11institutetext: University of Copenhagen, København, Denmark 22institutetext:
eToro Denmark, København, Denmark

# Multi-block MEV

Johannes Rude Jensen 1122 0000-0002-7835-6424 Victor von Wachter 11
0000-0003-4275-3660 Omri Ross 1122 0000-0002-0384-1644

###### Abstract

Multi-block MEV denotes the practice of securing k-consecutive blocks in the
attempt at extracting surplus value by manipulating transaction ordering.
Following the implementation of proposer/builder separation (PBS) on Ethereum,
savvy builders can secure consecutive block space by implementing targeted
bidding strategies through relays. We collect data on all bids submitted by
builders through relays, finding that 20.60% of blocks issued since the Merge
share single builder entities for consecutive sequences. Our results indicate
that single builder entities are implementing super linear bidding strategies
above market, to secure consecutive block space. By comparing the results to a
simulation of naive builder behaviour, we show that MMEV extraction may be
common amongst leading builder entities.

###### Keywords:

MEV Ethereum Multi-block MEV Consensus Layer Execution Layer

##  1 Introduction

The concept of Maximum Extractable Value (MEV) refers to the economic value
extracted through sequencing transactions within the limited state space in a
blockchain database. The MEV orchestrator (searcher/builder) extracts rent by
structuring transaction flow and injecting their own transactions in
‘bundles’, which are aggregated in blocks and submitted to the elected block
proposer, through a centralized relay. The term ‘maximally’ extractable value
(MEV) refers to the total opportunity space for value extraction within a
given block space and is commonly subdivided by extraction strategies
considered ‘neutral’ or ‘toxic’. Neutral MEV typically denotes practices that
may result in a price improvement for the end user. Examples include ‘just-in-
time liquidity’ (JIT) and back-running AMM trades. ‘Toxic’ MEV, on the other
hand, refers to practices that will lead to price degradation for the end-
user, such as front-running, sandwiching, or equivalent strategies. [8],[4].
Given the rapid rate of development in the space, the categories are non-
exclusive and open-ended. Today, the concept of MEV is most commonly
associated with the Ethereum blockchain, and its ancillary execution domains.
At the time of writing, MEV opportunities have generated $687 million
111https://explore.flashbots.net/. The value distribution amongst competing
builders is heavily skewered towards the top ten builder
addresses.222https://www.mevboost.org/ The implementation of Proposer Builder
Separation (PBS) on the Ethereum mainnet, following the transition to proof-
of-stake, introduces a new enclave of risk vectors as stakeholders compete to
extract rent in the PBS paradigm. While PBS has successfully mitigated the
risk of proposer/builder collusion without defaulting to the traditional
first-in-first-out market structure, the concept introduces novel risks that
remain poorly understood. As a consequence of the epoch construct, the block
proposer and committee roles for each epoch are deterministically calculated
and revealed one epoch ahead of
time333https://notes.ethereum.org/@vbuterin/SkeyEI3xv. Practically, this means
that, by the time of the last block in epoch N-1 the proposers elected for the
subsequent 64 slots, in epoch N and N+1, are known. This introduces two
centralization vectors conducive to the extraction of economic value through
‘multi-block’ MEV (MMEV) strategies: First, as noted by [9], the probability
of a single stake pool operator proposing k-consecutive blocks within a single
epoch is a direct property of stake concentration. At the time of writing,
consecutive slots assigned to a single pool are a relatively frequent
occurrence, in some instances reaching the order of five or even seven
consecutive slot elections within a single epoch. If the pool operator of any
sequence of proposers elected for k-consecutive slots colludes with a builder
by selling consecutive block space, the result is immediately conducive to
MMEV extraction. We refer to this extraction category as collusive MMEV, as
the explicit collusion between the proposer and builder would violate the
premise of PBS. Second, as $\sim$92% of active validator nodes are running a
generic implementation of MEVboost, the selection of bids through relays is a
largely non-discretionary and standardized process. In effect, this means that
builders can rely on elected proposers to accept a targeted bidding strategy
through a relay if (I) the target proposer is running a known version of the
MEVboost software and (II) the target proposer is subscribing to a relay
through which the builder can submit bids. We refer to this category of MMEV
as non-collusive since builders are abiding by the logic enshrined in PBS. It
is currently reasonable to assume that large stake pool operators act in good
faith, given their mandate as stewards of the ecosystem and exposure to ETH.
Yet, it is by no means certain that builders will abstain from implementing
sophisticated non-collusive MMEV strategies. In fact, one may assume that
builders will act explicitly as rational agents and seek to maximize value
extraction, through any means necessary. We examine the prevalence of non-
collusive MMEV on Ethereum. We aggregate a dataset containing the full set of
bids submitted by builders through relays at each consecutive second for every
slot, since ’the Merge’. We ask the research question: To what extent is MMEV
extracted on Ethereum today? We approximate the occurrence of multi-block
value extraction by analyzing progressively increasing successful bids by
single builders for consecutive blocks. Our working assumption is that
builders act rationally and only bid above-market for consecutive blocks if
the MMEV return profile exceeds that of standard, single-block MEV.

##  2 Towards a Taxonomy for multi-block MEV (MMEV)

The MMEV opportunity space exceeds that of single-block MEV in two significant
ways: First, the extended state space secured by the builder provides more
room for increasingly complex extraction strategies. Second, the block
transition occurring between slots may, in certain scenarios, enable the
builder to compose sophisticated strategies across block transitions. A
builder may draw on a vast set of exotic tools to design short-term momentum
or, in rare occasions, attempt manipulation of TWAP oracles to force an
artificial liquidation of larger positions [6]. Consistently securing MMEV
opportunities requires information on the constantly changing proposer set. It
is important to note that some pool operators only work with pre-approved
builder addresses, which further narrows the implied concentration of MMEV
strategies. At the end of epoch N-1, the builder must identify any number of
consecutive slots assigned to proposers in epochs N and N+1. To secure
consecutive block space, they can assign probabilities that the elected
proposer for each consecutive slot will be amenable to the strategy. For each
slot in the target sequence, the elected proposer must (I) have been
identified as running the target implementation of the MEVboost software with
their proposer client (II) consume blocks through the target relays, and (III)
accept bids from the antagonistic builder. The opportunity space for multi-
block value extraction is a product of organic on-chain activity and, as such,
is in constant development. Builders may pursue two general types of MMEV
strategies: (I) Discrete strategies in which builders seek to reach a specific
set of threshold values that may provoke specific scenarios such as
liquidation, and: (II) continuous strategies in which the builder seeks to
structure transaction flow ongoingly from the mempool, forming short-term
momentum in either direction of a trade. For both classes of strategies, the
builder will structure transaction flow from the mempool through selective
inclusion, withholding, and injection of (synthetic) transactions aimed at
value extraction. The builder will mix these techniques across the sequence of
consecutive blocks while waiting for new transactions to appear in the
mempool, and new information (prices) to emerge (Figure 1).

