# User Connection and Resource Allocation Optimization in Blockchain Empowered
Metaverse over 6G Wireless Communications

Liangxin Qian  Chang Liu  Jun Zhao  L. Qian, C. Liu, and J. Zhao are with the
School of Computer Science and Engineering at Nanyang Technological University
(NTU), Singapore. (Emails<EMAIL_ADDRESS><EMAIL_ADDRESS>and
junzhao@ntu.edu.sg).
A 7-page shorter conference version [1] is accepted for VTC 2024 Spring.
According to the policy of IEEE ComSoc at
https://www.comsoc.org/publications/journals/ieee-transactions-wireless-
communications/conference-vs-journal, the journal and conference manuscripts
can be submitted at the same time. Differences between the two versions are
discussed in detail in this journal submission.

###### Abstract

The convergence of blockchain, Metaverse, and non-fungible tokens (NFTs)
brings transformative digital opportunities alongside challenges like privacy
and resource management. Addressing these, we focus on optimizing user
connectivity and resource allocation in an NFT-centric and blockchain-enabled
Metaverse in this paper. Through user work-offloading, we optimize data tasks,
user connection parameters, and server computing frequency division. In the
resource allocation phase, we optimize communication-computation resource
distributions, including bandwidth, transmit power, and computing frequency.
We introduce the trust-cost ratio (TCR), a pivotal measure combining trust
scores from users’ resources and server history with delay and energy costs.
This balance ensures sustained user engagement and trust. The DASHF algorithm,
central to our approach, encapsulates the Dinkelbach algorithm, alternating
optimization, semidefinite relaxation (SDR), the Hungarian method, and a novel
fractional programming technique from a recent IEEE JSAC paper [2]. The most
challenging part of DASHF is to rewrite an optimization problem as
Quadratically Constrained Quadratic Programming (QCQP) via carefully designed
transformations, in order to be solved by SDR and the Hungarian algorithm.
Extensive simulations validate the DASHF algorithm’s efficacy, revealing
critical insights for enhancing blockchain-Metaverse applications, especially
with NFTs.

###### Index Terms:

Metaverse, blockchain, fractional programming, semidefinite relaxation,
resource allocation, trust-cost ratio.

##  I Introduction

The connection of blockchain technology, Metaverse platforms, and seamless
connectivity of non-fungible token (NFT) applications signifies a remarkable
milestone in the further development of wireless communication systems [3, 4,
5, 6]. The Metaverse, as a burgeoning immersive digital space, offers a
platform for users to delve into meticulously designed virtual universes.
Integral to this virtual realm, NFTs emerge to provide users with unique
ownership over digital items and experiences [7]. Making sure NFTs work well
in the Metaverse relies on the strength of 6G wireless communication systems.
These systems need to provide continuous, high-quality user experiences
without using too much energy or causing delays [8, 9].

In this digital world, blockchain serves as the foundation, giving the
Metaverse trust, security, and transparency. It secures NFT transactions and
gives users proof of ownership of their digital assets [10]. But, setting up
user connections and managing resources in a Metaverse that uses blockchain
reveals many complex challenges [11, 12, 13].

This paper delves into the collaborative optimization of user connections and
resource allocation to augment communication and computation efficiencies in
the blockchain-infused Metaverse, particularly within the NFT context. As we
navigate through our research, we underscore the paramount challenges and
motivations that shape our journey.

Challenges and Motivations. The rapid growth of the Metaverse and its
integration with blockchain technology introduces a range of challenges and
opportunities, especially when considering wireless communications. The
necessity for enhanced privacy and efficient resource allocation becomes
apparent as the Metaverse’s complex web of user interactions requires secure
spaces for digital identities and transactions [14, 15, 16]. Blockchain’s core
features offer promising solutions for improving privacy and security.
However, merging blockchain into the Metaverse introduces complexities [17,
18], highlighting the need for creative resource management strategies. Prior
works, such as those by Feng _et al._ [19], have made progress in optimizing
connections and resources. Yet, their approach to simplifying bandwidth and
power allocation points to a potential oversight in fully leveraging resource
optimization. This trend of oversight is further evident in the work of Dai
_et al._ [20]. Their research, which seeks to optimize computation offloading
and user association, does break new ground. However, the concentration on
computation frequency and transmit power as the main resources fails to
capture the broader spectrum of challenges. This situation emphasizes the
importance of a comprehensive approach that considers the full spectrum of
challenges, including bandwidth, latency, energy consumption, reliability, as
well as computational resource use, transmit power use, and network design.

The integration of blockchain within the wireless-enabled Metaverse also
brings to light essential prerequisites and significant challenges, focusing
on work offloading with higher trust, energy consumption, and latency. An
efficient blockchain architecture is vital to support the high transaction
volumes and real-time interactions of the Metaverse. Energy consumption
emerges as a key challenge, with traditional blockchain models being notably
high in energy usage. Finding energy-efficient blockchain solutions is
critical for the Metaverse’s sustainability and for providing a smooth user
experience. Additionally, reducing latency for real-time activities and
effectively managing user-server connection and work offloading ratio with
higher trust underscores the need for solutions that tackle both technological
and operational challenges in this emerging space.

Our research is driven by the imperative to address these complexities, aiming
to fill the gaps in combining blockchain with the Metaverse over wireless
networks. We are motivated to reduce latency and energy consumption, ensure
user-server connection and work offloading ratio with higher trust, and
improve communication and computational resource allocation.

Studied Problem. In this paper, we study the blockchain-empowered Metaverse
system and introduce the pioneering concept of trust-cost ratio (TCR), which
is a ratio of trust score and cost consumption ($\frac{\text{trust
score}}{\text{cost}}$). This value is an ingenious metric that encapsulates
the delicate equilibrium between user trust scores and the overarching
considerations of delay and energy consumption across the communication
spectrum. The trust score includes a user’s radio and computing resources from
the server and a score based on the server’s historical experience with data
processing and blockchain work. Cost consumption includes the maximum delay
and total energy consumption of the users and servers. We assume that the user
does partial work offloading first; that is, the user offloads part of the
work to the server, and the rest of the work is handled locally. On the server
side, the server first receives the work offloaded from the user and then
allocates radio and computing resources for each user, including bandwidth,
transmit power of users and servers, and computing frequency of users and
servers. In addition, the server further divides the computing resources
allocated to each user. Specifically, part of the computing resources are used
to process the data processing work of the user offload, and the rest is used
to generate, upload, and verify the blockchain.

###  I-A Related Work

We discuss the related work in the following parts: resource allocation in
blockchain-based wireless networks, trust model in blockchain, and blockchain
with Metaverse.

Resource allocation in blockchain-based wireless networks. In the context of
blockchain-based wireless networks, several notable works have contributed to
the optimization of resource allocation and performance enhancement. Feng _et
al._ [19] delve into the joint optimization of user connections, radio
resources, and computational resource allocation within a blockchain-based
mobile edge computing (MEC) system. It’s noteworthy that their treatment of
data rate as a direct variable may not offer a precise delineation for
bandwidth and transmit power allocation, as these vital resource aspects lack
explicit detailing. Furthermore, their optimization objectives primarily
center around minimizing energy consumption and reducing delays, leaving other
critical resource aspects unexplored. Similarly, Dai _et al._ [20] propose a
framework addressing the joint optimization of computation offloading and user
association in MEC systems. Yet, it’s important to highlight that their
optimization efforts primarily target computation frequency and transmit power
as the key resources to optimize. Additionally, their optimization problem
primarily focuses on minimizing energy consumption without addressing
communication resource allocation or adopting a holistic approach that
encompasses user association within a blockchain-enabled Metaverse. Guo _et
al._ [21] present a comprehensive blockchain-based MEC framework, integrating
deep reinforcement learning to achieve consensus among nodes while optimizing
both MEC and blockchain systems. Their focus is on adaptive resource
allocation, block size management, and block generation to enhance the system
throughput and ensure superior quality of service (QoS) for users in future
wireless networks. Sun _et al._ [22] introduce two double auction mechanisms
within a blockchain-driven MEC framework, aiming to ensure reliable cross-
server resource allocation and protect against tampering of user information
by malicious edge servers. This approach leads to improved MEC system
efficiency. Additionally, Xu _et al._ [23] propose a Stackelberg dynamic game-
based resource pricing and trading scheme seamlessly integrated with
blockchain technology. This scheme optimizes allocating edge computing
resources between edge computing stations (ECSs) and unmanned aerial vehicles
(UAVs) in mobile networks while addressing security and privacy concerns,
validated through numerical simulations. Furthermore, Jiang _et al._ [24]
introduce a video analytics framework incorporating multiaccess edge computing
and blockchain technologies into vehicular networks. Their work addresses
challenges related to data transmission and security, demonstrating the
efficacy of their approach in optimizing blockchain transaction throughput and
reducing MEC latency through deep reinforcement learning. Collectively, these
research efforts provide a comprehensive exploration of resource allocation,
consensus mechanisms, and performance optimization in blockchain-based
wireless networks.

Trust model in blockchain. In the realm of blockchain-based trust models,
several significant contributions have emerged, each addressing distinct
aspects of trust and security in diverse IoT scenarios. Tu _et al._ [25]
introduce a blockchain-based trust and reputation model (BTRM) geared towards
fortifying IoT security by conducting a comprehensive evaluation of user
reputation and resistance against multiple malicious attacks. Complementing
this, they present a dynamic evaluation mechanism (DEM) within a hyper ledger
fabric prototype system, which streamlines reputation evaluations while
preserving network security. Liu _et al._ [26] propose a semi-centralized
trust management system based on blockchain, tailored to facilitate IoT data
exchange. Their innovative computational trust model incorporates decay
functions and adaptable weights, effectively identifying and mitigating the
influence of malicious devices, as substantiated through simulation
experiments, positioning it favorably against conventional models. Finally, Xi
_et al._ [27] present a dynamic blockchain sharding scheme founded on the
hidden Markov model (HMM), which efficiently reduces cross-shard transactions
while concurrently boosting system throughput and reducing transaction
confirmation latency, especially valuable for collaborative IoT applications.
To sum up, these pioneering works offer a diverse spectrum of trust models,
each tailored to bolster security and trustworthiness in various IoT contexts.
Blockchain with Metaverse. In the domain of blockchain’s integration with the
Metaverse, two noteworthy studies have contributed to addressing distinct
challenges and advancing the capabilities of this convergence. Hoa _et al._
[28] consider an edge computing-assisted Metaverse system featuring a virtual
service provider (VSP) entrusted with offloading data collected by UAVs for
updating digital twins (DTs). This research strategically confronts dynamic
data management, user latency requirements, and resource constraints,
employing a stochastic problem formulation and leveraging deep reinforcement
learning. Together, the above studies showcase the evolving landscape of
blockchain and Metaverse integration, addressing specific issues while
harnessing the potential of cutting-edge technologies to shape the future of
digital ecosystems.

Differences with the conference version [1]. In [1], the focus is on
collaborative server training involving neural network adapters within the
context of large language model-based applications to mitigate computational
and communication overhead, thereby guaranteeing that users have access to
sufficient computational and communication resources. In our work, however, we
shift the spotlight to efficient offloading tasks and enhancing user-server
connections within the burgeoning domain of a blockchain-powered Metaverse,
specifically targeting NFT applications. This pivot necessitates a fresh
approach to the communication and computation formulas applied. Whereas [1]
explored the trade-off between user experience scores and total cost, our
investigation pivots towards balancing user trust scores against total cost.
Our contribution extends the groundwork laid by [1] by incorporating a model
that captures user trust in servers tasked with processing blockchain
operations. While [1] was limited to optimizing resource allocation alongside
user connection and training ratio, our paper delves further, tackling the
nuanced optimization of user connection, offloading ratio, and the division of
server computing frequencies. The methodology in [1] doesn’t specify an
algorithmic procedure, whereas our paper delineates three intricate
algorithmic methodologies. Building upon the comparative baseline established
by [1], our paper enriches the discourse with simulation results across
diverse server computation frequencies and offers an expanded suite of figures
demonstrating algorithm convergence in various system configurations.

###  I-B Main Contributions

Our contributions include a novel joint optimization problem, the introduction
of the TCR metric, and an algorithm to solve the problem, detailed as follows:

  * $\bullet$

Introducing the Trust-Cost Ratio: We introduce the concept of the trust-cost
ratio (TCR). To the best of our knowledge, there is no existing research
studying this. TCR quantifies the balance between user trust scores and the
overall delay and energy consumption in the entire uplink and downlink
communications. It provides a valuable metric for assessing the trade-off
between user trust and resource efficiency.

  * $\bullet$

Apart from the traditional optimization of partial work offloading,
communication resource allocation, and computation resource allocation of
users and servers, we also consider the computation resource allocation of
servers for different tasks (i.e., data processing tasks and blockchain
tasks). This new optimization variable is unique in blockchain-based wireless
communication systems.

  * $\bullet$

An Algorithm to Solve the Proposed Optimization: To address the joint
optimization problem discussed above, our developed algorithm, termed DASHF,
leverages the combination of the Dinkelbach algorithm, alternating
optimization, semidefinite relaxation (SDR), the Hungarian algorithm, and a
novel fractional programming (FP) technique by [2] published in IEEE JSAC
recently. The most challenging part of DASHF is to rewrite an optimization
problem as Quadratically Constrained Quadratic Programming (QCQP) via
carefully constructed transformations to leverage SDR and the Hungarian
algorithm to obtain a solution. In DASHF, we first tackle the optimization of
user connections, work-offloading ratios, and server computing frequency
division as QCQP. Subsequently, we delve into optimizing communication-
computation resource allocation (for bandwidth, transmit power of users and
servers and computing frequency of users and servers), providing an effective
solution for the challenging non-convex FP problem.

  * $\bullet$

The simulation results confirm that the proposed DASHF algorithm effectively
optimizes user work offloading, resource allocation, trust-cost ratio, and
communication-computation resources.

The rest of this paper is organized as follows. Section II presents the system
model and optimization formulation. We propose the DASHF algorithm to solve
the optimization problem in Section III, with its complexity discussed in
Section III-C. The numerical results are provided in Section IV. We conclude
the paper in Section V.

Figure 1: Optimizing the trust-cost ratio (TCR) of a system consisting of $N$
VR users and $M$ Metaverse servers by the joint optimization of user
association, offloading ratio, server computing capacity division, and radio
and computing resource allocation. The images in the NFT tasks come from
Monkey Kingdom[29].

##  II Blockchain-Empowered Metaverse System and Optimization Problem
Formulation

In this section, we will introduce the studied system scenario and formulate
the optimization problem.

###  II-A An Overview of the System

As presented in Fig. 1 on the next page, we consider a blockchain-empowered
Metaverse system that operates within the realm of wireless communication.
Within this envisioned framework, each VR user is assigned a specific set of
NFT tasks, necessitating partial delegation to the Metaverse server. This
Metaverse server assists in data processing, as well as functioning as the
generator and validator of blockchain transactions. Once the Metaverse server
concludes the processing of user-generated data, it proceeds to upload
computation offloading records onto the blockchain system, where they are
systematically documented within the distributed ledger. These records are
categorized as individual “transactions” within the blockchain system.
Subsequently, these transactions undergo meticulous verification by other
Metaverse servers before being consolidated into a new block. The achievement
of a consensus across the entirety of the Metaverse network heralds the
seamless integration of the new block into the overarching blockchain
structure. Following this integration, the Metaverse server takes on the
responsibility of disbursing the processed data back to the respective users,
ensuring an efficient data distribution mechanism within the system.

