# PERFORMANCE MODELING OF PUBLIC PERMISSIONLESS BLOCKCHAINS: A SURVEY

Molud Esmaili  University of South FloridaUSA and  Ken Christensen  University
of South FloridaUSA

###### Abstract.

Public permissionless blockchains facilitate peer-to-peer digital
transactions, yet face performance challenges specifically minimizing
transaction confirmation time to decrease energy and time consumption per
transaction. Performance evaluation and prediction are crucial in achieving
this objective, with performance modeling as a key solution despite the
complexities involved in assessing these blockchains. This survey examines
prior research concerning the performance modeling blockchain systems,
specifically focusing on public permissionless blockchains. Initially, it
provides foundational knowledge about these blockchains and the crucial
performance parameters for their assessment. Additionally, the study delves
into research on the performance modeling of public permissionless
blockchains, predominantly considering these systems as bulk service queues.
It also examines prior studies on workload and traffic modeling,
characterization, and analysis within these blockchain networks. By analyzing
existing research, our survey aims to provide insights and recommendations for
researchers keen on enhancing the performance of public permissionless
blockchains or devising novel mechanisms in this domain.

Public permissionless blockchain, Performance evaluation, Performance
modeling, Queueing theory, Blockchain consensus.

††doi: XXXXXXX.XXXXXX††copyright: none††ccs: Networks Network performance
modeling

##  1\. Introduction

A distributed ledger known as blockchain is formed by linking a sequence of
blocks within a peer-to-peer network (Xu et al., 2023). The foundational
concept of blockchain was initially introduced by Harber et al. in 1991 (Haber
and Stornetta, 1991). Their paper outlines the idea of a distributed ledger,
where each timestamp contains a hash of the previous timestamp. In 2008,
Nakamoto presented a paper on Bitcoin (Nakamoto, 2008), which served as the
pioneering implementation of blockchain as a peer-to-peer electronic cash
system. Reaching an agreement within a blockchain network is a critical
undertaking. In this regard, numerous consensus mechanisms have been designed
and developed to maintain validity in a trustless and unreliable network, and
the variety of these mechanisms continues to grow. The most renowned consensus
approach is Proof of Work (PoW) (Nakamoto, 2008), initially introduced by
Nakamoto and employed in Bitcoin. However, due to concerns regarding
throughput, energy efficiency, and mining equipment cost limitations
associated with PoW, alternative consensus mechanisms like Proof of Stake
(PoS) have been proposed. PoS relies on the concept of selecting the next
block creator through a combination of random selection, the stake amount
owned by a validator, and stake’s age or the duration it has been held by the
validator, enhancing scalability. This idea was first introduced in 2012 with
the Peercoin cryptocurrency (King and Nadal, 2012) and later utilized in other
projects such as Nxt (Community, 2104) and Blackcoin (Vasin, 2014). An
improvement to PoS is Delegated Proof of Stake (DPoS), introduced by Larimer
in 2014 (Larimer, 2014), wherein nodes choose representatives through voting
to validate blocks. Another consensus mechanism that can be employed in
blockchain networks is Practical Byzantine Fault Tolerance (PBFT), which was
introduced by Castro in 1999 (Castro et al., 1999) to address the Byzantine
generals’ problem (Lamport et al., 2019). Over time, several other variants of
these mechanisms have been proposed to enhance the consensus process in
blockchain networks. Examples of such variants include Proof of Elapsed Time
(Lake, 2017), Delegated Byzantine Fault Tolerance (NEO, 2014), Proof of
Activity (Bentov et al., 2014), and Proof of Burn (Karantias et al., 2020).
Blockchain extends beyond cryptocurrency and finds applications in different
industries like healthcare (Liang et al., 2017), supply chain management (Xu
et al., 2021), digital marketplaces (Ahmadi et al., 2023), streaming platforms
(Esmaili and Javidan, 2020) and more (Mohanta et al., 2019).

Blockchain systems can be broadly categorized as either permissioned or
permissionless. Regarding a permissioned blockchain, only authorized users are
permitted to serve as consensus nodes and access the blockchain’s data.
Conversely, in a permissionless blockchain, any user has the freedom to join
or exit the network (Swan, 2015). The other categorization for blockchain
systems is as public and private blockchains. Public blockchains are openly
accessible, permitting anyone to view the blockchain’s data. In contrast,
private ledgers restrict access solely to those who were approved beforehand
(Fan et al., 2020). It’s worth noting that existing public permissionless
blockchain systems, primarily based on variants of PoW and PoS, experience
performance issues such as limited throughput, high communication overhead,
and transaction confirmation latency (Peng et al., 2021).

Cryptocurrencies, specifically Bitcoin, have much slower transaction rates
compared to VISA’s network which handles thousands per second (Casino et al.,
2019). Addressing this latency is crucial for widespread cryptocurrency
adoption. To address the performance-related issues of blockchain systems,
numerous researchers have focused on the performance evaluation of private and
public blockchain systems. Furthermore, extensive research has been conducted
on scalability challenges and potential solutions in public blockchains. Table
1 provides a classification of survey papers that reviewed research on
evaluating and improving performance and scalability in blockchain systems.

The pioneering studies addressing the bulk service queuing problem, namely
(Chaudhry and Templeton, 1983; Bailey, 1954; Downton, 1955; Jaiswal, 1960b, a;
Neuts, 1967) drew inspiration from Kendall’s fundamental research in 1953
(Kendall, 1953). Simulations and analytical modeling are now commonly used
approaches for understanding the features of blockchain systems (Memon et al.,
2019). Furthermore, analytical modeling can be applied when a mathematical
model offers a closed-form solution, providing a simplified representation of
system operation through a set of equations (Smetanin et al., 2020).
Consequently, analytical modeling approaches are highly attractive for
blockchain analysis and evaluation, leading to numerous researchers reviewing
models to analyze blockchain systems (Kang et al., 2020; Rico-Pena et al.,
2023; Huang et al., 2021; Cao et al., 2022). The various approaches employed
to evaluate the performance of blockchain systems are depicted in Fig 1.

Figure 1. Taxonomy of performance evaluation methods in blockchain systems

###  1.1. Motivation

“Performance modeling” involves developing abstract representations or
mathematical constructs that mimic how a system, application, or process
behaves in terms of performance parameters. These models help predict and
evaluate how well the systems will perform under various workloads (Menasce et
al., 2004). When it comes to blockchain systems, performance modeling
approaches allow for the assessment and analysis of critical performance
metrics such as throughput and latency. Additionally, given the substantial
network traffic and decentralization in public permissionless blockchains, the
evaluation and enhancement of performance metrics assume heightened
significance. In addition, several new and unreviewed blockchain performance
models have been conducted in recent years (Ramezan et al., 2020; Kasahara,
2021; Fang and Liu, 2020; Vesely et al., 2021; Park et al., 2020; Ling et al.,
2020; Fralix, 2020; Lian et al., 2020).

Upon conducting an overview, as presented in Tables 1, it becomes evident that
a comprehensive survey of performance models of public permissionless
blockchain systems is still missing. Such a survey would serve a dual purpose:
(i) aiding researchers in selecting the most suitable performance modeling
approach to evaluate public permissionless blockchains, and (ii) assisting
researchers in characterizing and analyzing the workload and traffic of public
permissionless blockchains for a more precise performance evaluation. This
motivation has prompted us to compile this survey, offering a recent review of
this increasingly important field. The related survey papers to our survey are
summarized in Table 1.

Table 1. The related survey papers What is surveyed | Ref | Year |  Main contributions
---|---|---|---
Consensus algorithms | (Mingxiao et al., 2017) | 2017 |  Performance analysis of consensus mechanisms
from a performance | (Dinh et al., 2018) | 2018 |  Distributed ledgers, cryptography, consensus protocols, and smart contracts
perspective | (Nguyen et al., 2019) | 2019 |  Proof-of-Stake-based mechanisms
| (Zheng et al., 2019) | 2019 |  Performance and security concerns of distributed ledgers, cryptographies, consensus mechanisms
| (Xiao et al., 2020) | 2020 |  Performance comparison of consensus mechanisms
| (Wan et al., 2020) | 2020 |  Performance and scalability comparative analysis
| (Lepore et al., 2020) | 2020 |  Performance analysis of PoW, PoS, Pure PoS
| (Ferdous et al., 2021) | 2021 |  Performance and security analysis
| (Kaur et al., 2021) | 2021 |  Performance analysis of PoW, PoS, DPoS, and PoA
| (Oyinloye et al., 2021) | 2021 |  Performance analysis of recently proposed mechanisms
| (Ge et al., 2022) | 2022 |  Proof of Stake-based consensus algorithms
| (Xu et al., 2023) | 2023 |  Review and performance analysis of consensus mechanisms
Blockchain systems’ | (Wang et al., 2019) | 2019 |  Blockchain benchmarking tools and optimization methods
performance and | (Dabbagh et al., 2021) | 2020 |  Empirical performance evaluations of permissioned blockchains
scalability approaches | (Smetanin et al., 2020) | 2020 |  Performance evaluation approaches in blockchain
and solutions | (Bamakan et al., 2020) | 2020 |  Consensus mechanisms performance evaluation criteria
| (Fan et al., 2020) | 2020 |  Performance evaluation approaches for blockchain systems
| (Kim et al., 2018a) | 2018 |  Scalability solutions in blockchains
| (Xie et al., 2019) | 2019 |  Scalability solutions in blockchains
| (Yang et al., 2020) | 2020 |  Scalability solutions in blockchains
| (Zhou et al., 2020) | 2020 |  Scalability solutions in blockchains
| (Hafid et al., 2020) | 2020 |  Scalability solutions in blockchains
| (Khan et al., 2021) | 2021 |  Scalability solutions in blockchains
| (Sanka and Cheung, 2021) | 2021 |  Scalability and performance issues and solutions in blockchains
Models of blockchains’ | (Kang et al., 2020) | 2020 |  Stochastic modeling approaches to analyze blockchain systems
features and phases | (Huang et al., 2021) | 2021 |  Modeling, theories, and tools to analyze blockchain systems
| (Rico-Pena et al., 2023) | 2023 |  Models used for characterization of blockchain systems’ features
| (Cao et al., 2022) | 2023 |  Stochastic process, game theory, and ML methods used to analyze blockchains

