Stochastic Analysis of Blockchain Systems

Supervision

Denis Villemonais (IRMA, Strasbourg)

Laboratory and team

IRMA, Strasbourg - Team "PROBA"

Subject description

A blockchain is a distributed ledger maintained by a peer-to-peer network, where the nodes apply a consensus algorithm to agree on the recording of new data. Three properties define these systems: efficiency (transaction throughput), decentralization (distribution of control across nodes), and security (resistance to adversarial attacks). The original consensus mechanism, Proof of Work (PoW), relies on computational competition among nodes to solve cryptographic puzzles—a process that incurs substantial energy costs. This PhD project focuses on Proof of Stake (PoS), the consensus protocol underpinning Ethereum and other next-generation blockchains. Unlike PoW, PoS selects validators probabilistically, with selection probabilities scaled by their staked cryptocurrency holdings. The chosen validator proposes the next block and earns a reward in return. While this design might intuitively favor wealthier participants, empirical and theoretical analyses reveal a counterintuitive property: the long-term stake distribution converges to a stable equilibrium on average—a phenomenon mathematically captured by Pólya urn models, a class of reinforced stochastic processes. Modern variants of Proof of Stake (such as Algorand or Ouroboros) lack a unified mathematical description.

The objective is to develop a general framework to model these algorithms, drawing on the theory of reinforced stochastic processes and multi-color (or even infinitely many-color) Pólya urns to represent the large number of participants. The goal is to study their limiting behaviors and fluctuations to identify conditions that ensure fair decentralization. The work will also extend current theoretical results to better account for the heterogeneous and dynamic structure of blockchains (temporal variations in activity or transactional preferences). The second research axis adopts a queueing-theoretic framework to quantify blockchain transactional efficiency. Here, pending transactions are modeled as customers in a G/G/1 queue, where blocks act as batch-service events. We will analyze the arrival-process dynamics (e.g., transaction submission rates) and service discipline (block propagation and validation delays) to derive key performance metrics, including Mean confirmation time (time-to-inclusion in a block) and System congestion (mempool size evolution). Our approach begins with tractable M/M/1 and M/D/1 models (exponential/inter-determined arrivals and service times) to establish baseline results, then extends to non-Markovian settings (e.g., heavy-tailed distributions for bursty traffic). This progression will yield closed-form approximations for waiting-time distributions and throughputs. To bridge the gap between theory and practice, the model will integrate Two features. Fee-based prioritization as transactions compete for block inclusion via dynamic gas markets (Ethereum) or fee-per-byte auctions (Bitcoin). Time-sensitive abandonment as nodes may discard transactions with suboptimal fees after exceeding empirically observed patience thresholds (e.g., 95th-percentile waiting times).

This project has two main objectives:
- A rigorous mathematical characterization of Proof-of-Stake mechanisms, focusing on decentralization properties through stochastic reinforcement models.
- A quantitative framework for blockchain efficiency, grounded in queueing theory to analyze transactional dynamics. By combining probabilistic modeling with empirical validation, the work will deliver both theoretical insights and practical metrics—essential for designing next-generation blockchains that balance fairness, scalability, and sustainability.

Related mathematical skills

Probability and stochastic processes: martingales, Markov processes, convergence in distribution, limit theorems, and possibly elements of branching processes
Queueing theory (M/M/1 models, Little’s law, queues with abandonment/priority)
Scientific programming language (Python, R, Julia, Rust, C or C++)
Statistical methods for stochastic processes (estimation, data-driven calibration, distribution fitting) 

Other public funding

Candidates recruited as PhDs will benefit from IRMIA++ funding and will have to follow the Graduate Program "Mathematics and Applications: Research and Interactions" (https://irmiapp.unistra.fr/training/presentation).

IRMIA++ is one of the 15 Interdisciplinary Thematic Institute (ITI) of the University of Strasbourg. It brings together a research cluster and a master-doctorate training program, relying on 12 research teams and 9 master tracks.

It encompasses all mathematicians at Université de Strasbourg, with partners in computer science and physics. ITI IRMIA++ builds on the internationally renowned research in mathematics in Strasbourg, and its well-established links with the socio-economic environment. It promotes interdisciplinary academic collaborations and industrial partnerships.

A core part of the IRMIA++ mission is to realize high-level training through integrated master-PhD tracks over 5 years, with common actions fostering an interdisciplinary culture, such as joint projects, new courses and workshops around mathematics and its interactions.

Selection will rely on the professional project of the candidate, his/her interest for interdisciplinarity and academic results.