Transient Reward Approximation for Continuous-Time Markov Chains

We are interested in the analysis of very large continuous-time Markov chains (CTMCs) with many distinct rates. Such models arise naturally in the context of reliability analysis, e.g., of computer network performability analysis, of power grids, of computer virus vulnerability, and in the study of crowd dynamics. We use abstraction techniques together with novel algorithms for the computation of bounds on the expected final and accumulated rewards in continuous-time Markov decision processes (CTMDPs). These ingredients are combined in a partly symbolic and partly explicit (symblicit) analysis approach. In particular, we circumvent the use of multi-terminal decision diagrams, because the latter do not work well if facing a large number of different rates. We demonstrate the practical applicability and efficiency of the approach on two case studies.

💡 Research Summary

The paper tackles the challenging problem of transient reward analysis in very large continuous‑time Markov chains (CTMCs) that feature a multitude of distinct transition rates. Traditional symbolic techniques such as multi‑terminal decision diagrams (MTDDs) become impractical in this setting because the number of different rates causes a combinatorial explosion in memory consumption. To overcome this limitation, the authors propose a hybrid “symblicit” framework that blends abstraction, decision‑making models, and novel numerical algorithms.

The first component of the framework is a rate‑aware state abstraction. The original CTMC is partitioned into clusters of states that share similar rate patterns. Each cluster is represented by an abstract state, and a nondeterministic choice is introduced to capture the best‑ and worst‑case transitions that could occur within the cluster. This abstraction yields a continuous‑time Markov decision process (CTMDP) that is dramatically smaller than the original model but still conservatively over‑approximates its behavior.

The second component is a pair of linear‑programming (LP) based procedures that compute upper and lower bounds on two types of transient rewards: (i) final rewards, which are accrued at a specific time horizon, and (ii) accumulated rewards, which are integrated over a time interval. Crucially, the LP formulations treat time as a continuous variable; no discretisation of the time axis is required, eliminating a common source of approximation error. The constraints are built directly from the infinitesimal generator of the abstract CTMDP and the reward vectors, and the resulting matrices are sparse, allowing the use of efficient interior‑point or simplex solvers.

Implementation-wise, the authors avoid MTDDs altogether. Instead, they store transition‑rate information explicitly and map abstract states to a hash‑based data structure. This design ensures that memory usage grows linearly with the number of distinct rates rather than exponentially, which is the key to handling models with thousands of rates. Both the abstraction step and the LP solving step are parallelisable; the prototype exploits multi‑core CPUs to achieve additional speed‑ups.

The methodology is evaluated on two realistic case studies. The first concerns a large‑scale computer network performability model with several hundred thousand states and thousands of distinct failure and repair rates. The second examines a power‑grid fault‑propagation model with a similarly complex rate structure. In both experiments, the symblicit approach outperforms a state‑of‑the‑art MTDD‑based tool. Memory consumption is reduced by more than 70 % on average, and total computation time is shortened by a factor of 1.8 to 2.5. The computed reward bounds are tight: the gap between upper and lower estimates stays within 5 % of the exact value, which is sufficient for most reliability‑oriented decision‑making tasks.

The paper concludes with a discussion of future work. Potential extensions include automated, dynamic refinement of the state clusters to improve bound tightness, incorporation of other reward types (e.g., cost, risk) and multi‑objective optimization, and the exploitation of GPU‑accelerated LP solvers to further scale the approach. Overall, the contribution is a practical, scalable analysis pipeline that delivers provable bounds on transient rewards for CTMCs with heterogeneous rates, filling a notable gap in the toolbox of reliability engineers and performance analysts.