# Performance Evaluation, Optimization and Dynamic Decision in Blockchain
Systems: A Recent Overview

Quan-Lin Li1, Yan-Xia Chang1 and Qing Wang2
1School of Economics and Management
Beijing University of Technology, Beijing 100124, China
2Monash Business School, Monash University
900 Dandenong Road, Caulfield East, VIC, 3145, Australia Corresponding author:
Y. X. Chang<EMAIL_ADDRESS>
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###### Abstract

With rapid development of blockchain technology as well as integration of
various application areas, performance evaluation, performance optimization,
and dynamic decision in blockchain systems are playing an increasingly
important role in developing new blockchain technology. This paper provides a
recent systematic overview of this class of research, and especially,
developing mathematical modeling and basic theory of blockchain systems.
Important examples include (a) performance evaluation: Markov processes,
queuing theory, Markov reward processes, random walks, fluid and diffusion
approximations, and martingale theory; (b) performance optimization: Linear
programming, nonlinear programming, integer programming, and multi-objective
programming; (c) optimal control and dynamic decision: Markov decision
processes, and stochastic optimal control; and (d) artificial intelligence:
Machine learning, deep reinforcement learning, and federated learning. So far,
a little research has focused on these research lines. We believe that the
basic theory with mathematical methods, algorithms and simulations of
blockchain systems discussed in this paper will strongly support future
development and continuous innovation of blockchain technology.

Keywords: Blockchain; Performance evaluation; Performance optimization;
Optimal control; Dynamic decision.

##  1 Introduction

Since Bitcoin was proposed by Nakamoto [93] in 2008, blockchain technology has
received tremendous attention from both practitioners and academics. So far,
blockchain has made remarkable progress by means of many interesting and
creative combinations of multiple key computer technologies, such as
distributed systems, consensus mechanism, network and information security,
privacy protection, encryption technology, peer-to-peer networks, edge
computing, Internet of Things, and artificial intelligence. At the same time,
some effective scalable frameworks and security designs of blockchain have
been further developed, for example, off-chain, side-chain, cross-chain,
shard, fault tolerant, and attack detection. However, compared with rapid
development of blockchain technology, mathematical modeling and analysis of
blockchain systems is relatively backward, thus it is clear that developing
blockchain technology extremely needs such important basic theory and
necessary mathematical methods.

In this paper, we review mathematical modeling and analysis methods in some
aspects (but no completeness) of blockchain technology, including some
important progress that can further drive developing new potential blockchain
technologies. To this end, our overview in this paper is listed as follows:
(1) Mining processes and management; (2) consensus mechanism; (3) performance
evaluation; (4) performance optimization; (5) optimal control and dynamic
decision; (6) machine learning; (7) blockchain economy and market; and (8)
blockchain ecology. Note that the eight survey points aim to setting up
stochastic models and associated mathematical methods to theoretically improve
blockchain’s performance, scalability, security, privacy protection, work
efficiency, and economic benefit. In what follows, we use Figures 1 to 8 to
describe and analyze the eight survey points (1) to (8) simply.

(1) Mining processes and management

For mathematical modeling and analysis on this research direction, we need to
discuss the key system factors or parameters that largely influence
performance, scalability, security, and privacy protection of blockchain
systems. For example, the miners, the mining pools, the difficulty of solving
the cryptographic puzzle, the transaction fee, the blockchain reward, the
competitive behavior, the tree with forked structure, the work efficiency, the
economic benefit; the attack strategies, the security, the vulnerability, the
fault tolerance, and privacy protection. See Figure 1 for more details.

Figure 1: The mining processes and management

(2) Consensus mechanism

For mathematical modeling and analysis on this research direction, we need to
discuss the random consensus-accomplished times for different consensus
protocols (or algorithms), such as PoW, PoS, DPoS, BFT, PBFT, and Raft.
Furthermore, we need to analyze the blockchain systems under different
consensus protocols and to study the throughput, security, privacy protection,
and scalability of blockchain systems. Our main concerns include a set of
basic factors, such as consensus types, efficiency, convergence, consistency,
network delay, and energy consumption. See Figure 2 for more details.

Figure 2: The consensus mechanism

(3) Performance evaluation

In this class of mathematical modeling and analysis, we need to set up
performance models of blockchain systems when considering different consensus
mechanisms or protocols or algorithms (PoW, PoS, DPoS, PBFT, DAG and so on),
different blockchain types (Bitcoin, Ethereum, side-chain, cross-chain, off-
chain and so on), and innovation and new network architectures of blockchain
systems. See Figure 3 for more details.

Figure 3: Performance modeling and analysis of blockchain systems

(4) to (6) Performance optimization, dynamic decision, and machine learning

In this class of mathematical modeling and analysis (4), we need to optimize
the performance measures of a blockchain system by means of linear
programming, nonlinear programming, integer programming, multi-objective
programming and so on.

In this class of mathematical modeling and analysis (5), we need to realize
optimal control and dynamic decision of a blockchain system by using the
Markov decision processes, sensitivity-based optimization, and stochastic
optimal control. See Figure 4 for more details.

For machine learning (6), we need to develop machine learning, deep
reinforcement learning, and federated learning. See Figure 4 for more details.

Figure 4: Performance optimization, dynamic decision, and machine learning

(7) Blockchain economy and market, and (8) blockchain ecology

For the blockchain economy and market (7) as well as the blockchain ecology
(8), readers may refer to Figures 5 and 6 for a simple introduction,
respectively.

Figure 5: The blockchain economy and market Figure 6: The blockchain ecology

With fast development of blockchain, new blockchain technology continue to
emerge. Thus, performance evaluation, performance optimization, optimal
control, and dynamic decision of blockchain systems become progressively. In
particular, performance modeling and analysis methods have been increasingly
lacking and insufficient up to now, and especially, for dealing with the newly
developing blockchain technology. Blockchain is a hierarchical comprehensive
database, and it operates under a consensus mechanism of distribute systems in
a peer-to-peer network. In addition, blockchain is an interesting and creative
combination of multiple computer technologies, such as encryption techniques,
consensus mechanism, security, privacy protection, and scalability; and
wireless, mobility, cloud computing, edge computing, Internet of Things, and
quantum. Therefore, blockchain is always a complicated stochastic system under
a strongly practical environment. In this situation, performance evaluation,
performance optimization, and dynamic decision of blockchain systems always
become interesting and challenging in their theoretical study.