###  1.2. Our Contribution

  * •

Our review encompasses an introduction to an exploration of selected public
permissionless consensus algorithms and an examination of performance
evaluation methods and criteria in blockchain systems.

  * •

We review and compare existing studies that employ modeling approaches to
investigate public permissionless blockchains’ performance. We analyze their
advantages and limitations to offer guidance for future researchers.

  * •

Ultimately, we extensively explored research focused on proposing models for
blockchain network topologies, examining traffic and workload characteristics,
and initiating the development of benchmarking workloads.

###  1.3. Organization

This survey is structured as follows: In Section 2, we provide an introduction
to the fundamentals of selected public permissionless blockchains, the
performance evaluation of blockchains, and the criteria for evaluating the
performance of blockchains. Section 3 involves a review of research focusing
on the modeling of public permissionless blockchain systems’ performance.
Moving to Section 4, we delve into the examination of the studies conducted on
workload modeling for public permissionless blockchains. Section 5 outlines
several challenges and suggests potential future research directions, and
lastly, Section 6 serves as the conclusion of this article.

##  2\. Background

###  2.1. Public permissionless blockchains

As previously noted, existing public permissionless blockchains utilizing PoW
and PoS encounter scalability and performance challenges. We have selected
three well-known public permissionless blockchains, Bitcoin (Nakamoto, 2008)
as a PoW-based blockchain, Ethereum 2.0 (167, 2023)based on PoS, and Algorand
(Gilad et al., 2017)as a combination of PoS and PBFT, offering a condensed
overview and a comparison of their performance metrics.

####  2.1.1. Proof of Work

##### Bitcoin

Consensus mechanism: Proof of Work (PoW) is Bitcoin’s pioneering consensus
mechanism. Nodes compete to calculate block header hash functions by finding a
value that must be equal to or less than a given "nonce" value. Validated
blocks are broadcasted and appended to all the nodes’ chains. The time-
consuming authentication process incentivizes miners with mining rewards,
consisting of the block reward and the included transaction fees (Casino et
al., 2019).

Bitcoin’s data organization centers on transactions, blocks, and nodes.
Transactions encompass various fields like version number, flags, inputs,
outputs, and lock time. Blocks, each 1MB, contain a magic number, block size,
header, transaction counter, and transactions(Nakamoto, 2008). Each node in
the Bitcoin network maintains a memory pool, known as the Mempool, which
serves as a temporary storage area for unconfirmed transactions. These
transactions have been broadcast to the network but have not yet been included
in a block by miners (Easley et al., 2019). Bitcoin utilizes the Unspent
Transaction Output (UTXO) set to monitor unspent transaction outputs, which
serve as inputs for new transactions. Full Bitcoin nodes maintain a UTXO copy
to validate and create transactions efficiently, avoiding the need to review
the entire blockchain (Delgado-Segura et al., 2019). Segregated Witness
(SegWit) (161, 2022) is a protocol upgrade implemented in the Bitcoin network
to improve the scalability and security of Bitcoin transactions by separating
the transaction signatures (witness data) from the transaction data (Singh et
al., 2020). SegWit increases block capacity without altering the base block
size but introduces a new unit of “Weight unit” and switches the block limit
size form 1 MB to 4 million weight units.

Bitcoin faces scalability issues due to block size limitations (1MB) and block
creation time (10 minutes), restricting transaction throughput. This hinders
the network’s capacity to handle increased transaction volume, posing a
scalability challenge (Monrat et al., 2019). The performance parameters of
Bitcoin are summarized and compared with those of several other public
permissionless blockchains in Table 2.

####  2.1.2. Proof of Stake

##### Ethereum 2.0.

The Ethereum blockchain, originally relying on a proof-of-work mechanism
similar to Bitcoin, underwent a significant upgrade with Ethereum 2.0 released
in 2022. This update aimed to enhance security, scalability, and energy
efficiency within the network (Cassez et al., 2022).

In Ethereum 2.0’s proof-of-stake (PoS) consensus mechanism, time is segmented
into slots (each lasting 12 seconds) and epochs (comprising 32 slots).
Validators are chosen at random to propose blocks in slots, while a validator
committee validates the proposed blocks’ authenticity. When a node initiates a
transaction using its private key, it attaches a gas fee as an incentive for
validators to verify the sender’s ETH balance and signature. Once validated,
the transaction enters the node’s local Mempool and gets broadcast to other
nodes. Subsequently, a randomly selected block proposer creates the next
block, including transactions from the Mempool (167, 2023).

In Ethereum 2.0’s PoS, consensus clients utilize the LMD-GHOST algorithm
(Buterin et al., 2020) to resolve discrepancies among validators’
perspectives. LMD-GHOST identifies the fork with the most substantial
historical weight of attestations, ensuring agreement on the chain’s head.
Ethereum also introduced the concept of accounts within its protocol, serving
as both the originator and recipient of transactions. Consequently,
transactions directly alter account balances rather than maintaining a state,
as seen in Bitcoin’s UTXOs. This approach enables the transfer of values,
messages, and data among accounts, leading to state transitions (Monrat et
al., 2019).

Ethereum 2.0’s performance limitation involves potential bottlenecks in shard
communication and data transfer, impacting overall scalability despite its
sharding improvements (167, 2023). Table 2 provides a summary of the
performance metrics for Ethereum 2.0.

##### Algorand.

Algorand (Gilad et al., 2017) operates as a cryptocurrency utilizing a
Byzantine Agreement (BA) protocol, merged with Verifiable Random Functions
(Micali et al., 1999). This amalgamation selects users as committee members
via a cryptographic process to engage in the Byzantine Agreement for
transaction consensus. In contrast to Proof of Work, Algorand demands minimal
computational power and maintains a low probability of forking (Xiao et al.,
2020). Additionally, once a block is integrated into the blockchain, it
attains finality.

Algorand’s consensus mechanism comprises several key steps: a) Block Proposal:
Each round allows anyone to propose a block. b) Random Participant Selection:
A small, stake-based group is chosen privately, ensuring energy efficiency and
decentralized involvement. c) Block Proposer Election: From this group, one
participant becomes the block proposer for that round. d) Block Proposal and
Voting: The proposer creates a block, shared across the network, followed by
nodes voting on its acceptance using a secure voting scheme. e) Fast and
Secure Consensus: Algorand employs "BA* Byzantine Agreement" for swift and
secure agreement within a fixed number of rounds.

If the network agrees upon the proposed block, it becomes validated as the
subsequent block in the blockchain. Should the network reject it, the
procedure restarts in the subsequent round, featuring a fresh random selection
of participants and a different individual responsible for proposing the block
(Gilad et al., 2017).

Algorand faces performance limitations due to block propagation delay and
network latency. While its consensus mechanism ensures scalability and
security, the speed of block dissemination across the network and
synchronization can impact transaction throughput, potentially causing delays
in confirming transactions (Gilad et al., 2017). Table 2 outlines and compares
the performance metrics of Algorand with several other public permissionless
blockchains.

###  2.2. Performance evaluation criteria in blockchain systems

In this subsection, we examined and compared performance evaluation criteria
and notable latencies across various popular public permissionless
blockchains; Bitcoin, Algorand, and Ethereum 2.0 illustrated in Table 2.

