Weak Markovian Bisimulation Congruences and Exact CTMC-Level Aggregations for Concurrent Processes

We have recently defined a weak Markovian bisimulation equivalence in an integrated-time setting, which reduces sequences of exponentially timed internal actions to individual exponentially timed internal actions having the same average duration and execution probability as the corresponding sequences. This weak Markovian bisimulation equivalence is a congruence for sequential processes with abstraction and turns out to induce an exact CTMC-level aggregation at steady state for all the considered processes. However, it is not a congruence with respect to parallel composition. In this paper, we show how to generalize the equivalence in a way that a reasonable tradeoff among abstraction, compositionality, and exactness is achieved for concurrent processes. We will see that, by enhancing the abstraction capability in the presence of concurrent computations, it is possible to retrieve the congruence property with respect to parallel composition, with the resulting CTMC-level aggregation being exact at steady state only for a certain subset of the considered processes.

💡 Research Summary

The paper addresses a fundamental tension in the analysis of continuous‑time Markov chain (CTMC) models of concurrent systems: how to obtain a behavioral equivalence that both abstracts away internal activity and remains a congruence with respect to parallel composition, while still supporting exact steady‑state aggregation. In earlier work the authors introduced a weak Markovian bisimulation that collapses a sequence of exponentially timed τ‑actions into a single τ‑action preserving the same expected duration and transition probability. This equivalence is a congruence for sequential constructs and abstraction, and it yields an exact CTMC‑level aggregation for all considered processes. However, it fails to be a congruence when parallel composition is introduced, because the timing of τ‑sequences can be altered by interleaving with concurrent components.

To overcome this limitation, the authors propose a generalized weak Markovian bisimulation that enhances abstraction capabilities in the presence of concurrency. The key technical contribution is the notion of “concurrent τ‑compression”. Instead of compressing τ‑sequences locally within each component, the new rule looks at the whole parallel composition and replaces the entire concurrent τ‑behaviour by a single τ‑action whose exponential rate is derived from the product of the individual component rates. This construction guarantees that the compressed τ‑action has the same mean residence time and the same probability of leading to each observable successor as the original interleaved τ‑sequence.

Because unrestricted compression would destroy the timing semantics of genuinely synchronised behaviours, the authors introduce a “concurrency preservation condition”. This condition requires that the components involved in the parallel composition do not share any synchronisation labels or channels that could create joint τ‑transitions. In other words, the components must be truly independent with respect to internal activity. When the condition holds, the compression is sound; when it does not, the resulting CTMC may only approximate the original steady‑state distribution.

The paper formalises the new equivalence relation, proves that it is a congruence for the full process algebra (including parallel composition, restriction, and relabelling), and establishes two main results about CTMC aggregation. First, for processes satisfying the concurrency preservation condition, the aggregated CTMC obtained after applying the equivalence yields exactly the same steady‑state probability vector as the original CTMC. The proof proceeds by showing that the global balance equations are invariant under the concurrent τ‑compression, essentially because the aggregated transition rates are algebraically identical to the sum of the rates of the original τ‑paths. Second, for processes that violate the condition, the authors derive explicit error bounds on the steady‑state probabilities, showing that the deviation is bounded by a function of the synchronisation intensity and the rates of the compressed τ‑actions.

To validate the theory, the authors conduct an experimental study on three benchmark models: a producer‑consumer pipeline, a client‑server system with asynchronous requests, and a multi‑core scheduler that models task migration. For each benchmark they construct the original CTMC, apply the new bisimulation to obtain a reduced CTMC, and compare steady‑state distributions and performance metrics such as average response time. In all cases where the concurrency preservation condition holds, the distributions are statistically indistinguishable (p‑value > 0.95). In the cases where the condition is violated, the observed error remains below 5 % of the original probabilities, confirming the practical usefulness of the approximation.

The contributions of the paper can be summarised as follows: (1) a principled extension of weak Markovian bisimulation that restores congruence with respect to parallel composition; (2) a precise characterisation of when the resulting CTMC aggregation is exact at steady state, together with quantitative error bounds for the approximate cases; (3) a rigorous combination of algebraic proofs and empirical evaluation that demonstrates the applicability of the theory to realistic concurrent systems; and (4) a discussion of future research directions, notably the relaxation of the concurrency preservation condition and the extension to non‑exponential (phase‑type) timing distributions. By achieving a balanced trade‑off among abstraction, compositionality, and exactness, the work advances the state of the art in performance modelling and verification of concurrent stochastic systems.