Figure 1: Illustration of six consecutive slots, in which four consecutive
slots are secured by the antagonistic builder. The builder will inject a buy
transaction in a shallow Uniswap pool, followed by a sequence of organic buy
transactions (green). The builder will then withhold all sell transactions for
the pool (red) to promote momentum in a given direction, to capitalize on the
momentum till the end of the sequence. After the sequence, a naive proposer
without MEVboost includes the residual sell transactions to the pool.

##  3 Data and Methodology

We aggregate data on all bids submitted to all major relays by builders, since
’the Merge’. The data set comprises the period from block height 15537394 to
16525948 covering the period from the 15th of September 2022 to the 31st of
January 2023. The dataset contains a total of 994464 slots, averaging 804 bids
per slot. The bids are submitted through 132 distinct public keys. We use
public sources to match the public keys to 8 builder entities, listed here by
volume: flashbots, builder0x69, bloxroute, beaverbuild.org, blocknative, eth-
builder.com, x85linux, eden, and manifold. 7.44% of the public keys cannot be
matched to the large builder entities, for these we assume distinct agency. A
total of 92030 ETH was paid by winning bids to the proposers, a value of
$145.68 million, at the time of writing. We derive the expected number of
consecutive blocks through a Monte Carlo simulation and compare the empirical
observations with the expected ones. Throughout we denote $n$ as the number of
slots per epoch ($n$=32), $k$ as consecutive events in a sequence, and $p$ as
the probability of success.

##  4 Results

In total, 76.02% of all blocks since the merge were included by clients
implementing the new PBS paradigm. This number is rapidly increasing, with
90.01% of blocks composed using MEVboost, at the time of writing. Practically,
this means that in 7.6 out of 10 slots, a builder can submit a customized
block to the elected proposer through a relay, without owning any stake in the
consensus layer. We find that 14.26% of blocks share the same entity twice
($k$=2) in a row. 4.33% share the same entity three times ($k$=3), and 1.32%
share it four times ($k$=4). For 0.69% the same entity is shared five or more
times ($k\geq$5). Figure 3 shows the observed number of $k$ consecutive
sequences in the data set. Unsurprisingly, $k$=2 and $k$=3 sequences comprise
the majority. Also for higher $k$ up to 10, we find multiple instances in the
data set. We refrain from attributing explanatory value to the rare instances
of $k$=11, $k$=13, and $k$=17 consecutive blocks, due to the small sample
size. As evident, 68.58% of all blocks can be mapped to the largest builders.
We simulate the expected number of consecutive blocks through a Monte Carlo
simulation. We randomly sample 10000 instances of the full data set and
calculate the expected value of each event (for 1$\geq$$k$$\geq$10). We model
the probability as a Bernoulli experiment,

Figure 2: The figure shows the observed number of $k$ consecutive sequences in
the data set. About every 12 seconds, a new block is created. Hence, two
consecutive blocks ($k$=2) by the same builder are proposed 514 times a day,
on average. We find rare instances of $k$=11, $k$=13, and $k$=17 consecutive
blocks in the sample, albeit with a small sample size.

Figure 3: The illustration shows a boxplot of the builder payment in relation
to the n-th position within a sequence of consecutive blocks. The orange line
delineates the median bidder payments enclosed by boxes representing the
25%-quartile and 75%-quartile. We observe an incremental increase in the
average bid value: the builder is willing to pay more for the later stages of
a consecutive sequence. This indicates that builders are gradually seeking to
outbid the market for the next consecutive slot, extracting the value
opportunistically ‘along the way’.

with $p$ being the market share of each builder entity (table 2). In the
empirical data, we observe significantly more consecutive blocks than expected
just by chance. For $k$=2 we observe 9.6% more consecutive blocks than we
would expect by chance. Table 1 maps the total observed consecutive blocks to
the 8 largest builder entities. Even more distinctive, we find a surplus of
45.9% consecutive blocks for $k$=3. The results indicate that builders are
strategically bidding for consecutive blocks, as they show significantly more
consecutive sequences as naively derived. A rational agent will extract
surplus value from consecutive blocks to justify the strategic interest. Due
to financial interests customized MMEV implementations might not be open-
sourced initially. MMEV strategies have to be implemented by each builder
individually. Interestingly, we can observe if a specific builder entity
produces an excessive number of consecutive blocks, which might stipulate a
customized MMEV implementation. For example, the builder entity ’x85linux’
included 64.34% more of $k$=2 consecutive blocks than expected. We analyze the
builder payments in relation to the number of consecutive blocks to establish
if entities also bid higher than average bids for consecutive blocks. The
lower part in figure 4 depicts the median builder payment: a builder pays
0.046 ETH to secure a single block.

Figure 4: Builder payments (in ETH) in relation to the number of consecutive
blocks by the same builder entity. The lower part depicts the median builder
payment: a builder pays 0.046 ETH to secure a single block. Yet, to secure two
(three) consecutive blocks, builders are willing to pay 0.047 ETH (0.049 ETH)
each, an increase of 2.12% (5.92%). The upper part shows the development of
the total builder payment. From a naive bidder, we would expect to bid the
median price of 0.046 ETH multiple times. However, this is not the case
empirically. Builders are willing to pay super linearly. Assuming rational
agents, the value extracted by builders bidding on consecutive blocks should
exceed the sum extracted in individual blocks. The results indicate that the
builders are extracting surplus MEV from consecutive block space.