Table 2. performance criteria in the selected blockchain systems Parameter | Definition | BTC | ETH2.0 | Algorand
---|---|---|---|---
TPS | The number of transactions processed per second |  $\approx$ 7 |  $\approx$ 100,000 |  $\approx$ 10,000
Transaction Confirmation Time | The time interval between transaction creation and transaction |  $\approx$ 10-30 min |  $\approx$ 5 min |  $\approx$ 4 sec
| permanently addition to the blockchain |  |  |
Block Generation Time | The time interval between creation times of two consecutive blocks |  $\approx$ 10 min |  $\approx$ 12 sec |  $\approx$ 3.3 sec
Block Size | The maximum amount of data can be included in a single block |  $\approx$ 5000 tx |  $\approx$ 1500 tx |  $\approx$ 25000 tx
Latencies | Consensus latency | The time it takes for the nodes to reach an agreement on the validity | high | low | very low
| of transactions |  |  |
Finality latency | The time it takes for a transaction to be irreversibly confirmed and | high | moderate | zero
| considered as “Fsinalized” |  |  |
Waiting latency | The time a tx spends in a queue or pending state before being | high | low | very low
| processed, validated, and added to the blockchain |  |  |
Tx/Block Propagation | The time it takes for newly created blocks or arrived transactions to | high | low | moderate
latency | be disseminated across the entire network |  |  |
Processing overheads | Computational power is needed for consensus mechanisms, and | very high | low | low
| handling any other protocols in the system |  |  |

##  3\. Performance modeling of public permissionless blockchains

In this section, our objective is to classify and review the research
conducted in the field of performance modeling of public permissionless
blockchains. Several stochastic models have been employed to assess the
performance and security vulnerabilities of blockchains. These models
encompass Markov chains, Queueing models, Stochastic Petri nets, and other
models such as Random graph model and stochastic network model(Fan et al.,
2020). Markov process has been utilized to model the general features of
blockchains, such as consensus algorithms and platforms (Huang et al., 2019;
Sukhwani et al., 2017; Zheng et al., 2018; Liu et al., 2019). Furthermore,
stochastic models have been applied to evaluate blockchain security in the
work presented by (Li et al., 2020). When it comes to performance evaluation,
Markov chains have been employed to model the performance of private or
permissioned blockchains like PBFT (Ma et al., 2022), Raft (Huang et al.,
2019), the tangle (Cao et al., 2019), and HLF (Sukhwani et al., 2018; Jiang et
al., 2020). Similarly, Stochastic Petri nets have found utility in modeling
the performance of private or permissioned blockchains, as seen in the work on
PBFT (Sukhwani et al., 2018; Yuan et al., 2020).

For the performance modeling of public permissionless blockchains, queueing
theory stands out as the most suitable modeling approach. Because one of the
most critical aspects of performance evaluation in public blockchains is
transaction latency and transaction confirmation rate, which can be
effectively modeled using queueing models. Therefore, in this paper, we will
concentrate on this specific modeling approach.

###  3.1. Performance modeling using queueing theory

In the past decade, queueing theory has seen broad application in modeling
computer networks and also blockchain systems (Fan et al., 2020). Within the
context of a blockchain system, user-initiated transactions must go through
queues where miners or validators process them, add them to a new block,
validate them, and finally confirm them. In blockchain networks, as you can
see in Figure 2, various phases of consensus and block generation can be
represented as queueing systems, including queues for arrival transactions,
transaction confirmations, and block verifications. Within queue theory, some
important performance parameters, including the expected number of
transactions within the system, transaction throughput, and average block
generation time can be assessed.

Multiple queue types have been examined and categorized using the standardized
Kendall’s notation (Kendall, 1953). In Kendall’s notation, queueing models are
denoted as A/S/C, where A represents the inter-arrival time for queue entries
(input), S indicates the distribution of service times (queue-discipline), and
c represents the number of service channels at the node (service mechanism).
Common queue types utilized in blockchain modeling include M/G/1, M/M/1,
G/M/1, and GI/GI/1. In this notation, M denotes Markovian behavior with
Poisson arrival times, signifying exponential inter-arrival times. G stands
for general distribution, GI denotes general independent distribution, and the
last number represents the count of servers in the system (Kendall, 1953).

Figure 2. The transaction confirmation process in blockchain systems

In the following sections, we will introduce common types of queues that are
used in blockchain modeling and review the research that has been conducted
using each type of queue system.

####  3.1.1. Modeling used M/G/1 queue.

In 2016, Kasahara and Kawase (Kasahara and Kawahara, 2016) developed a model
representing Bitcoin’s transaction-confirmation process, utilizing a single-
server $M/G/1$ queue with batch service, a priority mechanism, and a non-work-
conserving discipline. The paper defines "Sojourn Time" or “Transaction-
confirmation Time” ($E[T]$) as the period between a user initiating a
transaction and confirmation of the block containing that transaction.
Categorizing transactions as high (H) and low (L) based on added fees, their
findings indicate that the average confirmation time for L-class transactions
surpasses that of H-class transactions. Additionally, they noted that when the
arrival rate of low-fee transactions increases fourfold compared to the
current rate, these transactions endure notably large confirmation times
within a 1 MB maximum block size scenario. Utilizing $f(\lambda)$ as a
function of lambda for $E[T]$, and $E[T_{H}]$ and $E[T_{L}]$ representing
transaction confirmation time for high and low priority transactions, they
derived equations (Kasahara and Kawahara, 2016):

(1) $E[T_{H}]=f(\lambda_{H})$ (2)
$E[T_{L}]=\lambda+1f(\lambda_{H}+\lambda_{L})-\lambda f(\lambda_{H}).$

Through validation via discrete-event simulation, the authors showed that the
block generation time in Bitcoin follows an exponential distribution. Their
examination of transaction confirmation times for H and L transactions
concluded that enlarging the maximum block size does not fundamentally address
transaction confirmation time and scalability concerns.

In 2017, Kawase and Kasahara (Kawase and Kasahara, 2017) explored Bitcoin’s
transaction confirmation time, employing queueing theory as a framework. They
modeled Bitcoin mining as an $M/G^{B}/1$ queue with batch service. The model
assumed that transaction arrivals followed a Poisson distribution, while
service time intervals were considered to have a general distribution. Within
the proposed model, incoming transactions couldn’t get immediate service, even
if there was room in the service unit, below the block size limit (the maximum
batch defined). Their research encompassed numerical analyses of service time
and transaction confirmation time- sojourn time- with a focus on how block
size or batch size influenced these metrics. The authors also conducted a
Monte Carlo simulation to validate their analytical results. Comparing these
results to measured data, they observed close alignment, with a relative error
of 1.35%. Real-world data from a 2-year period from blockchain.info (86, 2023)
informed network simulations, revealing variations between analytical and
simulated results for small block sizes but strong agreement as block sizes
increased. Their work delved into the complexities of joint distribution and
differential-difference equations but left a unique solution to future
research.

In 2019, Ricci et al. (Ricci et al., 2019) presented a framework that combines
machine learning and queueing theory to analyze Bitcoin transaction
confirmation times. This framework incorporates several key factors into the
queueing model, including Activity time, Mean time between transactions, Mean
time between block confirmations, Number of active blocks, and Transaction
delay. To determine which transactions are likely to be confirmed, they
employed machine learning techniques to classify transactions as confirmed or
not. The framework introduces a threshold, below which it’s advisable to wait
for natural validation. If the transaction gets confirmed by the expected
time, it’s considered an early confirmation. If not, a classifier is utilized
to predict whether the transaction will ultimately be confirmed, taking into
account the time it has already spent in the system. Additionally, the
researchers employed an $M/G/1$ queueing model to investigate factors
associated with transaction confirmation time. They found that there was a
probability $p$ that the users might need to wait for an additional block for
final confirmation. According to (Ricci et al., 2019) transaction delay $D$
can be expressed as:

(3) $E(D)=pE(B)+E(B_{r})$

which $B$ is the time interval between block confirmations and $B_{r}$ denotes
the residual time of the inter-block time. Ricci et al.’s findings using data
measurements across more than 3000 blocks in 2016 indicated that unconfirmed
transactions typically had lower average fees compared to confirmed ones.
Interestingly, the size of transactions did not influence their likelihood of
confirmation. However, the probability of confirmation was positively
correlated with the transaction value; larger transactions had a higher chance
of being confirmed. Approximately 90% of all transactions were confirmed
within 33 minutes, but typical users experienced delays exceeding the time
between block generations.

In (Misic et al., 2019) and (Mišić et al., 2019), Misic et al. introduced a
comprehensive analytical model aimed at understanding Bitcoin’s distribution
network. They employed branching processes to reflect random node
connectivity. Further, the authors applied the Jackson network model, treating
individual nodes as priority M/G/1 queuing systems and characterizing data
arrival as a non-homogeneous Poisson process derived from the data delivery
protocol’s analytical model. While presenting performance results, the study
highlights various metrics such as block and transaction distribution times,
node response times, forking probabilities, network partition sizes, and
ledger inconsistency period duration.