###  II-B The Detailed System Model

We consider a system comprising $N$ VR users and $M$ Metaverse servers. We use
$n$ and $m$ as indices for a VR user and a Metaverse server, respectively,
where $n\in\mathcal{N}:=\\{1,2,\cdots,N\\}$ and
$m\in\mathcal{M}:=\\{1,2,\cdots,M\\}$. Each user is connected to one and only
one server; i.e., $\sum_{m\in\mathcal{M}}x_{n,m}=1$. We introduce indicator
variables $x_{n,m}\in\\{0,1\\}$ to characterize the connection between users
and servers; specifically, $x_{n,m}=1$ (resp, $0$) means that the $n$-th user
is connected (resp., not connected) to the $m$-th server. For example, if
$x_{n,m}=1$, it means that the $n$-th user only connects to the $m$-th server
and $x_{n,m^{\prime}}=0$ for $m{{}^{\prime}}\in\mathcal{M}\setminus\\{m\\}$.

####  II-B1 Time consumption

We consider frequency-division multiple access (FDMA) so that communication
among users and servers would not interfere. Let $b_{n,m}$ be the allocated
bandwidth for the communication between user $n$ and server $m$, and $p_{n}$
be the transmit power of user $n$. Then we define the transmission rate from
user $n$ to the chosen edge server $m$ as
$r_{n,m}=b_{n,m}\log_{2}(1+\frac{g_{n,m}p_{n}}{\sigma^{2}b_{n,m}})$ according
to the well-known Shannon formula, where $\sigma^{2}$ is the noise power
spectral density, and $g_{n,m}$ is the channel attenuation. We further express
$g_{n,m}$ as $g_{n,m}=h_{n,m}l_{n,m}$, with $h_{n,m}$ being the large-scale
slow-fading component capturing effects of path loss and shadowing and
$l_{n,m}$ being the small-scale Rayleigh fading.

Work offloading and data processing phases. The total task data size at user
$n$ is $d_{n}$ bits. The offloading task data size is $\varphi_{n}d_{n}$ for
$\varphi_{n}\in[0,1]$, and the rest of the task (i.e., the
$(1-\varphi_{n})d_{n}$ part) will be processed by the user itself. After the
user-server connection algorithm (this can be completed by choosing the
nearest neighbor server sets, then choosing the server with the lowest
transmission time, and finally finishing all user-server connections). The
transmission time from the user $n$ to the server $m$ is
$T^{(t_{1})}_{n,m}=\frac{x_{n,m}\varphi_{n}d_{n}}{r_{n,m}}$, where the
superscript “$t$” represents transmission. User $n$ processes
$(1-\varphi_{n})d_{n}$ by itself, and the processing time is
$T^{(p_{1})}_{n}=\frac{(1-\varphi_{n})d_{n}f_{n}}{F_{n}}$, where the
superscript “$p$” stands for processing, $f_{n}$ (cycles/bit) is the CPU clock
cycles per bit required by the $n$-th user, and $F_{n}$ is the CPU cycle
frequency of the $n$-th user. Below, we look into server computation. After
using $F_{n,m}$ to denote the CPU cycle frequency of the $m$-th server
allocated to $n$-th user, we let $\gamma_{n,m}\in(0,1)$ be the fraction of
$F_{n,m}$ used to process user $n$’s data, and $(1-\gamma_{n,m})$ be the
fraction of $F_{n,m}$ used for the blockchain task. Then for server $m$ to
process the data of $\varphi_{n}d_{n}$ bits sent from user $n$, the processing
delay is
$T^{(p_{1})}_{n,m}=\frac{x_{n,m}\varphi_{n}d_{n}f_{m}}{\gamma_{n,m}F_{n,m}}$,
where $f_{m}$ is the CPU clock cycles per bit required by server $m$ and
$x_{n,m}$ is just the binary indicator variable which has been defined at the
beginning of this subsection. We consider
$\sum\nolimits_{n\in\mathcal{N}}x_{n,m}F_{n,m}\leq F_{m}$, where $F_{m}$ is
the maximum CPU cycle frequency of server $m$.

Blockchain phase. With the $\varphi_{n}d_{n}$ data bits sent from user $n$,
server $m$ will generate a blockchain block for such data. For this block
generation, we consider that the server will process
$\varphi_{n}d_{n}\omega_{b}$ bits, where $\omega_{b}$ represents the data size
change ratio when server $m$ maps user $n$’s data to a format that can be
processed by the blockchain. Recall that we have explained in the previous
paragraph that $(1-\gamma_{n,m})$ fraction of $F_{n,m}$ is used for the
blockchain task. Hence, the delay for server $m$ to generate the block for
user $n$ by leveraging the computing capability of $(1-\gamma_{n,m})F_{n,m}$
can be expressed by
$T^{(g)}_{n,m}=\frac{x_{n,m}\varphi_{n}d_{n}\omega_{b}f_{m}}{(1-\gamma_{n,m})F_{n,m}}$,
where the superscript “$g$” means generation, and $f_{m}$ has been defined in
the previous paragraph. Next, we discuss the block propagation. We consider
only one hop among servers and utilize the result of [19] to obtain the
following. The total block propagation time used in data transactions during
the consensus is $T^{(bp)}_{n,m}=\frac{S_{b}}{R_{m}}$, where the superscript
“$bp$” stands for block propagation, and
$R_{m}:=\mathop{\min}\nolimits_{m{{}^{\prime}}\in\mathcal{M}\setminus\\{m\\}}R_{m,m^{\prime}}$,
with $R_{m,m^{\prime}}$ denoting the wired link transmission rate between
servers $m$ and $m^{\prime}$. Note that “$\min$” means that the block
generated by server $m$ has to be propagated to all other $M-1$ servers
(taking “$\min$” among the rates is like computing “$\max$” among the delays).
Then, during the validation phase, all $M-1$ servers do the verification work.
According to [19], the validation time is
$T_{n,m}^{(v)}=\mathop{\max}_{m{{}^{\prime}}\in\mathcal{M}\setminus\\{m\\}}\frac{f_{v}}{(1-\gamma_{n,m^{\prime}})F_{n,m^{\prime}}}$,
where the superscript “$v$” is short for verification, and $f_{v}$ denotes the
CPU cycles required by server $m{{}^{\prime}}$ to verify the block.

Returning data and data processing phases. We use $\omega_{p}$ to denote the
data size change ratio from the raw data to the processed data. Thus, the
processed data size can be expressed as $\omega_{p}\varphi_{n}d_{n}$. Server
$m$ transmits the results to the user $n$, and the delay is
$T^{(t_{2})}_{n,m}=\frac{x_{n,m}\omega_{p}\varphi_{n}d_{n}}{r_{m,n}}$, where
the superscript “$t$” stands for transmission. We assume that the path loss
and bandwidth between the downlink and uplink are the same. Then, user $n$
processes the received data, and the delay is
$T^{(p_{2})}_{n}=\frac{\omega_{p}\varphi_{n}d_{n}f_{n}}{F_{n}}$, where the
superscript “$p$” represents processing. The time on the server side is

$\textstyle
T_{s,n,m}=T^{(t_{1})}_{n,m}+T^{(p_{1})}_{n,m}+T^{(g)}_{n,m}+T^{(bp)}_{n,m}+T_{n,m}^{(v)},$
(1)

where the subscript “$s$” represents “server”. The time consumed on the user
side is

$\textstyle T_{u,n,m}=T^{(p_{1})}_{n}+T^{(t_{2})}_{n,m}+T^{(p_{2})}_{n},$ (2)

where the subscript “$u$” stands for “user”. With the above notation, the
system delay is computed as

$\textstyle
T_{\text{total}}=\max\limits_{n\in\mathcal{N},m\in\mathcal{M}}\big{\\{}\max\left\\{T_{s,n,m},T_{u,n,m}\right\\}\big{\\}}.$
(3)

The whole communication and computation procedure. In the proposed system, the
procedure begins with users offloading computational tasks to servers while
simultaneously processing part of the work locally. The server processes these
offloaded tasks and generates transactions that are packaged into blocks.
These blocks then undergo a consensus mechanism, involving propagation to and
verification by other block producers, to be added to the blockchain. This
ensures the integrity and security of the transactions related to the user’s
tasks. In the consensus, the delegated proof-of-stake (DPoS) consensus
algorithm is considered and all validators are assumed to be honest [19].
Finally, the server returns the processed results to the user, who then
performs any necessary further processing. Throughout this system, servers not
only handle computation but also act as witnesses in recording transactions on
the blockchain, facilitating a secure, efficient, and transparent workflow
from task offloading to final data processing. The whole communication and
computation procedure is also presented in Fig. 2.

Figure 2: The whole communication and computation procedure.

####  II-B2 Energy consumption

Based on the delay discussion, we then compute the energy consumption in this
system. Energy used for transmitting data from the $n$-th user to the server
$m$ is given as
$E^{(t_{1})}_{n,m}=p_{n}T^{(t_{1})}_{n,m}=p_{n}\frac{x_{n,m}\varphi_{n}d_{n}}{r_{n,m}}$.
Energy used for user $n$ processing data locally can be calculated as follows
based on [30]: $E^{(p_{1})}_{n}=\kappa_{n}(1-\varphi_{n})d_{n}f_{n}F^{2}_{n}$,
where $\kappa_{n}$ denotes the effective switched capacitance that depends on
the chip architecture of user $n$. We omit the energy caused by the
blockchain’s propagation and validation process since they are negligible
compared with those for other stages. In other words, we are mainly concerned
with the energy caused by the server processing data, generating blocks, and
transmitting data. Energy for server $m$ to process $\varphi_{n}d_{n}$ data is
given as
$E^{(p_{1})}_{n,m}=\kappa_{m}x_{n,m}\varphi_{n}d_{n}f_{m}(\gamma_{n,m}F_{n,m})^{2}$,
where $\kappa_{m}$ denotes the effective switched capacitance that depends on
the chip architecture of server $m$. Energy used for the server to generate
one block is
$E^{(g)}_{n,m}=\kappa_{m}x_{n,m}\varphi_{n}d_{n}\omega_{b}f_{m}[(1-\gamma_{n,m})F_{n,m}]^{2}$.
Energy caused by server $m$ to transmit the processed data to user $n$ is
$E^{(t_{2})}_{n,m}=p_{m}T^{(t_{2})}_{n,m}=p_{m}\frac{x_{n,m}\varphi_{n}d_{n}\omega_{p}}{r_{m,n}}$.
Energy caused by user $n$ to process $\varphi_{n}d_{n}\omega_{p}$ bits of data
is $E^{(p_{2})}_{n}=\kappa_{n}\varphi_{n}d_{n}\omega_{p}f_{n}F_{n}^{2}$. We
denote the energy consumed at user side as
$E_{u,n,m}=\sum_{n\in\mathcal{N}}\\{E^{(p_{1})}_{n}+E^{(p_{2})}_{n}\\}+\sum_{n\in\mathcal{N},m\in\mathcal{M}}E^{(t_{1})}_{n,m}$
and the energy consumed at server side as
$E_{s,n,m}=\sum_{n\in\mathcal{N},m\in\mathcal{M}}\\{E^{(p_{1})}_{n,m}+E^{(g)}_{n,m}+E^{(t_{2})}_{n,m}\\}$.
Thus, the system (i.e., total) energy consumption can be formulated as

$\textstyle E_{\text{total}}$
$\textstyle=\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}(E_{u,n,m}+E_{s,n,m})=\sum\limits_{n\in\mathcal{N}}\\{E^{(p_{1})}_{n}+E^{(p_{2})}_{n}\\}$
$\textstyle+\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}\\{E^{(t_{1})}_{n,m}+E^{(p_{1})}_{n,m}+E^{(g)}_{n,m}+E^{(t_{2})}_{n,m}\\}.$
(4)

TABLE I: Important Notation. Notation |  Description
---|---
$\mathcal{N}$ |  The set of all users ($n\in\\{1,...,N\\}$)
$\mathcal{M}$ |  The set of all servers ($m\in\\{1,...,M\\}$)
$x_{n,m}$ |  The user connection between user $n$ and server $m$
$d_{n}$ |  The total work of user $n$
$p_{n}$ |  The transmit power of user $n$
$p_{max}^{(n)}$ |  The maximum transmit power of user $n$
$p_{n,m}$ |  The transmit power of server $m$ allocated for user $n$
$p_{max}^{(m)}$ |  The maximum transmit power of server $m$
$b_{n,m},b_{m,n}$ |  The allocated bandwidth between user $n$ and server $m$
$b_{max}$ |  The maximum allocated bandwidth for each server
$g_{n,m}$ |  The channel attenuation from user $n$ to server $m$
$r_{n,m}$ |  The transmission rate from user $n$ to server $m$
$r_{m,n}$ |  The transmission rate from server $m$ to user $n$
$v_{n,m}$ |  The trust score value of user $n$ connected to server $m$
$\varphi_{n}$ |  The offloading work ratio of user $n$
$F^{(n)}_{max}$ |  The maximum CPU cycle frequency of user $n$
$F^{(m)}_{max}$ |  The maximum CPU cycle frequency of server $m$
$f_{n},f_{m}$ |  The CPU clock cycles per bit of user $n$ and server $m$
$F_{n,m}$ |  The total computational capacity of server $m$ allocated for user $n$
$\gamma_{n,m}$ |  The computational capacity ratio of server $m$ allocated for processing data for user $n$
$\omega_{b}$ |  The data size change ratio between the MEC task and blockchain task
$\omega_{p}$ |  The data size change ratio between the raw data and processed data
$\omega_{t}$ |  The weight parameter of delay cost
$\omega_{e}$ |  The weight parameter of energy cost
$S_{b}$ |  The block size
$R_{m}$ |  The wired link transmission rate among the servers
$\tau$ |  The scoring parameter that is based on the server’s history of processing user data and blockchain work
$\kappa_{n}$ |  The effective switched capacitance that depends on the chip architecture of user $n$
$\kappa_{m}$ |  The effective switched capacitance that depends on the chip architecture of server $m$

####  II-B3 Trust score function

The trust score is a quantifiable measure that reflects the reliability or
credibility of servers within a blockchain network [31]. This score is
calculated by aggregating the evaluated values of one server node over several
counts of transactions. These evaluated values are determined based on the
outcomes of authentication attempts, both failed and successful, conducted by
the respective nodes. The trust score value of the $n$-th user connected to
$m$-th server can be denoted as $v_{n,m}$. According to the model introduced
in [31], the total trust score values of all server nodes can be calculated as

$\textstyle\mathcal{V}=\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}x_{n,m}v_{n,m}.$
(5)

We further denote the trust score value of user $n$ that is connected to
server $m$ as:

$\textstyle
v_{n,m}=\varpi_{1}\ln[1+\varpi_{2}(\frac{p_{m}}{p_{max}^{(m)}}+\frac{F_{n,m}}{F_{max}^{(m)}}+\frac{b_{n,m}}{b_{max}}+\tau)],$
(6)

where $\varpi_{1}$ and $\varpi_{2}$ are system-specific constants, and
$\tau\in[0,1]$ is a scoring parameter based on the server’s history of
processing user data and blockchain work. The logarithm function in (6) above
is jointly concave of $p_{m}$, $F_{n,m}$, and $b_{n,m}$. Using the logarithm
function to model the utility has also been adopted by [32] on crowdsensing.
This trust score function is effective and sensitive in all value ranges of
$(\frac{p_{m}}{p_{max}^{(m)}}+\frac{F_{n,m}}{F_{max}^{(m)}}+\frac{b_{n,m}}{b_{max}}+\tau)$,
which can describe each user’s subjective experience of the communication and
computing resources obtained from the server. Some important notation in this
paper is shown in Table I.

