# Structured Spectrum Allocation and User Association in Heterogeneous
Cellular Networks

Wei Bao and Ben Liang
Department of Electrical and Computer Engineering, University of Toronto,
Canada
Email: {wbao<EMAIL_ADDRESS>
###### Abstract

We study joint spectrum allocation and user association in heterogeneous
cellular networks with multiple tiers of base stations. A stochastic geometric
approach is applied as the basis to derive the average downlink user data rate
in a closed-form expression. Then, the expression is employed as the objective
function in jointly optimizing spectrum allocation and user association, which
is of non-convex programming in nature. A computationally efficient Structured
Spectrum Allocation and User Association (SSAUA) approach is proposed, solving
the optimization problem optimally when the density of users is low, and near-
optimally with a guaranteed performance bound when the density of users is
high. A Surcharge Pricing Scheme (SPS) is also presented, such that the
designed association bias values can be achieved in Nash equilibrium.
Simulations and numerical studies are conducted to validate the accuracy and
efficiency of the proposed SSAUA approach and SPS.

##  I Introduction

Traditional single-tiered macro-cellular networks provide wide coverage for
mobile user equipments (UEs), but they are insufficient to satisfy the
exploding demand driven by modern mobile traffic, such as multimedia
transmissions and cloud computing tasks. One efficient means to alleviate this
problem is to install a diverse set of small-cells (e.g., picocells and
femtocells), overlaying the macrocells, to form a multi-tiered heterogeneous
cellular network [1]. Each small-cell is equipped with a shorter-range and
lower-cost base station (BS), to provide nearby UEs with higher-quality
communication links with lower power usage.

However, in the presence of multiple tiers of BSs in a cellular network, user
association control becomes more challenging. A most direct approach is
association by maximum received power, in which UEs are associated with the BS
(in any tier) with the largest received power. However, in this case, because
small-cell BSs are transmitting at lower power levels, only the UEs very close
to them will connect with them, while most other UEs are still crowding in
macrocells, leading to degraded performance. An example is shown in Fig. 1(a),
in which many UEs are occupying the macrocells, while some small-cells are
nearly empty.

In order to resolve this issue, a flexible user association approach may be
employed [2, 3], in which each tier of BSs is assigned a user association bias
value, and a UE is associated with a BS with the largest received power
multiplied by the bias value. If small-cell BSs are assigned with larger
association bias values, the small-cells are “expanded” accordingly. This can
result in a more balanced mobile traffic pattern and thus better network
performance. Fig. 1(b) shows an example of flexible user association. However,
if the association bias values for small-cell BSs are too large, it will cause
improper expansions of small-cells such that UEs at their cell-edge may suffer
from inadequate received power. As a consequence, the association bias values
should be properly designed so that the overall network performance is
optimized.

Further complicating the resource management problem in a multi-tier cellular
network, the radio spectrum licensed by the network operator needs to be
shared by BSs of widely different power and coverage areas. How to optimally
allocate spectrum among different tiers is an important open problem. In order
to avoid cross-tier interference, and the prohibitive complexity in tracking
and provisioning for such interference especially with unplanned deployment of
small cells, a disjoint spectrum mode is commonly advocated [4, 5, 6], where
different tiers of BSs are allocated non-overlapping portions of the spectrum.
Even so, it is still a challenging problem to properly divide the spectrum for
optimal network performance.

(a) Association by maximum received power.

(b) Flexible user association.

Figure 1: An example of a three-tier cellular network. Macrocell BSs, picocell
BSs, and femtocell BSs are represented by squares, circles, and triangles
respectively; UEs are represented by dots; blue lines show cell boundaries.

In this work, our objective is to study jointly optimal spectrum allocation
and user association in a heterogeneous celluar network with multiple tiers of
BSs. First, we develop a stochastic geometric model to study the network
performance analytically. A closed-form expression for the average downlink UE
data rate is derived, which is then employed as the objective function for
jointly optimizing the spectrum allocation among tiers and the user
association bias values.

This resultant optimization problem is of non-convex programming in nature and
cannot be solved with a standard method. Instead, we explore two important
structures in solving the problem. Referred to as the density thresholding
structure, we show that the problem can be studied separately over sparse UE
and dense UE scenarios, divided by a parameter specific threshold. Referred to
as the priority ordering structure, we show that a tier with higher BS density
should have higher priority in spectrum allocation. Based on these
observations, we propose a computationally efficient Structured Spectrum
Allocation and User Association (SSAUA) approach to solve the problem
optimally in the sparse UE scenario, and near-optimally in the dense UE
scenario with a quantified performance bound.

Finally, toward practical implementation of SSAUA, we propose a Surcharge
Pricing Scheme (SPS), such that the designed association bias values can be
achieved in Nash equilibrium. Hence, each UE is incentivized to adopt the
proposed design with individual rationality.

The rest of the paper is organized as follows. In section II, we discuss the
relation between our work and prior works. In Section III, we present the
system model. In Sections IV, V, and VI, we present our contributions in UE
data rate derivation, SSAUA design, and SPS, respectively. In Section VII, we
present numerical results. Finally, conclusions are given in Section VIII.

##  II Related Works

###  II-A Stochastic Geometry as Analytical Basis

Stochastic geometry [7, 8, 9, 10] is a powerful mathematical modeling tool to
analyze the performance (e.g., outage probability and data rate) of cellular
networks with random spatial patterns of UEs and BSs. In this work, we focus
on the downlink user data rate as performance measure. Pioneering works on
downlink performance analysis using stochastic geometry include [11], [12],
and [13], for either the single-tier or the multi-tier case. None of them
considered spectrum allocation or user association.

###  II-B Spectrum Allocation and User Association

Most prior studies considered either spectrum allocation or user association
separately. For example, [6, 14] studied optimal spectrum allocation, in
cellular networks limited to two tiers of BSs, without flexible user
association.

Assuming a fixed number of UEs and BSs and without considering their random
spatial patterns, [15, 16, 17, 18, 19] studied optimal user association with
deterministic utility optimization. With a stochastic geometric approach, [2]
proposed the flexible user association model with bias values, which is
adopted in our work. It also derived the coverage probability and UE data
rate, considering cross-tier interference, but in non-closed forms. [2] did
not provide any design insights to optimize the derived performance metrics.
It was later extended in [3] to study optimal user association in a network
with two tiers of BSs, without considering spectrum allocation.

Joint spectrum allocation and user association was studied in [20]. It was
limited to two tiers of BSs. [20] presented a qualitative study on the optimal
network performance in terms of coverage probability and data rate, without
further providing analytical details in solving the optimization problems. A
similar problem was also studied in [21], with frequency reuse instead of
tiered spectrum division as the approach for spectrum sharing. It could
accommodate more than two tiers, but it provided only conditionally optimal
user association given frequency reuse factors or conditionally optimal
frequency reuse factors given user association bias values. Joint optimization
remained an open problem.

Compared with the above studies, we consider multiple tiers of BSs with
disjoint spectrum and provide optimal and analytically bounded near-optimal
solutions for joint spectrum allocation and user association.

##  III System Model

###  III-A Multi-tier Cellular Network

We consider a heterogeneous cellular network with randomly spatially
distributed $K\geq 2$ tiers of BSs. As in conventional stochastic geometric
modeling of multi-tier cellular networks [2, 20, 3, 12, 21], each tier of BSs
independently forms a homogeneous Poisson point process (PPP) in two
dimensional Euclidean space $\mathbb{R}^{2}$. Let $\Phi_{k}$ denote the PPP
corresponding to tier-$k$ BSs, with intensity $\lambda_{k}$. Without loss of
generality, we assume that $\lambda_{1}<\lambda_{2}\ldots<\lambda_{K}$. (If
$\lambda_{i}=\lambda_{j},i\neq j$ in reality, we may approximate
$\lambda_{j}=\lambda_{i}+\xi$, where $\xi$ is arbitrarily close to $0$.) UEs
are also modeled as a homogeneous PPP $\Psi$ with intensity $\mu$, independent
of all BSs. We assume each BS is connected to the core network by separate
high-capacity wired or wireless links that have no influence on our
performance analysis. In addition, because we focus on downlink analysis, we
assume that the downlink and uplink of the system are operated in different
spectrum, so that the uplink interference and capacity have no influence on
the downlink analysis.

###  III-B Power and Pathloss Model

We define the tiers of BSs by their transmission power. Let $P_{k}$ be the
transmission power of tier-$k$ BSs, which is a given parameter. If
$P_{t}(\mathbf{x})$, $P_{t}(\mathbf{x})\in\\{P_{1},P_{2},\ldots,P_{K}\\}$, is
the transmission power from a BS at $\mathbf{x}$ and $P_{r}(\mathbf{y})$ is
the received power at $\mathbf{y}$, we have
$P_{r}(\mathbf{y})=\frac{P_{t}(\mathbf{x})h_{\mathbf{x},\mathbf{y}}}{\alpha|\mathbf{x}-\mathbf{y}|^{\gamma}}$,
where $\alpha|\mathbf{x}-\mathbf{y}|^{\gamma}$ is the propagation loss
function with predetermined constants $\alpha$ and $\gamma$ (where the path
loss exponent $\gamma>2$ in practice), and $h_{\mathbf{x},\mathbf{y}}$ is the
fast fading term. Corresponding to common Rayleigh fading with power
normalization, $h_{\mathbf{x},\mathbf{y}}$ is independently exponentially
distributed with unit mean. Let $H(\cdot)$ be the cumulative distribution
function of $h_{\mathbf{x},\mathbf{y}}$.

###  III-C Spectrum Allocation

In order to avoid cross-tier interference, different tiers of BSs are
allocated separated spectrum. Assume the total spectrum bandwidth is $W$. The
network operator allocates $\eta_{k}W$ to each tier-$k$ BS, where $\eta_{k}$
is the spectrum allocation factor and $\sum_{k=1}^{K}\eta_{k}=1$. Let
$\bm{\eta}=(\eta_{1},\eta_{2},\ldots,\eta_{K})$. Note that BSs in the same
tier are operated on the same spectrum.

We additionally consider the possible constraints
$\eta_{\min,k}\leq\eta_{k}\leq\eta_{\max,k}$, for $k=1,2,\ldots,K$. Clearly,
we have $\sum_{k=1}^{K}\eta_{\min,k}\leq 1\leq\sum_{k=1}^{K}\eta_{\max,k}$.
Furthermore, we assume that
$0<\eta_{\min,1}\leq\eta_{\min,2}\ldots\leq\eta_{\min,K}$ and
$0<\eta_{\max,1}\leq\eta_{\max,2}\ldots\leq\eta_{\max,K}$, i.e., the network
operator is likely (but not necessarily) to allocate more spectrum to a tier
with higher BS density. Note that this is a general condition that contains
the special case where there is no constraint on $\bm{\eta}$.

Given a specific tier-$k$ BS, it is common to assume that all its associated
UEs are equally allocated spectrum [21, 2, 6]. Hence, the per-UE assigned
spectrum bandwidth is $\beta_{k}=\eta_{k}W/N_{k}$, where $N_{k}$ is a random
variable denoting the number of UEs associated with the BS.

###  III-D Coverage Probability and UE Data Rate

Let $T$ denote the minimum required Signal-to-Interference Ratio (SIR) of UEs.
The coverage probability of a UE is defined as the probability that its SIR is
no less than $T$ [7]. As in conventional wireless modeling [21, 2, 6], if a UE
experiences coverage probability $P^{\prime}$ and is allocated spectrum
bandwidth $\beta^{\prime}$, its data rate is $\beta^{\prime}\log(1+T)$ if the
SIR is no less than $T$, and its data rate is $0$ if the SIR is less than $T$
(i.e., outage occurs). Thus, the overall data rate of the UE is
$\beta^{\prime}\log(1+T)P^{\prime}$. Note that, unless otherwise stated,
$\log$ represents $\log_{2}$. Also, the thermal noise is assumed to be
negligible compared with interference.