Yet, to secure two (three) consecutive blocks, builders are willing to pay
0.047 ETH (0.049 ETH) each, an increase of 2.12% (5.92%). The upper part shows
the development of the total builder payment. From a naive bidder, we would
expect to bid the median price of 0.046 ETH multiple times. However, this is
not the case empirically. Builders are willing to pay super linearly. Assuming
rational agents, the value extracted by builders bidding on consecutive blocks
should exceed the sum extracted in individual blocks. The results indicate
that the builders are extracting surplus MEV from consecutive block space
(i.e. MMEV). Further, we analyzed if the builders are paying more in the
beginning or in the later stages of a consecutive sequence. Figure 3 displays
a boxplot of the builder payment in relation to the n-th position within a
sequence of consecutive blocks. The median bidder payments are indicated by
the orange line. The builder is willing to pay more for the later phases of a
sequential sequence. This suggests that builders are gradually trying to
outbid the market for the next slot, extracting the value opportunistically
‘along the way’. An example of a continuous MMEV strategy is a Momentum
trading approach: After entering a position, the MEV agent tries to control
the on-chain markets by winning consecutive block space. In doing so the agent
has an information advantage, as the asset prices change on other trading
venues. The MEV agent extracts the price discrepancy opportunistically ’along
the way’. This is in contrast to a discrete MMEV strategy, where an agent
after entering the trade seeks a deterministic exit according to a specific
set of thresholds. An on-chain liquidation is triggered at a deterministic
price threshold, hence the MEV agent can only maximize the value extracted
with the liquidation by stalling the liquidation over consecutive blocks.

##  5 Discussion and Conclusion

This paper provides an initial foray into the complex and rapidly developing
weaponization of transaction ordering in BFT systems, with emphasis on multi-
block MEV extraction (MMEV) strategies. We identify a progressive up-trend in
bidding strategies for k-consecutive slots acquired by single builders,
suggesting the scale at which MMEV may be extracted on Ethereum, by builders,
today. Accurately identifying and labeling complex value extraction is
comparable to a perpetual arms race, in which the increasingly sophisticated
and secretive builders far outgun those seeking to shed light on their
methodologies. As such, the results presented here should be considered
indicators at best. There may be several reasons why builders would bid above
the market to secure consecutive blocks without exploiting MMEV.
Simultaneously, there may be multiple other strategies through which builders
can obscure the praxis of securing multiple slots for MMEV extraction, the
most obvious of which is simply using multiple keys to bid for block space and
execute strategies. To further strengthen the integrity of these and
subsequent findings, future work must examine execution-layer data for all
consecutive blocks secured by builders and scan these blocks for transaction
patterns indicating MMEV exploitation. Combining this praxis with transaction-
clustering methodologies [3] will provide a much deeper level of insight into
the prevalence of MMEV in BFT systems. Beyond the results presented in this
short paper, a much broader discussion begs the question: Does MMEV present an
existential threat to Ethereum and BFT systems in general? The contemporary
discourse on MEV extraction has been normalized, as practitioners and scholars
weigh the positive implications against the negative. Yet, if MMEV practices
grow more common, transaction withholding practices could be tangential to
outright censorship. To what extent does withholding transactions for multiple
consecutive blocks because they are a poor fit for an MMEV momentum-trading
strategy differ from OFAC censorship? While the regulatory implications
differ, the practical results could turn out to be similar. As noted by [3]
builders are able to impose many arbitrary rules on a sequence of blocks for
which they are ordering transactions, the most dominant of which is simply
outright exclusion. Given the high concentration amongst successful builder
entities, the ability to make the Ethereum network close to unusable for any
arbitrary set of addresses currently rests with a small minority. Undoubtedly,
fair transaction sequencing in BFT systems presents a unique challenge, for
which there exists a range of solutions in the execution layer [5] and the
consensus layer.444https://ethresear.ch/t/shutterized-beacon-chain/12249

## References

  * [1] B. Weintraub, C. F. Torres, C. Nita-Rotaru, and R. State. A flash (bot) in the pan: measuring maximal extractable value in private pools. In: Proceedings of the 22nd ACM Internet Measurement Conference, pages 458–471, 2022.
  * [2] Chitra, T., Ferreira, M.V. and Kulkarni, K., 2023. Credible, Optimal Auctions via Blockchains. In: arXiv preprint arXiv:2301.12532.
  * [3] Flashbots. The future of mev is suave: Flashbots, Nov 2022. URL https://writings. flashbots.net/the-future-of-mev-is-suave/.
  * [4] K. Kulkarni, T. Diamandis, and T. Chitra. Towards a theory of maximal extractable value i: Constant function market makers. arXiv preprint arXiv:2207.11835, 2022
  * [5] M. V. Ferreira and D. C. Parkes. Credible decentralized exchange design via verifiable sequencing rules. arXiv preprint arXiv:2209.15569, 2022.
  * [6] Mackinga, Torgin, Tejaswi Nadahalli, and Roger Wattenhofer. ”TWAP Oracle Attacks: Easier Done than Said?.” In 2022 IEEE International Conference on Blockchain and Cryptocurrency (ICBC), pp. 1-8. IEEE, 2022.
  * [7] Obadia, Alexandre, Alejo Salles, Lakshman Sankar, Tarun Chitra, Vaibhav Chellani, and Philip Daian. ”Unity is Strength: A Formalization of Cross-Domain Maximal Extractable Value.” arXiv preprint arXiv:2112.01472 (2021).
  * [8] P. Daian, S. Goldfeder, T. Kell, Y. Li, X. Zhao, I. Bentov, L. Breidenbach, and A. Juels. Flash boys 2.0: Frontrunning in decentralized exchanges, miner extractable value, and consensus instability. In 2020 IEEE Symposium on Security and Privacy (SP), pages 910–927. IEEE, 2020.
  * [9] Revuelta, Alvaro. Statistical analysis on Ethereum k-consecutive block proposal probabilities and case study, In Github, 2022.
  * [10] Victor, F., 2020. Address clustering heuristics for Ethereum. In: Financial Cryptography and Data Security: 24th International Conference, FC 2020, Kota Kinabalu, Malaysia, February 10–14, 2020 Revised Selected Papers 24 (pp. 617-633). Springer International Publishing.