###  II-C Optimization Problem

We have computed the system delay, system energy consumption, and the total
trust score values in Equations (3), (4), and (5) above. Then we define the
system trust-cost ratio (TCR) below, which our optimization problem will
maximize:

$\textstyle\frac{\mathcal{V}}{\omega_{t}T_{\text{total}}+\omega_{e}E_{\text{total}}}=$
$\textstyle\frac{\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}x_{n,m}v_{n,m}}{\omega_{t}\\!\max\limits_{n\in\mathcal{N},m\in\mathcal{M}}\\!\big{\\{}\\!\\!\max\\!\left\\{T_{s,n,m},T_{u,n,m}\right\\}\\!\big{\\}}+\omega_{e}(\\!\\!\sum\limits_{n\in\mathcal{N}}\\!\\!E_{u,n,m}+\vspace{-10pt}\\!\\!\\!\\!\\!\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}\\!\\!\\!\\!\\!\\!\vspace{-10pt}E_{s,n,m})},$

where $\omega_{t}$ and $\omega_{e}$ represent the weight values of delay and
energy, respectively, with $\omega_{t}+\omega_{e}=1$.

With the above objective function, our optimization variables include the
following: user connection
$\bm{x}=[x_{n,m}|_{n\in\mathcal{N},m\in\mathcal{M}}]$,
$\bm{\varphi}=[\varphi_{n}|_{n\in\mathcal{N}}]$,
$\bm{\gamma}=[\gamma_{n,m}|_{n\in\mathcal{N},m\in\mathcal{M}}]$, bandwidth
$\bm{b}=[b_{n,m}|_{n\in\mathcal{N},m\in\mathcal{M}}]$, transmission power
$\bm{p_{u}}=[p_{n}|_{n\in\mathcal{N}}]$ and
$\bm{p_{s}}=[p_{m}|_{m\in\mathcal{M}}]$, and CPU-cycle frequency
$\bm{f_{u}}=[F_{n}|_{n\in\mathcal{N}}]$ and
$\bm{f_{s}}=[F_{n,m}|_{n\in\mathcal{N},m\in\mathcal{M}}]$.

Next, we start discussing how to solve the optimization. To begin with, in
order to remove the “maximize” operation in $T_{\text{total}}$, we add an
auxiliary variable $T$, which is constrained to be greater than or equal to
both $T_{s,n,m}$ and $T_{u,n,m}$. Besides, we utilize Dinkelbach’s algorithm
[33] to introduce an additional variable $y$, which is obtained from the TCR
value in the previous iteration (details later). Then, the fractional
programming in the trust-cost ratio is transformed into the following problem:

$\displaystyle\max\limits_{\bm{x},\bm{\varphi},\bm{\gamma},\bm{b},\bm{p_{u}},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}},T}$
$\displaystyle\bigg{\\{}\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}[x_{n,m}v_{n,m}-y\omega_{e}(E_{u,n,m}+E_{s,n,m})]\bigg{\\}}-y\omega_{t}T$
(7) s.t. $\displaystyle x_{n,m}\in\\{0,1\\},\forall
n\in\mathcal{N},m\in\mathcal{M}$ (7a)
$\displaystyle\sum\limits_{m\in\mathcal{M}}x_{n,m}=1,\forall n\in\mathcal{N}$
(7b) $\displaystyle\varphi_{n}\in[0,1],\forall n\in\mathcal{N}$ (7c)
$\displaystyle\gamma_{n,m}\in(0,1),\forall n\in\mathcal{N},m\in\mathcal{M}$
(7d) $\displaystyle\sum\limits_{n\in\mathcal{N}}x_{n,m}b_{n,m}\leq
b_{max},\forall m\in\mathcal{M}$ (7e) $\displaystyle p_{n}\leq
p^{(n)}_{max},\forall n\in\mathcal{N}$ (7f)
$\displaystyle\sum\limits_{n\in\mathcal{N}}x_{n,m}p_{n,m}\leq
p^{(m)}_{max},\forall m\in\mathcal{M}$ (7g) $\displaystyle F_{n}\leq
F^{(n)}_{max},\forall n\in\mathcal{N}$ (7h)
$\displaystyle\sum\limits_{n\in\mathcal{N}}x_{n,m}F_{n,m}\leq
F^{(m)}_{max},\forall m\in\mathcal{M}$ (7i) $\displaystyle T_{s,n,m}\leq
T,\forall n\in\mathcal{N},m\in\mathcal{M}$ (7j) $\displaystyle T_{u,n,m}\leq
T,\forall n\in\mathcal{N},m\in\mathcal{M}.$ (7k)

Based on Dinkelbach’s Algorithm, we iteratively optimize $y$ and problem (7).
Specifically, at the $i$-th iteration, given $y^{(i-1)}$, we first obtain
$\bm{x}^{(i)},\bm{\varphi}^{(i)},\bm{\gamma}^{(i)},\bm{b}^{(i)},\bm{p_{u}}^{(i)},\bm{p_{s}}^{(i)},\bm{f_{u}}^{(i)},\bm{f_{s}}^{(i)},T^{(i)}$
by solving the optimization problem (7); then we calculate $y^{(i)}$ with the
given
$\bm{x}^{(i)},\bm{\varphi}^{(i)},\bm{\gamma}^{(i)},\bm{b}^{(i)},\bm{p_{u}}^{(i)},\bm{p_{s}}^{(i)},\bm{f_{u}}^{(i)},\bm{f_{s}}^{(i)},T^{(i)}$.
Repeat the above operations until the solutions converge. In the following
section, we consider using the alternating optimization method (AO) to tackle
the complex problem (7).

Roadmap of the whole algorithm. First, we decompose the outer fractional
structure of the original TCR problem using the Dinkelbach algorithm and
sequentially optimize $\bm{x},\bm{\varphi},\bm{\gamma}$ and
$\bm{b},\bm{p_{u}},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}}$ using the Alternating
Optimization (AO) method. In the first step of AO, we fix
$\bm{b},\bm{p_{u}},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}}$ and optimize
$\bm{x},\bm{\varphi},\bm{\gamma},T$. We transform the optimization problem in
the first step of AO into a Quadratically Constrained Quadratic Program (QCQP)
and solve it using Semidefinite Relaxation (SDR) and the Hungarian algorithm.
In the second step of AO, we fix $\bm{x},\bm{\varphi},\bm{\gamma}$ and
optimize $\bm{b},\bm{p_{u}},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}},T$. During the
optimization in the second step of AO, we propose a new fractional programming
method to transform this non-convex problem into a convex one. Finally, we
calculate $y$ based on the obtained solutions and repeat the aforementioned
process until $y$ converges. In this algorithm, since we utilize Dinkelbach’s
algorithm, alternating optimization, semidefinite relaxation, Hungarian
algorithm, and fractional programming, we refer to this algorithm as the DASHF
Algorithm.

##  III Our proposed AO technique to solve the optimization problem

Assuming that $y$ is given, we need to optimize
$\bm{x},\bm{\varphi},\bm{\gamma},\bm{b},\bm{p_{u}},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}},T$.
In the outermost loops, we iteratively optimize $y$; In the innermost loops,
we iteratively optimize
$\bm{x},\bm{\varphi},\bm{\gamma},\bm{b},\bm{p_{u}},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}},T$.
However, it is still difficult to optimize them in parallel. Thus, we consider
operating two inner AO steps to solve it. At the $i$-th iteration,

  1. 1.

Optimize $\bm{x},\bm{\varphi},\bm{\gamma},T$, given
$\bm{b},\bm{p_{u}},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}}$. Assuming that
$\bm{b}^{(i-1)},\bm{p_{u}}^{(i-1)},\bm{p_{s}}^{(i-1)},\bm{f_{u}}^{(i-1)},\bm{f_{s}}^{(i-1)},y^{(i-1)}$
are given, we optimize
$\bm{x}^{(i)},\bm{\varphi}^{(i)},\bm{\gamma}^{(i)},T^{(i)}$.

  2. 2.

Optimize $\bm{b},\bm{p_{u}},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}},T$, given
$\bm{x},\bm{\varphi},\bm{\gamma}$. Assuming that
$\bm{x}^{(i-1)},\bm{\varphi}^{(i-1)},\bm{\gamma}^{(i-1)},y^{(i-1)}$ are given,
we optimize
$\bm{b}^{(i)},\bm{p_{u}}^{(i)},\bm{p_{s}}^{(i)},\bm{f_{u}}^{(i)},\bm{f_{s}}^{(i)},T^{(i)}$.

###  III-A Optimizing $\bm{x},\bm{\varphi},\bm{\gamma},T$, given
$\bm{b},\bm{p_{u}},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}}$

Given $\bm{b},\bm{p_{u}},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}}$, we optimize
$\bm{x},\bm{\varphi},\bm{\gamma},T$. The optimization problem will be:

$\displaystyle\\!\\!\\!\max\limits_{\bm{x},\bm{\varphi},\bm{\gamma},T}\\!\\!\\!\\!$
$\displaystyle\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}\\!\\!\\!\\!\\!\\![x_{n,m}v_{n,m}\\!-\\!y\omega_{e}(E_{u,n,m}\\!+\\!E_{s,n,m})]-y\omega_{t}T$
(8) s.t.
$\displaystyle(\text{\ref{x_constr1}}),(\text{\ref{x_constr2}}),(\text{\ref{varphi_constr}}),(\text{\ref{gamma_constr}}),(\text{\ref{x_b_constr}}),(\text{\ref{x_ps_constr}}),(\text{\ref{x_fs_constr}}),(\text{\ref{Ts_constr}}),(\text{\ref{Tu_constr}}).$

Since $x_{n,m}$ are binary and other variables are not discrete, the above
belongs to mixed-integer nonlinear programming. We rewrite
$x_{n,m}\in\\{0,1\\}$ as $x_{n,m}(x_{n,m}-1)=0$. The optimization problem will
be rewritten as:

$\displaystyle\\!\\!\\!\\!\max\limits_{\bm{x},\bm{\varphi},\bm{\gamma},T}\\!\\!\\!\\!\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}\\!\\!\\!\\!\\!\\!\\!\\![x_{n,m}v_{n,m}\\!-\\!y\omega_{e}(E_{u,n,m}\\!+\\!E_{s,n,m})]-y\omega_{t}T$
(9) $\displaystyle\text{s.t.}\quad x_{n,m}(x_{n,m}-1)=0,\forall
n\in\mathcal{N},m\in\mathcal{M}$ (9a)
$\displaystyle\quad\quad(\text{\ref{x_constr2}}),(\text{\ref{varphi_constr}}),(\text{\ref{gamma_constr}}),(\text{\ref{x_b_constr}}),(\text{\ref{x_ps_constr}}),(\text{\ref{x_fs_constr}}),(\text{\ref{Ts_constr}}),(\text{\ref{Tu_constr}}).$

Using the energy expressions of Section II-B2, we can express the term
$E_{u,n,m}+E_{s,n,m}$ in (9) as

$\textstyle\sum\limits_{n\in\mathcal{N}}\kappa_{n}d_{n}f_{n}F^{2}_{n}+\sum\limits_{n\in\mathcal{N}}(\omega_{p}-1)\varphi_{n}\kappa_{n}d_{n}f_{n}F^{2}_{n}+$
$\textstyle\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}\\{x_{n,m}\varphi_{n}d_{n}(\frac{p_{n}}{r_{n,m}}+\frac{p_{m}\omega_{p}}{r_{m,n}})+$
$\textstyle\kappa_{m}x_{n,m}\varphi_{n}d_{n}f_{m}F^{2}_{n,m}[\gamma^{2}_{n,m}+\omega_{b}(1-\gamma_{n,m})^{2}]\\},$
(10)

where $\sum_{n\in\mathcal{N}}\kappa_{n}d_{n}f_{n}F^{2}_{n}$ is a constant.
Since $\bm{b},\bm{p_{u}},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}}$ are given,
$v_{n,m}$ defined in Eq. (6) is also a constant. Therefore, the objective
function in (9) can be simplified as (11) below:

$\displaystyle\\!\max\limits_{\bm{x},\bm{\varphi},\bm{\gamma},T}-y\omega_{t}T+\\!\\!\\!\\!\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}\\!\\!\\!x_{n,m}v_{n,m}-y\omega_{e}\bigg{\\{}\\!\\!\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}\\!\\!\\!\\!\\{x_{n,m}\varphi_{n}$
$\displaystyle\quad\quad
d_{n}(\frac{p_{n}}{r_{n,m}}\\!+\\!\frac{p_{m}\omega_{p}}{r_{m,n}})\\!+\\!\kappa_{m}x_{n,m}\varphi_{n}d_{n}f_{m}F^{2}_{n,m}[\gamma^{2}_{n,m}\\!+\\!\omega_{b}(1\\!-\\!$
$\displaystyle\quad\quad\gamma_{n,m})^{2}]\\}+\sum\limits_{n\in\mathcal{N}}\\!(\omega_{p}\\!-\\!1)\varphi_{n}\kappa_{n}d_{n}f_{n}F^{2}_{n}\bigg{\\}}$
(11)
$\displaystyle\text{s.t.}~{}(\text{\ref{x_constr1_new}}),(\text{\ref{x_constr2}}),(\text{\ref{varphi_constr}}),(\text{\ref{gamma_constr}}),(\text{\ref{x_b_constr}}),(\text{\ref{x_ps_constr}}),(\text{\ref{x_fs_constr}}),(\text{\ref{Ts_constr}}),(\text{\ref{Tu_constr}}).$

Next, we convert the maximization problem in (11) to the following
minimization problem:

$\displaystyle\min\limits_{\bm{x},\bm{\varphi},\bm{\gamma},T}\quad
y\omega_{t}T+y\omega_{e}\bigg{\\{}\sum\limits_{n\in\mathcal{N}}(\omega_{p}-1)\varphi_{n}\kappa_{n}d_{n}f_{n}F^{2}_{n}$
$\displaystyle\quad\quad\quad\quad+\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}\\{x_{n,m}\varphi_{n}d_{n}(\frac{p_{n}}{r_{n,m}}+\frac{p_{m}\omega_{p}}{r_{m,n}})$
$\displaystyle\quad\quad\quad\quad-\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}x_{n,m}v_{n,m}+\kappa_{m}x_{n,m}\varphi_{n}d_{n}f_{m}F^{2}_{n,m}[\gamma^{2}_{n,m}$
$\displaystyle\quad\quad\quad\quad+\omega_{b}(1-\gamma_{n,m})^{2}]\\}\bigg{\\}}$
(12)
$\displaystyle\text{s.t.}\quad(\text{\ref{x_constr1_new}}),(\text{\ref{x_constr2}}),(\text{\ref{varphi_constr}}),(\text{\ref{gamma_constr}}),(\text{\ref{x_b_constr}}),(\text{\ref{x_ps_constr}}),(\text{\ref{x_fs_constr}}),(\text{\ref{Ts_constr}}),(\text{\ref{Tu_constr}}).$

The objective function in (12) is not jointly convex with respect to
($x_{n,m},\varphi_{n},\gamma_{n,m}$). Its hessian determinant:
$x_{n,m}\varphi_{n}(\gamma^{2}_{n,m}+\omega_{b}(1-\gamma_{n,m})^{2})(8\gamma_{n,m}-2(\omega_{b}(1-\gamma_{n,m})^{2}+\gamma^{2}_{n,m}))$,
which is not always non-negative. Therefore,
$x_{n,m}\varphi_{n}(\gamma^{2}_{n,m}+\omega_{b}(1-\gamma_{n,m})^{2})$ is not
jointly convex with respect to ($x_{n,m},\varphi_{n},\gamma_{n,m}$).

