This article has been accepted by IEEE Transactions on Vehicular Technology.

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Z. Ye, C. Pan, H. Zhu and J. Wang,“Tradeoff Caching Strategy of Outage
Probability and Fronthaul Usage in Cloud-RAN,” in _IEEE Transactions on
Vehicular Technology_ , vol. PP, no. 99, pp. 1-1.

doi: 10.1109/TVT.2018.2797957

https://doi.org/10.1109/TVT.2018.2797957

###### Abstract

In this paper, tradeoff content caching strategy is proposed to jointly
minimize the cell average outage probability and fronthaul usage in cloud
radio access network (Cloud-RAN). At first, an accurate closed form expression
of the outage probability conditioned on the user’s location is presented, and
the cell average outage probability is obtained through the composite
Simpson’s integration. The caching strategy for jointly optimizing the cell
average outage probability and fronthaul usage is then formulated as a
weighted sum minimization problem, which is a nonlinear 0-1 integer problem.
Two heuristic algorithms are proposed to solve the problem. Firstly, a genetic
algorithm (GA) based approach is proposed. Numerical results show that the
performance of the proposed GA-based approach with significantly reduced
computational complexity is close to the optimal performance achieved by
exhaustive search based caching strategy, and the GA-based approach can
improve the performance by up to 47.5% on average than the typical
probabilistic caching strategy. Secondly, in order to further reduce the
computational complexity, a mode selection approach is proposed. Numerical
results show that this approach can achieve near-optimal performance over a
wide range of the weighting factors through a single computation.

###### Index Terms:

Caching strategy, Cloud-RAN, joint optimization, outage probability, fronthaul
usage.

##  I Introduction

THE combination of network densification and coordinated multipoint
transmission is a major technical trend in the fifth generation (5G) wireless
mobile systems to improve the overall system performance [2, 3, 4, 5]. In the
traditional radio access network (RAN) architecture, each cell has its own
base station (BS), where the radio functionality is statically assigned to the
base band processing module. Adding more base stations and introducing
multiple input multiple output (MIMO) technology will increase the complexity
of the network and result in higher total cost of ownership (TCO) for the
mobile operators [6, 7].

Consisting of centralized base band processing resources, known as base band
unit (BBU) pool, and distributed remote radio heads (RRHs) or remote antenna
units [8, 9, 10], cloud radio access network (Cloud-RAN) becomes a new type of
RAN architecture to support multipoint transmission and access point
densification required by 5G systems[6, 7, 11, 12]. The scalable, virtualized,
and centralized BBU pool is shared among cell sites, and its computing
resources can be dynamically allocated to different cells according to their
traffic. The RRHs are responsible for the radio processing task, and they are
connected with the BBU pool through fronthauls, while the BBU pool performs
the base band processing task and it is connected to the core network through
backhauls. Thanks to the novel architecture, Cloud-RAN has many advantages
such as cost effective, lower energy consumption, higher spectral efficiency,
scalability and flexibility etc., which makes itself a promising candidate for
the 5G deployment. Particularly, Cloud-RAN is also a competitive solution for
the heterogeneous vehicular networks, which can provide better quality of
service (QoS) to intense vehicular users in an urban environment [13, 14].
However, existing fronthaul/backhaul of Cloud-RAN cannot meet the requirements
of the emerging huge data and signaling traffic in terms of transmission
bandwidth requirements, stringent latency constraints and energy consumption
etc. [15, 16, 17], which has become the bottleneck of the evolution towards
5G.

Statistics showed that a large amount of the data traffic is generated from a
small amount of most popular content files. These popular files are requested
by a large amount of users, which results in duplicated transmissions of the
same content files on the fronthaul and backhaul links. Therefore, content
caching in RAN can be a promising solution to significantly reduce the
fronthaul/backhaul traffic [18, 19, 20]. During off-peak times, popular
content files can be transferred to the cache-enabled access points (macro
base station, small cell, relay node etc.). If the files requested by mobile
users are cached in the access points of the RAN, the files will be
transmitted directly from the RAN’s cache without being fetched from the core
network, which can significantly reduce the fronthaul/backhaul traffic and
meanwhile shorten the access latency of the files, thus improve users’ quality
of experience (QoE). In Cloud-RAN, thanks to the ongoing evolution of
fronthaul technology and function splitting between the BBU and RRHs [16, 21],
there comes possibility to realize content caching in RRHs, which allows users
fetching required content files directly from RRHs and thus can further reduce
fronthaul traffic.