###  III-E Flexible User Association

Given that a UE is located at $\mathbf{y}$, it associates itself with the BS
that provides the maximum biased received power [2, 3, 21] as follows:

$\displaystyle\mathcal{BS}(\mathbf{y})=\arg\max_{\mathbf{x}\in\Phi_{k},\forall
k}B_{k}P_{k}|\mathbf{x}-\mathbf{y}|^{-\gamma},$ (1)

where $\mathcal{BS}(\mathbf{y})$ denotes the location of the BS associated
with the UE, and $P_{k}|\mathbf{x}-\mathbf{y}|^{-\gamma}$ is the received
power from a tier-$k$ BS located at $\mathbf{x}$, and $B_{k}$ is the
association bias, indicating the connecting preference of a UE toward tier-$k$
BSs. In this case, the resultant cell splitting forms a generalized Dirichlet
tessellation, or weighted Poisson Voronoi [22], shown in Fig. 1(b). Note that
for $B_{1},B_{2},\ldots,B_{K}$, their effects remain the same if we multiply
all of them by the same positive constant. Thus, without loss of generality,
we normalize them such that $\sum_{k=1}^{K}B_{k}=1$. Let
$\mathbf{B}=(B_{1},B_{2},\ldots,B_{K})$.

Let $A_{k}$ denote the probability that a UE associates itself with a tier-$k$
BS, and $\mathbf{A}=(A_{1},A_{2},\ldots,A_{K})$. As derived in [2], we have

$\displaystyle
A_{k}=\frac{\lambda_{k}(P_{k}B_{k})^{\frac{2}{\gamma}}}{\sum_{j=1}^{K}\lambda_{j}(P_{j}B_{j})^{\frac{2}{\gamma}}},$
(2)

and thus

$\displaystyle
B_{k}=\frac{P_{k}^{-1}(A_{k}/\lambda_{k})^{\frac{\gamma}{2}}}{\sum_{j=1}^{K}P_{j}^{-1}(A_{j}/\lambda_{j})^{\frac{\gamma}{2}}}.$
(3)

Hence, there is a one-to-one mapping between $\mathbf{A}$ and $\mathbf{B}$, so
we can view them interchangeably.

###  III-F Problem Statement

We first aim to derive a closed-form expression for the average UE data rate.
Then, our objective is to maximize the average UE data rate by jointly
optimizing the spectrum allocation factors $\bm{\eta}$ and the user
association bias values $\mathbf{B}$ (or equivalently $\mathbf{A}$). Finally,
we give a pricing scheme to incentivize each UE to adopt the designed
$\mathbf{B}$.

##  IV Closed-form Average UE Data Rate

In this section, we derive the average UE data rate via stochastic geometric
analysis. Consider a reference UE, termed as the typical UE, communicating
with its BS, termed as the typical BS. We are interested in the typical UE
since the average UE performance in the system is the same as the performance
of the typical UE [7]. Furthermore, due to the stationarity of UEs and BSs,
throughout this section we will re-define the coordinates so that the typical
UE is located at $\mathbf{0}$.

First, we study the coverage probability given that the typical UE is
associating with a tier-$k$ BS and their distance is $d$. In this case, the
overall interference to the typical UE is the sum interference from all
tier-$k$ BSs other than the typical BS. Let $I_{k}(d)$ denote such
interference. Then

$\displaystyle
I_{k}(d)=\sum_{\mathbf{x}\in\Phi_{k}^{\prime}}\frac{P_{k}h_{\mathbf{x},\mathbf{0}}}{\alpha|\mathbf{x}|^{\gamma}}.$
(4)

where $\Phi_{k}^{\prime}$ is the Palm point process corresponding to all
tier-$k$ BSs other than the typical BS, given that the typical BS is located
at a distance of $d$ from the typical UE. It can be shown that
$\Phi_{k}^{\prime}$ is a PPP with intensity $0$ in $\mathcal{B}(\mathbf{0},d)$
and intensity $\lambda_{k}$ in
$\mathbb{R}^{2}\backslash\mathcal{B}(\mathbf{0},d)$, where
$\mathcal{B}(\mathbf{0},d)$ denotes the disk region centered at $\mathbf{0}$
with radius $d$ [7].

The distribution of $I_{k}(d)$ is derived through its Laplace transform as
follows:

$\displaystyle\mathcal{L}_{I_{k}}(d,s)=\mathbf{E}\left[\exp\left(-\sum_{\mathbf{x}\in\Phi_{k}^{\prime}}\frac{sP_{k}h_{\mathbf{x},\mathbf{0}}}{\alpha|\mathbf{x}|^{\gamma}}\right)\right]$
$\displaystyle=$
$\displaystyle\exp\left(-\lambda_{k}\int_{\mathbb{R}^{2}\backslash\mathcal{B}(\mathbf{0},d)}\left(1-\int_{\mathbb{R}^{+}}e^{-\frac{sP_{k}h}{\alpha|\mathbf{x}|^{\gamma}}}H(\mathrm{d}h)\right)\mathrm{d}\mathbf{x}\right)$
(5) $\displaystyle=$
$\displaystyle\exp\left(-\lambda_{k}\int_{\mathbb{R}^{2}\backslash\mathcal{B}(\mathbf{0},d)}\frac{\frac{sP_{k}}{\alpha|\mathbf{x}|^{\gamma}}}{\frac{sP_{k}}{\alpha|\mathbf{x}|^{\gamma}}+1}\mathrm{d}\mathbf{x}\right)$
(6) $\displaystyle=$
$\displaystyle\exp\left(-2\pi\lambda_{k}\int_{d}^{\infty}\frac{\frac{sP_{k}r}{\alpha}}{\frac{sP_{k}}{\alpha}+r^{\gamma}}\mathrm{d}r\right),$
(7)

where (5) is obtained from the Laplace functional of PPP $\Phi_{k}^{\prime}$
[7], (6) is because $h$ is exponentially distributed with unit mean, and (7)
is through a transformation to polar coordinates.

Let $P_{cover,k}(d)$ denote the conditional coverage probability of the
typical UE (given $k$ and $d$). Then

$\displaystyle P_{cover,k}(d)=$
$\displaystyle\mathbf{P}\left(\frac{P_{k}h_{\mathbf{x}_{B},\mathbf{0}}}{\alpha
d^{\gamma}}\geq TI_{k}(d)\right)$ $\displaystyle=$
$\displaystyle\mathcal{L}_{I_{k}}(d,s)|_{s=\frac{T\alpha d^{\gamma}}{P_{k}}},$
(8)

where $\mathbf{x}_{B}$ is the coordinate of the typical BS, and
$|\mathbf{x}_{B}|=d$. Substituting (7) into (8), we have

$\displaystyle P_{cover,k}(d)=$
$\displaystyle\exp\left(-2\pi\lambda_{k}\int_{d}^{\infty}\frac{Td^{\gamma}r}{Td^{\gamma}+r^{\gamma}}\mathrm{d}r\right)$
$\displaystyle\overset{t=\frac{r^{2}}{T^{2/\gamma}d^{2}}}{=}$
$\displaystyle\exp\left(-\pi\lambda_{k}T^{\frac{2}{\gamma}}d^{2}\int_{(\frac{1}{T})^{\frac{2}{\gamma}}}^{\infty}\frac{1}{1+t^{\frac{\gamma}{2}}}\mathrm{d}t\right).$
(9)

Furthermore, the probability density function of the distance between the
typical UE and its associated tier-$k$ BS is

$\displaystyle f_{k}(d)=$
$\displaystyle\frac{2\pi\lambda_{k}}{A_{k}}d\exp\left(-\pi
d^{2}\sum_{j=1}^{K}\lambda_{j}\left(\frac{P_{j}B_{j}}{P_{k}B_{k}}\right)^{\frac{2}{\gamma}}\right)$
(10) $\displaystyle=$
$\displaystyle\frac{2\pi\lambda_{k}}{A_{k}}d\exp\left(-\pi
d^{2}\frac{\lambda_{k}}{A_{k}}\right),$ (11)

where (10) is derived in [2], and (11) is by substituting (2) into (10).

Hence, the coverage probability $P_{cover,k}$ of the typical UE associated
with a tier-$k$ BS can be computed as

$\displaystyle P_{cover,k}=$
$\displaystyle\int_{0}^{\infty}f_{k}(d)P_{cover,k}(d)\mathrm{d}d$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\frac{2\pi\lambda_{k}}{A_{k}}d\exp\left(-\pi
d^{2}\frac{\lambda_{k}}{A_{k}}\right)$
$\displaystyle\qquad\exp\left(-\pi\lambda_{k}\left(T\right)^{\frac{2}{\gamma}}d^{2}\int_{\left(\frac{1}{T}\right)^{\frac{2}{\gamma}}}^{\infty}\frac{1}{1+t^{\gamma/2}}\mathrm{d}t\right)\mathrm{d}d$
$\displaystyle=$ $\displaystyle\frac{1}{A_{k}}\frac{1}{\frac{1}{A_{k}}+C},$
(12)

where
$C=\left(T\right)^{\frac{2}{\gamma}}\int_{\left(\frac{1}{T}\right)^{\frac{2}{\gamma}}}^{\infty}\frac{1}{1+t^{\gamma/2}}\mathrm{d}t$
is a system-level constant only related to $\gamma$ and $T$.

Let $\mathbf{E}_{0}(\beta_{k})$ denote the expected spectrum bandwidth
allocated to the typical UE (connecting to a tier-$k$ BS). Following the model
in Section III-C, $\mathbf{E}_{0}(\beta_{k})$ equals the spectrum bandwidth
allocated to the typical tier-$k$ BS divided by the average number of UEs
associated with it conditioned on the typical UE, which is
$A_{k}\mu/\lambda_{k}+1$. Hence,

$\displaystyle\mathbf{E}_{0}(\beta_{k})=\frac{\eta_{k}W}{A_{k}\mu/\lambda_{k}+1}.$
(13)

Then, by Section III-D, the conditional expected data rate of the typical UE,
given it is associated with a tier-$k$ BS, can be computed as [2, 21]111By
doing so, we slightly underestimate the average data rate because the coverage
event and $\beta_{k}$ are not completely independent. Although some efforts
have been made to approximate their correlation [3, 23], all of them are
inexact but result in tremendous mathematical complexity. In Section VII, we
show that the resultant analysis is close to actual performance via
simulations.

$\displaystyle\overline{R}_{k}=\mathbf{E}_{0}(\beta_{k})\log(1+T)P_{cover,k}.$
(14)

Finally, the average data rate of the typical UE, and hence the average data
rate per UE in the system, is

$\displaystyle\mathbf{F}=$
$\displaystyle\sum_{k=1}^{K}A_{k}\overline{R}_{k}=\sum_{k=1}^{K}A_{k}\mathbf{E}_{0}(\beta_{k})\log(1+T)P_{cover,k}$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{K}\frac{\eta_{k}W\log(1+T)}{(A_{k}\mu/\lambda_{k}+1)(\frac{1}{A_{k}}+C)}.$
(15)

Note that stochastic geometric analysis often leads to non-closed forms
requiring numerical integrations (e.g., [11, 12, 2, 3]), due to the integral
form of the Laplace functional or generating functional of PPPs applied in
analysis [7, 10]. Fortunately, our derived closed-form expression for the
average UE data rate facilitates the tractability of the resultant
optimization problem.