Zhao et al. (Zhao et al., 2019) expanded the traditional blockchain queue
model by considering a potential zero-transaction scenario. They introduced a
M/G/1 queueing model with limited batch service to replicate how transactions
get confirmed. This model assumed that transactions arrived randomly,
following a Poisson distribution. In their model, they likened the mining
process to a vacation, and the block verification process to a service. The
time taken for mining (V) and block verification (S) were both i.i.d and
followed general distributions. The authors derived the average number of
transactions in the system and the average confirmation time of transactions
using this model. The study concluded by validating their model using both
analysis and simulations, assessing how well it portrayed the blockchain
system’s performance.

In Bitcoin, the transactions are stored in a transaction queue named Mempool
to await processing. Miners select transactions from this queue to form new
blocks. Ramezan et al. (Ramezan et al., 2020) conducted a study to investigate
the miner’s transaction selection process and its impact on the average
waiting time for transactions. In the proposed $M/G/1$ queueing model depicted
in Fig 3 , transaction arrivals are denoted as $\lambda$, following a Poisson
process. The average time required for mining has an exponential distribution
and is denoted by $\mu$, which remains independent of the number of users.

This paper introduces a novel mining strategy where the miner waits for D
transactions to accumulate in the queue before initiating a new round of
mining. This new approach is labeled $Wait-Min(D)$, in contrast to the
traditional method known as $Wait-Min(1)$. Denoting the average queue length
by $L(D)$ and utilizing Little’s law, the average waiting time per transaction
can be calculated as $\overline{W}(D)=\frac{\overline{L}(D)}{\lambda}$. The
authors in (Ramezan et al., 2020) derived the formula for the average waiting
time per transaction in traditional mining as:

(4)
$\overline{W}(1)=\frac{\lambda(\lambda+\mu)}{\mu(\lambda\mu+\mu^{2}+\lambda^{2})}$

To validate the newly proposed strategy, the authors conducted simulations and
compared the results obtained with their analytical findings to assess the
performance of the new method. Their investigation revealed that the optimal
value for D, which minimizes the average waiting time, is approximately
$0.9\lambda/\mu$. The results showed that the average transaction waiting time
can be reduced by 10% when the miner does not immediately start the next
mining process, using all the stored transactions in the queue.

Figure 3. The M/G/1 queueing model proposed in (Ramezan et al., 2020)

Fang and Liu (Fang and Liu, 2020) proposed a queueing analytical model to
allocate mining resources in PoW-based blockchain networks, utilizing Lyapunov
optimization techniques and emphasizing a tradeoff between mining energy and
queueing delay. Notably, the proposed model avoided statistical assumptions on
arrivals and services, ensuring a dynamic allocation algorithm without
requiring knowledge of arrival probability distributions or prioritization
mechanisms. The proposed algorithm in this paper, parameterized by $K$,
balances mining energy cost and queueing delay, showcasing an $O(1/K)$
proximity to optimality and a worst-case delay scaling as $O(K)$.

In 2021, Kasahara (Kasahara, 2021) analyzed three queuing models including
$M/G^{b}/1$ model, $M/G^{b}/1$ model with priority discipline, and $G/M^{b}/1$
to find the most accurate approach to model the Bitcoin transaction
confirmation process. Kasahara found that the $M/G^{b}/1$ model without
transaction priority yielded a transaction confirmation time half of the
measured value, suggesting that arriving transactions were not included in the
mining block. The $M/G^{b}/1$ model with transaction priority, classifying
transactions into Low (L) and High (H) priority, revealed a non-work-
conserving queuing mechanism, indicating intentional oversight of low-fee
transactions by miners. Lastly, the $G/M^{b}/1$ queue model, with transaction
interarrival times following a general distribution and block generation time
following an exponential distribution with rate µ, showed no significant
difference between measured and estimated values using the EM algorithm
(Khayari et al., 2003). Data-driven simulation results exhibited a significant
transaction confirmation time difference with a small block size, diminishing
as the block size increased, highlighting the third queuing model provided
more accurate transaction confirmation time results under the current block
size of 1 MB.

####  3.1.2. Modeling used M/M/1 queue.

In 2019, Srivastava (Srivastava, 2019) employed an $M/M/1$ queueing model to
examine transaction confirmation time and applied a Markov queueing model to
regulate block approval rates within the blockchain. The model’s accuracy was
verified through experimental outcomes. Using this model, Srivastava derived
the following formula for determining the average waiting time for a block to
be added to the blockchain at time t, W(t), in which $\lambda$ is the block
arrival rate and µ is the block departure rate (Srivastava, 2019).

(5)
$W(t)=\frac{1}{\mu}\left[\frac{\lambda}{\mu}\left(1-\frac{\lambda}{\mu}\right)\right]$

In (Memon et al., 2019), the authors presented a queuing theory-based model to
enhance understanding of blockchain performance. Performance parameters,
including the number of transactions in a block, Mining time, Throughput,
Memory pool count, Waiting time in the memory pool, the number of unconfirmed
transactions, total transactions, and generated blocks, were systematically
examined. In the model, the Memory pool, housing unconfirmed transactions, is
represented as an M/M/1 queue, and the miners’ pool, composed of miners
collaborating to solve mining puzzles, is modeled as an M/M/c queue with
multiple servers. The policy of queue is assumed as FCFS, and transaction
arrival rates follow a Poisson distribution. The model is depicted in Fig 4.

Figure 4. The proposed model for blockchain systems in (Memon et al., 2019)

Cao et al. (Cao et al., 2020) investigated the performance of PoW, PoS, and
DAG consensus algorithms by analyzing Block generation time, Confirmation
delay, Transaction per second, and Confirmation failure probability. It
assumed transaction arrivals follow a Poisson distribution and mining times as
service times follow an exponential distribution. The study revealed that
network resources such as computation power, coin age, and queue size, along
with network load conditions like transaction arrival rate, significantly
impact consensus algorithm performance. PoW and PoS are more sensitive to
network resource parameters, while DAG is more influenced by network load
conditions. The authors derived the following formulas for the mentioned
parameters, assuming $D$ as difficulty, $U$ as a nonce target, $r_{i}$ as the
hash power of node i, $\lambda$ as transaction arrival rate, $L$ as block
size, $Q_{i}$ as queue length at node i, and $m$ as the number of cumulative
blocks required to confirm a transaction (Cao et al., 2020).

Table 3. The formulas presented in (Cao et al., 2020) for the performance parameters of PoW and PoS-based blockchains Parameter | Equation for PoW | Equation for PoS
---|---|---
Block Time | $T_{b}^{pow}=\frac{D}{\sum_{i=1}^{n}r_{i}}$ | $T_{b}^{pos}=\frac{D}{\sum_{i=1}^{n}bal_{i}\times t_{i}}$
Confirmation delay | $T_{c}^{pow}=m^{pow}\times T_{b}^{pow}$ | $T_{c}^{pos}=m^{pos}\times T_{b}^{pos}$
TPS | $TPS^{pow}=\begin{cases}\sum_{i=1}^{n}\lambda_{i}&\sum_{i=1}^{n}\lambda_{i}\leq\frac{L}{T_{b}^{pow}}\\\ \frac{L}{T_{b}^{pow}}&\sum_{i=1}^{n}\lambda_{i}>\frac{L}{T_{b}^{pow}}\end{cases}$ | $TPS^{pos}=\begin{cases}\sum_{i=1}^{n}\lambda_{i}&\sum_{i=1}^{n}\lambda_{i}\leq\frac{L}{T_{b}^{pos}}\\\ \frac{L}{T_{b}^{pos}}&\sum_{i=1}^{n}\lambda_{i}>\frac{L}{T_{b}^{pos}}\end{cases}$

To explore block broadcasting in wireless networks and evaluate the
performance of blockchain radio access networks (B-RAN) (Ling et al., 2019),
Ling et al. (Ling et al., 2020) employed a time-homogeneous Markov chain to
establish an M/M/1 queueing model. They segmented the service process of a
valid request in B-RAN into four stages: 1) waiting for inclusion into a
block; 2) waiting for confirmations; 3) waiting for service; and 4) being in
service. The authors established multiple queues to model these phases.
However, they noted that during the third stage, requests within the same
block arrive simultaneously, and the number of requests correlates with the
block generation time. Consequently, this queue becomes non-Markovian because
prior events, apart from the current queue state, may influence its future
state. Furthermore, Ling et al. derived steady-state probabilities for various
latency-based metrics. By validating the proposed model with experimental
data, they demonstrated its potential to offer valuable insights for future
designs of B-RAN.

To address the question of whether increasing the block size improves
transaction waiting time in Bitcoin, Vesely et al. (Vesely et al., 2021) used
an M/M/1 queue model to scrutinize the transaction mining process. This
proposed model encompasses two stages including the "Block generation" stage
and the "Blockchain building" stage similar to the mode proposed in (Li et
al., 2018a). The model is employed to compare the number of transactions
awaiting processing in two scenarios: a block size of 1 MB and a block size of
5 MB. Through this comparison, the authors demonstrated that enlarging the
block size from 1 MB to 5 MB decelerates the growth of the transaction queue
length in Bitcoin. Nevertheless, this enlargement in block size results in
reduced server utilization in Bitcoin and poses challenges for users dealing
with the storage of a larger blockchain.

