This paper investigates the concept of "commitment lifetime" within the context of decentralized finance (DeFi) protocols, specifically focusing on the variability in the duration a user's commitment to a collateralized asset remains valid. The commitment lifetime, denoted as *LC(P)*, is defined as the difference between the release time and the birth time of the commitment, where *trelease* and *tbirth* represent the respective timestamps. The core objective is to analyze the probability distribution of commitment lifetimes, denoted as *P(LC > t)*, for different algorithmic approaches used in managing collateralized positions. We demonstrate that despite potentially differing underlying mechanisms, various DeFi protocols can arrive at identical conclusions regarding commitment lifetimes. This highlights the importance of understanding the mathematical representation of commitment lifetime distributions, which we term the "commitment lifetime distribution." This analysis provides a foundational understanding for risk management and parameter optimization within DeFi systems.
Ch Sree Kumar, Jatindra Kumar Dash, K. Hemant Kumar Reddy
Blockchain-enabled healthcare infrastructures demand latency-aware, privacy-preserving, and scalable transaction management due to the stochastic and high-volume nature of clinical data processing in decentralized environments. In this study, we propose a blockchain-aware Modified M/M/C (Mo M/M/C) queueing framework specifically designed for NFT-enabled healthcare systems integrated with Zero-Knowledge Proof (ZKP)-based verification. Unlike classical queueing models that assume single-stage service, the proposed model incorporates a multi-stage transaction pipeline consisting of medical processing, NFT tokenization, ZKP verification, and blockchain validation. Accordingly, an effective service rate formulation and blockchain-adjusted arrival rate are analytically derived to capture cryptographic overhead, consensus latency, and smart contract execution delays within a unified stochastic framework. Patient records, prescriptions, and diagnostic data are securely encapsulated as NFTs to ensure immutability, traceability, and decentralized ownership, while ZKP protocols enable privacy-preserving authentication without exposing sensitive medical information. The model further integrates dynamic priority-aware scheduling and validation-aware utilization to optimize resource allocation under heterogeneous healthcare workloads. Extensive discrete-event simulations conducted over scalable transaction volumes (1,000–100,000) evaluate key performance metrics including throughput, average waiting time, system response time, and latency. The results demonstrate that the proposed Mo M/M/C framework significantly improves queue stability, reduces congestion, and enhances throughput compared to FIFO, LIFO, SIRO, and standard M/M/C models. Overall, the revised framework provides a mathematically grounded, cryptography-aware, and blockchain-consistent solution for secure and real-time healthcare transaction management
Bitcoin transaction fees will become more important as the block subsidy declines, but fee formation is hard to study with blockchain data alone because the relevant queueing environment is unobserved. We develop and estimate a structural model of Bitcoin fee choice that treats the mempool as a market for scarce blockspace. We assemble a novel, high-frequency mempool panel, from a self-run Bitcoin node that records transaction arrivals, exits, block inclusion, fee-bumping events, and congestion snapshots. We characterize the fee market as a Vickery-Clarke-Groves mechanism and derive an equation to estimate fees. In the first-stage we estimate a monotone delay technology linking fee-rate priority and network state to expected confirmation delay. We then estimate how fees respond to that delay technology and to transaction characteristics. We find that congestion is the main determinant of delay; that the marginal value of priority is priced in fees, which is increasing in the gradient of confirmation time reduction per movement up in the fee queue; and that transactor choice of RBF, CPFP, and block conditions have economically important effects on fees.
The frozen SUPT-CA phase-coherence probe (α = 0.01, zero free parameters) was applied to live blockchain data from Bitcoin, Ethereum, Solana, Cardano, and Polkadot. Consensus mechanism design directly determines geometric regime: deterministic hardware clocking (Solana, Polkadot) produces deep-lock distributions; regulated proof-of-stake with fee targeting (Ethereum, Cardano) produces coherence-zone distributions; probabilistic proof-of-work (Bitcoin) produces clutch-band timing with sub-floor transaction variability. A validated congestion oracle signal is identified for Ethereum: transaction count d_ij crossing 1.0 in a rolling 150-block window marks network congestion onset, confirmed against the May 2024 memecoin congestion event. All data from live public RPC endpoints, April 15, 2026. No parameters adjusted.