##  V Joint Optimization Problem and SSAUA

We aim to maximize the average UE data rate $\mathbf{F}$ with respect to
$\bm{\eta}$ and $\mathbf{B}$. As there is a one-to-one mapping between
$\mathbf{A}$ and $\mathbf{B}$, we study the optimization problem over
$(\bm{\eta},\mathbf{A})$ instead for analytical convenience. This is formally
stated as optimization problem $\mathbf{P}$ as follows:

$\displaystyle\underset{\bm{\eta},\mathbf{A}}{\text{maximize}}$
$\displaystyle\mathbf{F}(\bm{\eta},\mathbf{A})=\sum_{k=1}^{K}\eta_{k}M_{k}(A_{k})$
subject to
$\displaystyle\sum_{k=1}^{K}\eta_{k}=1,\quad\eta_{\min,k}\leq\eta_{k}\leq\eta_{\max,k},\forall
k,$ $\displaystyle\sum_{k=1}^{K}A_{k}=1,\quad A_{k}\geq 0,\forall k,$ (16)

where $M_{k}(A_{k})$ is defined as

$\displaystyle
M_{k}(A_{k})=\frac{1}{\left(A_{k}\mu/\lambda_{k}+1\right)\left(\frac{1}{A_{k}}+C\right)}.$
(17)

Problem $\mathbf{P}$ is non-convex and cannot be solved through a standard
method. Instead, we investigate into two important structures of the optimal
solution, termed density thresholding and priority ordering, based on which we
propose a computationally efficient Structured Spectrum Allocation and User
Association (SSAUA) approach to solve the problem.

###  V-A Density Thresholding Structure

First, we define an important parameter

$\displaystyle a_{k}\triangleq\sqrt{\lambda_{k}/(\mu C)}.$ (18)

Note that $M_{k}(A_{k})$ is increasing on $[0,a_{k}]$ and decreasing on
$[a_{k},\infty)$. We further observe several useful properties of
$M_{k}(A_{k})$, which are presented in Appendix -A. Based on these properties,
we obtain the following lemma, whose proof is given in Appendix -B.

###### Lemma 1

Consider a potential solution $(\bm{\eta}^{**},\mathbf{A}^{**})$ to Problem
$\mathbf{P}$. If $\exists i\neq j$, such that $A^{**}_{i}<a_{i}$ and
$A^{**}_{j}>a_{j}$, then $(\bm{\eta}^{**},\mathbf{A}^{**})$ is not an optimal
solution.

Lemma 1 suggests that, in an optimal solution, every $A_{k}$ must be on the
same side of $a_{k}$. This directly leads to the following theorem, which is
fundamental to our optimization solution.

###### Theorem 1

(Density Thresholding) Let $(\bm{\eta}^{*},\mathbf{A}^{*})$ be an optimal
solution to Problem $\mathbf{P}$. If $\sum_{k=1}^{K}a_{k}>1$, then $\forall
k,A^{*}_{k}\leq a_{k}$; if $\sum_{k=1}^{K}a_{k}<1$, then $\forall
k,A^{*}_{k}\geq a_{k}$; if $\sum_{k=1}^{K}a_{k}=1$, then $\forall
k,A^{*}_{k}=a_{k}$.

###### Proof:

If $\sum_{k=1}^{K}a_{k}>1$, because $\sum_{k=1}^{K}A^{*}_{k}=1$, $\exists l$
such that $A^{*}_{l}<a_{l}$. This leads to $A^{*}_{k}\leq a_{k}$, $\forall k$,
according to Lemma 1. The cases where $\sum_{k=1}^{K}a_{k}<1$ and
$\sum_{k=1}^{K}a_{k}=1$ are similar. ∎

Note that, the condition $\sum_{i=1}^{K}a_{i}>1$ is equivalent to
$\sqrt{\frac{1}{C}}\left(\sum_{i=1}^{K}\sqrt{\lambda_{i}}\right)>\sqrt{\mu}$,
implying the density of UEs is sparse (compared with that of BSs). Thus, we
refer to the case $\sum_{i=1}^{K}a_{i}>1$ as the _sparse UE scenario_. On the
other hand, we refer to the case $\sum_{i=1}^{K}a_{i}<1$, which is equivalent
to
$\sqrt{\frac{1}{C}}\left(\sum_{i=1}^{K}\sqrt{\lambda_{i}}\right)<\sqrt{\mu}$,
as the _dense UE scenario_. If $\sum_{i=1}^{K}a_{i}=1$, Problem $\mathbf{P}$
can be trivially solved and is ignored in the rest of our discussion. Note
that because $a_{k}$ can be computed directly from the given parameters, one
can judge which scenario Problem $\mathbf{P}$ falls within before solving the
problem. Next, the solution to $\mathbf{P}$ will be investigated separately in
the sparse UE and dense UE scenarios.

###  V-B SSAUA in the Sparse UE Scenario

In this case, the original Problem $\mathbf{P}$ becomes Problem $\mathbf{P1}$
as follows:

$\displaystyle\underset{\bm{\eta},\mathbf{A}}{\text{maximize}}$
$\displaystyle\mathbf{F}(\bm{\eta},\mathbf{A})=\sum_{k=1}^{K}\eta_{k}M_{k}(A_{k})$
subject to
$\displaystyle\sum_{k=1}^{K}\eta_{k}=1,\quad\eta_{\min,k}\leq\eta_{k}\leq\eta_{\max,k},\forall
k,$ $\displaystyle\sum_{k=1}^{K}A_{k}=1,\quad 0\leq A_{k}\leq a_{k},\forall
k.$ (19)

We first observe an important ordering property of the optimal solution to
$\mathbf{P1}$, as shown in the following lemma, whose proof is given in
Appendix -C.

###### Lemma 2

(Ordering Property) Let $\mathbf{A}^{*}$ be optimal for $\mathbf{P1}$, then
$M_{1}(A^{*}_{1})\leq M_{2}(A^{*}_{2})\leq\ldots\leq M_{K}(A^{*}_{K})$.

Next, by sequentially computing $\bm{\eta}^{*}$ as follows:

$\displaystyle\left\\{\begin{array}[]{l}\eta^{*}_{K}=\min(1-\sum_{k=1}^{K-1}\eta_{\min,k},\eta_{\max,K}),\\\
\eta^{*}_{K-1}=\min(1-\eta^{*}_{K}-\sum_{k=1}^{K-2}\eta_{\min,k},\eta_{\max,K-1}),\\\
\ldots,\\\
\eta^{*}_{l}=\min(1-\sum_{k=l+1}^{K}\eta^{*}_{k}-\sum_{k=1}^{l-1}\eta_{\min,k},\eta_{\max,l}),\\\
\ldots,\\\ \eta^{*}_{1}=\min(1-\sum_{k=2}^{K}\eta^{*}_{k},\eta_{\max,1}),\\\
\end{array}\right.$ (26)

we have the following theorem:

###### Theorem 2

(Priority Ordering) Let $\mathbf{A}^{*}$ be optimal for Problem $\mathbf{P1}$,
then $(\bm{\eta}^{*},\mathbf{A}^{*})$, where $\bm{\eta}^{*}$ is computed in
(26), is an optimal solution to $\mathbf{P1}$.

###### Proof:

Consider Problem $\mathbf{P1A}$ as follows:

$\displaystyle\underset{\bm{\eta}}{\text{maximize}}$
$\displaystyle\sum_{k=1}^{K}\eta_{k}M_{k}(A^{*}_{k})$ subject to
$\displaystyle\sum_{k=1}^{K}\eta_{k}=1,\quad\eta_{\min,k}\leq\eta_{k}\leq\eta_{\max,k},\forall
k.$ (27)

Then $\mathbf{P1A}$ is a simple linear programming problem with ordered linear
coefficients in the objective, since $M_{1}(A^{*}_{1})\leq
M_{2}(A^{*}_{2})\leq\ldots\leq M_{K}(A^{*}_{K})$ due to Lemma 2. Note that
$\bm{\eta}^{*}$ does not depend on the exact values of $\mathbf{A}^{*}$; it
only requires the ordering property as shown in Lemma 2. Also, $\bm{\eta}^{*}$
is in the feasible region due to $\sum_{k=1}^{K}\eta_{\min,k}\leq
1\leq\sum_{k=1}^{K}\eta_{\max,k}$. It is easy to verify that,
$(\bm{\eta}^{*},\mathbf{A}^{*})$ is an optimal solution to $\mathbf{P1}$. ∎

Equation (26) indicates the priority ordering structure in spectrum
allocation. We see that tier-$K$ has the highest priority in spectrum
allocation, followed by tier-$(K-1)$, and so forth.

Theorem 2 provides a means to derive an optimal $\bm{\eta}^{*}$ regardless of
the $\mathbf{A}^{*}$ values. We need one further step to derive the
corresponding optimal $\mathbf{A}^{*}$ by solving the following Problem
$\mathbf{P1B}$:

$\displaystyle\underset{\mathbf{A}}{\text{maximize}}$
$\displaystyle\sum_{k=1}^{K}\eta^{*}_{k}M_{k}(A_{k})$ subject to
$\displaystyle\sum_{k=1}^{K}A_{k}=1,\quad 0\leq A_{k}\leq a_{k},\forall k.$
(28)

Note that $\mathbf{P1B}$ is a convex programming problem, since $M_{k}(A_{k})$
is concave on $[0,a_{k}]$. Thus, $\mathbf{A}^{*}$ can be computed by a
computationally efficient algorithm, such as the interior point method. Hence
the both steps to compute the jointly optimal solution
$(\bm{\eta}^{*},\mathbf{A}^{*})$ have low computational complexity.

###  V-C SSAUA in the Dense UE Scenario with Performance Bound

In this case, the original Problem $\mathbf{P}$ becomes Problem $\mathbf{P2}$
as follows:

$\displaystyle\underset{\bm{\eta},\mathbf{A}}{\text{maximize}}$
$\displaystyle\mathbf{F}(\bm{\eta},\mathbf{A})=\sum_{k=1}^{K}\eta_{k}M_{k}(A_{k})$
subject to
$\displaystyle\sum_{k=1}^{K}\eta_{k}=1,\quad\eta_{\min}\leq\eta_{k}\leq\eta_{\max},\forall
k,$ $\displaystyle\sum_{k=1}^{K}A_{k}=1,\quad A_{k}\geq a_{k},\forall k.$ (29)

Problem $\mathbf{P2}$ is more complicated compared with Problem $\mathbf{P1}$,
as $M_{k}(A_{k})$ is not concave, but an S-shaped function, in the feasible
region. Hence, $\mathbf{P2}$ generally incurs high computational complexity
even if an optimal $\bm{\eta}^{*}$ is given [24, 25].

Therefore, instead of directly solving $\mathbf{P2}$, we first approximate
$M_{k}(A_{k})$ by $\widetilde{M}_{k}(A_{k})$ defined as follows:

$\displaystyle\widetilde{M}_{k}(A_{k})=\frac{1}{\left(A_{k}\mu/\lambda_{k}\right)\left(\frac{1}{A_{k}}+C\right)}.$
(30)

Note that this approximation is reasonable because $A_{k}\mu/\lambda_{k}$ is
much larger than $1$ when $\mu$ is large (i.e., the dense UE scenario). This
observation is also supported by the small performance gap as derived in
Section V-C2. Some useful properties of $\widetilde{M}_{k}(A_{k})$ are shown
in Appendix -D.

The approximated problem is referred to as Problem $\mathbf{P2A}$, where we
simply replace the objective function of $\mathbf{P2}$ by the following:

$\displaystyle\mathbf{F^{\prime}}(\bm{\eta},\mathbf{A})=\sum_{k=1}^{K}\eta_{k}\widetilde{M}_{k}(A_{k}).$
(31)

####  V-C1 Solution to $\mathbf{P2A}$

The important ordering property still holds for Problem $\mathbf{P2A}$, as
formalized in the following lemma, whose proof is given in Appendix -E.

###### Lemma 3

(Ordering Property) Let
$(\widetilde{\bm{\eta}}^{*},\widetilde{\mathbf{A}}^{*})$ be an optimal
solution to $\mathbf{P2A}$, then
$\widetilde{M}_{1}(\widetilde{A}^{*}_{1})\leq\widetilde{M}_{2}(\widetilde{A}^{*}_{2})\leq\ldots\leq\widetilde{M}_{K}(\widetilde{A}^{*}_{K})$.

