Reasoning About Proportional Lumpability

Abstract

In this paper we reason about the notion of proportional lumpability, that generalizes the original definition of lumpability to cope with the state space explosion problem inherent to the computation of the performance indices of large stochastic models. Lumpability is based on a state aggregation technique and applies to Markov chains exhibiting some structural regularity. Proportional lumpability formalizes the idea that the transition rates of a Markov chain can be altered by some factors in such a way that the new resulting Markov chain is lumpable. It allows one to derive exact performance indices for the original process. We prove that the problem of computing the coarsest proportional lumpability which refines a given initial partition is well-defined, i.e., it has always a unique solution. Moreover, we introduce a polynomial time algorithm for solving the problem. This provides us further insights on both the notion of proportional lumpability and on generalizations of partition refinement techniques.

Efficient Algorithm for Proportional Lumpability and Its Application to Selfish Mining in Public Blockchains

“…This work extends our prior investigations outlined in [25,26]. Expanding upon our earlier research on proportional lumpability [25], we introduce a pivotal, novel, third characterization in this paper.…”

“…This work extends our prior investigations outlined in [25,26]. Expanding upon our earlier research on proportional lumpability [25], we introduce a pivotal, novel, third characterization in this paper. This result allows us to enhance the efficiency of our previously proposed algorithm, resulting in a groundbreaking approach to proportional lumpability.…”

“…The first one, proven in [21], provides an efficient method to determine if a partition of a Markov chain's state space is induced by proportional lumpability. The second characterization, outlined in [25], has been utilized to describe an algorithm for computing the coarsest proportional lumpability of a given Markov chain in O(|S | 4 ) time. The third characterization is new, and it allows us to improve on the above complexity of obtaining an algorithm running in O(|S | 2 log |S|).…”

“…The algorithm now boasts a remarkable time complexity of O(|S | 2 log |S|), where |S| denotes the cardinality of the state space of the analyzed model, offering the potential to substantially enhance the scalability and applicability of proportional lumpability analyses. Furthermore, we expand on the findings in [25] by offering a thorough case study. This study is inspired by our previous work [26], where a strongly lumpable model for a concrete blockchain problem was presented.…”

“…The first, proven in [21], enables efficient verification of whether a partition of the state space is induced by an equivalence relation representing proportional lumpability. The second, proven in [25], is used to design an algorithm to calculate the coarsest proportional lumpability of a given Markov chain in time O(|S | 4 ) with |S| being the cardinality of the state space of the analyzed model. The third characterization is a novel contribution, and it allows us to improve on the above complexity by obtaining an algorithm for proportional lumpability running in O(|S | 2 log |S|).…”