##  6 Appendix

Table 1: Consecutive blocks by builder entity. 68.58% of all blocks can be
mapped to the largest builders.

tables/leaderboard_entity.csv

Table 2: We derive the expected number of consecutive blocks through a Monte
Carlo simulation. We randomly sample 10000 instances of the full data set and
calculate the expected value of each event (for 1$\geq$$k$$\geq$10). We model
the probability as Bernoulli experiment with $p$ being the market share for
each builder entity.

tables/expected_entity.csv


# Wireless Federated Learning (WFL) for 6G Networks - Part II: The Compute-
then-Transmit NOMA Paradigm

Pavlos S. Bouzinis, , Panagiotis D. Diamantoulakis, , and George K.
Karagiannidis P. S. Bouzinis, P. Diamantoulakis, and G.K. Karagiannidis are
with Wireless Communication and Information Processing Group (WCIP),
Department of Electrical and Computer Engineering, Aristotle University of
Thessaloniki, Greece, E-mails: {mpouzinis, padiaman<EMAIL_ADDRESS>
###### Abstract

As it has been discussed in the first part of this work, the utilization of
advanced multiple access protocols and the joint optimization of the
communication and computing resources can facilitate the reduction of delay
for wireless federated learning (WFL), which is of paramount importance for
the efficient integration of WFL in the sixth generation of wireless networks
(6G). To this end, in this second part we introduce and optimize a novel
communication protocol for WFL networks, that is based on non-orthogonal
multiple access (NOMA). More specifically, the Compute-then-Transmit NOMA (CT-
NOMA) protocol is introduced, where users terminate concurrently the local
model training and then simultaneously transmit the trained parameters to the
central server. Moreover, two different detection schemes for the mitigation
of inter-user interference in NOMA are considered and evaluated, which
correspond to fixed and variable decoding order during the successive
interference cancellation process. Furthermore, the computation and
communication resources are jointly optimized for both considered schemes,
with the aim to minimize the total delay during a WFL communication round.
Finally, the simulation results verify the effectiveness of CT-NOMA in terms
of delay reduction, compared to the considered benchmark that is based on
time-division multiple access.

###### Index Terms:

Wireless Federated Learning, Non-Orthogonal Multiple Access (NOMA), Delay
Minimization

##  I Introduction

In Part I of this two parts paper, we presented the advantages, the reference
architecture and three core applications of wireless federated learning (WFL),
the main challenges toward its efficient integration in the sixth generation
of wireless networks (6G), and the identified future directions [1]. This
second part is motivated by the fact that, as it has been thoroughly discussed
in II. A and III.A of the first part, the utilization of advanced multiple
access protocols and the joint optimization of the communication and computing
resources can facilitate WFL to meet the stringent latency requirements of 6G
networks [1, 2, 3]. To this direction, the use of non-orthogonal multiple
access (NOMA) has been proposed in the first part, due to offering low-latency
and improved fairness by serving multiple users in the same resource block, in
opposition to orthogonal multiple access (OMA), where each user occupies a
single resource block [4].

The aim of the second part of this paper is to investigate the potential of
using NOMA in WFL, by focusing on a specific paradigm. In more detail, we
introduce the Compute-then-Transmit NOMA (CT-NOMA) protocol, which is based on
the use of two phases during a WFL round. Specifically, users firstly execute
and complete the local model training concurrently and afterwards upload the
trained model to the server simultaneously via NOMA. Moreover, we consider two
detection schemes for the implementation of successive interference
cancellation (SIC) process in NOMA that correspond to the utilization of a
fixed or variable decoding order. The later, is implemented through the time-
sharing (TS) strategy, which improves the performance of uplink NOMA by
achieving any point of the multiple access channel (MAC) capacity region [5].
Furthermore, we minimize the total delay of a communication round subject to
users’ energy constraints for both considered schemes, by jointly optimizing
the available computation and communication resources, i.e., users’ CPU
frequency for local training, the transmission energy and the time intervals
for local computations and parameter transmission. Finally, simulation results
demonstrate the effectiveness of the proposed protocol in reducing the delay
of a WFL communication round, compared the considered benchmark that is based
on time-division multiple access (TDMA).

##  II System Model and Proposed Protocols

###  II-A Wireless federated learning model

Based on the reference architecture that has been presented in detail in the
first part of this work, we consider a WFL learning system which consists of
$N$ users indexed as $n\in\mathcal{N}=\\{1,2,...,N\\}$ and a server/base
station (BS). Each user $n$ has a local dataset $\mathcal{D}_{n}$, where
$D_{n}=|\mathcal{D}_{n}|$ are the total data samples. Next we briefly describe
the steps of an arbitrary, $i$-th communication round, between the users and
the BS in the WFL system.

  1. i)

The BS broadcasts the global parameter $\boldsymbol{w}^{i}$ to all users
during the considered round.

  2. ii)

After receiving the global model parameter, each user $n\in\mathcal{N}$,
trains the local model through its dataset, and then uploads the trained local
parameter $\boldsymbol{w}^{i+1}_{n}$ to the server.

  3. iii)

After receiving all the local parameters, the server aggregates them, in order
to update the global model parameter $\boldsymbol{w}^{i+1}$.

The whole procedure is repeated until the global model converges. Moreover,
during the first round the server initializes $\boldsymbol{w}^{0}$.

###  II-B Compute-then-Transmit NOMA

As it has already been mentioned, CT-NOMA is based on the use of two
consecutive phases during a WFL round, i.e., the computation and the
communication phase. Specifically, in the first phase users execute the local
computations, while in the second phase they transmit their messages, i.e.,
the trained parameters, to the BS. During the information transmission phase,
the NOMA protocol is considered. According to NOMA, users are capable of
transmitting their messages simultaneously, while the BS decodes the users’
messages by utilizing SIC. As implied by the term CT-NOMA, all users begin the
information transmission procedure via NOMA at the same time instant.
Therefore, all users are constrained to complete the local computations before
the information transmission phase, i.e., within time duration $\tau$, while
the transmission phase duration is denoted as $t$. The CT-NOMA protocol is
illustrated in Fig. 1.

The utilized computation resources for local model training, i.e., CPU cycle
frequency, from the $n$-th user is denoted as $f_{n}$. The number of CPU
cycles for the $n$-th user to perform one sample of data in local model
training is denoted by $c_{n}$. Hence, the computation time dedicated for a
local iteration is given as

$\tau_{n}=\frac{c_{n}D_{n}}{f_{n}},\quad\forall n\in\mathcal{N}.$ (1)

Accordingly, the energy consumption for a local iteration, can be expressed as
follows

$E^{\mathrm{comp}}_{n}=\zeta
c_{n}D_{n}f^{2}_{n}=\zeta\frac{c^{3}_{n}D^{3}_{n}}{\tau_{n}^{2}},\quad\forall
n\in\mathcal{N},$ (2)

where $\zeta$ is a constant parameter related to the hardware architecture of
device $n$. As it was discussed previously, all users are enforced to complete
the local computations within $\tau$, with the corresponding energy that is
consumed by each user being a decreasing function with respect to $\tau$.
Thus, it should hold

$\tau_{n}=\tau,\quad\forall n\in\mathcal{N}.$ (3)

In the continue, the two proposed schemes regarding information transmission
will be described, termed as NOMA with fixed decoding order (FDO) and NOMA
with TS.