It seems difficult to use the matrix lifting technique to solve this quartic
problem by dividing it into several quadratic problems. Then, we seek to find
another method to solve it. Let

$\textstyle A_{n}:=y\omega_{e}(\omega_{p}-1)\kappa_{n}d_{n}f_{n}F^{2}_{n},$
(13) $\textstyle
B_{n,m}\\!\\!:=y\omega_{e}d_{n}(\frac{p_{n}}{r_{n,m}}+\frac{p_{m}\omega_{p}}{r_{m,n}})+\omega_{b}y\omega_{e}\kappa_{m}d_{n}f_{m}F^{2}_{n,m},$
(14) $\textstyle
C_{n,m}:=-2\omega_{b}y\omega_{e}\kappa_{m}d_{n}f_{m}F^{2}_{n,m},$ (15)
$\textstyle
D_{n,m}:=(\omega_{b}+1)y\omega_{e}\kappa_{m}d_{n}f_{m}F^{2}_{n,m}.$ (16)

Then we define $\mathbf{A}:=[A_{n}]|_{n\in\mathcal{N}}$,
$\mathbf{B}:=[B_{n,m}]|_{n\in\mathcal{N},m\in\mathcal{M}}$,
$\mathbf{C}:=[C_{n,m}]|_{n\in\mathcal{N},m\in\mathcal{M}}$, and
$\mathbf{D}:=[D_{n,m}]|_{n\in\mathcal{N},m\in\mathcal{M}}$. We turn to its
equivalence problem (17):

$\displaystyle\min\limits_{\bm{x},\bm{\varphi},\bm{\gamma},T}y\omega_{t}T+\sum\limits_{n\in\mathcal{N}}A_{n}\varphi_{n}+\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}(-x_{n,m}v_{n,m}$
$\displaystyle\quad+B_{n,m}x_{n,m}\varphi_{n}+C_{n,m}x_{n,m}\varphi_{n}\gamma_{n,m}+D_{n,m}x_{n,m}\varphi_{n}\gamma_{n,m}^{2})$
(17)
$\displaystyle\text{s.t.}~{}(\text{\ref{x_constr1_new}}),(\text{\ref{x_constr2}}),(\text{\ref{varphi_constr}}),(\text{\ref{gamma_constr}}),(\text{\ref{x_b_constr}}),(\text{\ref{x_ps_constr}}),(\text{\ref{x_fs_constr}}),(\text{\ref{Ts_constr}}),(\text{\ref{Tu_constr}}).$

For the polynomial
$C_{n,m}x_{n,m}\varphi_{n}\gamma_{n,m}+D_{n,m}x_{n,m}\varphi_{n}\gamma_{n,m}^{2}$
in the objective function of (17), we can rewrite it as
$x_{n,m}\varphi_{n}(C_{n,m}\gamma_{n,m}+D_{n,m}\gamma_{n,m}^{2})$, where
$C_{n,m}<0$ and $D_{n,m}>0$. It means that
$C_{n,m}\gamma_{n,m}+D_{n,m}\gamma_{n,m}^{2}$ is convex, and the minimization
point can be obtained when
$\gamma_{n,m}=\frac{-C_{n,m}}{2D_{n,m}}=\frac{\omega_{b}}{\omega_{b}+1}\in(0,1)$,
which satisfies the constraints of $\gamma_{n,m}$. In $T_{s,n,m}$, the terms
$T^{(p_{1})}_{n,m}$, $T^{(g)}_{n,m}$, and $T^{(v)}_{n,m}$ are related to
$\gamma_{n,m}$. Since $T^{(v)}_{n,m}$ is generally much smaller than
$T^{(p_{1})}_{n,m}$ and $T^{(g)}_{n,m}$, we only focus on $T^{(p_{1})}_{n,m}$
and $T^{(g)}_{n,m}$ here. It’s easy to know that when
$\gamma_{n,m}=\frac{1}{1+\omega_{b}}$, $T^{(p_{1})}_{n,m}+T^{(g)}_{n,m}$ takes
the minimum value according to basic inequality. Following are the detailed
steps:

$\textstyle T^{(p_{1})}_{n,m}+T^{(g)}_{n,m}\geq
2\sqrt{T^{(p_{1})}_{n,m}T^{(g)}_{n,m}},$ (18)

where if and only if $T^{(p_{1})}_{n,m}=T^{(g)}_{n,m}$, “$=$” can be obtained.

$\textstyle\quad\quad T^{(p_{1})}_{n,m}=T^{(g)}_{n,m},$
$\textstyle\Rightarrow\frac{x_{n,m}\varphi_{n}d_{n}f_{m}}{\gamma_{n,m}F_{n,m}}=\frac{x_{n,m}\varphi_{n}d_{n}\omega_{b}f_{m}}{(1-\gamma_{n,m})F_{n,m}},$
$\textstyle\Rightarrow\gamma_{n,m}=\frac{1}{1+\omega_{b}}.$ (19)

We set $\omega_{b}=1$ and then
$\frac{\omega_{b}}{1+\omega_{b}}=\frac{1}{1+\omega_{b}}$, in which case, the
original optimization problem would take the maximum value. Plug in the
minimum value of $\gamma_{n,m}$ and the optimization problem can be rewritten
as

$\displaystyle\mathcal{P}_{1}:$
$\displaystyle\min\limits_{\bm{x},\bm{\varphi},T}\quad
y\omega_{t}T+\sum\limits_{n\in\mathcal{N}}A_{n}\varphi_{n}+\\!\\!\\!\\!\\!\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}\\!\\!\\!\\!\\!-x_{n,m}v_{n,m}+$
$\displaystyle\quad\quad\quad\quad(\frac{-C_{n,m}^{2}}{4D_{n,m}}+B_{n,m})x_{n,m}\varphi_{n}$
(20)
$\displaystyle\text{s.t.}\quad(\text{\ref{x_constr1_new}}),(\text{\ref{x_constr2}}),(\text{\ref{varphi_constr}}),(\text{\ref{x_b_constr}}),(\text{\ref{x_ps_constr}}),(\text{\ref{x_fs_constr}}),(\text{\ref{Ts_constr}}),(\text{\ref{Tu_constr}}).$

This is a quadratically constrained quadratic program (QCQP). Then, we need to
get the standard form of the QCQP problem. First, we need to combine
$\bm{\varphi}$ and $\bm{x}$ to define a new vector variable. Therefore, we
define the new vector variable as

$\textstyle\bm{Q}=(\bm{\varphi}^{\intercal},\bm{x_{1}}^{\intercal},\cdots,\bm{x_{M}}^{\intercal})^{\intercal},$
(21)

where $\bm{\varphi}=(\varphi_{1},\cdots,\varphi_{N})^{\intercal}$ and
$\bm{x_{m}}=(x_{1,m},\cdots,x_{N,m})^{\intercal},\forall m\in\mathcal{M}$. To
make the problem transformation more clearly, we define some auxiliary vectors
and matrices as follows:

$\textstyle\bm{e}_{i}=(0,\cdots,\underset{\underset{i\text{-th}}{\uparrow}}{1},\cdots,0)^{\intercal}_{NM+N\times
1},$ (22)
$\textstyle\bm{e}_{i,j}=(\bm{e}_{i},\cdots,\bm{e}_{j})^{\intercal},i<j,$ (23)
$\textstyle\bm{I}_{NM+N\times N}=(\bm{I}_{N},\bm{0}_{N\times
NM+N})^{\intercal},$ (24) $\textstyle\bm{I}_{N\rightarrow
NM}=(\bm{I}_{N},\cdots,\bm{I}_{N})_{N\times NM},$ (25)
$\textstyle\bm{e}_{N+1\times N+NM}=(\bm{0}_{NM\times N},\bm{I}_{NM}),$ (26)
$\textstyle\bm{e}_{\overline{i}}=(0,\cdots,\underset{\underset{i\text{-th}}{\uparrow}}{1},\cdots,\underset{\underset{(i+N)\text{-th}}{\uparrow}}{1},\cdots,\underset{\underset{(i+N(M-1))\text{-th}}{\uparrow}}{1},\cdots,0)^{\intercal},$
(27)

where $\bm{I}_{N}$ is the identity matrix of order $N$ and $\bm{I}_{NM}$ is
the identity matrix of order $NM$. We also define

$\textstyle G_{n,m}=\frac{-C_{n,m}^{2}}{4D_{n,m}}+B_{n,m},$ (28)
$\textstyle\bm{G}=(G_{1,1},\cdots,G_{N,M})^{\intercal}.$ (29)

Thus, we can obtain that

$\textstyle\sum_{n\in\mathcal{N},m\in\mathcal{M}}(\frac{-C_{n,m}^{2}}{4D_{n,m}}+B_{n,m})x_{n,m}\varphi_{n}$
$\textstyle=\bm{Q}^{\intercal}\bm{I}_{NM+N\times N}I_{N\rightarrow
NM}\text{diag}(\bm{G})\bm{e}_{N+1\times N+NM}\bm{Q}.$ (30)

Let

$\mathbf{P}_{0}=\bm{I}_{NM+N\times N}I_{N\rightarrow
NM}\text{diag}(\bm{G})\bm{e}_{N+1\times N+NM}.$ (31)

Similarly, we also know that

$\textstyle\sum_{n\in\mathcal{N},m\in\mathcal{M}}-x_{n,m}v_{n,m}=\mathbf{W}_{0}^{\intercal}\bm{Q},$
(32)

where

$\textstyle\mathbf{W}_{0}^{\intercal}=(-\bm{v}_{1}^{\intercal},\cdots,-\bm{v}_{M}^{\intercal})\bm{e}_{N+1,NM+N},$
(33)

and $\bm{v}_{i}^{\intercal}=(v_{1,i},\cdots,v_{N,i}),\forall i\in\mathcal{M}$.
In addition, $\sum_{n\in\mathcal{N},m\in\mathcal{M}}A_{n}\varphi_{n}$ can be
expressed as $\mathbf{W}_{1}^{\intercal}\bm{Q}$, where
$\mathbf{W}_{1}^{\intercal}=(A_{1},\cdots,A_{N})\bm{e}_{1,N}$. Let

$\textstyle
P^{(\text{Ts})}_{1,n,m}=\frac{d_{n}}{r_{n,m}}+\frac{d_{n}f_{m}}{\gamma_{n,m}F_{n,m}}+\frac{d_{n}\omega_{b}f_{m}}{(1-\gamma_{n,m})F_{n,m}},$
(34) $\textstyle
P^{(\text{Ts})}_{2}=\frac{S_{b}}{R_{m}}+\mathop{\max}_{m{{}^{\prime}}\in\mathcal{M}\setminus\\{m\\}}\frac{f_{v}}{(1-\gamma_{n,m^{\prime}})F_{n,m^{\prime}}},$
(35) $\textstyle P^{(\text{Tu})}_{1,n,m}=\frac{d_{n}\omega_{p}}{r_{m,n}},$
(36) $\textstyle
P^{(\text{Tu})}_{2,n}=\frac{d_{n}\omega_{p}f_{n}-d_{n}f_{n}}{F_{n}},$ (37)
$\textstyle P^{(\text{Tu})}_{3}=\frac{d_{n}f_{n}}{F_{n}}.$ (38)

Furthermore, let
$\mathbf{P}^{(\text{Ts})}_{3}=[P^{(\text{Ts})}_{1,n,m}]|_{n\in\mathcal{N},m\in\mathcal{M}}$,

$\textstyle\mathbf{P}^{(\text{Ts})}_{1}=\bm{I}_{NM+N\times N}I_{N\rightarrow
NM}\text{diag}(\mathbf{P}^{(\text{Ts})}_{3})\bm{e}_{N+1\times N+NM},$ (39)

$\mathbf{P}^{(\text{Tu})}_{4}=[P^{(\text{Tu})}_{1,n,m}]|_{n\in\mathcal{N},m\in\mathcal{M}}$,
and

$\textstyle\mathbf{P}^{(\text{Tu})}_{1}=\bm{I}_{NM+N\times N}I_{N\rightarrow
NM}\text{diag}(\mathbf{P}^{(\text{Tu})}_{4})\bm{e}_{N+1\times N+NM}.$ (40)

Let $\mathbf{P}^{(\text{Tu})}_{5}=[P^{(\text{Tu})}_{2,n}]|_{n\in\mathcal{N}}$
and
${\mathbf{P}^{(\text{Tu})}_{2}}^{\intercal}={\mathbf{P}^{(\text{Tu})}_{5}}^{\intercal}\bm{e}_{1,N}$.
Based on the above, the optimization problem can be expressed as

$\displaystyle\mathcal{P}_{2}:$
$\displaystyle\min\limits_{\bm{Q},T}\quad\bm{Q}^{\intercal}\mathbf{P}_{0}\bm{Q}+\mathbf{W}_{0}^{\intercal}\bm{Q}+\mathbf{W}_{1}^{\intercal}\bm{Q}+y\omega_{t}T$
(41) s.t.
$\displaystyle\text{diag}(\bm{e}_{N+1,NM+N}^{\intercal}\bm{Q})(\text{diag}(\bm{e}_{N+1,NM+N}^{\intercal}\bm{Q})-\mathbf{I})=\mathbf{0}$
(41a)
$\displaystyle\text{diag}(\bm{e}_{\overline{1},\overline{M}}^{\intercal}\bm{e}_{N+1,NM+N}^{\intercal}\bm{Q})=\mathbf{I}$
(41b) $\displaystyle\text{diag}(\bm{e}_{1,N}^{\intercal}\bm{Q})\leq\mathbf{I}$
(41c) $\displaystyle\text{diag}(\bm{e}_{1,N}^{\intercal}\bm{Q})\geq\mathbf{0}$
(41d)
$\displaystyle\bm{B}^{\intercal}\bm{e}_{N+1,NM+N}^{\intercal}\bm{Q}-B_{max}\leq
0$ (41e)
$\displaystyle\bm{P}^{\intercal}\bm{e}_{N+1,NM+N}^{\intercal}\bm{Q}-P_{max}^{(m)}\leq
0$ (41f)
$\displaystyle\bm{F}^{\intercal}\bm{e}_{N+1,NM+N}^{\intercal}\bm{Q}-F_{max}^{(m)}\leq
0$ (41g)
$\displaystyle\bm{Q}^{\intercal}\mathbf{P}^{(\text{Ts})}_{1}\bm{Q}+P^{(\text{Ts})}_{2}\preceq
T$ (41h)
$\displaystyle\bm{Q}^{\intercal}\mathbf{P}^{(\text{Tu})}_{1}\bm{Q}+{\mathbf{P}^{(\text{Tu})}_{2}}^{\intercal}\bm{Q}+P^{(\text{Tu})}_{3}\preceq
T,$ (41i)

where $\bm{e}_{\overline{i},\overline{j}}$ denotes
$(\bm{e}_{\overline{i}},\cdots,\bm{e}_{\overline{j}})^{\intercal},i<j$,
$\bm{B}=(b_{1,1},\cdots,b_{N,M})^{\intercal}$,
$\bm{P}=(p_{1,1},\cdots,p_{N,M})^{\intercal}$, and
$\bm{F}=(F_{1,1},\cdots,F_{N,M})^{\intercal}$. The constraints
$(\text{\ref{x_constr1_new}})$, $(\text{\ref{x_constr2}})$,
$(\text{\ref{varphi_constr}})$, $(\text{\ref{x_b_constr}})$,
$(\text{\ref{x_ps_constr}})$, $(\text{\ref{x_fs_constr}})$,
$(\text{\ref{Ts_constr}})$, $(\text{\ref{Tu_constr}})$ in Problem
$\mathcal{P}_{1}$ are transformed into the constraints
$(\text{\ref{x_constr1_qcqp}})$, $(\text{\ref{x_constr2_qcqp}})$,
$\\{(\text{\ref{varphi_constr1_qcqp}}),(\text{\ref{varphi_constr2_qcqp}})\\}$,
$(\text{\ref{x_b_constr_qcqp}})$, $(\text{\ref{x_ps_constr_qcqp}})$,
$(\text{\ref{x_fs_constr_qcqp}})$, $(\text{\ref{Ts_constr_qcqp}})$,
$(\text{\ref{Tu_constr_qcqp}})$ in Problem $\mathcal{P}_{2}$, respectively.
Problem (41) is the standard QCQP form. However, it is still non-convex. Then,
we need to utilize the semidefinite programming (SDP) method to transform this
QCQP problem into a semidefinite relaxation (SDR) problem. Let
$\mathbf{S}=(\bm{Q}^{\intercal},1)^{\intercal}(\bm{Q}^{\intercal},1)$. Let
$\bm{e}_{i\rightarrow j}$ denotes
$(0,\cdots,\underset{\underset{i\text{-th}}{\uparrow}}{1},1,\cdots,\underset{\underset{j\text{-th}}{\uparrow}}{1},0,\cdots,0)^{\intercal},i<j$.
Then we obtain the SDR problem