We observe that with the same ordering property, (26) can again be adopted as
an optimal solution to $\mathbf{P2A}$ in the dense UE scenario, leading to the
following theorem:

###### Theorem 3

(Priority Ordering) Let $\widetilde{\mathbf{A}}^{*}$ be optimal for Problem
$\mathbf{P2A}$, then $(\widetilde{\bm{\eta}}^{*},\widetilde{\mathbf{A}}^{*})$,
where $\widetilde{\bm{\eta}}^{*}$ is computed the same way as $\bm{\eta}^{*}$
in (26), is an optimal solution to $\mathbf{P2A}$.

###### Proof:

The proof is similar to that of Theorem 2. ∎

Given an optimal $\widetilde{\bm{\eta}}^{*}$ for $\mathbf{P2A}$, we find the
corresponding optimal $\widetilde{\mathbf{A}}^{*}$ for $\mathbf{P2A}$ by
solving the following Problem $\mathbf{P2B}$:

$\displaystyle\underset{\mathbf{A}}{\text{maximize}}$
$\displaystyle\sum_{k=1}^{K}\widetilde{\eta}^{*}_{k}\widetilde{M}_{k}(A_{k})$
subject to $\displaystyle\sum_{k=1}^{K}A_{k}=1,\quad A_{k}\geq a_{k},\forall
k.$ (32)

Unlike in the sparse UE scenario, here we have an explicit solution, as stated
in the following theorem:

###### Theorem 4

Given an optimal $\widetilde{\bm{\eta}}^{*}$ for $\mathbf{P2A}$ (computed the
same way as $\bm{\eta}^{*}$ in (26)), the corresponding optimal
$\widetilde{\mathbf{A}}^{*}$ can be expressed as follows:

$\displaystyle\left\\{\begin{array}[]{ll}\widetilde{A}^{*}_{k}=a_{k},&k\geq
2\\\ \widetilde{A}^{*}_{1}=1-\sum_{l=2}^{K}\widetilde{A}^{*}_{l}.\\\
\end{array}\right.$ (35)

###### Proof:

See Appendix -F. ∎

Note that both (26) and (35) can be computed with low computational
complexity.

####  V-C2 Bounding the Performance Gap

Since $(\widetilde{\bm{\eta}}^{*},\widetilde{\mathbf{A}}^{*})$ is optimal for
$\mathbf{P2A}$ rather than $\mathbf{P2}$, we next quantify the performance gap
between $(\widetilde{\bm{\eta}}^{*},\widetilde{\mathbf{A}}^{*})$ and an
optimal solution $(\bm{\eta}^{*},\mathbf{A}^{*})$ to $\mathbf{P2}$.

The performance gap is defined as

$\displaystyle
E=\mathbf{F}(\bm{\eta}^{*},\mathbf{A}^{*})-\mathbf{F}(\widetilde{\bm{\eta}}^{*},\widetilde{\mathbf{A}}^{*}).$
(36)

Because
$\mathbf{F}(\widetilde{\bm{\eta}}^{*},\widetilde{\mathbf{A}}^{*})\leq\mathbf{F}(\bm{\eta}^{*},\mathbf{A}^{*})\leq\mathbf{F}^{\prime}(\bm{\eta}^{*},\mathbf{A}^{*})\leq\mathbf{F}^{\prime}(\widetilde{\bm{\eta}}^{*},\widetilde{\mathbf{A}}^{*})$,
we have

$\displaystyle
E\leq\mathbf{F}^{\prime}(\widetilde{\bm{\eta}}^{*},\widetilde{\mathbf{A}}^{*})-\mathbf{F}(\widetilde{\bm{\eta}}^{*},\widetilde{\mathbf{A}}^{*})\triangleq
E^{\prime}.$ (37)

Substituting $\widetilde{\bm{\eta}}^{*}$ and $\widetilde{\mathbf{A}}^{*}$ into
$E^{\prime}$, we have

$\displaystyle E^{\prime}=$
$\displaystyle\sum_{k=1}^{K}\frac{\widetilde{\eta}^{*}_{k}}{\frac{\widetilde{A}^{*}_{k}\mu}{\lambda_{k}}\left(\frac{\widetilde{A}^{*}_{k}\mu}{\lambda_{k}}+1\right)\left(\frac{1}{\widetilde{A}^{*}_{k}}+C\right)}.$
(38)

Therefore, the relative performance gap is bounded by

$\displaystyle\epsilon\triangleq$
$\displaystyle\frac{E}{\mathbf{F}(\bm{\eta}^{*},\mathbf{A}^{*})}\leq\frac{E^{\prime}}{\mathbf{F}(\widetilde{\bm{\eta}}^{*},\widetilde{\mathbf{A}}^{*})}$
(39) $\displaystyle=$
$\displaystyle\frac{\sum_{k=1}^{K}\frac{\widetilde{\eta}^{*}_{k}}{\frac{\widetilde{A}^{*}_{k}\mu}{\lambda_{k}}\left(\frac{\widetilde{A}^{*}_{k}\mu}{\lambda_{k}}+1\right)\left(\frac{1}{\widetilde{A}^{*}_{k}}+C\right)}}{\sum_{k=1}^{K}\frac{\widetilde{\eta}^{*}_{k}}{\left(\frac{\widetilde{A}^{*}_{k}\mu}{\lambda_{k}}+1\right)\left(\frac{1}{\widetilde{A}^{*}_{k}}+C\right)}}$
(40) $\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}$
$\displaystyle\max_{k}\frac{\lambda_{k}}{\widetilde{A}^{*}_{k}\mu}\leq\max_{k}\frac{\lambda_{k}}{a_{k}\mu}$
(41) $\displaystyle=$ $\displaystyle\sqrt{\lambda_{K}C/\mu},$ (42)

where inequality (a) is obtained by observing the common factor in the
summations in the numerator and denominator of (40). This implies that
$\epsilon$ scales as $O\left(\sqrt{\lambda_{K}/\mu}\right)$. Note that because
we are considering the dense UE scenario (i.e.,
$\sqrt{\frac{1}{C}}\left(\sum_{i=1}^{K}\sqrt{\lambda_{i}}\right)<\sqrt{\mu}$),
$O\left(\sqrt{\lambda_{K}/\mu}\right)$ is small by definition.

###  V-D Estimation for Complexity of Exhaustive Search

In this subsection, we briefly discuss the complexity of the exhaustive search
approach to solve Problem $\mathbf{P}$. First, as explained in details in our
technical report [26], we observe that at least one of the optimal solutions
to $\mathbf{P}$, $(\bm{\eta}^{*},\mathbf{A}^{*})$, has the following property:
there is at most one $k\in\\{1,2,\ldots K\\}$ such that
$\eta_{\min,k}<\eta^{*}_{k}<\eta_{\max,k}$; $\forall j\neq k$, either
$\eta^{*}_{j}=\eta_{\min,j}$ or $\eta^{*}_{j}=\eta_{\max,j}$ (i.e., at the
boundary). Thus, the search for $\bm{\eta}^{*}$ needs to be performed only at
these boundary cases, leading to a complexity of $\Omega(2^{K})$. Furthermore,
in the dense UE scenario, a numerical search over all locally optimal
$\mathbf{A}$ (at least $2^{K}$ of them) is required, leading to another fold
of $\Omega(2^{K})$, i.e., $\Omega(4^{K})$ overall, in complexity. Numerical
studies on the computational complexity will be presented in Section VII.

##  VI Nash Equilibrium for SSAUA

Individual UEs may behave selfishly to derive unfair advantage despite our
design of $\mathbf{B}^{*}$ (or equivalently $\mathbf{A}^{*}$). Thus, in this
section, we propose a Surcharge Pricing Scheme (SPS), such that the designed
$\mathbf{B}^{*}$ is the natural outcome of a Nash equilibrium. Note that the
designed spectrum allocation factors $\bm{\eta}^{*}$ can be maintained by the
network operator and is beyond our concern.

We consider a reference individual UE, whose association bias values are
$\mathbf{B}^{\prime}=(B_{1}^{\prime},B_{2}^{\prime},\ldots,B_{K}^{\prime})$.
Let
$\mathbf{A}^{\prime}=(A_{1}^{\prime},A_{2}^{\prime},\ldots,A_{K}^{\prime})$ be
its corresponding association probabilities. For the other UEs, suppose they
all obey the association bias values $\mathbf{B}^{*}$ assigned by the network
operator. Similar to the discussions in Section III and IV, the average data
rate of the reference UE is

$\displaystyle\mathbb{F}=\sum_{k=1}^{K}\frac{\eta^{*}_{k}W\log(1+T)}{(A^{*}_{k}\mu/\lambda_{k}+1)(\frac{1}{A_{k}^{\prime}}+C)}.$
(43)

If the reference UE performs an optimization on $\mathbb{F}$ with respect to
$\mathbf{A}^{\prime}$, the resultant optimal
$\mathbf{A}^{\prime*}=(A^{\prime*}_{1},A^{\prime*}_{2},\ldots,A^{\prime*}_{K})$
is unlikely to be the same as $\mathbf{A}^{*}$. Therefore, we add the
following Surcharge Pricing Scheme: the network operator applies a surcharge
$c_{k}$ to each UE associated with a tier-$k$ BS. Let
$\mathbf{c}=(c_{1},c_{2},\ldots,c_{K})$. In this case, the average surcharge
for the reference UE is $\sum_{k=1}^{K}c_{k}A_{k}^{\prime}$. Accordingly, the
reference UE will perform the following optimization Problem $\mathbf{P3}$:

$\displaystyle\underset{\mathbf{A}^{\prime}}{\text{maximize}}$
$\displaystyle\mathbb{F}^{\prime}=\sum_{k=1}^{K}\left(\frac{\eta^{*}_{k}W\log(1+T)}{\left(A^{*}_{k}\mu/\lambda_{k}+1\right)\left(\frac{1}{A_{k}^{\prime}}+C\right)}-c_{k}A_{k}^{\prime}\right)$
subject to $\displaystyle\sum_{k=1}^{K}A_{k}^{\prime}=1,\quad
A_{k}^{\prime}\geq 0.$ (44)

Different from $\mathbf{P}$, it can be shown that $\mathbf{P3}$ is a standard
convex optimization problem. By the KKT conditions, its optimal solution
$\mathbf{A}^{\prime*}$ satisfies

$\displaystyle\frac{H_{k}}{(1+CA^{\prime*}_{k})^{2}}-c_{k}-\nu+\theta_{k}=0,$
(45) $\displaystyle\theta_{k}A^{\prime*}_{k}=0,\quad\theta_{k}\geq 0,$ (46)

where $H_{k}=\frac{\eta^{*}_{k}W\log(1+T)}{A^{*}_{k}\mu/\lambda_{k}+1}$,
$\theta_{k}$ is a Lagrange multiplier corresponding to the inequality
constraint $A_{k}^{\prime}\geq 0$, and $\nu$ is a Lagrange multiplier
corresponding to the equality constraint $\sum_{k=1}^{K}A_{k}^{\prime}=1$.

Setting $A^{\prime*}_{k}=A^{*}_{k}$, we have

$\displaystyle c_{k}=\begin{cases}\infty,&\textrm{if }A^{*}_{k}=0,\\\
\frac{H_{k}}{(1+CA^{*}_{k})^{2}}-\nu,&\textrm{otherwise}.\end{cases}$ (47)

Note that $\nu$ could be set arbitrarily due to the equality constraint.
Without loss of generality, we set
$\nu=\min_{k}\frac{H_{k}}{(1+CA^{*}_{k})^{2}}$ so that the minimum surcharge
among tiers is $0$. As a consequence, a Nash Equilibrium is achieved where
every UE adopts the assigned $\mathbf{B}^{*}$.

##  VII Numerical Study

Figure 2: Average UE data rate under different UE density $\mu$.