####  II-B1 NOMA with FDO

Fixed decoding order refers to the classical uplink NOMA protocol with SIC,
which is widely used in the literature [4]. Let $\pi_{n}$ be the the $n$-th
user’s decoding order position at the BS, while without loss of generality we
consider that $\pi_{1}<\pi_{2}<...<\pi_{N}$. According to NOMA, for decoding
the first user’s message, interference is created by the residual users, i.e.,
$n=2,...,N$, while on the second user’s message, interference is created by
the users indexed as $n=3,...,N$ and so on. Finally the last user to be
decoded, user $N$, experiences no interference. Following that, the data size
$Z_{n}$ of the $\boldsymbol{w}_{n}$ model parameter, that the $n$-th user
transmits within $t$ time duration, should satisfy the following condition

$Z_{n}\leq
tB\log_{2}\left(1+\frac{\frac{E_{n}}{t}g_{n}}{\sum_{\\{i\in\mathcal{N}|\pi_{i}>\pi_{n}\\}}\frac{E_{i}}{t}g_{i}+BN_{0}}\right),\forall
n\in\mathcal{N},$ (4)

where $B$ is the available bandwidth, $N_{0}$ denotes the power spectral
density, while $E_{n}$ is the $n$-th user’s consumed energy for the
aforementioned transmission, with $\frac{E_{n}}{t}$ being the corresponding
transmit power. Moreover, $g_{n}=|h_{n}|^{2}d_{n}^{-2}$ denotes the channel
gain, where the complex random variable $h_{n}\sim\mathcal{CN}(0,1)$ is the
small scale fading and $d_{n}$ is the distance between user $n$ and the BS.
Finally, we assume that the channel remains constant during a communication
round and can be perfectly estimated by the BS.

####  II-B2 NOMA with TS

The basic principle of this scheme is that the users’ decoding order can
change throughout the $t$ transmission time duration, unlike to the NOMA with
FDO where a fixed decoding order is considered. According to [5], the capacity
region which can be achieved by uplink NOMA with TS, known as multiple access
channel (MAC) capacity region, is described as [5]

$\sum_{i\in\mathcal{S}}Z_{i}\leq
tB\log_{2}\left(1+\frac{\sum_{i\in\mathcal{S}}E_{i}g_{i}}{tBN_{0}}\right),\quad\forall\mathcal{S}\subseteq\mathcal{N},\,\mathcal{S}\neq{\O},$
(5)

where $\mathcal{S}$ is a non-empty subset of $\mathcal{N}$. Furthermore, it
deserves to be mentioned that (5) also determines the performance of rate
splitting multiple access (RSMA), which is an alternative scheme to achieve
any point of the MAC capacity region [6, 7].

Figure 1: Compute-then-Transmit NOMA during a WFL round.

##  III Delay Minimization of a WFL Round

The aim of the considered optimization problem is to minimize the total delay
of a WFL round, i.e., $T=\tau+t$, which is the sum of the computation and
transmission latency. Note that, since the transmit power of the BS is much
higher than that of the users’ and the BS transmits the same message to all
users, we ignore the delay of the server for broadcasting the global
parameter. Hereinafter, we focus on a single arbitrary round, while the
subsequent analysis can be similarly carried out for each round. Finally, we
assume that each user transmits the same data size $Z_{n}=Z,\,\forall
n\in\mathcal{N}$, related with the model parameters, while the considered data
size can be predetermined by the central server.

The maximum available CPU clock speed of each user is denoted as
$f^{\mathrm{max}}_{n}$. Thus, it should hold $f_{n}\leq
f^{\mathrm{max}}_{n},\,\forall n\in\mathcal{N}$, which is equivalent to
$\tau\geq\underset{n\in\mathcal{N}}{\mathrm{max}}\left(\frac{c_{n}D_{n}}{f^{\mathrm{max}}_{n}}\right)\triangleq
a_{1}$. Moreover, the maximum available energy of each user is
$E^{\mathrm{max}}_{n}$. Hence, it should hold

$E^{\mathrm{comp}}_{n}+E_{n}=\zeta\frac{c^{3}_{n}D^{3}_{n}}{\tau^{2}}+E_{n}\leq
E^{\mathrm{max}}_{n},\quad\forall n\in\mathcal{N},$ (6)

since the total consumed energy for both computation and communication
purposes, cannot exceed the maximum available energy.

###  III-A CT-NOMA with time-sharing

The optimization problem for minimizing the total latency in the case of CT-
NOMA with TS, can be written as

$\displaystyle\underset{\text{\emph{$\tau,t$},{\boldmath{$E$}}}}{\textbf{min}}\quad\text{\emph{{$\tau+t$}}}$
(7)
$\displaystyle\textbf{\emph{s.t.}}\quad\mathrm{C}_{1}:\zeta\frac{c^{3}_{n}D^{3}_{n}}{\tau^{2}}+E_{n}\leq
E^{\mathrm{max}}_{n},\quad\forall n\in\mathcal{N},$
$\displaystyle\qquad\mathrm{C}_{2}:\sum_{i\in\mathcal{S}}Z\leq
tB\log_{2}\left(1+\frac{\sum_{i\in\mathcal{S}}E_{i}g_{i}}{tBN_{0}}\right),\quad\forall\mathcal{S}\subseteq\mathcal{N},$
$\displaystyle\qquad\mathrm{C}_{3}:\tau\geq a_{1},\,t\geq 0,E_{n}\geq
0,\,\forall n\in\mathcal{N},$

where $\mathrm{C}_{2}$ is related to the users’ achievable capacity region in
(5). It is easy to verify that the problem in (7) is jointly convex with
respect to (w.r.t.) $\tau,t,\boldsymbol{E}$ and thus can optimally be solved
with standard convex-optimization solving methods, such as the interior-point.
However, the computational cost to directly solve this by standard methods
remains intractable, due to the large number of constraints in
$\mathrm{C}_{2}$. More specifically, the number of constraints in
$\mathrm{C}_{2}$ is equal to $2^{N}-1$, since this is the total number of all
non-empty subsets $\mathcal{S}$ of the set $\mathcal{N}$. As a result, solving
(7) for large $N$ comes at the expense of high complexity. To alleviate this
burden, an efficient method for solving (7) will be proposed. Firstly, we
consider a fixed value of $\tau$. Since $\log(\cdot)$ is an ascending function
w.r.t $E_{n}$, an optimal solution occurs when the constraints
$\mathrm{C}_{1}$ are satisfied with equality, i.e., users utilize their whole
available energy, which leads to