Людмила Василівна Ковальчук, Микола Юрійович Кузнєцов, Алла Антонівна Шумська
Розглядається модель функціонування блокчейну, що ґрунтується на протоколі консенсусу Proof-of-Stake, а саме — протоколі Ouroboros Praos. Кожен створений блок стає видимим усім стейкхолдерам з певною затримкою, або часом синхронізації, що є випадковою величиною з натуральними значеннями та заданим дискретним розподілом. Можливе розгалуження блокчейну внаслідок того, що слотлідер може не бачити останній створений блок. Тому в разі утворення нового блока формується посилання на раніше створений видимий блок. Ще однією причиною потенційного розгалуження є наявність декількох слотлідерів у одному й тому самому таймслоті, оскільки у разі застосування протоколу Ouroboros Praos така можливість є. Множина всіх блоків блокчейну, які були будь-коли створені (зважаючи й на ті, що не увійшли до історії блокчейну згідно з правилом «довшої гілки»), утворює дерево, гілками якого є ланцюги блоків, пов’язаних між собою посиланнями. Запропоновано застосовувати алгоритм методу Монте-Карло для оцінки таких характеристик дерева: середня довжина найдовшої гілки, середня ефективність процедури генерації блоків, середня довжина найдовшого розгалуження, а також розподіли відповідних випадкових величин. Числові приклади ілюструють високу точність отриманих оцінок за незначних витрат часу на обчислення.
In recent years, blockchains have been attracting attention because they are decentralized networks with transparency and trustworthiness. Generally, transactions on blockchain networks with higher transaction fees are processed preferentially compared to others. The processing fee varies significantly depending on other transactions; it is difficult to predict the fee, and it may be significantly high. These are major barriers to blockchain utilization. Although several consensus algorithms have been proposed to solve these problems, their performance has not been fully evaluated. In this study, we model a blockchain system with a base fee, such as in Ethereum, via a priority queueing model. To assess the model’s performance, we derive the stability condition, stationary probability, average number of customers, and average waiting time for each type of customer. In deriving the stability conditions, we propose a method that uses the theoretical values of the partial models. These theoretical values match well with those obtained from Monte Carlo simulations, confirming the validity of the analysis.
Abstract. The purpose of a consensus protocol is to keep a distributed network of nodes “in sync,” even in the presence of an unpredictable communication network and adversarial behavior by some of the participating nodes. In the permissionless setting relevant to modern blockchain protocols, these nodes may be operated by a large number of unknown players, with each player free to use multiple identifiers and to start or stop running the protocol at any time. Establishing that a permissionless consensus protocol is “secure” thus requires both a distributed computing argument (that the protocol guarantees consistency and liveness unless the fraction of adversarial participation is sufficiently large) and an economic argument (that carrying out an attack would be prohibitively expensive for a potential attacker). There is a mature toolbox for assembling arguments of the former type; the goal of this paper is to lay the foundations for arguments of the latter type. For example, the Ethereum protocol is oft-claimed to be “more economically secure” after “the merge,” meaning in its current proof-of-stake incarnation relative to the (proof-of-work) original. What, formally, does this assertion mean? Is it true? Could there be alternative protocols that are “still more economically secure” than Ethereum? How do the answers depend on the assumptions imposed on, for example, the reliability of message delivery or the active participation of non-malicious players? An ideal permissionless consensus protocol would, in addition to satisfying standard consistency and liveness guarantees, render consistency violations prohibitively expensive for the attacker without collateral damage to honest participants—for example, by programatically confiscating an attacker’s resources without reducing the value of honest participants’ resources, as is the intention for slashing in a proof-of-stake protocol. We make this idea precise with our notion of the EAAC (expensive to attack in the absence of collapse) property and prove the following results: (1) In the synchronous and dynamically available setting (in which the communication network is reliable but nonmalicious players may be periodically inactive), with an adversary that controls at least one-half of the overall resources, no protocol can be EAAC. In