So far, a few survey papers have discussed blockchain technology with a simple
introduction to performance analysis of blockchain systems. See Table 1 for
more details. From Table 1, it is easy to see that those survey papers focused
on several key perspectives: Performance, scalability, security, and privacy
protection.

Table 1: Survey papers for performance evaluation of blockchain systems Year |  Surveys or reviews |  Research scope
---|---|---
2018 |  Kim et al. [61] |  Scalability solutions
2019 |  Rouhani and Deters [110] |  Security, performance, and applications of smart contracts
2019 |  Wang et al. [132] |  Performance benchmarking tools; optimization methods
2019 |  Zheng et al. [145] |  Challenges and progresses in blockchain from a performance and security perspective
2020 |  Smetanin et al. [118] |  Effective simulation and modeling approaches
2020 |  Singh et al. [117] |  Side-chains for improving scalability, privacy protection, security of blockchain
2020 |  Zhou et al. [146] |  Scalability of blockchain
2020 |  Yu et al. [143] |  Sharding for blockchain scalability
2020 |  Fan et al. [28] |  Stochastic models for blockchain systems: Game theory, performance optimization, machine learning, etc.
2021 |  Cao et al. [13] |  Mathematical models for blockchain such as stochastic process, game theory, optimization, and machine learning
2021 |  Huang et al. [52] |  Performance models, and analysis tools of blockchain systems or networks

To open the scope of our survey research on performance evaluation,
performance optimization, optimal control, and dynamic decision of blockchain
systems, this paper chooses a collection of research materials from major
scientific journals, international conferences, and preprint sites including
IEEE Xplore, ACM digital library, Elsevier, SpringerLink, MPDI, arXiv, HAL,
and so on. Based on these research materials, we provide a detailed review and
analysis from the literature with respect to research on performance
evaluation, performance optimization, and dynamic decision of multiple
blockchain systems, including the consensus mechanism or protocols or
algorithms (PoW, PoS, DPoS, PBFT, DAG and so on), the blockchain types
(Bitcoin, Ethereum, side-chain, cross-chain, off-chain, and so on), and the
new network architecture of blockchain. At the same time, we provide how to
set up stochastic models and to develop effective methods or algorithms for
dealing with performance evaluation, performance optimization, optimal
control, and dynamic decision. Note that such a study of blockchain technology
is interesting and challenging in not only the basic theory but also many
practical applications.

Based on the above analysis, we summarize the main contributions of this paper
as follows:

  * 1.

We provide a basic overview for the available mathematical methods (in
particular, stochastic analysis), which greatly support performance modeling
and computation in performance evaluation, performance optimization, optimal
control, and dynamic decision.

  * 2.

We provide a clear outline and structure for performance evaluation and
performance optimization of blockchain systems. Important mathematical methods
and techniques include (a)performance evaluation: Markov processes, queueing
theory, Markov reward process, random walk, fluid and diffusion
approximations, martingale theory; and (b)performance optimization: Linear
programming, nonlinear programming, integer programming, and multi-objective
programming.

  * 3.

We summarize optimal control and dynamic decision of blockchain systems by
means of, for example, (c) Markov decision process, sensitivity-based
optimization, and stochastic optimal control; and (d) machine learning, deep
reinforcement learning, and federated learning. These issues are interesting
and challenging with greater potential in future study.

The remainder of this paper is organized as follows. Section 2 reviews the
recent literature on the performance evaluation of blockchain systems by means
of the queueing theory, the Markov processes, and the Markov reward processes.
Complementing Section 2, Section 3 provides some further methods for
performance evaluation of blockchain systems by using, for example, the random
walks, the fluid approximation, the diffusion approximation, and the
martingale theory. Section 4 reviews performance optimization of blockchain
systems by means of the linear programming, the nonlinear programming, the
integer programming, and the multi-objective programming. Section 5 focuses on
the overview of applications of the Markov decision processes to find the
optimal dynamic strategy of blockchain systems. Section 6 summarizes
applications of machine learning (e.g., deep reinforcement learning and
federated learning) to performance optimization and dynamic decision of
blockchain systems. Section 7 highlights some concluding remarks.

##  2 Performance Evaluation

In this section, we summarize performance evaluation models of blockchain
systems by means of queueing theory and the Markov process. Note that some
other mathematical methods for performance evaluation are left for the next
section.

###  2.1 Queueing theory

Queueing theory is a key mathematical tool to set up performance measures and
performance evaluation of blockchain systems. Applying queueing theory to
performance analysis of blockchain systems is interesting but challenging
since each blockchain system not only is a complicated stochastic system but
also has multiple key factors and a physical structure with different levels.
Specifically, the key factors include (1) transactions arrivals, (2)
transaction fees, (3) block size, (4) network delay, (5) block generated
process (e.g., mining process and voting process), (6) the pegging process of
a block (or a sub-chains), (7) mining competition among multiple mining pools
(e.g., a tree structure), (8) mining reward, (9) computing power distribution.
The physical structure contains (1) consensus mechanism (e.g., PoW, PoS, DPoS,
PBFT, and DAG), and (2) scalable structure (e.g., side-chain, cross-chain, and
off-chain). The research objectives of blockchain systems are designed as, for
example, (a) performance: Throughput, confirmation time; (b) security; (c)
privacy protection; and (d) scalability. Based on these specific examples, we
can see that it is useful and necessary to apply queueing theory to set up
performance models and to analyze performance measures in the study of
blockchain systems.

Understanding a blockchain system and its physical structure is not always
simple. Li et al. [71] may be the first one to provide a simple diagram of the
physical structure of the PoW blockchain system with a miner (or a mining
pool). See Figure 7 for more details.

Figure 7: A simple physical structure of the PoW blockchain system

For the other blockchain systems (e.g., PoS, DPoS, BFT, PBFT, and Raft), Chang
et al. [17] provided a queueing platform to evaluate their performance
measures once the voting processes are determined by using the Markov modeling
technology. Based on this, the first step is to study the voting processes,
and the second step is to set up a queueing platform through the voting
processes are regarded as the service processes. See Figure 8 for more
details. In this queueing platform, it first needs to determine the two random
variables: The block-generated time and the orphan-block-generated time, which
can be related to the arrival and service times in a queueing model
$\mathrm{M}\oplus\mathrm{M}^{\mathrm{b}}/\mathrm{M}^{\mathrm{b}}/1$ or
$\mathrm{M}\oplus\mathrm{PH}^{\mathrm{b}}/\mathrm{PH}^{\mathrm{b}}/1$.