(a) Sparse UE scenario. (b) Dense UE scenario.

Figure 3: Average UE data rate under different path loss exponent $\gamma$.
Figure 4: Designed association bias values under different UE density $\mu$.
Figure 5: Pricing values under different UE density $\mu$. Figure 6:
Comparison of run time.

In this section, we present numerical studies on the performance of SSAUA. We
label the SSAUA solution as
$(\widehat{\bm{\eta}}^{*},\widehat{\mathbf{B}}^{*})$ and
$(\widetilde{\bm{\eta}}^{*},\widetilde{\mathbf{B}}^{*})$ in the sparse and
dense UE scenarios, respectively. Note that
$(\widehat{\bm{\eta}}^{*},\widehat{\mathbf{B}}^{*})$ is optimal in the sparse
UE scenario. We also compare
$(\widetilde{\bm{\eta}}^{*},\widetilde{\mathbf{B}}^{*})$ with an optimal
solution $(\bm{\eta}^{*},\mathbf{B}^{*})$ obtained from exhaustive search in
the dense UE scenario.

First, we study the network performance under different UE density $\mu$. The
network parameters are as follows: $K=3$, $\lambda_{1}=1$ units/km2,
$\lambda_{2}=5$ units/km2, $\lambda_{3}=10$ units/km2, $P_{1}=56$ dBm,
$P_{2}=46$ dBm, $P_{3}=36$ dBm, $\eta_{\min,1}=0.2$, $\eta_{\min,2}=0.25$,
$\eta_{\min,3}=0.3$, $\eta_{\max,1}=0.35$, $\eta_{\max,2}=0.4$,
$\eta_{\max,3}=0.45$, $\gamma=4$, $W=200$ MHz, and $T=0.2$. In each round of
simulation, UEs and BSs are generated on a $10$ km $\times$ $10$ km square,
and the UEs in the central $5$ km $\times$ $5$ km square are sampled for
performance evaluation (in order to remove the edge effect). Each simulation
data point is averaged over all sampled UEs during 100 rounds of simulations.

The results are shown in Fig. 2. A vertical line indicates the threshold value
of $\mu$, as given in Theorem 1, separating the sparse and dense UE scenarios.
For both scenarios, we show results of the analytical and simulated
performance of SSAUA, and analytical and simulated performance of a “max
power” approach, which employs equal spectrum allocation and user association
based on the maximum received power. Since SSAUA is not optimal in the dense
UE scenario, we also add two sets of results accordingly: the analytical
optimal performance $\mathbf{F}(\bm{\eta}^{*},\mathbf{B}^{*})$ through
exhaustive search and its analytical upper bound
$\mathbf{F^{\prime}}(\widetilde{\bm{\eta}}^{*},\widetilde{\mathbf{B}}^{*})$.

Fig. 2 illustrates that the SSAUA achieves near-optimal solution in the dense
UE scenario. Furthermore, SSAUA substantially outperforms the max power
solution. Finally, the analytical performance is only slightly smaller than
the simulated performance, matching our discussions in Section IV.

Fig. 3 shows the optimal network performance under different path loss
exponent $\gamma$. The network parameters are the same as those used in Fig. 2
except $\mu$ is fixed at $100$ (i.e., sparse UE) and $500$ (i.e., dense UE)
units/km2 in Fig. 3 (a) and (b) respectively. This figure further confirms the
observations from Fig. 2. Furthermore, it shows that SSAUA is effective for a
wide range of path loss conditions.

Fig. 4 shows $\widehat{\mathbf{B}}^{*}$, $\widetilde{\mathbf{B}}^{*}$, and
$\mathbf{B}^{*}$; and Fig. 5 shows their corresponding prices
$\widehat{\mathbf{c}}^{*}$, $\widetilde{\mathbf{c}}^{*}$, and
$\mathbf{c}^{*}$, under different $\mu$. We observe that
$\widetilde{\mathbf{B}}^{*}$ and $\widetilde{\mathbf{c}}^{*}$ computed based
on SSAUA approach are close to their counterparts $\mathbf{B}^{*}$ and
$\mathbf{c}^{*}$.

Finally, a run time experiment is conducted to compare the computational
complexity of SSAUA with that of exhaustive search. The experiment is executed
by Matlab R2011a on an ASUS PC with Intel i7-3610QM 2.3GHz processor and 4GB
RAM. The results are averaged over 1000 runs for SSAUA and 10 runs for
exhaustive search (both with randomly generated parameters). Fig. 6 shows that
the run time of SSAUA is almost negligible compared with exhaustive search.
Note that the $y$-axis is in log scale. When $K$ increases, the run time of
exhaustive search exhibits an exponential growth tendency, while SSAUA remains
computationally efficient.

##  VIII Conclusions

In this work, we provide a theoretical framework to study the joint
optimization of spectrum allocation and user association in heterogeneous
cellular networks. We establish a stochastic geometric model that captures the
random spatial patterns of BSs and UEs, and a closed-form expression of the
analytical average UE data rate is derived. We then consider the problem of
maximizing the average UE data rate by jointly optimizing the spectrum
allocation factors $\bm{\eta}$ and user association bias values $\mathbf{B}$,
which is non-convex programming in nature. We propose the SSAUA approach to
solve this problem with low computational complexity. We show that the SSAUA
approach is optimal in the sparse UE scenario and near-optimal in the dense UE
scenario, with a quantified tight bound scaling as
$O(\sqrt{\lambda_{K}/\mu})$. We also propose the SPS such that the designed
association bias values can be achieved in Nash equilibrium.

###  -A Useful Properties of $M_{k}(A_{k})$

(M-1)

$M_{k}(A_{k})$ is increasing on $[0,a_{k}]$ and decreasing on
$[a_{k},\infty)$.

(M-2)

$M_{k}(A_{k})$ is concave on $[0,a^{\prime}_{k}]$ and convex on
$[a^{\prime}_{k},\infty)$, where $a^{\prime}_{k}$ is some threshold value
$a^{\prime}_{k}>a_{k}$. $M_{k}(A_{k})$ is concave on $[0,a_{k}]$.

(M-3)

If $\lambda_{i}<\lambda_{j}$, then $M_{i}(A)<M_{j}(A),\forall A>0$.

(M-4)

If $\lambda_{i}<\lambda_{j}$, then $M_{j}(A)-M_{i}(A)$ is strictly increasing
on $[0,a_{j}]$.

###  -B Proof of Lemma 1

###### Proof:

Suppose $\mathbf{A}^{**}$ is optimal, $A^{**}_{i}<a_{i}$, and
$A^{**}_{j}>a_{j}$. Consider that we increase $A^{**}_{i}$ by a small value
$\Delta>0$ and decrease $A^{**}_{j}$ by $\Delta$. According to property (M-1),
$\eta^{**}_{i}M_{i}(A^{**}_{i})+\eta^{**}_{j}M_{j}(A^{**}_{j})<\eta^{**}_{i}M_{i}(A^{**}_{i}+\Delta)+\eta^{**}_{j}M_{j}(A^{**}_{j}-\Delta)$.
Thus, through replacing $A^{**}_{i}$ and $A^{**}_{j}$ by $A^{**}_{i}+\Delta$
and $A^{**}_{j}-\Delta$ respectively, we find a better solution to
$\mathbf{P}$, leading to a contradiction. ∎

###  -C Proof of Lemma 2

###### Proof:

Suppose $\exists i<j$ such that $M_{i}(A^{*}_{i})>M_{j}(A^{*}_{j})$. This
implies that $A^{*}_{i}>A^{*}_{j}$. (Otherwise, suppose $A^{*}_{i}\leq
A^{*}_{j}$, then we have $M_{i}(A^{*}_{i})\leq
M_{i}(A^{*}_{j})<M_{j}(A^{*}_{j})$, leading to a contradiction.) A
corresponding diagram is shown in Fig. 7.

Case 1: $\eta^{*}_{i}\leq\eta^{*}_{j}$.

Let $\widehat{A}^{*}_{j}=A^{*}_{i}$ and $\widehat{A}^{*}_{i}=A^{*}_{j}$, then
we have

$\displaystyle[\eta^{*}_{i}M_{i}(\widehat{A}^{*}_{i})+\eta^{*}_{j}M_{j}(\widehat{A}^{*}_{j})]-[\eta^{*}_{i}M_{i}(A^{*}_{i})+\eta^{*}_{j}M_{j}(A^{*}_{j})]$
$\displaystyle=$
$\displaystyle[\eta^{*}_{i}M_{i}(A^{*}_{j})+\eta^{*}_{j}M_{j}(A^{*}_{i})]-[\eta^{*}_{i}M_{i}(A^{*}_{i})+\eta^{*}_{j}M_{j}(A^{*}_{j})]$
$\displaystyle=$
$\displaystyle\eta^{*}_{j}[M_{j}(A^{*}_{i})-M_{j}(A^{*}_{j})]+\eta^{*}_{i}[M_{i}(A^{*}_{j})-M_{i}(A^{*}_{i})]$
$\displaystyle\geq$
$\displaystyle\eta^{*}_{i}[M_{j}(A^{*}_{i})-M_{j}(A^{*}_{j})+M_{i}(A^{*}_{j})-M_{i}(A^{*}_{i})]>0,$
(48)

where (48) is due to property (M-4).

As a consequence, if $A^{*}_{i}$ and $A^{*}_{j}$ are replaced by
$\widehat{A}^{*}_{i}$ and $\widehat{A}^{*}_{j}$ respectively, we obtain a
larger $\mathbf{F}$, leading to a contradiction.

Case 2: $\eta^{*}_{i}>\eta^{*}_{j}$.

Let $\widehat{A}^{*}_{j}=A^{*}_{i}$, $\widehat{A}^{*}_{i}=A^{*}_{j}$,
$\widehat{\eta}^{*}_{j}=\eta^{*}_{i}$, and
$\widehat{\eta}^{*}_{i}=\eta^{*}_{j}$. (Note that because
$\eta_{\min,i}\leq\eta_{\min,j}$ and $\eta_{\max,i}\leq\eta_{\max,j}$,
$\widehat{\eta}^{*}_{j}$ and $\widehat{\eta}^{*}_{i}$ are guaranteed to be in
the feasible region.)

$\displaystyle[\widehat{\eta}^{*}_{i}M_{i}(\widehat{A}^{*}_{i})+\widehat{\eta}^{*}_{j}M_{j}(\widehat{A}^{*}_{j})]-[\eta^{*}_{i}M_{i}(A^{*}_{i})+\eta^{*}_{j}M_{j}(A^{*}_{j})]$
$\displaystyle=$
$\displaystyle[\eta^{*}_{j}M_{i}(A^{*}_{j})+\eta^{*}_{i}M_{j}(A^{*}_{i})]-[\eta^{*}_{i}M_{i}(A^{*}_{i})+\eta^{*}_{j}M_{j}(A^{*}_{j})]$
$\displaystyle=$
$\displaystyle\eta^{*}_{i}[M_{j}(A^{*}_{i})-M_{i}(A^{*}_{i})]+\eta^{*}_{j}[M_{i}(A^{*}_{j})-M_{j}(A^{*}_{j})]$
$\displaystyle>$
$\displaystyle\eta^{*}_{j}[M_{j}(A^{*}_{i})-M_{i}(A^{*}_{i})+M_{i}(A^{*}_{j})-M_{j}(A^{*}_{j})]>0.$
(49)

Thus, if $A^{*}_{i}$, $A^{*}_{j}$, $\eta^{*}_{i}$, and $\eta^{*}_{j}$ are
replaced by $\widehat{A}^{*}_{i}$, $\widehat{A}^{*}_{j}$,
$\widehat{\eta}^{*}_{i}$, and $\widehat{\eta}^{*}_{j}$ respectively, we can
find a larger $\mathbf{F}$, leading to a contradiction. ∎

Figure 7: Diagram of $M_{i}(\cdot)$ and $M_{j}(\cdot)$.