particular, this result rules out EAAC for all typical longest-chain protocols (be they proof-of-work or proof-of-stake). (2) In the partially synchronous and quasi-permissionless setting (in which resource-controlling non-malicious players are always active but the communication network may suffer periods of unreliability), with an adversary that controls at least one-third of the overall resources, no protocol can be EAAC. In particular, slashing in a proof-of-stake protocol cannot achieve its intended purpose if message delays cannot be bounded a priori. (3) In the synchronous and quasi-permissionless setting, there is a proof-of-stake protocol with slashing that, provided the adversary controls less than two-thirds of the overall stake, satisfies the EAAC property. Thus, while only “classical security” is possible in the dynamically available or partially synchronous settings, proof-of-stake protocols with slashing can obtain additional “economic security” in the quasi-permissionless and synchronous settings. All three results are optimal with respect to the size of the adversary. With respect to Ethereum, our work formalizes the potential security benefits of proof-of-stake sybil-resistance coupled with slashing and the common belief that the merge has increased Ethereum’s economic security. Our work also provides mathematical justifications for several key design decisions behind the post-merge Ethereum protocol, ranging from long cooldown periods for unstaking to economic penalties for inactivity.
Ivan Malakhov, Andrea Marin, Sabina Rossi, Daniel Sadoc Menasché
Blockchains based on Proof-of-Work (PoW) have introduced a new paradigm for distributed ledgers on the Web. In these systems, transactions compete to obtain a position inside the new blocks by offering a fee to be confirmed before others. A finite amount of memory is devoted to store unconfirmed transactions, called Mempool. When new transactions arrive and the Mempool is full, silent droppings of the cheapest unconfirmed transactions occur, without any notification to the owners. This challenge becomes particularly pressing as users have the freedom to append various types of data to the blockchain, including large media files, leading to swift Mempool depletion. In this paper, we study the reliability of PoW blockchains from a user perspective. We provide a numerical model to answer the question:What is the probability of confirmation for a transaction offering a fee$f$when the system is in a certain state?Our model allows blockchain-based applications to analyse the tradeoff between running costs and reliability, i.e., fees offered for the transactions and probability that the transactions will be eventually confirmed. The proposed method is proactive and does not require historical data on dropped transactions that, in fact, are not logged anywhere in the blockchain. This paper presents significant contributions, summarized as follows: (i) the introduction of a stochastic model and its efficient solution for analyzing dropping probability in blockchain systems; (ii) validation of the model through real traces extracted from the Bitcoin blockchain.
Byzantine fault-tolerant consensus protocols have provable safety and liveness properties for static validator sets. In practice, however, the validator set changes over time, potentially eroding the protocol's security guarantees. For example, systems with accountable safety may lose some of that accountability over time as adversarial validators exit. As a result, protocols must rate limit entry and exit so that the set changes slowly enough to ensure security. Here, the system designer faces a fundamental trade-off. Slower exits increase friction, making it less attractive to stake in the first place. Faster exits provide more utility to stakers but weaken the protocol's security. This paper provides the first systematic study of exit queues for Proof-of-Stake blockchains. Given a collection of validator-set consistency constraints imposed by the protocol, the social planner's goal is to provide a constrained-optimal mechanism that minimizes disutility for the participants. We introduce the MINSLACK mechanism, a dynamic capacity first-come-first-served queue in which the amount of stake that can exit in a period depends on the number of previous exits and the consistency constraints. We show that MINSLACK is optimal when stakers equally value the processing of their withdrawal. When stakers values are heterogeneous, the optimal mechanism resembles a priority queue with dynamic capacity. However, this mechanism must reserve exit capacity for the future in case a staker with a much higher need for liquidity arrives. We conclude with a survey of known consistency constraints and highlight the diversity of existing exit mechanisms.