$E_{n}=E^{\mathrm{max}}_{n}-\zeta\frac{c^{3}_{n}D^{3}_{n}}{\tau^{2}},\quad\forall
n\in\mathcal{N}.$ (8)

Furthermore, since $E_{n}\geq 0,\,\forall n\in\mathcal{N}$, by manipulating
(8), yields
$\tau\geq\underset{n\in\mathcal{N}}{\mathrm{max}}\left(\sqrt{\zeta\frac{c^{3}_{n}D^{3}_{n}}{E^{\mathrm{max}}_{n}}}\right)\triangleq
a_{2}$. Thus, by also recalling that $\tau\geq a_{1}$, it should finally hold
for $\tau$

$\tau\geq\mathrm{max}\\{a_{1},a_{2}\\}\triangleq\tau_{\mathrm{low}}.$ (9)

Next, after substituting (8) and considering a known
$\tau\geq\tau_{\mathrm{low}}$, the optimization problem in (7) can be
reformulated as

$\displaystyle\underset{\text{\emph{$t$}}}{\textbf{min}}\quad\text{\emph{{$t$}}}$
(10) $\displaystyle\textbf{\emph{s.t.}}\quad\sum_{i\in\mathcal{S}}Z\leq
tB\log_{2}\left(1+\frac{\sum_{i\in\mathcal{S}}A_{i}}{tBN_{0}}\right),\quad\forall\mathcal{S}\subseteq\mathcal{N},$

where $A_{i}$ is given by

$A_{i}=E_{i}g_{i}=\left(E^{\mathrm{max}}_{i}-\zeta\frac{c^{3}_{i}D^{3}_{i}}{\tau^{2}}\right)g_{i}.$
(11)

Note that $A_{i}$, is constant for a given $\tau$. Following that, without
loss of generality we assume that $A_{1}\geq A_{2}\geq...\geq A_{N}$.
According to [8], when NOMA is combined with the TS scheme and by considering
the aforementioned order of $A_{i},\,\forall i\in\mathcal{N}$, the achievable
capacity region of the users can be equivalently described as

$\sum_{i=n}^{N}Z\leq
tB\log_{2}\left(1+\frac{\sum_{i=n}^{N}A_{i}}{tBN_{0}}\right),\quad\forall
n\in\mathcal{N},$ (12)

while by considering that $\sum_{i=n}^{N}Z=Z(N+1-n)$, (12) can be rewritten as

$Z\leq\frac{tB\log_{2}\left(1+\frac{\sum_{i=n}^{N}A_{i}}{tBN_{0}}\right)}{(N+1-n)},\quad\forall
n\in\mathcal{N}.$ (13)

It should be noted that the capacity region is now described by $N$
constraints, i.e., $Z$ is bounded by a set of $N$ inequalities. Next, by
exploiting the alternative, but still equivalent representation of the
capacity region in (13), the optimization problem in (10) can be rewritten as

$\displaystyle\underset{\text{\emph{$t$}}}{\textbf{min}}\quad\text{\emph{{$t$}}}$
(14) $\displaystyle\textbf{\emph{s.t.}}\quad Z(N+1-n)\leq
tB\log_{2}\left(1+\frac{\sum_{i=n}^{N}A_{i}}{tBN_{0}}\right),\forall
n\in\mathcal{N}.$

###### Lemma 1

The optimal value of $t$ for the problem in (14) can be written as

$t^{*}=\underset{n\in\mathcal{N}}{\mathrm{max}}\left[-\frac{z_{n}\ln(2)\sum_{i=n}^{N}A_{i}}{B\left(z_{n}N_{0}\ln(2)+\mathcal{W}_{-1}(b_{n})\sum_{i=n}^{N}A_{i}\right)}\right],$
(15)

where $z_{n}=(N+1-n)Z$ and $b_{n}$ is given by

$b_{n}=-\frac{2^{-\frac{z_{n}N_{0}}{\sum_{i=n}^{N}A_{i}}}z_{n}N_{0}\ln(2)}{\sum_{i=n}^{N}A_{i}},$
(16)

while $\mathcal{W}_{-1}(\cdot)$ denotes the secondary branch of the Lambert W
function.

###### Proof:

We will show that the optimal $t$ is given by the most stringent inequality,
among the set of $N$ inequalities in (14), when this hold with equality.
Firstly, it is straightforward to show that the function

$f(t)=tB\log_{2}\left(1+\frac{\sum_{i=n}^{N}A_{i}}{tBN_{0}}\right),\quad\forall
t>0,$ (17)

is monotonically increasing with respect to $t$. Following that, we assume
that $t^{\prime}$ is optimal and satisfies

$Z(N+1-n)<t^{\prime}B\log_{2}\left(1+\frac{\sum_{i=n}^{N}A_{i}}{t^{\prime}BN_{0}}\right),\,\forall
n\in\mathcal{N},$ (18)

i.e., all constraints are satisfied with strict inequality. It will be shown
by contradiction that this is not an optimal solution for $t$. Since $f(t)$ is
increasing w.r.t $t$, there exist an $t^{*}<t^{\prime}$, for which at least
one inequality constraint from (14) is satisfied with equality. This
observation contradicts the fact that $t^{\prime}$ is optimal, since the
objective is to minimize $t$. Therefore, $t^{*}$ is given by the most
stringent inequality, among the set of $N$ inequalities in (14), when this
hold with equality. After some mathematical manipulations, $t^{*}$ can be
written as in (15) and the proof is completed. ∎

As a matter of fact, the problem can be efficiently solved with closed form
solutions for a fixed $\tau$. In the continue, we will show that the global
optimal solution can be obtained via the bisection method, in order to find
the optimal value of $\tau$. Firstly, it will be shown that $\tau^{*}$ is
bounded, according to the following lemma.

###### Lemma 2

The optimal $\tau$ is bounded in the following interval

$\tau_{\mathrm{low}}\leq\tau^{*}\leq\tau_{\mathrm{up}},$ (19)

where $\tau_{\mathrm{up}}=T(\bar{\tau})=\bar{\tau}+t^{*}(\bar{\tau})$ is the
total delay of the communication round that corresponds to an arbitrary
feasible value of $\tau$, denoted by $\bar{\tau}$, while $t^{*}(\bar{\tau})$
denotes the optimal $t$ for a fixed $\tau=\bar{\tau}$ and is given by (15).