Figure 8: A queueing platform of different blockchain systems

Kawase and Kasahara [59] may be the first to apply queueing theory to study
the PoW blockchain system with a miner, and a further paper by Kasahara and
Kawahara [58] considered a single-server queue with batch service and priority
mechanism to analyze the transaction-confirmation time. Because the block-
generation time (note that it also includes block-pegged time) is a general
probability distribution, the system of differential-difference equations
given in the two papers by using the supplementary variable method will be
unsolvable. For this reason, Li et al. [71] provided a Markov queue with two
stages (the block-generated time and the block-pegged time) to analyze the PoW
blockchain system with a miner. Li et al. [71] may be the first paper that
clearly describes and expresses the physical structure with multiple key
factors of the PoW blockchain system, as seen in Figure 7. For the two-stage
queue of the PoW blockchain system, the matrix geometric solution was applied
to give a complete solution of this system such that the performance
evaluation of the PoW blockchain system was established in a simple form and
was analyzed by means of a more detailed numerical analysis. In later studies,
Li et al. [72] relaxed the model assumptions of Li et al. [71] to a more
general case that the transaction arrivals are a Markovian arrival process
(MAP), and the block-generated time and the block-pegged time are all of phase
type (PH). Obviously, computing the mean transaction-confirmation time becomes
very difficult and complicated due to the complicated blockchain structure, as
suggested by Li et al. [72].

Kawase and Kasahara [59] and Li et al. [71] have inspired numerous later
strain of literature to use the queueing theory in performance evaluation of
blockchain systems. Now, we list some literature as follows:

Geissler et al. [35] neglected the information propagation delays and assumed
the immediate distribution of transactions and blocks to all the peers in the
network. They developed a discrete-time queueing model that allows performance
evaluation of a blockchain system, such as the transaction waiting time
distribution.

Zhao et al. [144] regarded the mining process as a vacation, and the block-
verification process as a service. Specially, they established a non-
exhaustive queueing model with a limited batch service and a possible zero-
transaction service and derived the average number of transactions and the
average confirmation time of a transaction in the blockchain system.

Krieger et al. [64] proposed a Markovian non-purging $(n,k)$ fork-join
queueing model to analyze the delay time of the synchronization process among
the miners, where a vote-based consensus procedure is used.

Ahmad et al. [2] presented an end-to-end blockchain system for dealing with
the audit trail applications, and analyzed the time, space, consensus, search
complexity, and security of this blockchain system by using the queueing
theory.

Mišić et al. [91] applied the Jackson network model to the entire network, in
which each individual node operates as a priority M/G/1 queue, and developed
an analytical model for analyzing the Bitcoin’s blockchain network.

Frolkova and Mandjes [33] proposed a G/M/$\infty$-like Bitcoin queueing model
to consider the propagation delay between two individual users. Fralix [32]
provided a further discussion for the infinite-server queue introduced in
Frolkova and Mandjes [33].

Seol et al. [114] proposed an embedded Markov chain to analyze a blockchain
system with a specific interest in Ethereum.

He et al. [49] introduced a queueing model with priority to incorporate the
operational feature of blockchain, the interplay between miners and users, and
the security issue associated with the decentralized nature of the blockchain
system.

Gopalan et al. [39] analyzed the stability and scalability of the DAG-based
blockchain system by using queueing networks.

Fang and Liu [29] proposed a dynamic mining resources allocation algorithm
(DMRA) to reduce the mining cost in the PoW blockchain networks through using
the logical queueing-based analytical model.

Meng et al. [90] proposed a queueing model for studying the three stages of
the consortium blockchain consensus, analyze the consistency properties of
consortium blockchain protocols, and provided performance evaluation for the
main stages of the blockchain consensus.

Sun et al. [126] provided a queueing system with three service stages, which
express the three-stage consensus process of the RC-chain and the building of
a new block. By using the queueing model, they obtained three key performance
measures: The average number of transactions in system, the average
transaction confirmation time, and the average transaction throughput.

Altarawneh et al. [3] set up a queueing model to compute the average waiting
time for the victim client transactions, and evaluated the security and
reliability of the blockchain system.

Wilhelmi et al. [135] proposed a batch-service queue model for evaluating the
network delay in a blockchain system. Furthermore, they provided some
simulations to assess the performance of the synchronous and asynchronous
mechanisms.

Ricci et al. [108] proposed a framework encompassing machine learning and a
queueing model M/G/1 to identify which transactions will be confirmed, and
characterized the confirmation time of confirmed transactions.

Li et al. [67] discussed a queueing game with a non-preemptive priority of a
blockchain system and considered both the miners’ mining rewards and the
users’ time costs.

Sukhwani et al. [125] presented a performance method of Hyperledger Fabric
v1.0+ by using a stochastic Petri net modeling (stochastic reward nets) to
compute the throughput, utilization, mean queue length at each peer, and the
critical processing stages within a peer.

For ease of reading, we summarize the queueing models of blockchain systems in
Table 2 .