Hyperledger Fabric (shortened to Fabric) is an open-source, enterprise-level, permissioned distributed ledger technology platform with a highly modular, configurable architecture. It supports writing smart contracts in general-purpose programing languages and has become the preferred choice for enterprise-level blockchain applications. However, the transaction throughput of the Fabric system remains a critical factor that restricts the further application of this technology in various fields. Therefore, it is necessary to evaluate and optimize the performance of the Fabric blockchain platform. Existing performance modeling methods need to be improved in terms of compatibility and effectiveness. To address this, we propose a performance-compatible modeling method for Fabric using queuing theory, which considers the limited transaction pool and the situation where node groups are attacked. Using the Fabric 2.0 version as an example, we have established a model of the transaction process in the Fabric network. By analyzing the model’s continuous 3D time Markov process, we solved the system stationary equation and obtained analytical expressions for performance indicators such as system throughput, system steady-state queue length, and system average response time. We conducted extensive analyses and simulations to verify the models’ and formulations’ accuracy and validity. We believe this approach can be extended to various scenarios in other blockchain systems.
Abstract Distributed ledgers, including blockchain and other decentralized databases, are designed to store information online where all trusted network members can update the data with transparency. The dynamics of a ledger’s development can be mathematically represented by a directed acyclic graph (DAG). In this paper, we study a DAG model that considers batch arrivals and random delay of attachment. We analyze the asymptotic behavior of this model by letting the arrival rate go to infinity and the inter-arrival time go to zero. We establish that the number of leaves in the DAG, as well as various random variables characterizing the vertices in the DAG, can be approximated by its fluid limit, represented as the solution to a set of delayed partial differential equations. Furthermore, we establish the stable state of this fluid limit and validate our findings through simulations.
Joachim Neu, Srivatsan Sridhar, Lei Yang, David Tse
Classic BFT consensus protocols guarantee safety and liveness for all clients if fewer than one-third of replicas are faulty. However, in applications such as high-value payments, some clients may want to prioritize safety over liveness. Flexible consensus allows each client to opt for a higher safety resilience, albeit at the expense of reduced liveness resilience. We present the first construction that allows optimal safety--liveness tradeoff for every client simultaneously. This construction is modular and is realized as an add-on applied on top of an existing consensus protocol. The add-on consists of an additional round of voting and permanent locking done by the replicas, to sidestep a sub-optimal quorum-intersection-based constraint present in previous solutions. We adapt our construction to the existing Ethereum protocol to derive optimal flexible confirmation rules that clients can adopt unilaterally without requiring system-wide changes. This is possible because existing Ethereum protocol features can double as the extra voting and locking. We demonstrate an implementation using Ethereum's consensus API.
This paper proposes a, namely, asynchronous chain and presents a Variable Bulk Arrival and Asynchronous Bulk Service (VBAABS) model of the type in order to provide a quantitative method to design an asynchronous chain as a basis in its initial theoretical design stage. Note that the proposed chain is precisely and namely asynchronous along with adaptive-sized blocks in a proactive manner versus the conventional chain that controls the block posting in a strictly synchronous manner to the fixed-sized blocks. The model of the type (i.e., VBAVBS) is considered as the theoretical and quantitative baseline model for the proposed model, namely, an adaptive chain model, yet with a rather reactively dynamic size of blocks, while the proposed asynchronous model carries a proactively dynamic (or adaptive) size of blocks. The proposed asynchronous chain model assumes variable bulk arrivals of transactions in Poisson distribution, i.e., , where represents the number of slots across all the mined transactions, and variable asynchronous services of transactions, each of which applies to a block potentially of different capacity in terms of the number of slots in it, in exponential time, i.e., , for being posted in the current block, namely, VBAABS. The major quantitative distinction between the adaptive model, i.e., VBAVBS, and the asynchronous model, i.e., VBAABS, is that in VBAVBS, every state , where , transitions back into while in VBAABS, every state , where , transitions back into , where , as well as back into , and is the state in which the transaction up in execution is to be discarded, in other words, that the current block capacity is in excess of the required capacity for the transaction in execution to result in a potentially excessive delay as much as than an otherwise ordinarily fully synchronous block posting delay that is as much as . VBAABS will reveal the performance advantages of the asynchronous chain versus the baseline chain, i.e., VBASBS [30] and the adaptive chain, i.e., VBAVBS, with respect to the average time for a slot to wait in the block and the average spatial requirement by the slots in the block, in a quantitative manner. Extensive numerical simulations are conducted on Matlab. Further, for feasibility validation purpose, an asynchronous chain algorithm will be developed and implemented by redesigning the Ethereum open source and analyzed.
