Markov processes in blockchain systems

Abstract

In this paper, we develop a more general framework of block-structured Markov processes in the queueing study of blockchain systems, which can provide analysis both for the stationary performance measures and for the sojourn times of any transaction or block. Note that an original aim of this paper is to generalize the two-stage batchservice queueing model studied in Li et al. [56] both "from exponential to phase-type" service times and "from Poisson to MAP" transaction arrivals. In general, the MAP transaction arrivals and the two stages of PH service times make our blockchain queue more suitable to various practical conditions of blockchain systems with crucial factors, for example, the mining processes, the block-generations, the blockchain-building and so forth. For such a more general blockchain queueing model, we focus on two basic research aspects: (1) By using the matrix-geometric solution, we first obtain a sufficient stable condition of the blockchain system. Then we provide simple expressions for the average stationary number of transactions in the queueing waiting room, and the average stationary number of transactions in the block. (2) However, comparing with Li et al. [56], analysis of the transaction-confirmation time becomes very difficult and challenging due to the complicated blockchain structure. To overcome the difficulties, we develop a computational technique of the first passage times by means of both the PH distributions of infinite sizes and the RG-factorizations. Finally, we hope that the methodology and results given in this paper will open a new avenue to queueing analysis of more general blockchain systems in practice, and can motivate a series of promising future research on development of blockchain technologies. 1

Performance Evaluation of Blockchain Systems: A Systematic Survey

“…Candidates are a mid-state transiting from follower to leader. The whole Raft consensus can be CTMC GI/M/1 queue [104] PoW Bitcoin mean stationary number of txs in the queue and in the block M/G/1 queue variant [105] PoW Bitcoin confirmation time and tx delay non-exhaustive queue [106] PoW NA mean number of txs and mean confirmation time of txs in the system Discrete-time GI/GI N /1 queue [107] Proof-of-Authority Ethereum system queue size and tx waiting time M/M B /1 queue [71] BFT-SMaRt HLF tx latency M/G/1 and M/M/1 queue [108] PBFT NA system delay (n,k) fork-join queue [109] vote-based consensus permissioned blockchain blocks commitment delay, block validation response time and synchronization processes among mining nodes.…”

“…It is too specific and not suitable for many practical conditions of blockchain systems. To generalize this model, in their more recent work [104], the authors changed the transaction arrivals from Poisson to Markov arrival process (MAP), the service times from exponential to phase-type (PH), and the service discipline from FCFS to service-in-random-order. Under the new assumptions, the blockchain queueing model description keeps the same.…”

“…This makes the computation of transaction-confirmation time very difficult. To overcome the challenge, the authors borrowed a computational technique by means of both the PH distributions and the RG factorizations [104].…”

Resilience Analysis for Double Spending via Sequential Decision Optimization

“…Other consensus protocols (see [9]) are also discussed in terms of interesting models (for instance in [10]), but are outside of our scope. For a detailed overview of literature about applications of stochastic methods to blockchains, we refer the interested reader to the recent work [11].…”

Performance Analysis of Blockchain Consensus System With Interference Factors and Sleep Stage

“…Li et al [31] designed a GI/M/1 queueing system with two different service stages, which simulated the processes of mining and building the new blockchain to obtain the average number of transactions in the queue, the average number of transactions in each block and the average confirmation time of transactions in the steady state of the system were obtained by using the matrix-geometric solution method. Li et al [32] extended the service time from the exponential distribution to PH distribution, the steady-state of the blockchain system was analysed by using the matrixgeometric solution method, and then the queueing waiting time and average transactions number of the blockchain system were obtained. Kasahara et al [33], [34] studied the transaction confirmation process based on an M/G b /1 queueing system with batch services and priority, the stationary distribution of the system was obtained by using supplementary variable method, and numerical examples were used to illustrate influence of transaction commissions and maximum block size on the confirmation time of transactions.…”