Table 2: The queueing models of blockchain systems Paper |  Year |  Queue type |  Research scope
---|---|---|---
[125] |  2018 |  Petri Nets model |  Throughput; utilization; mean queue length at each peer; critical processing stages within a peer
[67] |  2018 |  A queueing game |  The miners’ mining rewards; the users’ time cost
[35] |  2019 |  GI/GX/1 |  Queue size; waiting time of transactions
[144] |  2019 |  M/G${}^{X}\oplus$ G/1 |  The average number of transactions; the average confirmation time of transactions
[64] |  2019 |  Fork-join queue |  The delay performance of the synchronization process among the miners
[2] |  2019 |  M/D/c |  The time, space, consensus, and search complexity; security
[91] |  2019 |  Jackson network model; M/G/1 |  Probability distributions of block and transaction distribution time; node response time; forking probabilities; network partition sizes; duration of ledger’s inconsistency period.
[108] |  2019 |  M/G/1 |  Identify which transactions will be confirmed; the confirmation time of confirmed transactions
[33] |  2019 |  GI/M/$\infty$ |  Propagation delay between two individual users
[32] |  2020 |  Infinite-server queue |  A further study of the infinite-server queue studied in [33]; related infinite-server queues have similar dynamics
[114] |  2020 |  MX/MX/1 |  The average number of slots; the average waiting time per slot; throughput
[49] |  2020 |  M/MX/1 with priority |  Users’ equilibrium behavior; total fee rate; confirmation latency; system equilibria
[39] |  2020 |  Monotone separable queuing models |  Stability and scalability of the DAG network
[29] |  2020 |  Logical queueing-based analytical model |  Mining resources allocation; mining cost; stability
[90] |  2021 |  M/H2/1; M/M/1; M/Er/1 |  The consistency and security of consortium blockchain protocols
[3] |  2021 |  M/M/1; M/M/$\infty$ |  The average waiting time for the victim client transactions; security; reliability
[126] |  2021 |  Three-phase service queuing process |  The average number of transactions; the average transaction confirmation time; the average transaction throughput.
[135] |  2021 |  A novel batch-service queue model |  The learning completion delay of blockchain-enabled federated learning; performance of synchronous and asynchronous mechanisms
[17] |  2022 |  $\mathrm{M}\oplus\mathrm{M}^{\mathrm{b}}/\mathrm{M}^{\mathrm{b}}/1$ |  Throughput of the dynamic PBFT blockchain system; the stationary rate (or probability) that a block is pegged on the blockchain; the stationary rate (or probability) that an orphan block is returned to the transaction pool

By means of the queueing theory, some papers have conducted research on the
simulation and empirical study of blockchain systems. Important examples
include among which Memon et al. [89] and Spirkina et al. [122] proposed a
queueing theory-based simulation model to understand the performance measures
of the blockchain system.

In the queueing models of blockchain systems, Bowden et al. [10] is a key work
because the generation time is related to the service time. They showed that
the generation time of a new block has some key statistical properties, for
example, the generation time is non-exponential, and it can also be affected
by many physical factors.

So far, many classes of blockchain systems have still been lacking research on
performance evaluation by using the queueing theory. For example, the PoW
blockchain system with multiple mining pools, the PBFT blockchain system of
dynamic nodes, the DAG-based blockchain systems, the Ethereum, and the large-
scale blockchain systems with either cross-chain, side-chain, or off-chain.
Therefore, the queueing models of blockchain systems are always interesting
and challenging in the future study of blockchain technology.

###  2.2 Markov processes and Markov reward processes

In performance evaluation of blockchain systems, Markov processes and Markov
reward processes are two effective mathematical methods. See Li [68] for a set
of Markov models and computational methods by using the RG-factorizations.
Note that Markov processes are used to evaluate throughput, confirmation time,
and security and privacy protection of blockchain systems; while the Markov
reward processes are applied to analyzing work efficiency, economic benefits,
and cost control of blockchain systems.

For the vulnerability and forked structure of the PoW blockchain systems with
two mining pools (honest and dishonest), Eyal and Sirer [27] proposed a
selfish mining strategy for the competitive mining process between two mining
pools, and set up a simple Markov process with a special reward structure to
discuss the competitive behavior between the two mining pools. By means of an
intuitive reward analysis, they indicated that the selfish miner can win a
higher mining reward through violating the honest agreement in the blockchain
system. However, Li et al. [69] showed that the Markov process with rewards
given in Eyal and Sirer [27] is not correct from the ordinary theory of Markov
processes.

For a PoW blockchain system with two mining pools (honest and dishonest), Li
et al. [69] showed the competitive behavior between the two mining pools by
means of Figure 9.

Figure 9: The competitive behavior between the two mining pools

When the two block branches forked at a common tree root, let $I(t)$ and
$J(t)$ be the numbers of blocks mined by the honest and dishonest mining pools
at time $t$, respectively. It is seen from Li et al. [69] that
$\left\\{\left(I(t),J(t)\right):t\geq 0\right\\}$ is a continuous-time Markov
process whose infinitesimal generator is given by

$Q=\left(\begin{array}[]{ccccccc}Q_{\widetilde{\mathbf{0}},\widetilde{\mathbf{0}}}&Q_{\widetilde{\mathbf{0}},0}&Q_{\widetilde{\mathbf{0}},1}&&&&\\\
Q_{0,\widetilde{\mathbf{0}}}&Q_{0,0}&Q_{0,1}&&&&\\\
Q_{1,\widetilde{\mathbf{0}}}&&Q_{1,1}&Q_{1,2}&&&\\\ B&&&A&C&&\\\ B&&&&A&C&\\\
\vdots&&&&&\ddots&\ddots\end{array}\right).$

For the PoW blockchain system with two mining pools (honest and dishonest),
Gőbel et al. [37] set up a two-dimensional Markov process with network
propagation delay and provided performance evaluation of the PoW blockchain
system. Javier and Fralix [55] further discussed the two-dimensional Markov
process given by Gőbel et al. [37] and developed a new computational method.
Li et al. [69] set up a new two-dimensional pyramidal Markov (reward) process
of the blockchain system, which leads to a novel theoretical framework for
performance evaluation of a PoW blockchain system with adding new random
factors by means of a new class of matrix geometric solutions.

Using the Markov process approach of Eyal and Sirer [27], Nayak et al. [94]
introduced a new type of mining strategy: The stubborn mining strategy and
also established its two extended forms: The equal-fork stubborn mining
strategy and the path stubborn mining strategy. Further important examples
include Wang et al. [133] and Liu et al. [80]. In addition, inspired by the
Markov process approach of Eyal and Sirer [27], the selfish mining strategy
was extended to the Ethereum system. Readers can see Grunspan and Pérez-Marco
[46] and Niu and Feng [97] for more details. Also, the impact of the selfish
mining behavior of multiple mining pools on the blockchain system has also
been paid widespread attention, e.g., see Bai et al. [7], Bai et al. [8],
Chang [16], Liu et al. [78], Marmolejo-Cossío et al. [88] and Xia et al.
[139].

From the ordinary theory of Markov processes, we summarize some works that use
the Markov processes or Markov reward processes to study other interesting
issues of blockchain systems as follows.