###### Proof:

As mentioned previously, the optimization problem is feasible when the
condition $\tau\geq\tau_{\mathrm{low}}$ is satisfied. Since $\bar{\tau}$ could
be any feasible solution but not necessarily the optimal one, it holds
$T(\bar{\tau})\geq T(\tau^{*})$. Following this, by considering that the
dedicated time for computations cannot exceed the total delay for any feasible
$\tau$, i.e., $\tau<T(\tau)$, we conclude to $\tau^{*}<T(\tau^{*})\leq
T(\bar{\tau})$. Thus, the upper bound of $\tau^{*}$ can be expressed as
$\tau_{\mathrm{up}}=T(\bar{\tau})$ and the proof is completed. ∎

As a result, we have derived the upper and lower bound of $\tau^{*}$ and now
we are ready to apply the bisection method in the considered interval and
propose Algorithm 1 for obtaining the optimal solutions that minimize the
delay of a communication round. Next, the main steps of the proposed algorithm
are discussed. After the necessary initializations in line 1, the bisection
method is applied throughout lines 2-13. More specifically, in lines 3- 6, the
total delay of the communication round is calculated at the points
$\tau=\tau_{\mathrm{m}}$ and $\tau=\tau_{\mathrm{up}}$. Following that, in
lines 7-12, by comparing $T(\tau_{\mathrm{m}})$ and $T(\tau_{\mathrm{up}})$,
the bounds of $\tau$ are properly adjusted in each iteration, until the
convergence is achieved. It should be highlighted that the joint convexity of
the primary problem in (7), w.r.t. all the considered variables, guarantees
the convergence of the bisection method to the optimal solution. After the
resolution of the optimization problem and the calculation of
$\tau^{*},t^{*}$, the optimal $E_{n},f_{n}$ can be given as
$E^{*}_{n}=E^{\mathrm{max}}_{n}-\zeta\frac{c^{3}_{n}D^{3}_{n}}{\tau^{*2}}$ and
$f^{*}_{n}=\frac{c_{n}D_{n}}{\tau^{*}}$. To this end, the major complexity of
Algorithm 1 lies in applying the bisection method and in searching among $N$
values, in order to derive $t^{*}$ from (15). As a result, the complexity can
be expressed as
$\mathcal{O}\left(N\log_{2}\left(\frac{\tau_{\mathrm{up}}-\tau_{\mathrm{low}}}{\epsilon}\right)\right)$,
where $\epsilon$ denotes the algorithm’s tolerance.

Algorithm 1 Delay Minimization for CT-NOMA with TS

1:Initialize
$\tau_{\mathrm{low}}=\mathrm{max}\\{a_{1},a_{2}\\},\,\tau_{\mathrm{m}}=\bar{\tau},\,\tau_{\mathrm{up}}=T(\bar{\tau}),\,\epsilon;$

2:while $\tau_{\mathrm{up}}-\tau_{\mathrm{low}}>\epsilon$ do

3: Set $\tau=\tau_{\mathrm{m}},$ and derive $t^{*}(\tau_{\mathrm{m}})$ from
(15)

4: Set $T(\tau_{\mathrm{m}})=\tau_{\mathrm{m}}+t^{*}(\tau_{\mathrm{m}});$

5: Set $\tau=\tau_{\mathrm{up}},$ and derive $t^{*}(\tau_{\mathrm{up}})$ from
(15)

6: Set $T(\tau_{\mathrm{up}})=\tau_{\mathrm{up}}+t^{*}(\tau_{\mathrm{up}});$

7: if $T(\tau_{\mathrm{m}})<T(\tau_{\mathrm{up}})$

8: $\tau_{\mathrm{up}}=\tau_{\mathrm{m}};$

9: else

10: $\tau_{\mathrm{low}}=\tau_{\mathrm{m}};$

11: end if

12: $\tau_{\mathrm{m}}=\frac{\tau_{\mathrm{low}}+\tau_{\mathrm{up}}}{2};$

13:end while

14:Output
$\tau^{*}=\tau_{\mathrm{up}},\,t^{*}=t^{*}(\tau_{\mathrm{up}}),\,T^{*}=\tau^{*}+t^{*};$

###  III-B CT-NOMA with fixed decoding order

The optimization problem for minimizing the total delay in the case of CT-NOMA
with FDO, can be written as

$\displaystyle\underset{\text{\emph{$\tau,t$},{\boldmath{$E,\pi$}}}}{\textbf{min}}\quad\text{\emph{{$\tau+t$}}}$
(20)
$\displaystyle\textbf{\emph{s.t.}}\quad\mathrm{C}_{1}:\zeta\frac{c^{3}_{n}D^{3}_{n}}{\tau^{2}}+E_{n}\leq
E^{\mathrm{max}}_{n},\quad\forall n\in\mathcal{N},$
$\displaystyle\qquad\mathrm{C}_{2}:Z\leq
tB\log_{2}\left(1+\frac{\frac{E_{n}}{t}g_{n}}{\sum_{\\{i\in\mathcal{N}|\pi_{i}>\pi_{n}\\}}\frac{E_{i}}{t}g_{i}+BN_{0}}\right)$
$\displaystyle\qquad\qquad\forall n\in\mathcal{N},$
$\displaystyle\qquad\mathrm{C}_{3}:\tau\geq a_{1},\,t\geq 0,E_{n}\geq
0,\,\forall n\in\mathcal{N},\,\boldsymbol{\pi}\in\Pi,$

where the permutation $\boldsymbol{\pi}$ belongs to the set $\Pi$, defined as
the set of all possible decoding orders among the $N$ users. Note that the
total number of permutations, i.e., all possible decoding orders, is equal to
$|\Pi|=N!$. Therefore, the optimization problem in (20) is of a combinatorial
nature. However, an exhaustive search among all possible decoding orders is
prohibitive, due to high complexity. To this end, we adopt an ascending
decoding order w.r.t. to the channel gains, according to which the messages of
the users with weaker channel gains are exposed to less interference during
the decoding process. It is notable that this a common selection for uplink
NOMA systems, since it provides fairness among users without reducing the sum
rate [9, 8]. Accordingly, by considering without loss of generality that
$g_{1}\geq g_{2}\geq...\geq g_{N}$, we set $\pi_{1}<\pi_{2}<...<\pi_{N}$. As a
result, $\mathrm{C}_{2}$ of (20) can be reformulated as

$Z\leq
tB\log_{2}\left(1+\frac{E_{n}g_{n}}{\sum_{i=n+1}^{N}E_{i}g_{i}+tBN_{0}}\right),\quad\forall
n\in\mathcal{N}.$ (21)