###  -D Useful Properties of $\widetilde{M}_{k}(A_{k})$

(M-1’)

$\widetilde{M}_{k}(A_{k})$ is a decreasing convex function.

(M-2’)

If $\lambda_{i}<\lambda_{j}$,
$\widetilde{M}_{i}(a_{i})<\widetilde{M}_{j}(a_{j})$.

(M-3’)

If $\lambda_{i}<\lambda_{j}$, then $\widetilde{M}_{j}(A)-\widetilde{M}_{i}(A)$
is a strictly decreasing function.

(M-4’)

$\widetilde{M}_{k}(A)-\widetilde{M}_{k}(A+D)>\widetilde{M}_{k}(A^{\prime})-\widetilde{M}_{k}(A^{\prime}+D)$,
for any $A^{\prime}>A\geq a_{k}$ and $D>0$.

(M-5’)

If $\lambda_{i}<\lambda_{j}$, then
$\widetilde{M}_{j}(a_{j})-\widetilde{M}_{j}(a_{j}+D)>\widetilde{M}_{i}(a_{i})-\widetilde{M}_{i}(a_{i}+D)$,
for any $D>0$.

(M-6’)

If $\lambda_{i}<\lambda_{j}$, then
$\widetilde{M}_{j}(a_{j})-\widetilde{M}_{j}(a_{j}+D)>\widetilde{M}_{i}(A^{\prime})-\widetilde{M}_{i}(A^{\prime}+D)$,
for any $D>0$ and $A^{\prime}>a_{i}$ (combining (M-4’) and (M-5’)).

###  -E Proof of Lemma 3

###### Proof:

Suppose that $\exists i<j$ (i.e., $\lambda_{i}<\lambda_{j}$) such that
$\widetilde{M}_{i}(\widetilde{A}^{*}_{i})>\widetilde{M}_{j}(\widetilde{A}^{*}_{j})$,
which also implies that
$a_{i}\leq\widetilde{A}^{*}_{i}<\widetilde{A}^{*}_{j}$. The corresponding
diagrams are shown in Fig. 8.

Case 1: $\widetilde{\eta}^{*}_{i}\leq\widetilde{\eta}^{*}_{j}$.

Case 1.1: $\widetilde{A}^{*}_{i}\geq a_{j}$.

Let $\widehat{A}^{*}_{j}=\widetilde{A}^{*}_{i}$ and
$\widehat{A}^{*}_{i}=\widetilde{A}^{*}_{j}$, then we have

$\displaystyle[\widetilde{\eta}^{*}_{i}\widetilde{M}_{i}(\widehat{A}^{*}_{i})+\widetilde{\eta}^{*}_{j}\widetilde{M}_{j}(\widehat{A}^{*}_{j})]-[\widetilde{\eta}^{*}_{i}\widetilde{M}_{i}(\widetilde{A}^{*}_{i})+\widetilde{\eta}^{*}_{j}\widetilde{M}_{j}(\widetilde{A}^{*}_{j})]$
$\displaystyle=$
$\displaystyle[\widetilde{\eta}^{*}_{i}\widetilde{M}_{i}(\widetilde{A}^{*}_{j})+\widetilde{\eta}^{*}_{j}\widetilde{M}_{j}(\widetilde{A}^{*}_{i})]-[\widetilde{\eta}^{*}_{i}\widetilde{M}_{i}(\widetilde{A}^{*}_{i})+\widetilde{\eta}^{*}_{j}\widetilde{M}_{j}(\widetilde{A}^{*}_{j})]$
$\displaystyle=$
$\displaystyle\widetilde{\eta}^{*}_{j}[\widetilde{M}_{j}(\widetilde{A}^{*}_{i})-\widetilde{M}_{j}(\widetilde{A}^{*}_{j})]+\widetilde{\eta}^{*}_{i}[\widetilde{M}_{i}(\widetilde{A}^{*}_{j})-\widetilde{M}_{i}(\widetilde{A}^{*}_{i})]$
$\displaystyle\geq$
$\displaystyle\widetilde{\eta}^{*}_{i}[\widetilde{M}_{j}(\widetilde{A}^{*}_{i})-\widetilde{M}_{j}(\widetilde{A}^{*}_{j})+\widetilde{M}_{i}(\widetilde{A}^{*}_{j})-\widetilde{M}_{i}(\widetilde{A}^{*}_{i})]>0,$
(50)

where (50) is due to property (M-3’).

Thus, if $\widetilde{A}^{*}_{i}$ and $\widetilde{A}^{*}_{j}$ are replaced by
$\widehat{A}^{*}_{i}$ and $\widehat{A}^{*}_{j}$ respectively, we obtain a
larger $\mathbf{F}^{\prime}$, leading to a contradiction.

Case 1.2: $\widetilde{A}^{*}_{i}<a_{j}$.

Let $\widehat{A}^{*}_{j}=a_{j}$, $D=\widetilde{A}^{*}_{j}-a_{j}$ and
$\widehat{A}^{*}_{i}=\widetilde{A}^{*}_{i}+D$, then we have

$\displaystyle[\widetilde{\eta}^{*}_{i}\widetilde{M}_{i}(\widehat{A}^{*}_{i})+\widetilde{\eta}^{*}_{j}\widetilde{M}_{j}(\widehat{A}^{*}_{j})]-[\widetilde{\eta}^{*}_{i}\widetilde{M}_{i}(\widetilde{A}^{*}_{i})+\widetilde{\eta}^{*}_{j}\widetilde{M}_{j}(\widetilde{A}^{*}_{j})]$
$\displaystyle=$
$\displaystyle[\widetilde{\eta}^{*}_{i}\widetilde{M}_{i}(\widetilde{A}^{*}_{i}+D)+\widetilde{\eta}^{*}_{j}\widetilde{M}_{j}(a_{j})]-[\widetilde{\eta}^{*}_{i}\widetilde{M}_{i}(\widetilde{A}^{*}_{i})+\widetilde{\eta}^{*}_{j}\widetilde{M}_{j}(\widetilde{A}^{*}_{j})]$
$\displaystyle=$
$\displaystyle\widetilde{\eta}^{*}_{j}[\widetilde{M}_{j}(a_{j})-\widetilde{M}_{j}(\widetilde{A}^{*}_{j})]+\widetilde{\eta}^{*}_{i}[\widetilde{M}_{i}(\widetilde{A}^{*}_{i}+D)-\widetilde{M}_{i}(\widetilde{A}^{*}_{i})]$
$\displaystyle\geq$
$\displaystyle\widetilde{\eta}^{*}_{i}[\widetilde{M}_{j}(a_{j})-\widetilde{M}_{j}(\widetilde{A}^{*}_{j})+\widetilde{M}_{i}(\widetilde{A}^{*}_{i}+D)-\widetilde{M}_{i}(\widetilde{A}^{*}_{i})]>0,$
(51)

where (51) is due to property (M-6’).

Thus, if $\widetilde{A}^{*}_{i}$ and $\widetilde{A}^{*}_{j}$ are replaced by
$\widehat{A}^{*}_{i}$ and $\widehat{A}^{*}_{j}$ respectively, we obtain a
larger $\mathbf{F}^{\prime}$, leading to a contradiction.

Case 2: $\widetilde{\eta}^{*}_{i}>\widetilde{\eta}^{*}_{j}$. Case 2.1:
$\widetilde{A}^{*}_{i}\geq a_{j}$. Case 2.2: $\widetilde{A}^{*}_{i}<a_{j}$.
The proof for these cases can be found in our technical report [26]. ∎

(a) Case 1.1 and 2.1.

(b) Case 1.2 and 2.2.

Figure 8: Diagrams of $\widetilde{M}_{i}(\cdot)$ and
$\widetilde{M}_{j}(\cdot)$.

###  -F Proof of Theorem 4

###### Proof:

Suppose $\exists k\geq 2$ such that $\widetilde{A}^{*}_{k}>a_{k}$. Let $l=1$,
$\widehat{A}^{*}_{k}=a_{k}$, $D=\widetilde{A}^{*}_{k}-a_{k}$, and
$\widehat{A}^{*}_{l}=\widetilde{A}^{*}_{l}+D$. Similar to the proof of Lemma
3, we can show that if we replace $\widetilde{A}^{*}_{l}$ and
$\widetilde{A}^{*}_{k}$ by $\widehat{A}^{*}_{l}$ and $\widehat{A}^{*}_{k}$
respectively, we find a better solution to Problem $\mathbf{P2A}$, which leads
to a contradiction. See [26] for details.

∎

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# Enhancing Trust and Security in the Vehicular Metaverse: A Reputation-Based
Mechanism for Participants with Moral Hazard

Lotfi Ismail1, Marwa Qaraqe1, Ali Ghrayeb2, Dusit Niyato3  1 Division of
Information and Computing Technology, College of Science and Engineering,
Hamad Bin Khalifa University, Qatar Foundation, Doha, Qatar.  2 Electrical and
Computer Engineering (ECE) department, Texas A&M University at Qatar, Doha,
Qatar.  3School of Computer Science and Engineering, Nanyang Technological
University, Singapore

###### Abstract

In this paper, we tackle the issue of moral hazard within the realm of the
vehicular Metaverse. A pivotal facilitator of the vehicular Metaverse is the
effective orchestration of its market elements, primarily comprised of sensing
internet of things (SIoT) devices. These SIoT devices play a critical role by
furnishing the virtual service provider (VSP) with real-time sensing data,
allowing for the faithful replication of the physical environment within the
virtual realm. However, SIoT devices with intentional misbehavior can identify
a loophole in the system post-payment and proceeds to deliver falsified
content, which cause the whole vehicular Metaverse to collapse. To combat this
significant problem, we propose an incentive mechanism centered around a
reputation-based strategy. Specifically, the concept involves maintaining
reputation scores for participants based on their interactions with the VSP.
These scores are derived from feedback received by the VSP from Metaverse
users regarding the content delivered by the VSP and are managed using a
subjective logic model. Nevertheless, to prevent “good” SIoT devices with
false positive ratings to leave the Metaverse market, we build a vanishing-
like system of previous ratings so that the VSP can make informed decisions
based on the most recent and accurate data available. Finally, we validate our
proposed model through extensive simulations. Our primary results show that
our mechanism can efficiently prevent malicious devices from starting their
poisoning attacks. At the same time, trustworthy SIoT devices that had a
previous miss-classification are not banned from the market.

###### Index Terms:

Auction theory, digital twin, metaverse, reputation mechanism.

##  I Introduction

The Metaverse, regarded as the next generation of the internet, is spreading
recently over several applications such as remote workplaces, vehicular
systems and online banking, amongst others [1, 2]. Several technologies such
as virtual reality (VR), augmented reality (AR) and artificial intelligence
(AI) are key enablers of the Metaverse. With the release of Meta’s Quest 3,
mixed reality (XR), another key enabler of the Metaverse, is closer to
realization than ever before111https://www.meta.com/quest/quest-3/. As the
majority of the Metaverse users are expected to be mobile, e.g., vehicular
Metaverse, next-generation mobile networks should be well designed to support
the different Metaverse services.

The first step towards exploring the benefits of vehicular Metaverse is the
rendering of the physical world into its digital twin. For this objective, the
VSP recruits sensing IoT (SIoT) devices (e.g., UAVs, intelligent cars and CCTV
cameras) in the field to collect data about the physical environment and then
deliver that data to the VSP. Nevertheless, as the majority of this collected
data consists of images and video scenes, the network load increases
exponentially and reaches its limit. Additionally, appropriate mechanisms
should be designed to incentivize SIoT devices to share their data with the
VSP. To address these challenges, existing work proposed the use of semantic
communication to reduce the data load and enable the VSP to collect more data
with high freshness levels [3, 2, 4]. The idea of semantic communication
consists of applying machine learning (ML) algorithms to extract the semantic
information from the raw data then delivers only the extracted information to
the VSP. To incentivize SIoT devices, auction theory and contract theory were
adapted to design mechanisms that guarantee the properties of individual
rationality (IR) and incentive compatibility (IC) [3, 2].