Song et al. [121] provided a Markov process theory for network growth
processes of DAG-based blockchain systems.

Chang et al. [17] applied a large-scale Markov process to study the dynamic-
PBFT blockchain system.

Carlsten [14] applied the Markov process to study the impact of transaction
fees on the selfish mining strategy of the blockchain.

Shi et al. [116] developed a new consensus protocol (Proof-of-Age, PoA) and
employed a continuous time Markov chain to show that the consensus protocol
can disincentivize the pooled mining.

Kiffer et al. [60] set up a Markov-chain to analyze the consistency properties
of blockchain protocols.

Huang et al. [51] established a Markov process with an absorbing state to give
performance analysis of the raft consensus algorithm in private blockchains.

Ma et al. [86] established a two-dimensional Markov process to provide
performance evaluation of PBFT blockchain systems.

Srivastava [124] computed the transaction confirmation time of blockchain by
using a Markov model.

Li et al. [76] established a Markov process to analyze performance and
security of the IoT ledgers with a directed acyclic graph.

Li et al. [75] established the Markov process to study the block access
control mechanism in the wireless blockchain network.

Piriou and Dumas [99] constructed a Markov process to analyze the blockchain
system and developed a simulation model of blockchain technology.

Nguyen et al. [95] applied the Markov process and deep reinforcement learning
to study the task offloading problem in the mobile blockchain with privacy
protection.

Jofré et al. [56] established a Markov process to study the convergence rate
of blockchain mining games.

Together, these studies outline a critical role of Markov processes and Markov
reward processes in the performance evaluation of blockchain systems. This
would be a potential area for future study.

##  3 Further Methods for Performance Evaluation

In this section, we summarize further methods for performance evaluation of
blockchain systems, including the random walk, the fluid approximation, the
diffusion approximation, and the martingale theory.

###  3.1 The random walk

The random walk is a key mathematical method in analyzing many stochastic
models, such as queueing systems and information and communication technology
(ICT) systems. See Spitzer [123], Prabhu [102], and Xia et al. [138] for more
details.

Recent, a few papers have studied blockchain systems by using the random walk,
and especially, analyzing the double-spending attacks of blockchain.

Goffard [38] refined a random walk model underlying the double-spending
problem and provided a fraud risk assessment of the blockchain system.

In contrast with Goffard’s model [38], Jang and Lee [54] proposed a new random
walk model to further study the probability distribution of catch-up time
spent for the fraudulent chain to catch up with the honest chain, which takes
into account the block confirmation. They discussed the profitability of the
double-spending attacks that manipulate a priori mined transaction in a
blockchain system.

Brown et al. [11] studied the duration and probability of success of a double-
spend attack in terms of the random walk.

Grunspan and Pérez-Marco [45] determined the minimal number of confirmations
requested by the recipient such that the double spend strategy is non-
profitable by means of the random walk.

###  3.2 The fluid and diffusion approximations

The fluid and diffusion approximations are two key mathematical methods in
analyzing many stochastic models with general random variables, such as
queueing systems, inventory models, supply chains, and communication networks.
The fluid and diffusion approximations describe a deterministic process that
aims to approximately analyze the evolution of stochastic processes, that is,
they can analyze the evolution of generalized stochastic processes by using
the idea of weak limits. Recently, fluid and diffusion approximations have
been widely used in analyzing of large-scale complex networks with the
tendency of scale expansion, complex structure, and dynamic state. See Chen
and Yao [20], Whitt [134], Dai et al. [24], Büke and Chen [12], Chen and
Shanthikumar [18] for more details.

So far, fluid and diffusion approximations have been applied to the analysis
of blockchain systems. Important examples include among which Frolkova and
Mandjes [33] developed a Bitcoin-inspired infinite-server model by means of a
random fluid limit. King [62] proposed a fluid approximation of the random
graph model and discussed the related technologies of shared ledgers and
distributed ledgers in blockchain systems. Ferraro et al. [31] studied the
stability of unverified transaction systems in the DAG-based distributed
ledgers by means of the fluid approximation. Koops [63] applied the diffusion
approximation to predict the confirmation time of Bitcoin transactions.

There are a few blockchain works that analyze the evolution of generalized
stochastic processes by using the idea of weak limits. For example, Corcino et
al. [23] discussed the mean square displacement of fluctuations of Bitcoin
unit prices over time on a daily basis by applying the method of Brownian
motion and Gaussian white noise analysis. Chevallier et al. [21] used the Lévy
jump diffusion Markov switching model to study the price fluctuation
characteristics of Bitcoin.

For the fluid and diffusion approximations of blockchain systems, it is
interesting and challenging to study the PoW blockchain systems with multiple
mining pools. See Li et al. [70] for a general tree representation of
complicated mining competition among multiple mining pools. Note that the
fluid and diffusion approximations can also provide performance evaluation of
blockchain systems, thus there exists a great potential and innovation in the
future research of many blockchain systems (e.g., PoS, DPoS, PBFT, and DAG).

###  3.3 The martingale theory

The martingale theory not only enriches the contents of probability theory but
also provides a powerful method for studying stochastic processes and
stochastic models, and it is widely applied in economics, networks, decision,
and control. Grunspan and Pérez-Marco applied the martingale theory to study
the profits of miners under different attacks of blockchain systems since
2018. Using the martingale theory, the research on common attacks in
blockchain systems is summarized in Table 3.

Table 3: Research on attacks of blockchain by using martingale theory Year |  Attack type |  Research scope |  Method or theory
---|---|---|---
2018 |  Selfish mining [40] |  Expected duration of attack cycles; the profitability model by using repetition games; improvement of Bitcoin protocol; the miner’s attraction to the selfish mining pools |  Martingale theory; Doob stopping time theorem
2018 |  Stubborn mining [41] |  The profitabilities of stubborn mining strategies |  Martingale theory; Catalan numbers and Catalan distributions
2018 |  Trailing mining [42] |  The revenue ratio of the trail stubborn mining strategy in the Bitcoin network; the profitability of other block-withholding strategies |  Martingale theory; classical analysis of hiker problems
2020 |  SM; LSM; EFSM and so on [43] |  The profitabilities of various mining strategies |  Martingale theory; Markov chains; Dyck words
2020 |  SM, intermittent SM and smart mining [44] |  The closed forms for the profit lag; the revenue ratio for the strategies “selfish mining” and “intermittent selfish mining” |  Martingale theory; foundational set-up from previous companion article
2021 |  Nakamoto double spend [45] |  The exact profitability for Nakamoto double spend strategy; the minimal number of confirmations to be requested by the recipient such that this double spend strategy is non-profitable |  Martingale theory; glambler ruin; random walk

##  4 Performance Optimization

In this section, we provide an overview for performance optimization of
blockchain systems by using different optimal methods.