####  3.1.3. Modeling used G/M/1 queue.

The authors in (Kawase and Kasahara, 2018) employed a G/M/1 queueing model
with batch service and general input to comprehend the stochastic nature of
the transaction confirmation process. They aimed to predict transaction-
confirmation times by comparing analytical outcomes with trace-driven
simulations in numerical examples. The findings indicated that, for the
current maximum block size limit of 1 MB, exponential distributions offered
the most precise estimation of the mean transaction confirmation time.
Moreover, the study explored the relationship between transaction arrival
rates, block size, and transaction confirmation times. Enhancing the block
size has the potential to incrementally boost transaction throughput (TPS);
however, this improvement doesn’t fundamentally resolve scalability
challenges. Additionally, it was noted that while the transaction confirmation
time gradually increased with the coefficient of variation of exponential
distribution, this effect became insignificant with larger block sizes.

In 2018, Li et al. (Li et al., 2018a) constructed a Markovian batch-service
queueing model with two stages to analyze transaction confirmation processes
in blockchains. Utilizing the matrix geometric solution, they identified
stability conditions for the system and assessed three key performance
metrics: 1) average queued transactions, 2) average transactions within a
block, and 3) average transaction confirmation time. Their proposed GI/M/1
queueing model features two stages: the "Block generation" stage with a
service rate of μ2 and the "Blockchain building" stage with a service rate of
μ1. Both mentioned times are i.i.d distributed, following an exponential
distribution. Similarly, blockchain building times in stage 2 follow an i.i.d
exponential distribution with a service rate μ1. The model also assumes
transactions are processed on an (FCFS) basis from the transaction pool.

Results indicated that the average transaction confirmation time decreases as
the block size increases. Furthermore, they observed a threshold value η,
where for b $\leq$ η, E[T] increases as λ increases, whereas for b > η, E[T]
increases as λ decreases. Although based on simplistic exponential or Poisson
assumptions, this model suggests future avenues for research in blockchain
queueing theory. The queueing model proposed in (Li et al., 2018a) can be
observed in Fig 5.

Figure 5. The Blockchain queueing system proposed in (Li et al., 2018a)

One year later, Li et al. extended the queueing model introduced in (Li et
al., 2019) by making several modifications. They altered the service
discipline from (FCFS) to service-in-random-order, transitioned from
exponential to phase-type (PH) service times—which included both blockchain
generation and building— and shifted from Poisson to Markov arrival process
(MAP) for transaction arrivals. Initially, they established a stable condition
for the blockchain system utilizing the matrix-geometric solution. To address
the challenges in calculating transaction confirmation times in (Li et al.,
2019), the authors devised a computational technique for the first passage
times, employing both infinite-sized phase-type distributions and RG
factorizations. Additionally, Li et al. delved into some key performance
metrics, such as the average number of transactions in the queueing waiting
room, the average number of transactions within a block, and the average
transaction confirmation time. Finally, they validated the computational
feasibility of the presented theoretical findings through numerical examples.

####  3.1.4. Modeling used GI/GI/1 queue.

Geissler et al. (Geissler et al., 2019) constructed a discrete-time
$GI/GI^{N}/1$ queueing model aimed at analyzing queue size and transaction
waiting times within Proof-based blockchains. This model facilitated the
prediction and assessment of queue size and waiting time distributions,
considering diverse input parameters. In an effort to simplify and generalize
the model, the authors made several assumptions, including an infinite
transaction pool, a discrete-time block generation process, a single server
without propagation delays among network nodes, and fixed service times.
Nonetheless, the proposed model deviates from real-world processes and lacks
the required accuracy and practical applicability.

####  3.1.5. Modeling used other queue types.

Frolkova et al. (Frolkova and Mandjes, 2019) undertook an examination of busy
periods’ distribution and service delays within the Bitcoin network. They
approached the network’s synchronization process by formulating it as an
infinite server $G/M/\infty$ model with batch departures. In their study, the
authors managed to derive an analytical solution for the model, enabling the
capture of the queue’s stationary distribution. Additionally, they introduced
a random-style fluid limit considering service latencies. The authors utilized
a connection between two particular service policies to acquire the steady-
state distribution for the model.

Expanding on Frolkova’s model, Fralix (Fralix, 2020) introduced an additional
perspective on modeling the blockchain transaction confirmation process as an
infinite-server queue. Fralix demonstrated that a variant of the
distributional Little’s law, which is time-dependent, serves as an effective
tool to investigate the time-varying characteristics inherent within this
model.

Lian et al. (Lian et al., 2020) devised a stock trading model utilizing an
$M/M/N/m$priority-based queue model characterized by $N$ channels and m users,
inclusive of three priority classes with arrival rates: $\lambda_{1}$,
$\lambda_{2}^{k=1}$, and $\lambda_{2}^{k=2}$. The model comprises two phases:
the block generation phase and the chain establishment phase. In the block
generation phase, incoming data flows queue within an infinite waiting room
and are categorized into high-priority and low-priority streams, awaiting
assembly into blocks. Subsequently, in the chain establishment phase, newly
formed blocks are concatenated to form a blockchain. Each consensus mechanism
operates within different delays, which the study does not explicitly discuss.
These stages facilitate the derivation of average waiting times and queue
lengths for various priority data streams within a blockchain. The proposed
queueing model underwent simulation and measurement, assessing throughput,
delay, and channel utilization, and compared with an FCFS-based queuing model.

###  3.2. Performance modeling using other modeling approaches

In their pioneering work detailed in (Gervais et al., 2016), the authors
extended the Markov Decision Process (MDP) to create a comprehensive framework
for evaluating the security and performance of blockchains. Additionally, they
integrated an MDP security model with a Bitcoin simulator to form a unified
framework, facilitating the assessment of blockchain security by utilizing the
simulator’s stale block rate as input for the MDP model. However, it’s
important to note that the framework primarily concentrates on propagation
delay and does not address the crucial processing delay inherent in solving
consensus puzzles within PoW-based systems. Furthermore, the framework does
not encompass queueing delay. The proposed framework depicted in (Gervais et
al., 2016) is illustrated in Fig 6.

Figure 6. The structure of the proposed framework in (Gervais et al., 2016)

Zhang et al. (Zhang et al., 2018) introduced a predictive model based on
Ethereum’s "World State" structure to anticipate contract execution
performance and storage requirements based on transaction volume.
Additionally, the research delved into Ethereum’s "World State" and its Merkle
Patricia Tree (MPT) implementation and analyzed its core impact on performance
and storage. Through an examination of the Modified Merkle Patricia Tree
(MPT), the study establishes a logarithmic relationship between MPT
performance/storage and transaction volume, offering valuable insights for
businesses in decision-making and early warnings regarding system performance
and storage needs. By comparing experimental data with predictions, the study
demonstrates the model’s relative accuracy.

A random network is characterized by the probability of an edge existing
between two nodes, which is not influenced by the nodes’ characteristics.
Essentially, the degree distribution $P(k)$ becomes the sole relevant function
in this scenario (Barrat et al., 2008). Two widely used models in various
applications are $G(n,m)$ proposed by Erdős and Rï¿œnyi (ERDdS and R&wi, 1959)
and $G(n,p)$ recommended by Edgar Gilbert (Gilbert, 1959). In the
$G(n,m)$model, all graphs consisting of n nodes and m edges are assigned equal
probabilities, whereas in the $G(n,p)$ model, each potential edge among n
nodes emerges independently with a constant probability p (Drobyshevskiy and
Turdakov, 2019).

In 2019, Shahsavari et al. (Shahsavari et al., 2019) introduced an analytical
model utilizing the Erdos-Renyi random graph (ERDdS and R&wi, 1959) to model
performance and analyze the inventory-based protocol for block dissemination
in Bitcoin. Validating their model using OMNet++ network simulator (94, 2023),
they compared its accuracy using data extracted from the Bitcoin network. The
findings highlighted a tradeoff among the default connections per node,
network bandwidth, and block size, crucial in determining the optimal block
propagation delay across the network. A comprehensive summary of performance
modeling studies regarding public permissionless blockchains is provided in
Table 4.