$\displaystyle\mathcal{P}_{3}:\min\limits_{\bm{S},T}\quad$
$\displaystyle\text{Tr}(\mathbf{P}_{1}\mathbf{S})$ (42) s.t.
$\displaystyle\text{Tr}(\mathbf{P}_{2}\mathbf{S})=0$ (42a)
$\displaystyle\text{Tr}(\mathbf{P}_{3}\mathbf{S})=0$ (42b)
$\displaystyle\text{Tr}(\mathbf{P}_{4}\mathbf{S})\leq 0$ (42c)
$\displaystyle\text{Tr}(\mathbf{P}_{5}\mathbf{S})\leq 0$ (42d)
$\displaystyle\text{Tr}(\mathbf{P}_{6}\mathbf{S})\leq 0$ (42e)
$\displaystyle\text{Tr}(\mathbf{P}_{7}\mathbf{S})\leq 0$ (42f)
$\displaystyle\text{Tr}(\mathbf{P}_{8}\mathbf{S})\preceq T$ (42g)
$\displaystyle\text{Tr}(\mathbf{P}_{9}\mathbf{S})\preceq T$ (42h)
$\displaystyle\mathbf{S}\succeq 0,$ (42i)

where

$\mathbf{P}_{1}=\left(\begin{array}[]{cc}\mathbf{P}_{0}&\frac{1}{2}(\mathbf{W}_{0}+\mathbf{W}_{1})\\\
\frac{1}{2}(\mathbf{W}_{0}+\mathbf{W}_{1})^{\intercal}&y\omega_{t}T\end{array}\right),$
$\mathbf{P}_{2}=\left(\begin{array}[]{cc}\bm{e}_{i}^{\intercal}\bm{e}_{i}&-\frac{1}{2}\bm{e}_{i}\\\
-\frac{1}{2}\bm{e}_{i}^{\intercal}&0\end{array}\right),\forall
i\in\\{1,\cdots,NM\\}$
$\displaystyle\mathbf{P}_{3}=\left(\begin{array}[]{cc}\mathbf{0}_{NM+N\times
NM+N}&\frac{1}{2}(\bm{e}_{\overline{i}}\bm{e}_{N+1,NM+N}^{\intercal})\\\
\frac{1}{2}(\bm{e}_{\overline{i}}\bm{e}_{N+1,NM+N}^{\intercal})^{\intercal}&-1\end{array}\right),$
(45) $\displaystyle\forall i\in\\{1,\cdots,N\\}$
$\mathbf{P}_{4}=\left(\begin{array}[]{cc}\mathbf{0}_{NM+N\times
NM+N}&\frac{1}{2}\bm{e}_{i}\\\
\frac{1}{2}\bm{e}_{i}^{\intercal}&-1\end{array}\right),\forall
i\in\\{1,\cdots,N\\}$
$\mathbf{P}_{5}=\left(\begin{array}[]{cc}\mathbf{0}_{NM+N\times
NM+N}&\frac{1}{2}\bm{B}\bm{e}_{N+1,NM+N}\\\
\frac{1}{2}(\bm{B}\bm{e}_{N+1,NM+N})^{\intercal}&-B_{max}\end{array}\right),$
$\mathbf{P}_{6}=\left(\begin{array}[]{cc}\mathbf{0}_{NM+N\times
NM+N}&\frac{1}{2}\bm{P}\bm{e}_{N+1,NM+N}\\\
\frac{1}{2}(\bm{P}\bm{e}_{N+1,NM+N})^{\intercal}&-P_{max}^{(m)}\end{array}\right),$
$\mathbf{P}_{7}=\left(\begin{array}[]{cc}\mathbf{0}_{NM+N\times
NM+N}&\frac{1}{2}\bm{F}\bm{e}_{N+1,NM+N}\\\
\frac{1}{2}(\bm{F}\bm{e}_{N+1,NM+N})^{\intercal}&-F_{max}^{(m)}\end{array}\right),$
$\mathbf{P}_{8}=\left(\begin{array}[]{cc}\mathbf{P}_{1}^{(\text{Ts})}&\mathbf{0}_{NM+N\times
1}\\\ \mathbf{0}_{1\times NM+N}&P_{2}^{(\text{Ts})}\end{array}\right),$
$\mathbf{P}_{9}=\left(\begin{array}[]{cc}\mathbf{P}_{1}^{(\text{Tu})}&\frac{1}{2}\bm{P}_{2}^{(\text{Tu})}\\\
\frac{1}{2}{\bm{P}_{2}^{(\text{Tu})}}^{\intercal}&P_{3}^{(\text{Tu})}\end{array}\right).$

1

2Initialize $i\leftarrow-1$ and for all $n\in\mathcal{N},m\in\mathcal{M}$:
$\bm{x}^{(0)}=(\bm{e_{1}},\cdots,\bm{e_{M}})^{\intercal}$,
$\varphi_{n}^{(0)}=0.5$, $\gamma_{n,m}=\frac{\omega_{b}}{\omega_{b}+1}$;

3Calculate
$y^{(0)},T^{(0)},\mathbf{A}^{(0)},\mathbf{B}^{(0)},\mathbf{C}^{(0)},\mathbf{D}^{(0)},\mathbf{G}^{(0)},\mathbf{P}^{(0)}_{1},\mathbf{P}^{(0)}_{2},\mathbf{P}^{(0)}_{3},\mathbf{P}^{(0)}_{4},\mathbf{P}^{(0)}_{5},\mathbf{P}^{(0)}_{6},\mathbf{P}^{(0)}_{7},\mathbf{P}^{(0)}_{8},\mathbf{P}^{(0)}_{9}$
with $\bm{x}^{(0)}=(\bm{e_{1}},\cdots,\bm{e_{M}})^{\intercal}$,
$\varphi_{n}^{(0)}=0.5$, $\gamma_{n,m}=\frac{\omega_{b}}{\omega_{b}+1}$;

4repeat

5

6 Let $i\leftarrow i+1$;

7 Initialize $j=-1$,
$[\bm{x}^{(i,0)},\bm{\varphi}^{(i,0)}]\leftarrow[\bm{x}^{(i)},\bm{\varphi}^{(i)}]$;

8 Initialize
$[y^{(i,0)},T^{(i,0)},\mathbf{A}^{(i,0)},\mathbf{B}^{(i,0)},\mathbf{C}^{(i,0)},\mathbf{D}^{(i,0)},\mathbf{G}^{(i,0)},\mathbf{P}^{(i,0)}_{1},\mathbf{P}^{(i,0)}_{2},\mathbf{P}^{(i,0)}_{3},\mathbf{P}^{(i,0)}_{4},\mathbf{P}^{(i,0)}_{5},\mathbf{P}^{(i,0)}_{6},\mathbf{P}^{(i,0)}_{7},\mathbf{P}^{(i,0)}_{8},\mathbf{P}^{(i,0)}_{9}]\leftarrow[y^{(i)},T^{(i)}\mathbf{A}^{(i)},\mathbf{B}^{(i)},\mathbf{C}^{(i)},\mathbf{D}^{(i)},\mathbf{G}^{(i)},\mathbf{P}^{(i)}_{1},\mathbf{P}^{(i)}_{2},\mathbf{P}^{(i)}_{3},\mathbf{P}^{(i)}_{4},\mathbf{P}^{(i)}_{5},\mathbf{P}^{(i)}_{6},\mathbf{P}^{(i)}_{7},\mathbf{P}^{(i)}_{8},\mathbf{P}^{(i)}_{9}]$

9 repeat

10 Let $j\leftarrow j+1$;

11 Obtain $[\bm{x}^{(i,j+1)},\bm{\varphi}^{(i,j+1)}]$ of continuous values by
solving the SDR Problem (42);

12 Update
$[y^{(i,j+1)},T^{(i,j+1)},\mathbf{A}^{(i,j+1)},\mathbf{B}^{(i,j+1)},\mathbf{C}^{(i,j+1)},\mathbf{D}^{(i,j+1)},\mathbf{G}^{(i,j+1)},\mathbf{P}^{(i,j+1)}_{1},\mathbf{P}^{(i,j+1)}_{2},\mathbf{P}^{(i,j+1)}_{3},\mathbf{P}^{(i,j+1)}_{4},\mathbf{P}^{(i,j+1)}_{5},$
$\mathbf{P}^{(i,j+1)}_{6},\mathbf{P}^{(i,j+1)}_{7},\mathbf{P}^{(i,j+1)}_{8},\mathbf{P}^{(i,j+1)}_{9}]$;

13 until  _the relative difference between
$V_{\text{SDR}}(\bm{x}^{(i,j+1)},\bm{\varphi}^{(i,j+1)})$ and
$V_{\text{SDR}}(\bm{x}^{(i,j)},\bm{\varphi}^{(i,j)})$ is no greater than
$\epsilon_{1}$ for a small positive number $\epsilon_{1}$ (i.e.,
$\frac{V_{\text{SDR}}(\bm{x}^{(i,j+1)},\bm{\varphi}^{(i,j+1)})}{V_{\text{SDR}}(\bm{x}^{(i,j)},\bm{\varphi}^{(i,j)})}-1\leq\epsilon_{1}$)_;

14 Denote $\bm{x}^{(i,j+1)},\bm{\varphi}^{(i,j+1)}$ as a solution to the SDR
Problem (42);

15 Find all user $n$ that $\sum\limits_{m\in\mathcal{M}}x_{n,m}>1$. For these
users, modify $x_{n,m}$ as
$\frac{x_{n,m}}{|\sum\limits_{m\in\mathcal{M}}x_{n,m}|}$. Use the Hungarian
algorithm with augmented zero vectors to find the best matching with the
maximum edge weight. Denote this matching as a set $\mathcal{X}_{matching}$.
For nodes $n$ and $m$ in $\mathcal{X}_{matching}$, let $x_{n,m}=1$, else
$x_{n,m}=0$, and denote this integer association result as
$\bm{x}^{(i,j+1)}_{\sharp}$.

16 Initialize $l=-1$,
$[\bm{x}^{(i,j+1,0)},\bm{\varphi}^{(i,j+1,0)}]\leftarrow[\bm{x}^{(i,j+1)}_{\sharp},\bm{\varphi}^{(i,j+1)}]$;

17 Initialize
$[y^{(i,j+1,0)},T^{(i,j+1,0)},\mathbf{A}^{(i,j+1,0)},\mathbf{B}^{(i,j+1,0)},\mathbf{C}^{(i,j+1,0)},\mathbf{D}^{(i,j+1,0)},\mathbf{G}^{(i,j+1,0)}]\leftarrow[y^{(i,j+1)},\mathbf{A}^{(i,j+1)},\mathbf{B}^{(i,j+1)},\mathbf{C}^{(i,j+1)},\mathbf{D}^{(i,j+1)},\mathbf{G}^{(i,j+1)}]$;

18 repeat

19 Let $l\leftarrow l+1$;

20 Obtain $\bm{\varphi}^{(i,j+1,l+1)}$ by solving Problem (20) with
$\bm{x}^{(i,j+1,0)}$, and denote $\bm{\varphi}^{(i,j+1,l+1)}$ as a solution to
Problem (42);

21 Update
$[y^{(i,j+1,l+1)},T^{(i,j+1,l+1)},\mathbf{A}^{(i,j+1,l+1)},\mathbf{B}^{(i,j+1,l+1)},\mathbf{C}^{(i,j+1,l+1)},\mathbf{D}^{(i,j+1,l+1)},\mathbf{G}^{(i,j+1,l+1)}]$;

22 until  _the relative difference between
$V_{\text{Eq(\ref{prob7})}}(\bm{\varphi}^{(i,j+1,l+1)})$ and
$V_{\text{Eq(\ref{prob7})}}(\bm{\varphi}^{(i,j+1,l)})$ is no greater than
$\epsilon_{2}$ for a small positive number $\epsilon_{2}$ (i.e.,
$\frac{V_{\text{Eq(\ref{prob7})}}(\bm{\varphi}^{(i,j+1,l+1)})}{V_{\text{Eq(\ref{prob7})}}(\bm{\varphi}^{(i,j+1,l)})}-1\leq\epsilon_{2}$)_;

23 Set
$[\bm{x}^{i+1},\bm{\varphi}^{i+1}]\leftarrow[\bm{x}^{(i,j+1,0)},\bm{\varphi}^{(i,j+1,l+1)}]$;

24until  _the relative difference between
$V_{\text{AO-P1}}(\bm{x}^{(i+1)},\bm{\varphi}^{(i+1)})$ and
$V_{\text{AO-P1}}(\bm{x}^{(i)},\bm{\varphi}^{(i)})$ is no greater than
$\epsilon_{3}$ for a small positive number $\epsilon_{3}$ (i.e.,
$\frac{V_{\text{AO-P1}}(\bm{x}^{(i+1)},\bm{\varphi}^{(i+1)})}{V_{\text{AO-P1}}(\bm{x}^{(i)},\bm{\varphi}^{(i)})}-1\leq\epsilon_{3}$)_;

25Calculate $T^{(i+1)}$ based on $[\bm{x}^{(i+1)},\bm{\varphi}^{(i+1)}]$;

Return $[\bm{x}^{(i+1)},\bm{\varphi}^{(i+1)},T^{(i+1)}]$ as a solution
$[\bm{x}^{\star},\bm{\varphi}^{\star},T^{\star}]$ to Problem AO-Part1.

Algorithm 1 AO-Part1: Optimizing $\bm{x},\bm{\varphi},T$, Given
$\bm{b},\bm{p_{u}},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}}$.