Performance optimization is to optimize performance measures of blockchain
systems by means of mathematical programming (e.g., linear programming,
nonlinear programming, integer programming, and multi-objective programming).
And it composes four elements: Optimization problem, optimization variables,
objective functions, and restrictive conditions. The optimization process
needs to accomplish such a task: When the restrictive conditions are
satisfied, the optimization variables are adjusted to make that these
objective functions go to a maximum or a minimum.

Performance optimization is necessary and important in the study of blockchain
systems, including design, organization, control, and management of blockchain
systems. Such a study will strongly support the overall development of
theoretical research and practical applications of blockchain technology.

So far, performance optimization of blockchain systems has been studied in at
least three aspects as follows:

(1) From consensus mechanism and network architecture of blockchain systems,
it is interesting to optimize performance (e.g., throughput and confirmation
time), work efficiency, economic benefit; improve scalability, security,
privacy protection and degree of decentralization; and balancing operations
costs and efficiency, and allocation of profits. Important examples include
Lundbaek and D’Iddio [83], Liang [77], Nguyen et al. [96], Wang et al. [131],
Saad et al. [111], Reddy and Sharma [107], Leonardos et al. [65], Liu et al.
[79], Varma and Maguluri [128], and Li et al. [66].

(2) From some key factors (e.g., operations costs, pricing, computing power,
transaction fee, network delay) of PoW blockchain systems, it is necessary to
consider the optimal strategies of dishonest miners, for example, how to pack
a transaction package from a transaction pool? How to incentive honest miners
to jump into the dishonest mining pool? How to incentive the dishonest miners
to keep mining in a round of competition? How to maximize miners’ economic
benefit or work efficiency? Important examples include Kang et al. [57],
Aggarwal et al. [1], Ramezan et al. [106], and Liu et al. [79].

Table 4: Performance optimization of blockchain systems Proposed for |  Optimization scope |  Optimization factors |  Methods
---|---|---|---
Governed blockchains [83] |  Solving the MINLP optimization problems for computing optimal Proof of Work configuration parameters that trade off potentially conflicting aspects such as availability, resiliency, security, and cost |  Expected availability; resiliency; security; cost |  Mixed integer nonlinear programming
A new system [77] |  Re-innovating all the core elements of the blockchain technology to achieve the best balance among scalability, security and decentralization |  Transaction confirmation time; information propagation latency |  Min-max optimization
Users, miners, and verifiers [57] |  Considering the tradeoff between the network delay of block propagation process and offered transaction fee from the blockchain user to jointly maximize utility of the blockchain user and individual profit of the miners |  Network delay of block propagation process; offered transaction fee from the blockchain user |  Nonlinear programming
A new sharding paradigm [96] |  Proposing OptChain that can minimize transactions and maintain a temporal balance among shards to improve the confirmation time and throughput |  Confirmation time; transaction throughput; cross-shard transactions minimization; temporal balancing |  Nonlinear programming
A new dynamic routing solution [131] |  Proposing a new dynamic routing solution Flash to strike a better tradeoff between path optimality and probing overhead |  Payment size; transaction fees; probing overhead; transaction throughput |  Linear programming
Miners [1] |  Demonstrating BTC’s robust stability, and find that the implemented design of emergency difficulty adjustment resulted in maximal miners’ profits |  Coinbase reward; competition cost reward; transaction fees; competition cost fees; mining cost; waiting cost; switching incentive; miners’ profits |  Mixed integer nonlinear programming
A new form of attacks [111] |  Studying a new form of attacks that can be carried out on the memory pools and proposing countermeasures that optimize the mempool size and help in countering the effects of DDoS attacks |  Attack cost; relay fee; mining fee; memorypool size |  Nonlinear programming
PoW blockchain and blockDAG [107] |  Proposing two models to scale the transaction throughput |  Block creation rate; transaction throughput; main chain block growth rate; propagation delay; risk |  Nonlinear programming
PoS protocols [65] |  Leveraging weighted majority voting rules that optimize collective decision making to improve the efficiency and robustness of the consensus mechanism |  Validators’ voting behavior; blockchain rewards; collective decision; collective welfare |  Mixed integer nonlinear programming
A new pricing mechanism [109] |  Presenting a pricing mechanism that aligns incentives of agents who exchange resources on a decentralized ledger to greatly increase transaction throughput with minimal loss of security |  Transaction pricing; expected transaction efficiency; block assembly; transaction throughput; security |  Integer linear programming
Miners [106] |  How should miners pick up transactions from a transaction pool to minimize the average waiting time per transaction |  Average waiting time per transaction |  Mixed integer nonlinear programming
Enterprises and users [147] |  Choosing the most effective platform from many blockchains to control costs and share data |  Technical, market and popularity indicators; improved global DEA-Malmquist measure |  Nonlinear programming
Lightning and Spider network [128] |  Setting up a two-sided queue model and propose a throughput optimal algorithm that stabilizes the system under any load within the capacity region |  Transaction throughput; arrival rate; capacity region; payment requests |  Linear programming
A new protocol [66] |  Proposing EntrapNet protocol and optimize EntrapNet to deal with the fundamental tradeoff between security and efficiency |  Security; efficiency |  Nonlinear programming
Protocol designer, users, and miners [79] |  Proposing a Fee and Waiting Tax (FWT) mechanism to improve the incentives for the miners’ participation and blockchain security, and to mitigate blockchain insufficient fee issue |  Storage costs of miners; users’ transaction fee; fee choices and waiting tax for users; transaction waiting time |  Multi-objective programming

(3) For users or enterprises with pricing, cost, transaction fee, and platform
selection, how to maximize user (or enterprise) utility? Important examples
include Kang et al. [57], Riehl and Ward [109], Zhou et al. [147], Varma and
Maguluri [128], and Liu et al. [79].