Table 4. Summary of performance models for public permissionless blockchain systems | Model | Ref | Platform | Performance metrics | Validation
---|---|---|---|---|---
Queueing theory | M/G/1 | (Ramezan et al., 2020) | BTC | Tx average waiting time | Simulation
| (Ricci et al., 2019) | BTC | Tx confirmation time/ Tx delay | Measured data
| (Kawase and Kasahara, 2017) | BTC | Tx sojourn (Response time) | Simulation
| (Kasahara and Kawahara, 2016) | BTC | Mean confirmation time | Numerical analysis
| (Kasahara, 2021) | BTC | Tx confirmation time | Measured data
| (Mišić et al., 2019) | BTC | Tx rate/ Block rate/ Node response time | Experimentation
| (Misic et al., 2019) | BTC | Node response time | Experimentation
| (Zhao et al., 2019) | BTC | Tx confirmation time/ Mean number of txs | Simulation
| (Fang and Liu, 2020) | PoW | Queue delay | Simulation
| (Qi et al., 2020) | BTC | Average response time | Simulation
M/M/1 | (Memon et al., 2019) | BTC/ETH | Number of txs per block/Throughput/ Mempool size/ tx waiting time | Actual data
| (Vesely et al., 2021) | BTC | The number of waiting txs in queue | Actual data
| (Park et al., 2020) | ETH | Tx confirmation time/ Block throughput | Simulation
| (Ling et al., 2020) | B-RAN | Latency | Simulation
| (Srivastava, 2019) | NA | Waiting time/ Tx confirmation time | Experimentation
| (Fan et al., 2021) | BTC | Average number of transactions/ Average confirmation time | Numerical analysis
| (Seol et al., 2020) | ETH | Average waiting time/ Throughput | Simulation
| (Cao et al., 2020) | BTC/ETH | Block generation time/Confirmation delay/TPS/Failure probability | Simulation
| (Ke and Park, 2021) | BTC/ETH | Transaction queue size/ Block waiting time | Simulation
G/M/1 | (Li et al., 2018a) | BTC | Average tx confirmation time/ Average number of txs in queue and a block | Numerical analysis
| (Li et al., 2019) | BTC | Mean stationary number of txs in queue and block | Numerical analysis
| (Kawase and Kasahara, 2018) | BTC | Tx confirmation time | Simulation
GI/GI/1 | (Geissler et al., 2019) | ETH | Tx waiting time/ Queue size | Measured data
GI/M/$\infty$ | (Frolkova and Mandjes, 2019) | BTC | Queue length | Numerical analysis
| (Fralix, 2020) | BTC | Time-dependent behavior | Numerical analysis
M/M/N | (Lian et al., 2020) | NA | Tx delay/ Channel utilization | Experimentation
Other models | Prediction model | (Zhang et al., 2018) | ETH | Executing contract’s Performance and storage prediction | Experimentation
Stochastic networks | (Papadis et al., 2018) | ETH | Block generation rate | Simulation
Random graph model | (Shahsavari et al., 2019) | BTC | Block propagation delay/ Traffic workload | Experimentation
Markov Decision | (Gervais et al., 2016) | BTC | Transaction per second Block propagation time | Simulation
Process (MDP) | (Varma and Maguluri, 2021) | BTC | Network stability | Simulation
Sleep queueing system | (Ma et al., 2020) | BTC | Average tx confirmation time | Experimentation
Game theory | (Li et al., 2018b) | BTC | Transactions/ waiting time in Mempool | Numerical analysis

Figure 7. a) Research papers published per performance evaluation method and
blockchain platform, b) Research papers published on public permissionless
blockchains per blockchain platform.

Figure 8. a) Research papers published on performance modeling per modeling
method, b) Research papers published on public permissionless blockchains’
performance modeling using queueing theory per type of queue.

##  4\. Workload and traffic modeling of blockchain systems

Understanding, analyzing, and modeling traffic and workload holds significant
importance within the realm of performance modeling and evaluation. A computer
system’s performance relies heavily on how it handles network traffic and the
workload it receives. For instance, optimal performance occurs when tasks are
evenly distributed, contrasting with unpredictable bursts that lead to
congestion (Feitelson, 2002). Therefore, reliable performance assessments
necessitate representative workloads to yield dependable outcomes. Achieving
this involves gathering data from actual workloads and devising statistical
models that encompass their key attributes. Pioneering studies have delved
into modeling, characterizing, and regenerating internet traffic (Barford and
Crovella, 1998; Jain and Routhier, 1986; Karagiannis et al., 2004; Leland et
al., 1994; Ferrari, 1984). The evaluation of performance through workload
modeling in computer networks is extensively discussed in (Feitelson, 2002;
Ferrari, 1972; Feitelson, 2015; Feitelson and Tsafrir, 2006; Liu et al.,
2001). To explore the specifics of network and traffic modeling within
blockchain networks, we categorized these studies into four groups: 1)
Modeling network topology, 2) Characterizing workloads, 3) Benchmarking via
workload, and 4) Analyzing and modeling traffic. A comprehensive overview of
this section is outlined in Table 5.

###  4.1. Network topology modeling

In 2020, Lao et al. examined existing traffic models in P2P and blockchain
systems and introduced a traffic model specifically tailored for IoT-
blockchain systems (Lao et al., 2020). In (Eisenbarth et al., 2021),
Eisenbarth et al. conducted a study focusing on the public nodes that
constitute the structure of the Bitcoin P2P network.

Numerous studies have concentrated on modeling and analyzing the topology of
blockchain networks. Here, we’ll delve into a subset of these papers,
particularly emphasizing studies pertaining to public permissionless
blockchains.

In 2014, Donet et al. (Donet Donet et al., 2014) conducted a comprehensive
data collection, identifying over 872,000 distinct Bitcoin nodes. This effort
aimed to provide insights into various aspects of the Bitcoin peer-to-peer
network, including its size, geographic distribution of nodes, network
stability, and data related to the transmission speed of information. In 2015,
Miller et al. (Miller et al., 2015) introduced efficient methods for mapping
the Bitcoin broadcast network and pinpointing nodes with significant influence
within this network. AddressProbe, a technique designed for uncovering peer-
to-peer connections in Bitcoin, is presented in this paper. Additionally, the
authors developed CoinScope, an infrastructure tailored to facilitate short
yet large-scale experiments in Bitcoin, aimed at supporting AddressProbe and
other related tools.

To assess the degree of decentralization in the prominent blockchains Ethereum
and Bitcoin, Gencer et al. (Gencer et al., 2018) conducted a measurement study
involving diverse decentralization metrics for both networks. They utilized
established internet measurement methods and employed the Falcon Relay Network
as a new tool to gather their data. In (Deshpande et al., 2018), Deshpande et
al. introduced BTCmap, a framework created specifically to explore the
topology of the Bitcoin network. This framework comprises two essential
modules: a sniffer, which retrieves the local addresses database from each
peer, and a Bitcoin peer emulator, responsible for selecting neighboring nodes
and constructing the topology of the network. To analyze the underlying peer-
to-peer network in Ethereum blockchain, Kim et al. in (Kim et al., 2018b)
created NodeFinder, an innovative monitoring tool offering an unparalleled
perspective of the network ecosystem in Ethereum. Imtiaz et al. in (Imtiaz et
al., 2019) discovered and provided experimental evidence for the previously
unrecognized impact of churn—intermittent network connectivity—on the Bitcoin
network. Their study involved a comprehensive analysis of churn, encompassing
the daily churn rate and statistical modeling of session durations for up and
down network connections. This statistical model is valuable for researchers
interested in analyzing, simulating, or replicating Bitcoin network behavior.
Leveraging this statistical characterization, they assessed how churn
influences the block propagation delay within the live Bitcoin network.

In (Essaid et al., 2019), the authors introduced a topology discovery system
aimed at creating an accurate graph-based representation of the Bitcoin
network’s structure. This system involves live gathering and examination of
Bitcoin’s peer-to-peer connections.. They utilized a modified version of the
PageRank algorithm (Sankaralingam et al., 2003)to aggregate incoming node
information, enabling more comprehensive graph analysis. This approach
facilitated insights into various aspects of the Bitcoin network, such as its
size, network stability, and data related to the geographic locations of
Bitcoin nodes. To understand the topological characteristics of the Ethereum
P2P network, Gao et al. (Gao et al., 2019) performed a measurement analysis of
the Ethereum P2P network.

Seres et al. in (Seres et al., 2020) aimed to measure and evaluate the
structural and topological characteristics of Bitcoin’s Lightning Network
(Poon and Dryja, 2016), a method to enhance Bitcoin’s scalability. They
contended that the current topological features of Bitcoin’s Lightning Network
could be enhanced to improve its security, thus enabling it to achieve its
full potential. In 2020, Lee et al. (Lee et al., 2020) explored various local
and global graph characteristics across four Ethereum blockchains
(ContractNet, TransactionNet, TraceNet and TokenNet), along with three notable
subnetworks (Binance Coin, Bancor, and Zilliqa). Their investigation
encompassed a comprehensive experimental assessment of these networks. In
(Bistarelli et al., 2023)the authors delved into analyzing the Bitcoin
protocol using an extended version of the probabilistic model checker PRISM.
By employing probabilistic modeling, the study explored the mechanisms behind
forks and their reliance on specific protocol parameters like crypto puzzle
difficulty and network communication delays. The research also investigated
the behavior of networks with churn miners, those exiting and later rejoining
the network, as well as various network topologies. Additionally, the study
includes simulations of the Bitcoin protocol, considering diverse network
structures to assess whether these alternative topologies exhibit equal or
greater resistance to forks compared to the original Bitcoin network.