There are two stages related with caching: _caching placement stage_ and
_caching delivery stage_ [20]. Caching placement, or known as caching
strategy, is the stage to determine which files should be stored in which
cache-enabled access points, and delivery stage refers to stage of
transmitting the requested files from access points to mobile users through
wireless channels. Among these two stages, caching placement is performed for
a relative long-timescale. Once a caching placement is carried out, it will
not change very frequently. The reason is that the popularity of the content
files will remain the same for a relative long period such as several hours,
one day, or even longer time. On the other hand, delivery stage runs in a
short-timescale. In a delivery stage, the wireless transmission scheme should
be able to adapt to the instantaneous channel state information (CSI) which
varies very rapidly.

There are many researches investigating the delivery stage with the target of
data association or/and energy consumption optimization under a given caching
strategy, such as [22, 23, 24]. On the other hand, caching strategy is of
importance because it is the initial step to perform caching and obviously it
will have an impact on the performance of the delivery stage. The researches
investigating caching strategies generally focus on reducing the file access
latency [25, 26, 27], or minimizing the transmission cost of the backhaul [28,
29], or both of them [30]. However, the wireless transmission characteristics
such as fading were not considered in the aforementioned researches, i.e., it
was assumed that the wireless transmission is error-free. The caching strategy
will affect the wireless transmission performance such as outage probability,
which is an important metric of the system’s performance. For the
fronthaul/backhaul traffic or average file access delay reduction, caching
different files in the RAN will be optimal, however there is no transmit
diversity to combat fading in the file delivery stage, which may decrease the
reliability of the wireless transmission. Hence, caching strategy should be
optimized by taking into consideration the wireless transmission performance.

There are some papers considering wireless fading characteristics when
designing caching strategy [31, 32]. The authors only considered small scale
Rayleigh fading by assuming that the user has the same large scale fading at
any location. However, in reality, several RRHs will jointly serve the user in
Cloud-RAN, and obviously the distance between each RRH and the user will not
be the same, so it is important to consider large scale fading in wireless
transmission. In addition, they focused on single-objective optimization
without considering the fronthaul/backhaul usage.

The aim of caching in RRHs of Cloud-RAN is to significantly reduce the
fronthaul traffic. Fronthaul usage, i.e., whether the fronthaul is used, is a
metric which can reflect not only the file delivery latency but also the
energy consumption of the fronthaul. For example, lower fronthaul usage
implies there are more possibilities that mobile user can access the content
files in near RRHs, which will shorten the file access latency, meanwhile the
fronthaul cost (i.e., the energy consumption) will be lower. On the other
hand, outage probability is an important performance metric of the system,
which reflects the reliability of the wireless transmission, i.e., whether the
requested content files can be successfully transferred to the user, and it
also reflects the utility of the wireless resources. If replicas of certain
content files are cached in several RRHs, the outage probability will be
reduced due to the transmit diversity in wireless transmissions, while the
fronthaul usage will become higher because the total number of different files
cached in the RRHs are reduced and there is a high possibility to fetch files
from the BBU pool. On the other hand, caching different files in the RRHs will
reduce the fronthaul usage, while the outage probability will become
relatively higher due to the decrease of wireless diversity. Therefore, there
exists tradeoff between fronthaul usage and outage probability.

In this paper, we investigate downlink transmission in a virtual cell in
Cloud-RAN, such as a hot spot area, shopping mall, or an area covered by the
Cloud-RAN based vehicular network etc. The tradeoff caching strategy is
proposed to jointly minimize the cell average outage probability and the
fronthaul usage. A realistic fading channel is adopted, which includes path
loss and small scale Rayleigh fading. The caching strategy is designed based
on the long-term statistics about the users’ locations and content file
request profiles. The major contributions of this paper are:

  1. 1)

Closed form expression of outage probability conditioned on the user’s
location is derived, and the cell average outage probability is obtained
through the composite Simpson’s integration. Simulation results show that the
analysis is highly accurate.

  2. 2)

The joint optimization problem is formulated as a weighted sum minimization of
cell average outage probability and fronthaul usage, which is a 0-1 integer
problem. Two heuristic algorithms are proposed to solve the problem:

    1. a)

An effective genetic algorithm (GA) based approach is proposed, which can
achieve nearly the same performance as the optimal exhaustive search, while
the computational complexity is significantly reduced.

    2. b)

In order to further reduce the computational complexity, a mode selection
approach is proposed. Simulation results show that it can achieve near-optimal
performance over a wide range of weighting factors through a single
computation.

The remainder of this paper is organized as follows. Section II reviews the
related works. System model is described in Section 1. The optimization
problem is formulated in Section IV and the cell average outage probability
and fronthaul usage are analyzed. The proposed GA-based approach and the mode
selection schemes are described in Section V. Numerical results are given in
Section VI and the conclusion is given in Section VII.