Based on the above analysis, we summarize performance optimization of
blockchain systems in Table 4. It is seen from Table 4 that most of the
research on performance optimization of blockchain systems focuses on
discussing the following issues:

(i) Does there exist a better network architecture or consensus mechanism such
that the blockchain system is more efficient, secure, and scalable?

(ii) Is there a better application scenario that makes blockchain more
consistent and less waste of resources?

(iii) Is there a more effective economic incentive mechanism that makes
blockchain more profitable and the cost of operations, verification and
communication lower?

(iv) Is there a better trading platform and a more favorable market
environment that make users in blockchain more usable and more credible among
users?

In a word, performance optimization of blockchain systems is an interesting
and hot frontier research topic, and also there exists a large capacity for
research innovation through discussing broad blockchain systems (e.g.,
consensus mechanism and network architectures), for example, cross-chain,
side-chain, off-chain, and interoperability of information and assets among
different chains; data synchronization, data security; pricing, cost, economic
benefit, and work efficiency; scalability, security, and privacy protection.

##  5 Markov Decision Processes

In this section, we apply the Markov decision processes (MDPs) to the study of
blockchain systems and provide some algorithms for computing the optimal
dynamic policy of such a Markov decision process. For the Markov decision
processes, readers may refer to Puterman [103] and Li et al. [73] for more
details.

The Markov decision processes are widely applied to deal with the selfish
mining attacks in the PoW blockchain systems because the selfish mining
process needs to choose a series of mining policies to be able to maximize the
reward or to minimize the cost.

When a PoW blockchain system has two different miners or mining pools (honest
and dishonest) to compete for a more mining reward, in which the dishonest
miner may adopt different mining policies based on the longest chain rule. The
dishonest miner can control the fork structure of block tree through releasing
some parts of blocks to obtain his maximum benefit. Accordingly, an
interesting topic focuses on how the dishonest miner finds an optimal mining
policy (i.e., how many mined blocks are released in a round of competition).
Important examples include among which Sapirshtein et al. [113], Sompolinsky
and Zohar [120] and Gervais et al. [36] introduced four different policies:
Adopt, cover, match, and wait for selfish miners, and they determined the
optimal selfish mining policy.

Zur et al. [148] studied the optimal selfish mining policy of the PoW
blockchain system by using the Markov decision process and proposed a new
method to solve the Markov decision process with an average reward criterion.

Bai et al. [8] applied the Markov process to study the PoW blockchain system
with multiple miners and used the Markov decision process with observable
information to find the optimal selfish mining policy for a special case with
two different miners.

Li et al. [74] discussed the PoW blockchain system by using the hidden Markov
decision process and proposed an improved selfish mining policy.

Ma and Li [87] analyzed the optimal selfish mining policy of the PoW
blockchain system with two mining pools through using the sensitivity-based
optimization theory.

In addition, the Markov decision processes are also applied to deal with other
blockchain control issues as follows:

Niu et al. [98] provided an incentive analysis for the Bitcoin-NG protocol by
using the Markov decision process, and showed that the Bitcoin-NG protocol can
maintain the incentive-compatible mining attacks.

Wüst [137] used the Markov decision process to study the data security in the
blockchain system.

Chicarino et al. [22] discussed the selfish mining inspection and tracking
attacks in the PoW blockchain network by means of the Markov decision
processes.

##  6 Machine Learning

In this section, we summarize the applications of machine learning (e.g., deep
reinforcement learning and federated learning) to performance optimization and
dynamic decision of blockchain systems.

Recent, machine learning (e.g., deep reinforcement learning and federated
learning) has been applied to study performance optimization and dynamic
decision of blockchain systems. Since the Markov decision process of a
blockchain system is always more complicated, it is difficult and challenging
to find the optimal policy of the Markov decision process, while the machine
learning can provide an approximate solution for such an optimal policy.
Therefore, it is interesting to develop approximate methods or algorithms to
find the optimal policy by using, artificial intelligence, machine learning,
deep reinforcement learning, and federated learning.

The survey papers: Liu et al. [81] provided a survey for the recent literature
that the blockchain technology is analyzed by means of machine learning and
discussed several interesting directions on this research line. Ekramifard et
al. [26] provided a systematic overview for applying artificial intelligence
to the study of blockchain systems, including the Markov decision process and
machine learning. Chen et al. [19] applied machine learning to performance
optimization and dynamic decision of blockchain systems and proposed several
interesting topics for future research. Shafay et al. [115] reviewed the
recent literature on applications of deep reinforcement learning to develop
the blockchain technology.

In what follows, we summarize the recent research on applications of machine
learning to the study of blockchain systems from several different aspects:
The mining policy, the mobile-edge computing, and the Internet of Things or
Industrial Internet of Things.

The mining policy: Considering the optimal policy of selfish mining attacks in
Bitcoin as well as the Nash equilibrium in block withholding attacks, Hou et
al. [50] proposed a SquirRL framework to apply deep reinforcement learning to
analyze the impact of attacks on the incentive mechanism of PoW blockchain.
Bar-Zur [9] used reinforcement learning to find the optimal policy for the
miners of different sizes through solving a Markov decision process problem
with an average reward criterion. Wang et al. [130] applied reinforcement
learning (machine learning) to find the optimal mining policy in the Bitcoin-
like blockchain and designed a new multi-dimensional reinforcement learning
algorithm to solve the mining MDP problem with a non-linear objective function
(rather than a linear objective function in the standard MDP problems).

When the growth of PoW blockchain is modeled as a Markov decision process, a
learning agent needs to make the optimal decisions over all the states of
Markov environment in every moment. To track the generation of new blocks and
their verification process (i.e., solving the mathematical puzzles), You [141]
set up the PoW consensus protocol (i.e., solving mathematical puzzles) through
dealing with a reinforcement learning problem. In this case, the verification
and generation of new blocks are designed as a deep reinforcement learning
iterative process.