In Ethereum, miners are responsible for expanding the blockchain ledger by appending new blocks of transactions in exchange for incentives. Within the current Ethereum incentive mechanism, miners can still receive a significant amount of reward when creating non-full or even empty blocks, despite their negative impact on the system performance. We provide an extensive data-driven analysis of the impact of non-full blocks on the system performance, with the help of the BlockSim simulation tool. We collect the data for 500,000 Ethereum blocks and fit the appropriate probability distributions to the data to provide input suitable for the simulator. We show that the performance of Ethereum can be improved by over 50% if all blocks were filled with transactions. We propose an adjustment to the current Ethereum incentive model to assure the received incentive is always proportional to the block utilization level. Using our proposed approach, the incentive for non-full blocks is significantly reduced, making this behavior less attractive for miners. This implies that miners would be enforced to fill their blocks with transactions, and thus the performance is pushed to its optimal level. We show that our approach can work in practice without any crucial security issues.
In blockchains , transaction fees are fixed by the users. The probability for a transaction to be processed quickly increases with the fee level. In this paper, we study the transaction fee optimization problem in the Ethereum blockchain. This problem consists of determining the minimum price a user should pay so that its transaction is processed with a given probability in a given amount of time. To reach this goal, we define a new solution method based on a Monte Carlo approach to predict the probability that a transaction will be mined within a given time limit. Numerical results on real data highlight the quality of the results.
Francesc Wilhelmi, Sergio Barrachina‐Muñoz, Paolo Dini
Due to the increasing interest in blockchain technology for fostering secure, auditable, decentralized applications, a set of challenges associated with this technology need to be addressed. In this letter, we focus on the delay associated with Proof-of-Work (PoW)-based blockchains, whereby participants validate the new information to be appended to a distributed ledger via consensus to confirm transactions. We propose a novel end-to-end latency model based on batch-service queuing theory that characterizes timers and forks for the first time. Furthermore, we derive an estimation of the optimal block size analytically. Endorsed by analytical and simulation results, we show that the optimal block size approximation is a consistent method that leads to close-to-optimal performance by significantly reducing the overheads associated with blockchain applications.
In recent years, the use of consensus mechanism to maintain the security of blockchain system has become a considerable concern of the community. Delegated proof of stake (DPoS) and practical Byzantine fault tolerant (PBFT) consensus mechanisms are key technologies in maintaining the security of blockchain system. First, this study proposes a consensus mechanism combining DPoS and PBFT, which can rapidly deal with malicious witness nodes and shorten the time of block verification. Second, the M/PH/1 queuing model is used to analyze the performance of the proposed consensus mechanism, and the performance of the improved practical Byzantine fault tolerant consensus mechanism is evaluated from steady-state conditions and key performance measure of the system. Third, the current study uses the theoretical method of open (Jackson) queuing network, combined with the blockchain consensus process, and provides theoretical analysis with special cases. Lastly, this research utilizes numerical examples to verify the computability of the theoretical results. The analytic method is expected to open a series of potentially promising research in queueing theory of blockchain systems.
We review the recent developments in dynamic inventory models with financial flow considerations. The focus is on the literature that introduces cash flow dynamics into the classic inventory models that do not explicitly consider the interactions between physical (or material) and financial flows. These augmented models serve two important purposes. First, they help understand the impact of financial flows on inventory dynamics and decisions. Second, with the connection to the classic inventory models, one can leverage the extant results to derive the optimal control policy or to evaluate/optimize the performance of any given type of policy and reveal insights. We summarize models for both single-stage and multi-stage inventory systems, and discuss the implications and applications to decentralized systems within a broader topic of supply chain finance.