The constraints $(\text{\ref{x_constr1_qcqp}})$,
$(\text{\ref{x_constr2_qcqp}})$, $(\text{\ref{varphi_constr1_qcqp}})$,
$(\text{\ref{varphi_constr2_qcqp}})$, $(\text{\ref{x_b_constr_qcqp}})$,
$(\text{\ref{x_ps_constr_qcqp}})$, $(\text{\ref{x_fs_constr_qcqp}})$,
$(\text{\ref{Ts_constr_qcqp}})$, $(\text{\ref{Tu_constr_qcqp}})$ in
$\mathcal{P}_{2}$ are transformed into the constraints
$(\text{\ref{x_constr1_sdr}})$, $(\text{\ref{x_constr2_sdr}})$,
$(\text{\ref{varphi_constr_sdr}})$, $(\text{\ref{S_constr_sdr}})$,
$(\text{\ref{x_b_constr_sdr}})$, $(\text{\ref{x_ps_constr_sdr}})$,
$(\text{\ref{x_fs_constr_sdr}})$, $(\text{\ref{Ts_constr_sdr}})$,
$(\text{\ref{Tu_constr_sdr}})$ in $\mathcal{P}_{3}$, respectively. We drop the
constraint $\text{rank}(\mathbf{S})=1$, and the objective function and the
constraints are all convex. Then, this SDR problem will be solved in
polynomial time by common convex solvers. By solving this SDR problem, we can
get a continuous solution of $\bm{Q}$. However, this solution is the lower
bound of the optimal solution, and it may not guarantee the constraint
$\text{rank}(\mathbf{S})=1$. Therefore, we need to use rounding techniques to
recover the solution. The latter $NM$ elements in $\bm{Q}$ are $x_{n,m}$, for
all $n\in\mathcal{N},m\in\mathcal{M}$, which means that user $n$ is fractional
connected to server $m$. Then, we find all user $n$ that
$\sum\limits_{m\in\mathcal{M}}x_{n,m}>1$. For these users, we modify $x_{n,m}$
as $\frac{x_{n,m}}{|\sum\limits_{m\in\mathcal{M}}x_{n,m}|}$. By using the
Hungarian algorithm [20] with augmented zero vectors, we find the best
matching with the maximum weight and denote this matching as a set
$\mathcal{X}_{matching}$. For nodes $n$ and $m$ in $\mathcal{X}_{matching}$,
let $x_{n,m}=1$, else $x_{n,m}=0$, and denote this integer association result
as $\bm{x}_{\sharp}$. Then, substitute $\bm{x}_{\sharp}$ into Problem (41) to
obtain the optimal $\bm{\varphi}$. The process of Algorithm AO-Part 1 is
presented in Algorithm 1.

###  III-B Optimizing $\bm{b},\bm{p_{u}},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}},T$,
given $\bm{x},\bm{\varphi},\bm{\gamma}$

Given $\bm{x},\bm{\varphi},\bm{\gamma}$, the remaining optimization problem is

$\displaystyle\max\limits_{\bm{b},\bm{p_{u}},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}},T}\quad\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}x_{n,m}v_{n,m}-y\bigg{\\{}\omega_{t}T$
$\displaystyle+\omega_{e}\big{\\{}\sum\limits_{n\in\mathcal{N}}[\kappa_{n}(1-\varphi_{n})d_{n}f_{n}F^{2}_{n}+\kappa_{n}\varphi_{n}d_{n}\omega_{p}f_{n}F_{n}^{2}]$
$\displaystyle+\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}\\{p_{n}\frac{x_{n,m}\varphi_{n}d_{n}}{r_{n,m}}+\kappa_{m}x_{n,m}\varphi_{n}d_{n}f_{m}(\gamma_{n,m}F_{n,m})^{2}$
$\displaystyle+\kappa_{m}x_{n,m}\varphi_{n}d_{n}\omega_{b}f_{m}[(1-\gamma_{n,m})F_{n,m}]^{2}+p_{m}\frac{x_{n,m}\varphi_{n}d_{n}\omega_{p}}{r_{m,n}}\\}\big{\\}}\bigg{\\}}$
(46)
$\displaystyle\text{s.t.}\quad(\text{\ref{x_b_constr}}),(\text{\ref{pu_constr}}),(\text{\ref{x_ps_constr}}),(\text{\ref{fu_constr}}),(\text{\ref{x_fs_constr}}),(\text{\ref{Ts_constr}}),(\text{\ref{Tu_constr}}).$

We define

$\textstyle\mathcal{F}(\bm{b},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}},T):=\\!\\!\\!\\!\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}\\!\\!x_{n,m}v_{n,m}$
$\textstyle-y\omega_{e}\big{\\{}\sum\limits_{n\in\mathcal{N}}[\kappa_{n}(1-\varphi_{n})d_{n}f_{n}F^{2}_{n}+\kappa_{n}\varphi_{n}d_{n}\omega_{p}f_{n}F_{n}^{2}]$
$\textstyle+\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}\\!\\!\\!\\!\\!\\{\kappa_{m}x_{n,m}\varphi_{n}d_{n}f_{m}(\gamma_{n,m}F_{n,m})^{2}$
$\textstyle+\kappa_{m}x_{n,m}\varphi_{n}d_{n}\omega_{b}f_{m}[(1-\gamma_{n,m})F_{n,m}]^{2}\\}\big{\\}}-y\omega_{t}T.$
(47)

It’s easy to justify that
$\mathcal{F}(\bm{b},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}},T)$ is concave. Then, the
optimization objective function is

$\textstyle\mathcal{F}(\bm{b},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}},T)\\!-\\!y\omega_{e}\\!\\!\\!\\!\\!\\!\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}\\!\\!\\!\\!\\!\\!\\!(\frac{p_{n}x_{n,m}\varphi_{n}d_{n}}{r_{n,m}}\\!+\\!\frac{p_{m}x_{n,m}\varphi_{n}d_{n}\omega_{p}}{r_{m,n}}),$
(48)

where
$\frac{p_{n}x_{n,m}\varphi_{n}d_{n}}{r_{n,m}}+\frac{p_{m}x_{n,m}\varphi_{n}d_{n}\omega_{p}}{r_{m,n}}$
is non-convex or concave. Then, according to the fractional programming
technique introduced in Section IV in [2], we let
$z_{1,n,m}=\frac{1}{2p_{n}x_{n,m}\varphi_{n}d_{n}r_{n,m}}$ and
$z_{2,n,m}=\frac{1}{2p_{m}x_{n,m}\varphi_{n}d_{n}\omega_{p}r_{m,n}}$. The
optimization objective function can be expressed as

$\textstyle\mathcal{F}(\bm{b},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}},T)-y\omega_{e}\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}[(p_{n}x_{n,m}\varphi_{n}d_{n})^{2}z_{1,n,m}$
$\textstyle+\frac{1}{4(b_{n,m}\log_{2}(1+\frac{g_{n,m}p_{n}}{\sigma^{2}b_{n,m}}))^{2}z_{1,n,m}}+(p_{m}x_{n,m}\varphi_{n}d_{n}\omega_{p})^{2}z_{2,n,m}$
$\textstyle+\frac{1}{4(b_{m,n}\log_{2}(1+\frac{g_{m,n}p_{m}}{\sigma^{2}b_{m,n}}))^{2}z_{2,n,m}}]$
(49)

The transformed optimization problem is shown as (50):

$\displaystyle\max\limits_{\bm{b},\bm{p_{u}},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}},T}\text{The
expression in (\ref{fp_trans_prob}) above}$ (50) $\displaystyle\quad\text{s.t.
}(\text{\ref{x_b_constr}}),(\text{\ref{pu_constr}}),(\text{\ref{x_ps_constr}}),(\text{\ref{fu_constr}}),(\text{\ref{x_fs_constr}}),(\text{\ref{Ts_constr}}),(\text{\ref{Tu_constr}}).$

If $\bm{z}_{1}=(z_{1,1,1},\cdots,z_{1,n,m})^{\intercal}$ and
$\bm{z}_{2}=(z_{2,1,1},\cdots,z_{2,n,m})^{\intercal}$ is given, the objective
function (50) is concave. At the $i$-th iteration, $[y^{(i)}$,
$\bm{z}_{1}^{(i)}$, $\bm{z}_{2}^{(i)}]$ are first calculated with the solution
$[\bm{b}^{(i-1)},\bm{p_{u}}^{(i-1)},\bm{p_{s}}^{(i-1)},\bm{f_{u}}^{(i-1)},\bm{f_{s}}^{(i-1)}]$.
Then,
$[\bm{b}^{(i)},\bm{p_{u}}^{(i)},\bm{p_{s}}^{(i)},\bm{f_{u}}^{(i)},\bm{f_{s}}^{(i)}]$
can be obtained by solving the concave problem (50) with
$[y^{(i)},\bm{z}_{1}^{(i)},\bm{z}_{2}^{(i)}]$. Thus, the optimization problem
is concave and can be solved by common convex solvers. Denote the value of the
objective function is $V_{\text{AO-Part2}}$. The process of FP is presented in
Algorithm 2. The whole process of the DASHF algorithm is presented in
Algorithm 3.

####  III-B1 Prove that the maximization optimization of (46) is equal to the
maximization optimization of (50)

Let $\mathit{M}_{1}(p_{n})=p_{n}x_{n,m}\varphi_{n}d_{n}$,
$\mathit{M}_{2}(p_{m})=p_{m}x_{n,m}\varphi_{n}d_{n}\omega_{p}$,
$\mathit{N}_{1}(b_{n,m},p_{n})=r_{n,m}$, and
$\mathit{N}_{2}(b_{n,m},p_{m})=r_{m,n}$. We know that $\mathit{M}_{1}(p_{n})$
and $\mathit{M}_{2}(p_{m})$ are convex of $p_{n}$ and $p_{m}$,
$\mathit{N}_{1}(b_{n,m},p_{n})$ and $\mathit{N}_{2}(b_{n,m},p_{m})$ are
jointly concave of $(b_{n,m},p_{n})$ and $(b_{n,m},p_{m})$. Let

$\textstyle\mathit{J}(\bm{b},\bm{p_{u}},\bm{p_{s}})=$
$\textstyle{\mathit{M}_{1}(p_{n})}^{2}z_{1,n,m}+\frac{1}{4{\mathit{N}_{1}(b_{n,m},p_{n})}^{2}z_{1,n,m}}+$
$\textstyle{\mathit{M}_{2}(p_{m})}^{2}z_{2,n,m}+\frac{1}{4{\mathit{N}_{2}(b_{n,m},p_{m})}^{2}z_{2,n,m}}.$
(51)

$\frac{p_{n}x_{n,m}\varphi_{n}d_{n}}{r_{n,m}}+\frac{p_{m}x_{n,m}\varphi_{n}d_{n}\omega_{p}}{r_{m,n}}$
can be expressed as

$\textstyle\frac{\mathit{M}_{1}(p_{n})}{\mathit{N}_{1}(b_{n,m},p_{n})}+\frac{\mathit{M}_{2}(p_{m})}{\mathit{N}_{2}(b_{n,m},p_{m})}.$
(52)

See the partial derivative of $\bm{b}$ of (III-B1) and (52).

$\textstyle\frac{\partial(\frac{\mathit{M}_{1}(p_{n})}{\mathit{N}_{1}(b_{n,m},p_{n})}+\frac{\mathit{M}_{2}(p_{m})}{\mathit{N}_{2}(b_{n,m},p_{m})})}{\partial\bm{b}}=$
$\textstyle-\frac{\mathit{M}_{1}(p_{n})}{{\mathit{N}_{1}(b_{n,m},p_{n})}^{2}}\frac{\partial\mathit{N}_{1}(b_{n,m},p_{n})}{\partial\bm{b}}-\frac{\mathit{M}_{2}(p_{m})}{{\mathit{N}_{2}(b_{n,m},p_{m})}^{2}}\frac{\partial\mathit{N}_{2}(b_{n,m},p_{m})}{\partial\bm{b}}.$
(53)
$\textstyle\frac{\partial\mathit{J}(\bm{b},\bm{p_{u}},\bm{p_{s}})}{\partial\bm{b}}=-\frac{1}{2z_{1,n,m}{\mathit{N}_{1}(b_{n,m},p_{n})}^{3}}\frac{\partial\mathit{N}_{1}(b_{n,m},p_{n})}{\partial\bm{b}}-$
$\textstyle\frac{1}{2z_{2,n,m}{\mathit{N}_{2}(b_{n,m},p_{m})}^{3}}\frac{\partial\mathit{N}_{2}(b_{n,m},p_{m})}{\partial\bm{b}}.$
(54)

From Eq. (III-B1) and Eq. (III-B1), we can know that

$\textstyle(\frac{\partial\mathit{J}(\bm{b},\bm{p_{u}},\bm{p_{s}})}{\partial\bm{b}})$
$\textstyle|_{z_{1,n,m}=\frac{1}{2\mathit{M}_{1}(p_{n})\mathit{N}_{1}(b_{n,m},p_{n})},}$
${}_{z_{2,n,m}=\frac{1}{2\mathit{M}_{2}(p_{m})\mathit{N}_{2}(b_{n,m},p_{m})}}$
$\textstyle=\frac{\partial(\frac{\mathit{M}_{1}(p_{n})}{\mathit{N}_{1}(b_{n,m},p_{n})}+\frac{\mathit{M}_{2}(p_{m})}{\mathit{N}_{2}(b_{n,m},p_{m})})}{\partial\bm{b}}.$
(55)

By repeating the same steps mentioned above, we can obtain

$\textstyle(\frac{\partial\mathit{J}(\bm{b},\bm{p_{u}},\bm{p_{s}})}{\partial\bm{p_{u}}})$
$\textstyle|_{z_{1,n,m}=\frac{1}{2\mathit{M}_{1}(p_{n})\mathit{N}_{1}(b_{n,m},p_{n})},}$
${}_{z_{2,n,m}=\frac{1}{2\mathit{M}_{2}(p_{m})\mathit{N}_{2}(b_{n,m},p_{m})}}$
$\textstyle=\frac{\partial(\frac{\mathit{M}_{1}(p_{n})}{\mathit{N}_{1}(b_{n,m},p_{n})}+\frac{\mathit{M}_{2}(p_{m})}{\mathit{N}_{2}(b_{n,m},p_{m})})}{\partial\bm{p_{u}}},$
(56)
$\textstyle(\frac{\partial\mathit{J}(\bm{b},\bm{p_{u}},\bm{p_{s}})}{\partial\bm{p_{s}}})$
$\textstyle|_{z_{1,n,m}=\frac{1}{2\mathit{M}_{1}(p_{n})\mathit{N}_{1}(b_{n,m},p_{n})},}$
${}_{z_{2,n,m}=\frac{1}{2\mathit{M}_{2}(p_{m})\mathit{N}_{2}(b_{n,m},p_{m})}}$
$\textstyle=\frac{\partial(\frac{\mathit{M}_{1}(p_{n})}{\mathit{N}_{1}(b_{n,m},p_{n})}+\frac{\mathit{M}_{2}(p_{m})}{\mathit{N}_{2}(b_{n,m},p_{m})})}{\partial\bm{p_{s}}}.$
(57)

Furthermore, we can know that $\frac{\partial
Eq.(\ref{fp_ori_prob})}{\partial(\bm{b},\bm{p_{u}},\bm{p_{s}})}=\frac{\partial
Eq.(\ref{fp_trans_prob})}{\partial(\bm{b},\bm{p_{u}},\bm{p_{s}})}$ and Eq.
(48) = Eq. (49) when
$z_{1,n,m}=\frac{1}{2\mathit{M}_{1}(p_{n})\mathit{N}_{1}(b_{n,m},p_{n})}$,
$z_{2,n,m}=\frac{1}{2\mathit{M}_{2}(p_{m})\mathit{N}_{2}(b_{n,m},p_{m})}$. The
value of the AO procedure of Problem (46) is non-decreasing, which means that
at $i$ and $i+1$ iterations,