As the pioneering study exploring the evolution and temporal characteristics
of the Ethereum network, Zhao et al. delved into investigating the
evolutionary patterns, temporal dynamics, and predictive elements of four
Ethereum blockchain interaction networks (TraceNet, ContractNet,
TransactionNet, and TokenNet)(Zhao et al., 2021). Subsequently, the authors
conducted an extensive empirical assessment of these networks’ graphical
representations. In the paper (Cortes-Goicoechea and Bautista-Gomez, 2021),
the authors showcased several findings derived from their analysis of the
Ethereum 2.0 peer-to-peer network. They obtained this data using their network
monitoring tool, Armiarma, which transparently collected information,
providing them with a comprehensive understanding of the network’s current
status. Maeng et al. in (Maeng et al., 2021) presented Search-node, an adapted
version of an Ethereum client for discovering nodes in the Ethereum main
network. They visualized and analyzed the Ethereum network’s topology,
exploring various network parameters and delving into node properties,
including relationships, connectivity, geographical distribution, security
implications, and potential attacks targeting influential nodes.

In (Shahsavari et al., 2020), the authors introduced a random graph model to
analyze the inventory-based protocol’s performance in Bitcoin’s block
propagation. By deriving explicit equations for block propagation delay and
traffic overhead, they provided insights into the impact of these parameters
on the network. Additionally, they evaluated the implications of deploying a
relay network and studied how the relay network size affects both network
performance and decentralization. To validate their model, the authors
conducted simulations and compared the results with real data from the Bitcoin
network. In their recent paper, Essaid et al. (Essaid et al., 2023) conducted
a study on the Bitcoin network’s behavior and dynamic topology from 2018 to
2022. They employed Node-Probe, a technique utilizing recursive scanning to
gather snapshots of the Bitcoin main network. The study focused on analyzing
network properties, community structures, and topology changes. Their findings
revealed significant alterations in the Bitcoin network’s characteristics over
time, particularly in the number of active, fresh, and permanent nodes.
Furthermore, their analytical exploration indicated intriguing insights,
suggesting that enhancing the propagation mechanism through master nodes could
potentially reduce propagation delays by approximately 25% compared to the
default Bitcoin protocol. Zheng et al. (Zheng et al., 2023) pioneered a study
delving into Algorand’s network and transactions analysis. They conducted an
examination of Algorand’s features using graph analysis. The team collected
external data via an Algorand indexer, establishing connections to the nodes
for the ledger’s data processing, followed by the construction of the graph.
Analyzing this graph yielded fresh observations about this blockchain system.

###  4.2. Workload characterization

Workload characterization is one of the essential field evaluate and analysis
of computer networks. Numerous investigations have concentrated on workload
characterization of networks, like (Calzarossa et al., 2000; Ferrari, 1972).
Additionally, there have been studies delving into specific workloads, such as
web server workload (Arlitt and Williamson, 1997, 1996; Hsu and Smith, 2003),
web services workload (Gill et al., 2007; Ihm and Pai, 2011), IOT workload
(Tadakamalla and Menascé, 2019), those examined workloads of social networks
(Ahn et al., 2007; Benevenuto et al., 2009), more. A comprehensive compilation
of these workload characterization works is available in survey papers within
this particular field (Calzarossa and Serazzi, 1993; Shishira et al., 2017;
Calzarossa et al., 2016).

The prediction of network behavior within a blockchain system to maintain
essential service benchmarks and network stability demands an understanding of
the network’s characteristics and its workload. The study by Spirkina et al.
(Spirkina et al., 2020) undertook an examination of Blockchain traffic,
investigating its features in information transmission across the network.
Additionally, the paper surveyed analytical and simulation modeling solutions,
offering an overview that enables the prediction of network load. In the paper
by Vladyko (Vladyko et al., 2021), another investigation into predicting
network behavior within blockchains is conducted. This study explored methods
for modeling these systems and advocates for the adoption of a simulation
system to evaluate network performance and its components. Furthermore, the
paper shared findings from comparative simulations. A master thesis authored
by Jidrot (Jidrot and Perumal, 2021) applied Blockchain technology to securely
gather and preserve data from distributed cloud resources, emphasizing the
necessity for data integrity over extended periods. The primary objectives
included implementing a Blockchain for distributed cloud monitoring,
characterizing the workload, and evaluating the Blockchain system’s
performance. The thesis defines an initial system model, develops an
implementation based on HLF blockchain, and conducts experiments to comprehend
how design factors and system inputs influence the system’s capabilities and
performance. Gebraselase et al. (Gebraselase et al., 2021) conducted an
extensive analysis of Bitcoin’s transaction characteristics within the Bitcoin
blockchain domain. They also provided an analysis by examining collected data
from a measurement setup, which encompassed blocks and transactions
information.

Table 5. Studies on network and traffic modeling in blockchain systems

Category | Platform | Ref | PP* | Validation | Description
---|---|---|---|---|---
Network | BTC | (Donet Donet et al., 2014) | ✓ | Measurement | Model and analyze various aspects of the Bitcoin p2p network
topology |  | (Miller et al., 2015) | ✓ | Measurement | A technique to map and analyze the Bitcoin broadcast network
modeling |  | (Deshpande et al., 2018) | ✓ | Implementation | A model to discover Bitcoin network topology
|  | (Imtiaz et al., 2019) | ✓ | Implementation | A statistical model to analyze the connectivity of the network
|  | (Essaid et al., 2019) | ✓ | Implementation | A topology discovery system to represent Bitcoin’s network
|  | (Seres et al., 2020) | ✓ | Measurement | A model to analyze network properties of BTC lightning network
|  | (Bistarelli et al., 2023) | ✓ | Simulation | A probabilistic model to analyze Bitcoin protocol
|  | (Shahsavari et al., 2020) | ✓ | Simulation | Model delay and traffic overhead using Random Graph
|  | (Essaid et al., 2023) | ✓ | Measurement | Model behavior of network and dynamic topology
| ETH | (Kim et al., 2018b) | ✓ | Measurement | A model to monitor and analyze Ethereum network
|  | (Gao et al., 2019) | ✓ | Measurement | A model to analyze topological characteristics of ETH
|  | (Lee et al., 2020) | ✓ | Implementation | A model to analyze and measure ETH P2P network
|  | (Zhao et al., 2021) | ✓ | Implementation | A model to study evolution and temporal characteristics of ETH
|  | (Maeng et al., 2021) | ✓ | Measurement | A node discovery model for Ethereum’s main network
| BTC/ETH | (Gencer et al., 2018) | ✓ | Measurement | A model to measure the decentralization degree of blockchains
| Algorand | (Zheng et al., 2023) | ✓ | Measurement | Analysis of network revolution and transaction activities
| ETH2.0 | (Cortes-Goicoechea and Bautista-Gomez, 2021) | ✓ | Measurement | A model to analyze ETH 2.0 P2P network
Workload | General | (Spirkina et al., 2020) | ✓ | Simulation | Network characterization to predict its behavior
characterization |  | (Vladyko et al., 2021) | ✓ | Simulation | Network behavior prediction and performance evaluation
| HLF | (Jidrot and Perumal, 2021) | ✕ | Implementation | Workload characterization and evaluation of a BC implementation
| BTC | (Gebraselase et al., 2021) | ✓ | Measurement | Characterization of Bitcoin transactions
Benchmarking | ETH/HLF/Parity | (Dinh et al., 2017) | ✕ | Implementation | Performance benchmarking using the workload of Blockbench
using | ETH/HLF | (Pongnumkul et al., 2017) | ✕ | Implementation | Performance analysis using varying numbers of transaction
workloads |  | (Saingre et al., 2020) | ✕ | Implementation | Performance benchmarking with BCTMark
| Parity/Multichain | (Oliveira et al., 2019) | ✕ | Implementation | Performance evaluation using realistic BTC workload
| HLF | (Thakkar et al., 2018) | ✕ | Implementation | Performance benchmarking using workload characterization
Traffic | General | (Elagin et al., 2020) | ✓ | Implementation | Model the network to predict the on-network load
analysis |  | (Danzi et al., 2018) | ✓ | Simulation | Model traffic generated by synchronization protocols
and | BTC | (Decker and Wattenhofer, 2013) | ✓ | Implementation | A model to discover propagation delay and forks
modeling |  | (Bianconi and Agrawal, 2018) | ✓ | Implementation | A model to predict the number of transactions of BTC
|  | (Bowden et al., 2018) | ✓ | Simulation | A model of the block arrival process
|  | (Zhang et al., 2021) | ✕ | Implementation | Mathematically model traffic between Blockchain and IoT devices