However, the integrity of the delivered data might be unsecured due to the
moral hazard problem. Although incentive mechanisms with IR and IC properties
guaranteed prevent adverse selection problem, the moral hazard problem remains
unsolved. While the adverse selection problem appears before the agreement on
the contract between the two parties (i.e., the VSP and the SIoT devices), the
moral hazard problem appears after the agreement. Specifically, the SIoT
device can deliver content that does not meet the requirements of the VSP as
agreed initially. With the emergence of adversarial attacks [5], it becomes
difficult to differentiate between legitimate content and malicious content.
Importantly, in the scenario of vehicular Metaverse, a malicious SIoT device
can launch a poisoning attack on the delivered semantic data to cause
perturbation of the Metaverse system, which can degrade the quality of service
(QoS) and the quality of experience (QoE) of the Metaverse users
significantly. Even more concerning is the potential for life-threatening
attacks capable of triggering accidents in the physical world [1].

Although the Metaverse has attracted the communication and networking
researchers recently, a number of works are already available to address a
number of security issues in these systems. In [6], a moving target defense
(MTD) strategy was proposed to secure digital twins over mobile networks where
the network characteristics are altered to disrupt various phases of the cyber
kill chain. The problem is then formulated using a semi-Markov decision
process and solved using a deep reinforcement learning algorithm. The idea
behind MTD schemes consists of dynamically modifying relevant network
properties, such as transmission routes and IP addresses, to disrupt the
adversary’s prior knowledge. However, the problem addressed in [6] is quite
different from our focus in this work. Specifically, the main problem
addressed in [6] is related to preventing distributed denial-of-service (DDoS)
attacks while our focus is on mitigating poisoning attacks on vehicular
Metaverse from SIoT devices with moral hazard.

To this end, the problem of moral hazard in the vehicular Metaverse should be
urgently addressed before their deployment. To the best of our knowledge, no
previous work has addressed the problem of moral hazard in Metaverse services.
We address this gap by proposing a greedy algorithm based on subjective logic
and reputation mechanism to incentivize SIoT devices to participate in the
Metaverse services truthfully at all times. In summary, our main contributions
are as follows:

  * •

We introduce a novel approach to address the issue of moral hazard within the
context of the vehicular Metaverse, providing a unique perspective on trust
and security concerns in this emerging technology. The proposed reputation-
based mechanism effectively deters dishonest behavior by participants,
enhancing the overall integrity of the system.

  * •

We use subjective logic to model the reputation of each SIoT device and
introduce a vanishing-like system to manage previous ratings. This innovative
approach helps prevent reputable SIoT devices from leaving the market due to
occasional false positive ratings.

  * •

The extensive simulations conducted to validate the proposed model provide
empirical evidence of its effectiveness. This empirical validation reinforces
the practical applicability and robustness of the proposed mechanism.

##  II System Model

###  II-A System Model Description

Fig. 1 describes our proposed system model. We examine a Metaverse market
comprising a VSP, a set of $\mathcal{N}=\\{1,\dots,i,\dots,N\\}$ IoT devices,
acting as data owners and a set of $\mathcal{M}=\\{1,\dots,j,\dots,M\\}$
vehicular Metaverse users (VMUs). The VSP’s role is to construct the digital
twin within the Metaverse using data gathered from the set of chosen SIoT
devices and then deliver the construct digital twin to the VMUs. Each SIoT
device is equipped with a suite of sensors for gathering geo-spatial data from
its immediate surroundings, along with machine learning (ML) models designed
to extract semantic information from the collected raw data. The SIoT devices
send the extracted semantic data to the VSP instead of all the raw data which
brings more efficiency to the network load [3]. Given multiple sellers
(representing the IoT devices) and a single buyer (the VSP), the choice of a
reverse auction, which essentially reverses the traditional roles of buyers
and sellers, aligning well with our system design [7]. Nevertheless, an
auction mechanism that guarantees an optimal solution and satisfies both IR
and IC conditions only addresses the problem of adverse selection which
prevents a malicious participant from selecting a bundle not dedicated for
them. The auction mechanism still faces another major security threat where
participants with moral hazards can inject a poisoning attack into the
Metaverse system. The VSP should be able to prevent such attacks in real-time
to avoid degradation of its services.

Figure 1: System model of anti-poisoning attacks vehicular Metaverse
ecosystem.

In what follows, we first start by highlighting the major type of poisoning
attacks that can be launched by malicious SIoT devices with moral hazard. We
then present an overview of our proposed system design to build a robust
Metaverse system that can efficiently reduce the occurrence of such poisoning
attacks.

###  II-B Poisoning Attack Models

The attackers are considered to follow a randomized strategy to attack the
vehicular Metaverse system. Specifically, a malicious SIoT device alternates
between sending legitimate semantic data and sending falsified semantic data
to lure the VSP. Different types of poisoning attacks can be launched by
malicious SIoT devices. Here, we highlight three major types of attacks:

  * •

Reducing service cost: As the continuous collection of sensing data and the
execution of semantic data extraction algorithms require continuous energy
consumption, a selfish SIoT device can choose to deliver semantic data that is
based on outdated sensing data. This action will cause the malicious SIoT
device to increase its utility (as lower energy consumption is required, see
(5)) while the rendered digital twin will not reflect real-time dynamics of
the physical world, causing degraded QoE of the VMUs.

  * •

Digital twin tampering: While the previous type of attack does not have an
objective of causing damages to the Metaverse services, the digital twin
tampering is initiated by a malicious SIoT device to cause severe degradation
of the Metaverse services. For instance, after extracting the semantic from
real-time images of the physical world, the attacker alters the position or
the type of certain object so as the experience of the Metaverse users is
degraded to the maximum. Interestingly, if the actions taken in the digital
twin are reflected in the physical world, serious safety issues can arise,
e.g., car accidents. Fig. 2 illustrates an instance of such attack.

Figure 2: Exemplary case of semantic attack where the attacker hides the
person on the right side of the image.

  * •

Man-in-the-middle attack: This attack can be launched from an SIoT device with
low security defense levels, e.g., data encryption is compromised. The
attacker transmits their altered data instead of the legitimate SIoT device
data. The VSP has no visibility of the existence of the attacker in the middle
of the transmission. Therefore, if the attacker is detected, the compromised
SIoT device is flagged as an untrustworthy device.

###  II-C System Overflow

  * •

First, as described in Fig. 1, the VSP initiates a request to collect physical
world data from SIoT devices in the field using a reverse auction mechanism.
Interested SIoT devices located in the region of the VSP’ interest express
their willingness to sell their data with their offered prices. Specifically,
SIoT device $i$ reports its type $t_{i}=\\{b_{i},\hat{s}_{i}\\}$ to the VSP,
where $b_{i}$ indicates the price offered for its data and $\hat{s}_{i}$
represent the semantic value of the offered data. The semantic value of the
data, $\hat{s}_{i}$, is calculated as in [3] and includes parameters such as
weather condition, number of detected objects and their positions. For
instance, in adverse weather conditions, the valuation of the derived semantic
information is expected to rise, indicating its heightened significance to the
VSP. Additionally, as SIoT devices require a specific number of channels to
deliver their semantic data based on their size, each SIoT device $i$ also
reports $C_{i}$, the number of channels required, to the VSP. Later, we show
how the VSP uses this information when selecting the set of winners (i.e., the
set of SIoT devices who won the auction and are allowed to deliver their data
to the VSP).

  * •

After receiving the offers from all SIoT devices, the VSP first uses the
reputation database it has constructed over the previous interactions with
SIoT devices and VMUs to decide which SIoT devices are more likely to be
trustworthy, which are denoted by $\mathcal{N}^{{}^{\prime}}$. The set of
$\mathcal{N}^{{}^{\prime}}$ SIoT devices is then used in the winner selection
algorithm to decide the final list of SIoT devices from which the VSP buys the
semantic data with the given payments.

  * •

Once the VSP receives the semantic data from the chosen SIoT devices, the
digital twin is then constructed in the vehicular Metaverse and then delivered
to the VMUs. Based on the quality of service experienced by the VMUs, feedback
ratings are sent by the VMUs to the VSP to update the reputation score of all
participating SIoT devices. In the next round of semantic data collection, the
VSP uses the updated reputation scores to select trustworthy SIoT devices.

##  III Problem Formulation And Methods

In this section, we first present the formulation of reputation management for
SIoT devices utilizing subjective logic. Subsequently, we detail our approach
to effectively filter reputations obtained from the VMUs. Finally, we
formulate the reverse auction mechanism as a social welfare maximization
problem.

###  III-A Subjective Logic Model for Reputation Calculation

We use $r_{i:j}=\\{b_{i:j},d_{i:j},u_{i:j},\alpha_{i}\\}$ to denote reputation
score received from Metaverse user $j$ to SIoT device $i$. $b_{i:j}$,
$d_{i:j}$, $u_{i:j}$ and $\alpha_{i}$ represent belief, disbelief,
uncertainty, and the effective degree of uncertainty effect on reputation of
SIoT device $i$, respectively [8, 9]. $b_{i:j}+d_{i:j}+u_{i:j}=1$ and
$b_{i:j}$, $d_{i:j}$, $u_{i:j}$, $\alpha_{i}\in[0,1]$. We define $R_{i:j}$ as
the reputation of SIoT device $i$ given by VMU $j$, which is given by

$R_{i:j}=b_{i:j}+\alpha_{i}u_{i:j}.$ (1)

As the VSP interacts with SIoT devices and get feedback from VMUs over several
time periods, the local reputation for an SIoT device $i$ constructed at the
VSP is expressed based on this period. We denote $\tau$ as the effective
period of interaction and divide it into $Y$ time periods, i.e.,
$\\{t_{1},\dots,t_{y},\dots,t_{Y}\\}$. The reputation value of SIoT device $i$
based on feedback from VMU $j$ at timestep $t_{y}$ is then calculated as

$\begin{cases}b^{t_{y}}_{i:j}=\frac{\omega_{1}p^{t_{y}}_{i:j}}{\omega_{1}p^{t_{y}}_{i:j}+\omega_{2}q^{t_{y}}_{i:j}+\kappa},\\\
d^{t_{y}}_{i:j}=\frac{\omega_{2}q^{t_{y}}_{i:j}}{\omega_{1}p^{t_{y}}_{i:j}+\omega_{2}q^{t_{y}}_{i:j}+\kappa},\\\
u^{t_{y}}_{i:j}=\frac{\kappa}{\omega_{1}p^{t_{y}}_{i:j}+\omega_{2}q^{t_{y}}_{i:j}+\kappa}.\end{cases}$
(2)

where $p^{t_{y}}_{i:j}$ and $q^{t_{y}}_{i:j}$ represent the number of positive
and negative interactions between the SIoT device $i$ and VMU $j$,
respectively. $\omega_{1}$ and $\omega_{2}$ are weighting factors for positive
and negative interactions, respectively, and are considered to sum-up to 1,
i.e., $\omega_{1}+\omega_{2}=1$. $\kappa$ is a constant value that reflects
the rate of uncertainty and is set to $\kappa=1$ [8, 9].