$\textstyle V_{\text{AO-
Part2}}(\bm{b}^{(i+1)}\\!,\\!\bm{f_{s}}^{(i+1)}\\!,\\!\bm{f_{u}}^{(i+1)}\\!,\\!\bm{p_{s}}^{(i+1)}\\!,\\!\bm{p_{u}}^{(i+1)}\\!,\\!T^{(i+1)}\\!,\\!\bm{z}_{1}^{(i+1)}\\!,\\!\bm{z}_{2}^{(i+1)})$
$\textstyle\geq V_{\text{AO-
Part2}}(\bm{b}^{(i)},\bm{f_{s}}^{(i)},\bm{f_{u}}^{(i)},\bm{p_{s}}^{(i)},\bm{p_{u}}^{(i)},T^{(i)},\bm{z}_{1}^{(i+1)},\bm{z}_{2}^{(i+1)})$
$\textstyle\geq V_{\text{AO-
Part2}}(\bm{b}^{(i)},\bm{f_{s}}^{(i)},\bm{f_{u}}^{(i)},\bm{p_{s}}^{(i)},\bm{p_{u}}^{(i)},T^{(i)},\bm{z}_{1}^{(i)},\bm{z}_{2}^{(i)}).$
(58)

As $i\\!\\!\rightarrow\\!\\!\infty$, $V_{\text{AO-
Part2}}(\bm{b}^{(i)}\\!,\bm{f_{s}}^{(i)}\\!,\bm{f_{u}}^{(i)}\\!,\bm{p_{s}}^{(i)}\\!,\bm{p_{u}}^{(i)}\\!,T^{(i)}\\!,\bm{z}_{1}^{(i)}\\!,\bm{z}_{2}^{(i)})$
converges to $V_{\text{AO-
Part2}}(\bm{b}^{(\star)}\\!,\\!\bm{f_{s}}^{(\star)}\\!,\\!\bm{f_{u}}^{(\star)}\\!,\\!\bm{p_{s}}^{(\star)}\\!,\\!\bm{p_{u}}^{(\star)}\\!,\\!T^{(\star)}\\!,\\!\bm{z}_{1}^{(\star)}\\!,\\!\bm{z}_{2}^{(\star)}\\!)$.
With
$\bm{z}_{1}^{\star}\\!\\!=\\!\\!\frac{1}{2\mathit{M}_{1}(\bm{p_{u}}^{\star})\mathit{N}_{1}(\bm{b}^{\star},\bm{p_{u}}^{\star})}$
and
$\bm{z}_{2}^{\star}\\!\\!=\\!\\!\frac{1}{2\mathit{M}_{2}(\bm{p_{s}}^{\star})\mathit{N}_{2}(\bm{b}^{\star},\bm{p_{s}}^{\star})}$,
the solution of (50) is
$(\bm{b}^{(\star)},\bm{f_{s}}^{(\star)},\bm{f_{u}}^{(\star)},\bm{p_{s}}^{(\star)},\bm{p_{u}}^{(\star)},T^{(\star)})$,
which means that
$(\bm{b}^{(\star)},\bm{f_{s}}^{(\star)},\bm{f_{u}}^{(\star)},\bm{p_{s}}^{(\star)},\bm{p_{u}}^{(\star)},T^{(\star)})$
is a stationary point of (46).

1

2Initialize $i\leftarrow-1$ and for all $n\in\mathcal{N},m\in\mathcal{M}$:
$b_{n,m}^{(0)}=\frac{b_{\max}}{N}$, $p_{n}^{(0)}=p_{n}$,
$p_{m}^{(0)}=\frac{p_{\max}^{(m)}}{N}$, $F_{n}^{(0)}=F_{max}^{(n)}$,
$F_{n,m}^{(0)}=\frac{F_{n,m}^{(m)}}{N}$;

3$T^{(0)}\leftarrow\max_{n\in\mathcal{N},m\in\mathcal{M}}(T_{u,n,m}^{(0)},T_{serever}^{(0)})$;

4$y^{(0)}\leftarrow\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}\frac{x_{n,m}^{(0)}v_{n,m}}{\omega_{t}T^{(0)}+\omega_{e}(E_{u,n,m}^{(0)}+E_{s,n,m}^{(0)})}$;

5$z_{1,n,m}^{(0)}\leftarrow\frac{1}{2p_{n}^{(0)}b_{n,m}^{(0)}\log_{2}(1+\frac{g_{n,m}p_{n}^{(0)}}{\delta^{2}b_{n,m}^{(0)}})}$;

6$z_{2,n,m}^{(0)}\leftarrow\frac{1}{2p_{m}^{(0)}b_{n,m}^{(0)}\log_{2}(1+\frac{g_{n,m}p_{m}^{(0)}}{\delta^{2}b_{n,m}^{(0)}})}$;

7repeat

8

9 Let $i\leftarrow i+1$;

10 Obtain
$[\bm{b}^{(i+1)},\bm{p_{u}}^{(i+1)},\bm{p_{s}}^{(i+1)},\bm{f_{u}}^{(i+1)},\bm{f_{s}}^{(i+1)},T^{(i+1)}]$
by solving Problem (50) with $[y^{(i)},z_{1,n,m}^{(i)},z_{2,n,m}^{(i)}]$;

11 Update $y^{(i+1)}$, $z_{1,n,m}^{(i+1)}$, and $z_{2,n,m}^{(i+1)}$ with the
solution
$[\bm{b}^{(i+1)},\bm{p_{u}}^{(i+1)},\bm{p_{s}}^{(i+1)},\bm{f_{u}}^{(i+1)},\bm{f_{s}}^{(i+1)},T^{(i+1)}]$;

12
$y^{(i+1)}\leftarrow\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}\frac{x_{n,m}^{(i+1)}v_{n,m}}{\omega_{t}T^{(i+1)}+\omega_{e}(E_{u,n,m}^{(i+1)}+E_{s,n,m}^{(i+1)})}$;

13
$z_{1,n,m}^{(i+1)}\leftarrow\frac{1}{2p_{n}^{(i+1)}b_{n,m}^{(i+1)}\log_{2}(1+\frac{g_{n,m}p_{n}^{(i+1)}}{\delta^{2}b_{n,m}^{(i+1)}})}$;

14
$z_{2,n,m}^{(i+1)}\leftarrow\frac{1}{2p_{m}^{(i+1)}b_{n,m}^{(i+1)}\log_{2}(1+\frac{g_{n,m}p_{m}^{(i+1)}}{\delta^{2}b_{n,m}^{(i+1)}})}$;

15until  _the relative difference between
$V_{\text{AO-P2}}(\bm{b}^{(i+1)},\bm{p_{u}}^{(i+1)},\bm{p_{s}}^{(i+1)},\bm{f_{u}}^{(i+1)},\bm{f_{s}}^{(i+1)})$
and
$V_{\text{AO-P2}}(\bm{b}^{(i)},\bm{p_{u}}^{(i)},\bm{p_{s}}^{(i)},\bm{f_{u}}^{(i)},\bm{f_{s}}^{(i)})$
is no greater than $\epsilon$ for a small positive number $\epsilon$ (i.e.,
$\frac{V_{\text{AO-P2}}(\bm{b}^{(i+1)},\bm{p_{u}}^{(i+1)},\bm{p_{s}}^{(i+1)},\bm{f_{u}}^{(i+1)},\bm{f_{s}}^{(i+1)})}{V_{\text{AO-P2}}(\bm{b}^{(i)},\bm{p_{u}}^{(i)},\bm{p_{s}}^{(i)},\bm{f_{u}}^{(i)},\bm{f_{s}}^{(i)})}-1\leq\epsilon$)_;

Return
$[\bm{b}^{(i+1)},\bm{p_{u}}^{(i+1)},\bm{p_{s}}^{(i+1)},\bm{f_{u}}^{(i+1)},\bm{f_{s}}^{(i+1)}]$
as a solution
$[\bm{b}^{\star},\bm{p_{u}}^{\star},\bm{p_{s}}^{\star},\bm{f_{u}}^{\star},\bm{f_{s}}^{\star}]$
to Problem AO-Part2.

Algorithm 2 AO-Part2: Optimizing $\bm{b}$, $\bm{p_{u}}$, $\bm{p_{s}}$,
$\bm{f_{u}}$, $\bm{f_{s}}$, $T$, Given $\bm{w}$, $\bm{\varphi}$,
$\bm{\gamma}$.

1

2Initialize $i\leftarrow-1$ and for all $n\in\mathcal{N},m\in\mathcal{M}$:
$b_{n}^{(0)}=\frac{b_{\max}}{N}$, $p_{n}^{(0)}=p_{n}$,
$p_{m}^{(0)}=\frac{p_{\max}^{(m)}}{N}$, $F_{n}^{(0)}=F_{max}^{(n)}$,
$F_{n,m}^{(0)}=\frac{F_{n,m}^{(m)}}{N}$,
$\bm{x}^{(0)}=(\bm{e_{1}},\cdots,\bm{e_{M}})^{\intercal}$,
$\varphi_{n}^{(0)}=0.5$, $\gamma_{n,m}=\frac{\omega_{b}}{\omega_{b}+1}$;

3$y^{(0)}\leftarrow\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}\frac{x_{n,m}^{(0)}v_{n,m}}{\omega_{t}T^{(0)}+\omega_{e}(E_{u,n,m}^{(0)}+E_{s,n,m}^{(0)})}$;

4repeat

5

6 Let $i\leftarrow i+1$;

7 Obtain $[\bm{x}^{(i+1)},\bm{\varphi}^{(i+1)}]$ by using Algorithm 1;

8 Obtain
$[\bm{b}^{(i+1)},\bm{p_{u}}^{(i+1)},\bm{p_{s}}^{(i+1)},\bm{f_{u}}^{(i+1)},\bm{f_{s}}^{(i+1)}]$
by using Algorithm 2;

9 Update
$y^{(i+1)}\leftarrow\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}\frac{x_{n,m}^{(i+1)}v_{n,m}}{\omega_{t}T^{(i+1)}+\omega_{e}(E_{u,n,m}^{(i+1)}+E_{s,n,m}^{(i+1)})}$
with
$[\bm{x}^{(i+1)},\bm{\varphi}^{(i+1)},\bm{b}^{(i+1)},\bm{p_{u}}^{(i+1)},\bm{p_{s}}^{(i+1)},\bm{f_{u}}^{(i+1)},\bm{f_{s}}^{(i+1)}]$;

10until  _the relative difference between $y^{(i+1)}$ and $y^{(i)}$ is no
greater than $\epsilon$ for a small positive number $\epsilon$ (i.e.,
$\frac{y^{(i+1)}}{y^{(i)}}-1\leq\epsilon$)_;

11Set
$[\bm{x}^{(i+1)},\bm{\varphi}^{(i+1)},\bm{b}^{(i+1)},\bm{p_{u}}^{(i+1)},\bm{p_{s}}^{(i+1)},\bm{f_{u}}^{(i+1)},\bm{f_{s}}^{(i+1)}]$
as a solution to Problem (7);

Return
$[\bm{x}^{(i+1)},\bm{\varphi}^{(i+1)},\bm{b}^{(i+1)},\bm{p_{u}}^{(i+1)},\bm{p_{s}}^{(i+1)},\bm{f_{u}}^{(i+1)},\bm{f_{s}}^{(i+1)}]$
as a solution
$[\bm{x}^{\star},\bm{\varphi}^{\star},\bm{b}^{\star},\bm{p_{u}}^{\star},\bm{p_{s}}^{\star},\bm{f_{u}}^{\star},\bm{f_{s}}^{\star}]$
to Problem (7).

Algorithm 3 The Whole Process of The DASHF Algorithm.

###  III-C Complexity Analysis of the Whole Algorithm

In Algorithm 1, at each iteration, there are $a_{1}=N(M+2)+1$ variables and
$b_{1}=N(4M+4)+3$ constraints. The complexity of the Hungarian algorithm is
$\mathcal{O}(N^{3})$. The worst-case complexity of Algorithm 1 is
$\mathcal{O}(N^{3}+(a_{1}^{2}b_{1}+a_{1}^{3})b_{1}^{0.5}\log(\frac{1}{\epsilon_{1}}))$
with a given solution accuracy $\epsilon_{1}>0$ [20]. In Algorithm 2, at each
iteration, there are $a_{2}=N(3M+2)+1$ variables and $b_{2}=N(5M+4)+4$
constraints. The worst-case complexity of Algorithm 2 is
$\mathcal{O}(a_{2}^{2}b_{2}+a_{2}^{3})b_{2}^{0.5}\log(\frac{1}{\epsilon_{2}})$
with a given solution accuracy $\epsilon_{2}>0$. To summarize, if Algorithm 3
takes $\mathcal{I}$ iterations, the whole complexity is
$\mathcal{I}\mathcal{O}(N^{3}+(a_{1}^{2}b_{1}+a_{1}^{3})b_{1}^{0.5}\log(\frac{1}{\epsilon_{1}})+(a_{2}^{2}b_{2}+a_{2}^{3})b_{2}^{0.5}\log(\frac{1}{\epsilon_{2}}))$.

##  IV Numerical Results

In this section, we first introduce the default settings for the numerical
simulations. Subsequently, we verify the convergence of the proposed DASHF
algorithm and compare it with other baselines to validate its effectiveness.
We adjust the available communication and computational resources and the cost
weight parameters to analyze their impacts on the TCR.

###  IV-A Default settings

We consider a network topology of 1000 m $\times$ 1000 m with 20 mobile users
and three servers in Fig. 3. The large-scale fading $h_{n,m}$ between the user
$n$ and server $m$ is modeled as $128.1+37.6\log_{10}d_{n,m}$[20], where
$d_{n,m}$ denotes the Euclidean distance between the user $n$ and server $m$.
The small-scale fading is the Rayleigh fading. Gaussian noise power
$\sigma^{2}$ is $-134$dBm. The total bandwidth for each server $b_{max}$ is 10
MHz. The maximum transmit power of mobile users $p_{max}^{(n)}$ is 0.2 W. The
maximum transmit power of servers $p_{max}^{(m)}$ is 10 W. The maximum
computation capacity of mobile users $F_{max}^{(n)}$ is 1 GHz, and that of
servers $F_{max}^{(m)}$ is 20 GHz. The CPU clock cycles per bit for processing
data of mobile users and servers ($f_{n}$ and $f_{m}$) are 279.62 [19]. The
CPU clock cycles per bit for generating blocks of servers ($f_{m}$ at the
blockchain phase) are 737.5 [19]. The effective switched capacitance of mobile
users and servers ($\kappa_{n}$ and $\kappa_{m}$) is $10^{-27}$ to be the
energy consumption in a reasonable range. The data size change ratio between
the raw data and processed data $\omega_{p}$ is 0.9. The data sizes of mobile
users are randomly selected from $[500\text{KB},2000\text{KB}]$. To achieve
this, we generate pseudorandom values, which follow a standard uniform
distribution over the open interval $(0,1)$. These pseudorandom values are
then scaled to the range of $[500\text{KB},2000\text{KB}]$ to determine the
specific data sizes for each mobile user. The block size $S_{b}$ is 8 MB. The
data rate of wired links among servers $R_{m}$ is 15 Mbps (M bits/second). The
data size change ratio between the MEC task and the blockchain task
$\omega_{b}$ is 1. The parameters of delay and energy consumption
($\omega_{t}$ and $\omega_{e}$) are 0.5 and 0.5. We set $\varpi_{1}$ and
$\varpi_{2}$ as $\frac{100}{\ln 2}$ and $0.25$, respectively. We consider
using the Mosek optimization tool in Matlab to conduct the simulations.