* Public permissionless

###  4.3. Benchmarking using workload

In 2017, the authors in (Dinh et al., 2017) introduced Blockbench as an
evaluation framework designed to analyze performance of private blockchains.
It enables the integration of any private blockchain through straightforward
APIs and facilitates benchmarking against workloads derived from genuine and
artificial smart contracts. Subsequently, the authors utilized Blockbench to
perform an analysis of three blockchain systems—private Ethereum, Parity, and
Hyperledger—employing macro benchmarks and micro benchmarks. Additionally, In
2020, Saingre et al. (Saingre et al., 2020) introduced BCTMark, a benchmarking
framework for blockchain systems that incorporates features such as load
generation and metrics collection. To demonstrate its versatility, they
conducted experiments on three blockchain systems—Ethereum Clique, Ethereum
Ethash and Hyperledger Fabric. In 2017, Pongnumkul et al. (Pongnumkul et al.,
2017) discussed a performance evaluation of Hyperledger Fabric and Ethereum,
examining their capabilities as two private blockchains under varying
workloads. In 2018, Thakkar et al. in (Thakkar et al., 2018) characterized the
transaction workload and performance of Hyperledger Fabric to analyze
performance parameters and bottlenecks under various block sizes, policies,
and network situations. Using realistic workload to evaluate blockchains’
performance, Oliveira et al. in (Oliveira et al., 2019) compared Parity and
Multichain, two private blockchains, focusing on transaction-related metrics
using a workload derived from the probability distribution of Bitcoin
transaction arrival times.

###  4.4. Traffic analysis and modeling

In (Decker and Wattenhofer, 2013), decker et al. examined Bitcoin’s
utilization of a multi-hop broadcast to disseminate transactions and blocks
across the network for ledger updates. Subsequently, they presented a model
elucidating the presence of blockchain forks. Employing this model, the
authors confirmed the hypothesis that the primary reason for blockchain forks
is the propagation delay within the network.mThe goal of Bianconi et al. in
(Bianconi and Agrawal, 2018) was to predict the forthcoming volume of
transactions within the Bitcoin network by leveraging its existing network
structure. This objective was similar to link prediction, but it specifically
aimed to estimate the expected quantity of connections (edges) within the
network over a defined time period. Modeling block intervals in Bitcoin,
Bowden et al. (Bowden et al., 2018) contradicted the initial claim made in the
Bitcoin paper proposing a homogeneous Poisson process for blockchain arrivals.
Their demonstration relied on empirical data and stochastic analysis of block
arrival patterns, revealing a different reality. In 2018, Danzi et al. (Danzi
et al., 2018) conducted a study focusing on synchronization among IoT devices
with the blockchain. They proposed two protocols to synchronize and manage
traffic between these networks, analyzing the necessary bandwidth and time
requirements for synchronization. To examine the key features of blockchain
networks traffic, Elagin et al. (Elagin et al., 2020) introduced an analysis
model that can predict the network load, aiming to contribute towards ensuring
secure and high-quality data exchanges within this network in the future.
Zhang et al. (Zhang et al., 2021) introduced a traffic model for blockchain
systems within the context of IoT. They conducted an extensive analysis of IoT
input, blockchain network traffic, and self-similarity. They deployed a
lightweight blockchain similar to Bitcoin, utilizing Proof of Work (PoW)
connected to multiple IoT simulators to facilitate a range of network traffic
experiments.

##  5\. Challenges and future research directions

In the preceding sections, we provide an overview of the performance modeling
landscape concerning public permissionless blockchains. Despite significant
advancements in the evaluation and modeling of blockchain systems, several
open questions persist. Using the information we’ve gathered, we will discuss
the challenges for performance modeling of public permissionless blockchains
in this section. We’ll also suggest some ideas for future research.

###  5.1. Performance modeling and evaluation issues and future directions

Performance modeling in public blockchains presents greater challenges
compared to private and permissioned blockchains due to the inherent
variability of network conditions and the decentralization of nodes. While
queueing models have been commonly used for performance modeling in these
networks, identifying the optimal modeling approach remains an open question.
This is primarily because queueing models encounter limitations when applied
to public blockchains, given their dynamic and unpredictable nature.
Overcoming these limitations necessitates the advancement of more
sophisticated modeling techniques capable of capturing the unique attributes
of public blockchain systems.

Additionally, selecting the most suitable queue type for modeling the
performance of public blockchains poses another significant challenge. For
instance, in a study by Kasahara (Kasahara, 2021), the researcher investigated
the process of transaction confirmation in Bitcoin by employing three types of
queues and assessing their accuracy through simulation outcomes.

One of the challenges in evaluating and modeling the performance of public
blockchains is the measurement of certain performance parameters like delays
and transaction confirmation time. These parameters are crucial for assessing
blockchain performance, yet they pose difficulties in measurement within
public blockchains. For instance, various studies have focused on analyzing
transaction confirmation time using modeling techniques. However, accurately
determining this parameter’s precise value in a real-world public blockchain
presents significant challenges.

As previously discussed, employing a modeling approach is considered the most
effective method for assessing the performance of public blockchains. In this
context, numerous researchers have conducted studies on the performance
modeling of Bitcoin and Ethereum, which are comprehensively reviewed in this
survey. However, the performance modeling of recently developed public
blockchains like Algorand and Ethereum 2.0 has received minimal attention.
Consequently, there exists a promising avenue for future research to delve
into the performance modeling of these two blockchains, aiming to identify
weaknesses and propose adjustments accordingly.

Another requirement in this area is the absence of a dedicated simulator
designed for Ethereum 2.0 and Algorand. While one simulator has been proposed
by the authors in (Hünseler and Lemke-Rust, 2021), Ethereum 2.0 has undergone
gradual changes since then, indicating an ongoing need for an updated Ethereum
2.0 simulator. Consequently, the development of simulation tools for newly
introduced public blockchains emerges as a prospective direction, offering
support to other researchers engaged in performance modeling of public
blockchains.

###  5.2. Performance improvement issues and future directions

As previously mentioned, transaction confirmation time stands out as a
critical performance metric in public blockchains. As illustrated in Figure 2,
this metric encompasses various components, such as transaction propagation
latency, transaction waiting latency, processing overheads, block propagation
latency, consensus latency, and finality delay, all of which vary across
different public blockchains. Modeling and pinpointing these delays within
public blockchains, to minimize each latency individually, remains an
unresolved issue in the realm of performance modeling for public blockchains.

Another promising avenue for research involves analyzing the tradeoff between
performance and security in public blockchains, particularly focusing on newly
developed ones such as Algorand and Ethereum 2.0. It is understood that, at
some point, enhancing the performance of public blockchains may require
compromising the security of the system. Investigating this balance and
forecasting the implications on network security when enhancing performance
parameters represents a significant research direction. Exploring this aspect
of public blockchains can potentially lead to the development of existing
public blockchains that maintain both acceptable performance and security
simultaneously. Furthermore, it would aid researchers in designing new public
blockchains with these desirable attributes.

###  5.3. Traffic and workload issues and future directions

During our review, we identified another promising avenue for research: the
characterization of transactions and traffic in newly developed public
blockchains. While certain papers have tackled similar challenges within
Bitcoin (e.g., (Gebraselase et al., 2021)), there remains a notable gap in
research about newly established public blockchains like Algorand and Ethereum
2.0. Understanding and characterizing transactions and incoming workloads
within a public blockchain is of paramount importance due to the substantial
number of nodes and the inherent variability of network conditions. Exploring
this research direction could enable researchers to create synthetic workloads
for modeling and assessing the performance of public blockchains in real-world
scenarios. Consequently, delving into the generation of synthetic traffic for
public blockchains emerges as another promising avenue in the realm of
performance evaluation for such systems.

##  6\. Conclusion

After conducting an initial overview of existing surveys concerning the
performance evaluation of blockchains, we identified a crucial gap.
Specifically, there is an absence of a dedicated survey that concentrates on
evaluating the performance of blockchain systems, specifically public
permissionless blockchain systems. To address this gap, we conducted an
extensive survey concentrating on the most advanced methods in performance
modeling of blockchains, particularly those models proposed for analyzing and
evaluating the performance of public permissionless blockchains. Within this
survey, we presented a taxonomy, ideas, and solutions for each category,
aiming to provide improved understanding and novel insights into utilizing
modeling approaches for predicting, analyzing, and evaluating the performance
of public permissionless blockchains. Additionally, recognizing the
significance of network topology, workload, and traffic characterization in
performance evaluation process, we delved into studies centered on modeling
blockchain network topology, analyzing traffic and workload characteristics,
and introducing blockchain benchmarking workloads for the first time. We are
of the opinion that our survey offers a valuable guidance regarding the
insights into evaluating blockchain performance, catering to blockchain
researchers, and a broader audience of readers.

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