Mobile-edge computing: Nguyen et al. [96] applied the Markov processes and
deep reinforcement learning to study the task offloading problem of mobile
blockchain under privacy protection. Qiu et al. [105] formulated the online
offloading problem as a Markov decision process and proposed a new model-free
deep reinforcement learning-based online computation offloading approach for
the blockchain-empowered mobile edge computing, in which both the mining tasks
and the data processing tasks are considered. Feng et al. [30] developed a
cooperative computation offloading and resource allocation framework for the
blockchain-enabled mobile-edge computing systems and designed a multi-
objective function to maximize the computation rate of mobile-edge computing
systems and the transaction throughput of the blockchain systems by means of
the Markov decision processes.

Asheralieva and Niyato [4] developed a hierarchical learning framework by
means of the Markov decision processes with the service provider and the
miners and studied resource management of edge computing to support the public
blockchain networks. By applying the Markov decision process, Asheralieva and
Niyato [5] presented a novel game-theoretic, Bayesian reinforcement learning
and deep learning framework to represent the interactions among the miners for
the public and consortium blockchains with mobile edge computing. Yuan et al.
[142] applied the Markov decision processes and deep reinforcement learning to
study the sharding technology for the blockchain-based mobile edge computing.

Internet of Things: Waheed et al. [129] provided a summary of the security and
privacy protection of blockchain technology in the Internet of Things by using
machine learning algorithms. Gao et al. [34] studied the task scheduling of
the mobile blockchain supporting applications of the Internet of Things by
means of deep reinforcement learning and Markov decision processes.

Industrial Internet of Things: Qiu et al. [104] and Luo et al. [84] studied
the blockchain-based software-defined Industrial Internet of Things by means
of a dueling deep Q-learning approach and the Markov decision processes. Yang
et al. [140] studied the energy-efficient resource allocation for the
blockchain-enabled Industrial Internet of Things by deep reinforcement
learning and Markov decision processes. Wu et al. [136] provided a review for
the deep reinforcement learning applied to the blockchain systems in the
Industrial Internet of Things.

##  7 Concluding Remarks

Since Nakamoto [93] proposed Bitcoin in 2008, research on blockchain has
attracted tremendous attention from both theoretical research and engineering
applications. With fast development of blockchain technology, blockchain has
developed many imaginative applicable modes through a series of innovative
combinations among distributed data storage, point-to-point transmission,
consensus mechanisms, encryption techniques, network and data security,
privacy protection, and other computer technologies. Also, their subversive
and imaginative features can further inspire endless technological innovations
of blockchain. Among them, the most representative technologies, such as
timestamp-based chain block structure, DAG-based network data structure,
distributed consensus mechanism, consensus mechanism-based economic
incentives, and flexible and programmable smart contracts, have increased
extremely rich colors to various practical applications. Important examples
include digital economy [15], Fintech [92], cloud services [47], reputation
systems [25], social security [127], e-commerce supply chain [82], artificial
intelligence [53], sharing economy [48], and supply chain management [112].

Performance evaluation, performance optimization, and dynamic decision are one
of the most basic theoretical research of blockchain systems, and they play a
key role in design, control, stability, improvement, and applications of
blockchain systems. So far, some blockchain pitfalls (e.g., low performance
and scalability, weak security and privacy protection, and inconvenient
interoperability among blockchain subsystems) have severely limited a wide
range of applications of blockchain technology. To resolve these blockchain
pitfalls, a few technologies or methods have been proposed and developed,
e.g., see off-chain [100], side-chain and cross-chain [6], sharding [85], and
DAG [101]. However, it is a key to deal with whether these novel technologies
and methods can effectively improve these pitfalls of the blockchain systems,
while such an interesting issue is to need to be sufficiently studied by means
of some strictly mathematical analysis. On the other hand, it is an
interesting topic to set up some useful mathematic relations among
performance, scalability, security, privacy protection and so forth. Some
intuitively understanding examples include among which increased security will
result in low throughput; increased scalability will result in high
throughput; increased security will result in strong privacy protection. Note
that the mathematic relationships can be set up by means of research on
performance evaluation, performance optimization, and dynamic decision of
blockchain systems.

It is easy to understand that practical applications will lead to the
innovation boundary of blockchain technology. That is, blockchain applications
are a main driving force of blockchain technology development. When a new
application of blockchain technology is launched, the interface between
technology and application is not always friendly, the performance and
stability are not always high, and there are also deficiencies in security and
privacy protection. Note that all the necessary improvements or increasing
maturity need some plentiful research on performance evaluation, performance
optimization, and dynamic decision of blockchain systems. In addition, for the
current blockchain technology, we need to actively create a social atmosphere
and ecological environment for both theoretical research and practical
applications of blockchain. Also, this can powerfully promote deep integration
between the blockchain technology and the key information technologies (such
as artificial intelligence, big data, and the Internet of Things).

For a large-scale blockchain system or a new blockchain technology, it is key
to find the bottleneck through analyzing vulnerability and fault tolerance of
network architecture by means of some new mathematical theory and methods
developed in research on performance evaluation, performance optimization, and
dynamic decision of blockchain systems. Thus, this motivates us in this paper
to provide a recent systematic overview of performance evaluation, performance
optimization, and dynamic decision of blockchain systems, which involves
mathematical modeling and basic theory of blockchain systems. Important
examples include (a) performance evaluation: Markov processes, queuing theory,
Markov reward processes, random walks, fluid and diffusion approximations, and
martingale theory; (b) performance optimization: Linear programming, nonlinear
programming, integer programming, and multi-objective programming; (c) optimal
control and dynamic decision: Markov decision processes, and stochastic
optimal control; and (d) machine learning: Deep reinforcement learning and
federated learning. We believe that the new basic theory with mathematical
methods, algorithms, and simulations discussed in this paper will strongly
support future development and continuous innovation of blockchain technology.

Based on the above analysis, we believe that there are still many interesting
research directions to be explored, such as smart contract, DAG-based
blockchain, cross-chain, side-chain, off-chain and other network
architectures; and some basic or new consensus protocols. Our future research
includes:

– Developing effective methods to compute and improve performance, stability,
and scalability of blockchain systems.

– Setting up a mathematical theoretical framework for security and privacy
protection of blockchain systems.

– Providing effective methods to optimize and dynamically control performance,
security and privacy protection of large-scale blockchain systems.

– Developing machine learning for performance optimization and dynamic
decision of blockchain systems.

–Developing a healthy ecological environment and reasonable operations
management in the blockchain community by means of research on performance
evaluation, performance optimization, and dynamic decision of blockchain
systems.

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