t \le \overline{X}_t) under the natural partial order. Crucially, the envelope update operator is closed: applying the same event to an envelope yields another envelope that still contains all possible successors. This closure property guarantees that if the two envelopes ever coincide, the entire family of trajectories has coupled, and the common state is an exact draw from the stationary distribution of the original chain.

The algorithm proceeds as follows. For each incoming event (E_t) the algorithm identifies the subset of pieces (\mathcal{K}(E_t)) that are affected. Because the event is homogeneous inside each piece, the update of the envelope reduces to a simple computation of new minima and maxima for those pieces. The rest of the envelope remains unchanged. After the update the algorithm checks whether the lower and upper envelopes are identical; if so, coupling has occurred and the algorithm terminates, returning the coupled state as a perfect sample.

The authors provide rigorous theoretical guarantees. Theorem 1 proves exactness: the state returned at the coupling time follows the stationary distribution of the original Markov chain. Theorem 2 establishes that, under the piecewise homogeneity assumption, the expected coupling time is finite and can be bounded by a function of the number of pieces, the maximal transition magnitude within each piece, and the probability that an event touches a given piece. Theorem 3 shows that the memory footprint is linear in the number of pieces (only two envelope vectors need to be stored) and that each event is processed in time proportional to the number of affected pieces, which is typically constant for many applications.

A substantial part of the paper is devoted to queueing networks, a class of models that naturally satisfies the piecewise homogeneity condition. The authors illustrate the method on several canonical networks:

• Jackson networks – each node’s queue length evolves by unit increments or decrements; the envelope for node (i) is simply an interval (