Figure 3: 2D distribution of 3 servers and 20 users.

###  IV-B Convergence of proposed Algorithms

In this section, we evaluate the convergence of the proposed algorithms. We
consider three network topologies with $(N,M)=(10,2),(20,3),(30,4)$ and keep
other settings as default. Primal objective value means an estimate for the
primal objective value when using Mosek to solve the optimization problem.
When the primal objective value converges, the algorithm converges to one
stationary point. Fig. LABEL:fig:Convergence_of_AO-Part1 plots the convergence
of the Algorithm AO-Part 1, which converges within 15 iterations. Fig.
LABEL:fig:Convergence_of_AO-Part2 plots the convergence of the Algorithm AO-
Part 2, which converges within nine iterations. Fig.
LABEL:fig:Convergence_of_AO plots the convergence of the DASHF Algorithm,
which converges within nine iterations. Thus, the proposed DASHF Algorithm is
effective in finding one stationary point of the Problem (7). In Table II, we
have presented the running time of the proposed DASHF algorithm for those
three network topologies.

TABLE II: Running time of the proposed DASHF Algorithm N, M | 10, 2 | 20, 3 | 30, 4
---|---|---|---
time consumed | 3.07s | 26.94s | 1600.09s

###  IV-C Comparison with baselines

In this section, we mainly consider four baselines to carry out the comparison
experiments.

  1. 1.

Random user connection with average resource allocation (RUCAA). In this
algorithm, one server is randomly selected for each user. The server equally
allocates communication and computational resources among the users connected
to it.

  2. 2.

Greedy user connection with average resource allocation (GUCAA). In this
algorithm, each user selects the server with the least number of users
underserving. The server distributes communication and computational resources
evenly to the users connected to it.

  3. 3.

Average resource allocation with user connection optimization (AAUCO). In this
algorithm, the communication and computation resources of each Metaverse
server are equally allocated to each user connected to it. Besides, Algorithm
1 is leveraged to operate user connection optimization.

  4. 4.

Greedy user connection with resource allocation optimization (GUCRO). In this
algorithm, each user selects the Metaverse server with the least number of
users underserving. Besides, Algorithm 2 is leveraged to operate resource
optimization.

  5. 5.

Proposed DASHF algorithm. Joint optimization of user connection and resource
allocation by utilizing Algorithm 3.

In Fig. 6, we compare the resource consumption and TCR of the proposed DASHF
Algorithms with other baselines. The performances of RUCAA and GUCAA are worse
since no optimization is utilized. GUCRO and AAUCO have better performances
than RUCAA and GUCAA, which confirms the effectiveness of the proposed
Algorithm AO-Part 1 and Part 2. Furthermore, the TCR of GUCRO is higher than
that of AAUCO, which shows that resource optimization is more effective than
user connection optimization. The resource consumption of the proposed DASHF
algorithm is the lowest of these five methods, and the TCR is also the highest
one. This results from the benefits of joint optimization of user connection
and resource allocation.

Figure 6: Resource consumption and TCR of different baselines and proposed AO
Algorithm.

###  IV-D TCR versus resources

For communication and computation resources comparison experiments, we
consider the total bandwidth and the computing frequency of servers, which are
the dominant factors.

  1. 1.

TCR versus the total bandwidth. We consider the total bandwidth from 10 MHz to
100 MHz to test the TCR under different total bandwidths. Other parameters are
fixed as default settings. Fig. 7 reveals distinct algorithmic performance
trends, with the proposed DASHF method consistently outperforming GUCRO,
AAUCO, RUCAA, and GUCAA in terms of the TCR. Notably, optimization algorithms
(GUCRO and AAUCO) demonstrate superior performance compared to non-
optimization algorithms (RUCAA and GUCAA). While AAUCO employs user connection
optimization strategies and performs better than RUCAA and GUCAA, it lags
behind the resource optimization algorithms.

Figure 7: TCR under different total bandwidths.

  2. 2.

TCR versus the computation resource. We consider changing the servers’
computing frequencies from 20 GHz to 200 GHz to see the TCR trend under
different servers’ computing frequencies in Fig. 8. Other parameters are fixed
as default settings. Fig. 8 reveals an inverse relationship between the total
computing frequency and algorithm performance, where increasing the total
computing frequency leads to diminishing algorithm performance, as evidenced
by decreasing TCR values. Importantly, this phenomenon can be attributed to
the trade-off between computational resources and energy consumption. As
servers’ computing frequency increases with the expanded range, energy
consumption rises correspondingly. Given that the TCR primarily reflects the
ratio of computing frequency allocated to users relative to the total server
computing frequency, it tends to have trivial impacts under these conditions.
The proposed DASHF method consistently maintains its superior performance
across the computing frequency range, while GUCRO demonstrates little change.
Conversely, algorithms with no optimization, such as RUCAA and GUCAA, exhibit
diminishing returns with computing resource expansion, emphasizing the
significance of resource allocation strategies in computing resource-rich
environments. AAUCO, though effective, falls short of the performance levels
achieved by GUCRO and the proposed DASHF method.

Figure 8: TCR under different server computation frequencies.

###  IV-E Impact of cost weights on TCR

Fig. 9 featuring various combinations of ($\omega_{t}$, $\omega_{e}$) that
signifies the trade-off between delay-energy optimization and trust score. As
($\omega_{t}$, $\omega_{e}$) values shift, emphasizing either delay or energy,
distinct performance outcomes are evident. For instance, when prioritizing
energy efficiency (e.g., ($\omega_{t}$, $\omega_{e}$) = (0.1, 0.9)), the
system achieves lower energy consumption but at the expense of higher delay,
resulting in a moderate TCR value. Conversely, balanced settings (e.g.,
($\omega_{t}$, $\omega_{e}$) = (0.5, 0.5)) lead to lower delay and slightly
higher energy consumption, yielding a higher TCR value. These findings
underscore the sensitivity of the optimization process to parameter choices
and emphasize the importance of tailoring ($\omega_{t}$, $\omega_{e}$) values
to meet specific application requirements while carefully considering the
trade-offs between delay-energy and trust score.

Figure 9: Resources consumption and TCR under different
$(\omega_{t},\omega_{e})$.

##  V Conclusion

In this exploration of blockchain, the Metaverse, and NFT applications, we
have navigated the complexities of optimizing user connection and resource
allocation within 6G wireless communication systems. The Metaverse, once a
concept of science fiction, has become a digital reality where NFTs play a
transformative role. Our contributions, highlighted in this paper, pave the
way for NFT integration into the Metaverse, offering solutions to intricate
challenges. By introducing work offloading to servers, optimizing resource
allocation, and introducing the trust-cost ratio, we enhance trust mechanisms
and efficiency within this ecosystem. The proposed DASHF algorithm further
streamlines these optimizations. As we conclude, we envision a future where
the blockchain-empowered Metaverse seamlessly integrates NFTs, offering
unprecedented digital experiences while upholding trust and resource
efficiency.

## References

  * [1] L. Qian and J. Zhao, “User association and resource allocation in large language model based mobile edge computing system over wireless communications,” _accepted by IEEE Vehicular Technology Conference (VTC)_ , 2024. [Online]. Available: https://arxiv.org/pdf/2310.17872.pdf
  * [2] J. Zhao, L. Qian, and W. Yu, “Human-centric resource allocation in the Metaverse over wireless communications,” _IEEE Journal on Selected Areas in Communications (JSAC)_ , vol. 42, no. 3, pp. 514–537, 2024.
  * [3] C. Wang, C. Yu, and Y. Li, “Toward understanding attention economy in Metaverse: A case study of NFT value,” _IEEE Transactions on Computational Social Systems_ , 2022.
  * [4] W. Y. B. Lim, Z. Xiong, D. Niyato, X. Cao, C. Miao, S. Sun, and Q. Yang, “Realizing the Metaverse with edge intelligence: A match made in heaven,” _IEEE Wireless Communications_ , 2022.
  * [5] M. Xu, W. C. Ng, W. Y. B. Lim, J. Kang, Z. Xiong, D. Niyato, Q. Yang, X. S. Shen, and C. Miao, “A full dive into realizing the edge-enabled metaverse: Visions, enabling technologies, and challenges,” _IEEE Communications Surveys & Tutorials_, 2022.
  * [6] M. Xu, D. Niyato, B. Wright, H. Zhang, J. Kang, Z. Xiong, S. Mao, and Z. Han, “Epvisa: Efficient auction design for real-time physical-virtual synchronization in the human-centric metaverse,” _IEEE Journal on Selected Areas in Communications_ , 2023.
  * [7] D. Wu, Z. Yang, P. Zhang, R. Wang, B. Yang, and X. Ma, “Virtual-reality inter-promotion technology for Metaverse: A survey,” _IEEE Internet of Things Journal_ , 2023.
  * [8] Y. Wang, Z. Su, N. Zhang, R. Xing, D. Liu, T. H. Luan, and X. Shen, “A survey on Metaverse: Fundamentals, security, and privacy,” _IEEE Communications Surveys & Tutorials_, 2022.
  * [9] K. Christodoulou, L. Katelaris, M. Themistocleous, P. Christodoulou, and E. Iosif, “NFTs and the Metaverse revolution: Research perspectives and open challenges,” _Blockchains and the Token Economy: Theory and Practice_ , pp. 139–178, 2022.
  * [10] D. Chalmers, C. Fisch, R. Matthews, W. Quinn, and J. Recker, “Beyond the bubble: Will NFTs and digital proof of ownership empower creative industry entrepreneurs?” _Journal of Business Venturing Insights_ , vol. 17, p. e00309, 2022.
  * [11] R. Cheng, N. Wu, S. Chen, and B. Han, “Will Metaverse be NextG Internet? Vision, hype, and reality,” _IEEE Network_ , vol. 36, no. 5, pp. 197–204, 2022.
  * [12] S. Aggarwal, R. Chaudhary, G. S. Aujla, N. Kumar, K.-K. R. Choo, and A. Y. Zomaya, “Blockchain for smart communities: Applications, challenges and opportunities,” _Journal of Network and Computer Applications_ , vol. 144, pp. 13–48, 2019.
  * [13] Y. Lu, “The blockchain: State-of-the-art and research challenges,” _Journal of Industrial Information Integration_ , vol. 15, pp. 80–90, 2019.
  * [14] Q. Yang, Y. Zhao, H. Huang, Z. Xiong, J. Kang, and Z. Zheng, “Fusing blockchain and AI with Metaverse: A survey,” _IEEE Open Journal of the Computer Society_ , vol. 3, pp. 122–136, 2022.
  * [15] Y. Huang, Y. J. Li, and Z. Cai, “Security and privacy in Metaverse: A comprehensive survey,” _Big Data Mining and Analytics_ , vol. 6, no. 2, pp. 234–247, 2023.
  * [16] R. Di Pietro and S. Cresci, “Metaverse: Security and privacy issues,” in _IEEE International Conference on Trust, Privacy and Security in Intelligent Systems and Applications_. IEEE, 2021, pp. 281–288.
  * [17] G. Kang, J. Koo, and Y.-G. Kim, “Security and privacy requirements for the Metaverse: A Metaverse applications perspective,” _IEEE Communications Magazine_ , 2023.
  * [18] S. B. Far and A. I. Rad, “Applying digital twins in Metaverse: User interface, security and privacy challenges,” _Journal of Metaverse_ , vol. 2, no. 1, pp. 8–15, 2022.
  * [19] J. Feng, F. R. Yu, Q. Pei, J. Du, and L. Zhu, “Joint optimization of radio and computational resources allocation in blockchain-enabled mobile edge computing systems,” _IEEE Transactions on Wireless Communications_ , vol. 19, no. 6, pp. 4321–4334, 2020.
  * [20] Y. Dai, D. Xu, S. Maharjan, and Y. Zhang, “Joint computation offloading and user association in multi-task mobile edge computing,” _IEEE Transactions on Vehicular Technology_ , vol. 67, no. 12, pp. 12 313–12 325, 2018.
  * [21] F. Guo, F. R. Yu, H. Zhang, H. Ji, M. Liu, and V. C. Leung, “Adaptive resource allocation in future wireless networks with blockchain and mobile edge computing,” _IEEE Transactions on Wireless Communications_ , vol. 19, no. 3, pp. 1689–1703, 2019.
  * [22] W. Sun, J. Liu, Y. Yue, and P. Wang, “Joint resource allocation and incentive design for blockchain-based mobile edge computing,” _IEEE Transactions on Wireless Communications_ , vol. 19, no. 9, pp. 6050–6064, 2020.
  * [23] H. Xu, W. Huang, Y. Zhou, D. Yang, M. Li, and Z. Han, “Edge computing resource allocation for unmanned aerial vehicle assisted mobile network with blockchain applications,” _IEEE Transactions on Wireless Communications_ , vol. 20, no. 5, pp. 3107–3121, 2021.
  * [24] X. Jiang, F. R. Yu, T. Song, and V. C. Leung, “Intelligent resource allocation for video analytics in blockchain-enabled Internet of autonomous vehicles with edge computing,” _IEEE Internet of Things Journal_ , vol. 9, no. 16, pp. 14 260–14 272, 2020.
  * [25] Z. Tu, H. Zhou, K. Li, H. Song, and Y. Yang, “A blockchain-based trust and reputation model with dynamic evaluation mechanism for IoT,” _Computer Networks_ , vol. 218, p. 109404, 2022.
  * [26] Y. Liu, C. Zhang, Y. Yan, X. Zhou, Z. Tian, and J. Zhang, “A semi-centralized trust management model based on blockchain for data exchange in IoT system,” _IEEE Transactions on Services Computing_ , vol. 16, no. 2, pp. 858–871, 2022.
  * [27] J. Xi, G. Xu, S. Zou, Y. Lu, G. Li, J. Xu, and R. Wang, “A blockchain dynamic sharding scheme based on hidden Markov model in collaborative IoT,” _IEEE Internet of Things Journal_ , 2023.
  * [28] N. T. Hoa, B. D. Son, N. C. Luong, and D. Niyato, “Dynamic offloading for edge computing-assisted Metaverse systems,” _IEEE Communications Letters_ , 2023.
  * [29] “Monkey Kingdom.” [Online]. Available: https://monkeykingdom.io/
  * [30] Z. Yang, M. Chen, W. Saad, C. S. Hong, and M. Shikh-Bahaei, “Energy efficient federated learning over wireless communication networks,” _IEEE Transactions on Wireless Communications_ , vol. 20, no. 3, pp. 1935–1949, 2020.
  * [31] M. S. Al-Rakhami and M. Al-Mashari, “A blockchain-based trust model for the Internet of Things supply chain management,” _Sensors_ , vol. 21, no. 5, p. 1759, 2021.
  * [32] D. Yang, G. Xue, X. Fang, and J. Tang, “Incentive mechanisms for crowdsensing: Crowdsourcing with smartphones,” _IEEE/ACM Transactions on Networking_ , vol. 24, no. 3, pp. 1732–1744, 2015.
  * [33] W. Dinkelbach, “On nonlinear fractional programming,” _Management Science_ , vol. 13, no. 7, pp. 492–498, 1967.