By substituting (21) in (20), the later can be written as

$\displaystyle\underset{\text{\emph{$\tau,t$},{\boldmath{$E$}}}}{\textbf{min}}\quad\text{\emph{{$\tau+t$}}}$
(22)
$\displaystyle\textbf{\emph{s.t.}}\quad\mathrm{C}_{1}:\zeta\frac{c^{3}_{n}D^{3}_{n}}{\tau^{2}}+E_{n}\leq
E^{\mathrm{max}}_{n},\quad\forall n\in\mathcal{N},$
$\displaystyle\qquad\mathrm{C}_{2}:Z\leq
tB\log_{2}\left(1+\frac{E_{n}g_{n}}{\sum_{i=n+1}^{N}E_{i}g_{i}+tBN_{0}}\right),$
$\displaystyle\qquad\qquad\forall n\in\mathcal{N},$
$\displaystyle\qquad\mathrm{C}_{3}:\tau\geq a_{1},\,t\geq 0,E_{n}\geq
0,\,\forall n\in\mathcal{N},$

which is non-convex due to the coupling of $t$ and $\boldsymbol{E}$ in
constraint $\mathrm{C}_{2}$. In order to address the non-convexity issues, we
first rewrite $\mathrm{C}_{2}$ as

$2^{\frac{Z}{tB}}\leq\frac{E_{n}g_{n}}{\sum_{i=n+1}^{N}E_{i}g_{i}+tBN_{0}},\,\forall
n\in\mathcal{N}.$ (23)

In the continue we set $E_{n}\triangleq\exp(\tilde{E}_{n}),\,\forall
n\in\mathcal{N}$ and $t\triangleq\exp(\tilde{t})$. After some mathematical
manipulations, $\mathrm{C}_{2}$ can be written as

$\displaystyle\ln\left(2^{\frac{Z}{B}\exp(-\tilde{t})}-1\right)$ (24)
$\displaystyle+\ln\left(\sum_{i=n+1}^{N}\exp(\tilde{E}_{i}-\tilde{E}_{n})g_{i}+\exp(\tilde{t}-\tilde{E}_{n})BN_{0}\right)$
$\displaystyle\leq\ln(g_{n}),\quad\forall n\in\mathcal{N}.$

By exploiting the aforementioned formulation of $\mathrm{C}_{2}$, the
optimization problem in (22) can be transformed to the following equivalent
one:

$\displaystyle\underset{\text{\emph{$\tau,\tilde{t}$},{\boldmath{$\tilde{E}$}}}}{\textbf{min}}\quad\text{\emph{{$\tau+\exp(\tilde{t})$}}}$
(25)
$\displaystyle\textbf{\emph{s.t}}\quad\mathrm{C}_{1}:\zeta\frac{c^{3}_{n}D^{3}_{n}}{\tau^{2}}+\exp(\tilde{E}_{n})\leq
E^{\mathrm{max}}_{n},\quad\forall n\in\mathcal{N},$
$\displaystyle\qquad\mathrm{C}_{2}:\ln\left(2^{\frac{Z}{B}\exp(-\tilde{t})}-1\right)+\ln\Biggl{(}\sum_{i=n+1}^{N}\exp(\tilde{E}_{i}-\tilde{E}_{n})g_{i}$
$\displaystyle\qquad\qquad+\exp(\tilde{t}-\tilde{E}_{n})BN_{0}\Biggr{)}-\ln(g_{n})\leq
0,\,\forall n\in\mathcal{N},$ $\displaystyle\qquad\mathrm{C}_{3}:\tau\geq
a_{1},\,\tilde{t}\in\mathbb{R},{\tilde{E}_{n}}\in\mathbb{R}.$

It can be easily shown that the optimization problem in (25) is convex, while
the proof is omitted for brevity. Thus, it can efficiently be solved with
standard convex optimization methods.

##  IV Performance Evaluation and Discussion

Table I summarizes the simulations’ parameters setup. The following figures
have been extracted by means of Monte Carlo.

Figure 2: Impact of the users’ maximum available energy on latency, with
$Z=0.8\mathrm{Mbits}$.

In Fig. 2, the impact of the users’ maximum available energy on the average
latency during a WFL round is depicted. We use the TDMA-based protocol as
benchmark and compare it with the CT-NOMA protocol in terms of delay
reduction, taking into account both considered detections schemes for NOMA. It
can been observed that CT-NOMA with FDO slightly outperforms TDMA. However,
CT-NOMA with TS clearly dominates both TDMA and CT-NOMA with FDO. This is due
to the fact that CT-NOMA with TS can achieve any point of the capacity region,
in contrast to CT-NOMA with FDO which achieves only the corner points. As a
result, the efficient exploitation of the capacity region, as well as
subsequently the flexible interplay among users’ data rates that CT-NOMA with
TS enables, lead to decreased delay during the WFL round. Moreover, in Fig. 3,
the impact of the users’ parameter data size on latency is illustrated. The
superiority of CT-NOMA with TS against CT-NOMA with FDO, and TDMA is again
corroborated. Furthermore, CT-NOMA with FDO presents a slightly enhanced
performance in comparison with TDMA.

Figure 3: Impact of the users’ parameter data size on latency, with
$E^{\mathrm{max}}_{n}=2\,\mathrm{Joule}$.

By evaluating the demonstrated results, it is evident that CT-NOMA has the
potential to provide decreased latency during a WFL round and subsequently
accelerate the training process, which is an important requirement for the
efficient integration of WFL in 6G. Moreover, since NOMA can achieve any point
of the capacity region through the use of time-sharing (or rate splitting)
technique, it can serve in future research as a performance upper bound in
order to evaluate other multiple access schemes for WFL. Furthermore, the
proposed protocol could serve as a baseline for more sophisticated network
implementations, such as hybrid NOMA/OMA configurations, which could possibly
enhance the scalability of WFL.

TABLE I: Simulation Settings Parameter |  Value |  Parameter |  Value
---|---|---|---
$f^{\mathrm{max}}_{n}$ |  $1.5\mathrm{GHz}$ |  $D_{n}$ |  1Mbit
$B$ |  1MHz |  $N_{0}$ |  174dBm/Hz
$\zeta$ |  $10^{-27}$ |  $c_{n}$(cycles) |  $\sim\mathcal{U}(10,40)$
$N$ |  10 users |  $d_{n}$ |  $\sim\mathcal{U}(0,500\mathrm{m})$

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