Motivated by the fact that recent interactions where malicious devices attempt
to attack the Metaverse system are more important than older interactions, we
consider the following formulation to show their importance in calculating the
final reputation score. Recent interaction events hold greater significance
due to their freshness, carrying a higher weight than those from the past.
Specifically, we define the fading function that reflects this behavior as
$\psi(t_{y})=\psi_{y}=z^{Y-y}$, where $z\in(0,1)$ is a given fading parameter
and $y$ is the current time slot that determines the freshness degree of the
reputation value. Therefore, (2) is reformulated as

$\begin{cases}b^{fin}_{i:j}=\frac{\sum^{Y}_{y=1}\psi_{y}b^{t_{y}}_{i:j}}{\sum^{Y}_{y=1}\psi_{y}},\\\
d^{fin}_{i:j}=\frac{\sum^{Y}_{y=1}\psi_{y}d^{t_{y}}_{i:j}}{\sum^{Y}_{y=1}\psi_{y}},\\\
u^{fin}_{i:j}=\frac{\sum^{Y}_{y=1}\psi_{y}u^{t_{y}}_{i:j}}{\sum^{Y}_{y=1}\psi_{y}}.\end{cases}$
(3)

Therefore, the final reputation value of SIoT device $i$ from VMU $j$ is
calculated as $R^{fin}_{i:j}=b^{fin}_{i:j}+\alpha_{i}u^{fin}_{i:j}$. Finally,
we calculate the final reputation value of a SIoT device $i$ as the average
value of the reputation values received from all VMUs, i.e.,

$R^{fin}_{i}=\frac{\sum^{M}_{j=1}R^{fin}_{i:j}}{M}.$ (4)

###  III-B Reputation Backpropagation

One of our key contributions in this paper is the _reputation backpropagation
mechanism_ which is based on the human-in-the-loop design and is presented in
Fig. 3. Specifically, as there is no direct mapping between the VMUs and the
SIoT devices, an important issue to solve is how to backpropagate the feedback
received by a VMU at the VSP server to a specific SIoT device. Motivated by
the idea of backpropagation in deep neural networks (DNN)[10], we develop a
new approach to backpropagate the feedbacks received by the VSP from the VMUs.
The reputation backpropagation is conducted as follows:

  * •

Step 1  3D Grid : As the area of coverage by the $\mathcal{M}$ SIoT devices is
regarded as a 3D grid containing $K$ cubes, VMUs send feedbacks on regions of
the grid in which they interacted with. We denote the subset of SIoT devices
who covered cube $k$ as $\mathcal{S}_{k}\subset\mathcal{M}$.

  * •

Step 2  Negative feedbacks only: However, as the space of interaction between
the VMUs and the Metaverse interface is large (e.g., touch screen), we
consider that the VMUs send negative feedbacks only on regions, i.e., cubes in
the 3D grid, in which they experienced a low QoE. Therefore, not receiving a
feedback about the other cubes is considered as a positive feedback in the
reputation mechanism.

  * •

Step 3  Feedback Filtering : However, a low QoE in a specific cube is likely
attributed to the misbehavior of just one of the SIoT devices, rather than all
of them. Consequently, some SIoT devices may receive an inaccurate negative
rating. To mitigate the impact of this issue, reputations undergo a final
filtering process outlined in Algorithm 1.

Figure 3: Representation of the cube under interest.

Input : Initialize an empty set $\mathcal{N}^{-}$ for SIoT devices with
negative rating;

Output : The filtered set of SIoT devices, $\mathcal{N}^{-}$, which deserves a
negative rating;

1 begin

2

3 foreach  _$k\in K$_ do

4 Calculate pairwise similarity of the content delivered by SIoT devices in
the subset $\mathcal{S}_{k}$;

5 SIoT device with the lowest similarity score with the others is added to
$\mathcal{N}^{-}$;

6

7 end foreach

8 SIoT devices not in $\mathcal{N}^{-}$ are assigned a positive reputation;

9

10 end

11

Algorithm 1 Feedback Filtering Algorithm

###  III-C Social Welfare Maximization Reverse Auction

The social welfare, defined as the system efficiency [11], is a key metric in
evaluating the effectiveness of the Metaverse market system (i.e., selling the
semantic data by the SIoT devices to the VSP). Therefore, maximizing social
welfare signifies an efficient operation of the system. The major objective of
auction-based mechanism design is to prevent market manipulation by malicious
SIoT devices which can lead to an unjustified increase in their utilities.
Detailed description of our developed reverse auction mechanism that addresses
these issues are omitted here and can be found in [3]. Here, we briefly define
the utility of the SIoT devices and the VSP to formulate the social welfare of
the auction mechanism, which is necessary for the performance evaluation.

The SIoT device $i$ utility is defined as

$u_{i}=p_{i}-c_{i},$ (5)

where $p_{i}$ is the payment received from the VSP, and $c_{i}$ is the service
cost which covers the communication cost, data collection cost and execution
of the semantic extraction algorithms. The VSP’s utility is defined as

$\hat{u}=\sum\limits_{i\in\mathcal{N}}\xi_{i}R_{i}^{S}-\sum\limits_{i\in\mathcal{N}}\xi_{i}p_{i}-\hat{c},$
(6)

where $\xi_{i}$ is a binary variable denoting whether a SIoT device $i$ is
selected amongst the winners or not, $R_{i}^{S}$ is the value of the semantic
data delivered to the VSP by SIoT device $i$ and $\hat{c}$ is the cost for
allocation the wireless channel from spectrum service providers by the VSP.
The social welfare of the system is quantified as the aggregate of utilities
derived from all entities within the system, including both the VSP and the
SIoT devices and is written as

$S(\xi)=\sum\limits_{i\in\mathcal{N}}\xi_{i}\left(R_{i}^{S}-c_{i}\right)-\hat{c}.$
(7)

Finally, the social welfare maximization problem can be written as an integer
linear programming (ILP) subject to the availability of channels as follows

$\displaystyle\begin{split}\max_{\xi}S(\xi)=\sum\limits_{i\in\mathcal{N}}\xi_{i}\left(R_{i}^{S}-c_{i}\right)-\hat{c},\end{split}$
(8a) $\displaystyle\begin{split}\hskip
28.45274pts.t.\sum_{i\in\mathcal{N}}\xi_{i}C_{i}\leq B\end{split}$ (8b)
$\displaystyle\begin{split}\hskip 28.45274pt\xi_{i}\in\\{0,1\\},\forall
i\in\mathcal{N}\end{split}$ (8c)

where $C_{i}$ is the number of channels requested by SIoT device $i$ and $B$
represents the number of channels offered by the VSP. The ILP presented in (8)
can be straightforwardly solved using a deterministic off-the-shelf solver,
e.g., _Gurobi optimizer_.

##  IV Experiments

In this section, we perform experiments using real-world data and provide
numerical results to assess the effectiveness of our proposed reputation-based
incentive mechanism for poisoning attack mitigation. Unless otherwise stated,
we consider that the number of SIoT devices is $N=20$ and that the channel’s
capacity for each SIoT device is fixed to $r=10kbps$.

###  IV-A Dataset and Simulations Settings

We carry out our experiments using the CARRADA dataset [12], a contemporary
open dataset featuring 30 scenes of synchronized sequences of camera and radar
images. The radar images encompass both the range-angle (RA) and the range-
Doppler (RD) views. More description on the dataset can be found in [3] and
are omitted here for brevity. The output of the semantic segmentation
algorithms yields an image with a white background, featuring the masked
objects (refer to Fig. 2). Additionally, a meta-data text file in JSON format
is generated. This file encompasses a mapping of every object within the
image, along with its associated class, as well as additional derived semantic
attributes like range and shape. The meta-data text file has a maximum size of
$d_{meta}=1kb$. To reflect the the real environment where different SIoT
devices capture the same scene but from different angles, we consider that
SIoT devices transmit a set of scenes that are very close in time (in $ms$).
At each episode, a different scene from the $30$ synchronized sequences is
used222The implementation of the 3D grid representation is left for the future
work.. In addition, we consider that the simulations last for $10$ episodes
where each episode has a length of $50$ time-steps. To model the different
types of SIoT devices that participate in the vehicular Metaverse market, we
group the SIoT devices into three types:

  * •

Type-1 SIoT: at every episode, an attack is launched with probability $p=0.05$
in each timestep. This type reflects normal SIoT device with no willingness to
attack and no man-in-the-middle attack.

  * •

Type-2 SIoT: at every episode, an attack is launched with probability $p=0.5$
in each timestep.

  * •

Type-3 SIoT: at the beginning of every episode, choose with probability
$p=0.5$ either to attack for the whole episode or not to attack.

Finally, the types of the participating SIoT devices are evenly distributed.

###  IV-B Results

We first start by comparing the case when no anti-poisoning attack is used and
the case where our algorithm is used, which is illustrated in Fig. 4. We
observe that with the use of our technique, the frequency of tampered digital
twin decreases significantly. In the case without reputation mechanism, the
VSP chooses SIoT devices from the different types with equal numbers. However,
when our developed reputation-based anti-poisoning attack mechanism is used,
SIoT devices dominate the set of selected SIoT devices for semantic data
delivery. Interestingly, we observe that the SIoT devices from type-3 have a
higher percentage compared to SIoT devices from type-2. We explain this
behavior by the fact that SIoT devices from type-3 choose to be non-malicious
for certain episodes for the whole period. By observing this behavior in real-
time, the VSP is able to benefit from these SIoT devices and collect their
semantic data. This further shows the benefits of collecting real-time
feedbacks from VMUs.

Figure 4: Acceptance rate of SIoT devices from different types with and
without reputation mechanism.

Next, to observe the impact of the reputation backpropagation filtering
Algorithm on our anti-poisoning mechanism, we plot in Fig. 5 social welfare of
the system averaged over all the episodes and the successful attack rate for
three cases: the case where no anti-poisoning attack mechanism is adopted
(denoted by “No reputation”), the case where our anti-poisoning attack
mechanism is used with and without the reputation backpropagation is adopted.
We observe that the average social welfare of the system without the
reputation mechanism is much higher than that when our reputation-based
mechanism is adopted, which seems to be an undesirable result. Higher value of
the social welfare is expected to reflect a higher system efficiency. This
behavior is justified by the fact that our anti-poisoning mechanism does not
address the social welfare maximization issue. Specifically, we should first
note that the social welfare captures only the system benefits with respect to
the defined utility functions in (5) and (6) without considering decrease in
the security performance (i.e., digital twin tampering). This is clearly
captured by the high successful attack rate in the case where no anti-
poisoning attack mechanism is used. As shown in Fig. 4, our proposed anti-
poisoning attack mechanism decreases the successful attack rate on the system
significantly.

Figure 5: Average social welfare and successful attack rates in different
scenarios.

Finally, we also observe from Fig. 5 the advantages of our feedback filtering
algorithm on the performance of our system. Specifically, although the
successful attack rate in the case where the filtering is used and that where
it is not used is the same, we observe a significant decrease in the average
social welfare of the system. This argument is supported by the notion that
when negative feedback is distributed regarding specific regions in the
delivered Metaverse content, non-malicious SIoT devices (SIoT devices with
false negative ratings) may be deterred from allocation in subsequent
iterations for content delivery. This is reflected in the decrease of the
value of the semantic data as formulated in (8).

##  V Conclusion And Future Works

In conclusion, this paper addresses the critical issue of moral hazard within
the domain of the vehicular Metaverse. By proposing an incentive mechanism
rooted in a reputation-based strategy, we aim to deter dishonest actors from
engaging in fraudulent behavior after receiving payments. This approach
involves assigning reputation scores to participants based on their
interactions with the VSP, informed by feedback from Metaverse users.
Furthermore, we implement a vanishing-like system for previous ratings to
prevent reputable SIoT devices with occasional false positive ratings from
exiting the market. This ensures that the VSP can make decisions based on the
most timely and accurate data available, upholding the integrity of the
Metaverse market and enabling trustworthy IoT devices to continue contributing
their sensing data. Ultimately, extensive simulations validate the
effectiveness of our proposed model. The results demonstrate the efficiency of
our mechanism in preventing malicious devices from launching poisoning
attacks, while also ensuring that reliable SIoT devices with prior
missclassifications are not unfairly excluded from the market. This research
provides a robust framework for enhancing security and trustworthiness in the
vehicular Metaverse ecosystem. For future works, it is worth exploring the
direction of identifying the ideal weights for diminishing the influence of
previous data.

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