_Notations_ : $\mathbb{E}(\cdot)$ denotes statistical expectation, and
$\operatorname{Re}(\cdot)$ denotes the real part of a complex number.
$\mathbf{A}^{L\times N}=\\{a_{l,n}\\}$ denotes $L\times N$ matrix, $a_{l,n}$
or $\mathbf{A}(l,n)$ represents the $(l,n)$-th entry of the matrix.
$\mathbb{R}^{+}$ denotes the set of positive real numbers, and
$\mathbb{Z}^{+}$ denotes positive integer set. $\mathcal{CN}(\mu,\sigma^{2})$
represents complex Normal distribution with mean $\mu$ and variance
$\sigma^{2}$, and $\chi^{2}(k)$ is the central Chi-squared distribution with
$k$ degrees of freedom.

##  II Related Works

From the caching point of view, there exists significant similarities between
Cloud-RAN, small cell network, marcocell network and some vehicular networks
etc. There are many researches investigating the delivery stage under a
certain caching strategy, and the main target was to optimize the data
association (e.g., RRH clustering and transmit beamforming), such as [22, 23,
24]. In [22] and [23], optimal base station clustering and beamforming were
investigated to reduce the backhaul cost and transmit power cost under certain
caching strategy. In addition, the performances of different commonly used
caching strategies, such as popularity-aware caching, random caching, and
probabilistic caching, were compared in [23]. In [24], assuming there are
several small base stations in an orthogonal frequency division multiple
access (OFDMA) macro cell, the optimal association of the users and small base
stations was investigated to reduce the long-term backhaul bandwidth
allocation. In these researches, the caching strategy was assumed to be fixed
when designing the delivery schemes, which is because the delivery stage runs
in a much shorter timescale than the caching placement stage.

On the other hand, caching strategy has attracted widely concern recently, the
related researches mainly focused on the reduction of file access latency [25,
26, 27], fronthaul/backhaul transmissions [28, 29], or both of them [30]. In
[25], a collaborative strategy of simultaneously caching in BS and mobile
devices was proposed to reduce the latency for requesting content files. The
proposed optimal strategy was to fill the BS’s cache with the most popular
files and then cache the remaining files of higher popularity in the mobile
devices. In [26], a distributed algorithm with polynomial-time and linear-
space complexity was proposed to minimize the expected overall access delay in
a cooperative cell caching scenario. The delay from different sources to the
user was modeled as uniformly distributed random variables within a certain
range. In [27], the probabilistic caching strategy was optimized in clustered
cellular networks, where the limited storage capacity of the small cells and
the amount of transferred contents within the cluster were considered as two
constraints to minimize the average latency. The optimized caching probability
of each content file was obtained.

In [28], a coded caching placement was proposed to minimize the backhaul load
in a small-cell network, where multicast was adopted. The file and cache sizes
were assumed to be heterogeneours. In [29], to minimize the total transmission
cost among the BSs and from the core network, each BS’s cache storage was
divided into two parts, the first part of all the BSs cached same files with
higher popularity ranks, while the second part of all the BSs stored different
files. The cache size ratio of the two parts was optimized through particle
swarm optimization (PSO) algorithm. In [30], caching strategy was investigated
in a Cloud-RAN architecture based networks, and the average content
provisioning cost (e.g., latency, bandwidth etc.) was analyzed and optimized
subjecting to the sum storage capacity constraint. Analytical results of the
optimal storage allocation (how to partition the storage capacity between the
control BS and traffic BS) and cache placement (decision on which file to
cache) were obtained. However, the aforementioned researches did not take
wireless transmission characteristics into consideration. In practice, the
caching strategy will have an impact on the performance of the delivery stage,
so wireless transmission performance should be considered in order to optimize
the caching strategy.

There are some papers considering wireless fading characteristics when
designing/investigating caching strategy [33, 34, 31, 32]. Stochastic geometry
was used to analyze large scale networks in [33, 34]. In [33], considering a
cache-enabled two-tier heterogeneous network with one macrocell BS and several
small-cell BSs, outage probability, throughput, and energy efficiency (EE)
were analyzed. Each of the BSs caches the most popular content files until the
storage is full filled. Numerical results showed that larger small-cell cache
capacity may leads to lower network energy efficiency when the density of the
small cells is low. In [34], the performance of probabilistic caching strategy
was analyzed and optimized in a small-cell environment, and the aim was to
maximize the successful download probability of the content files. However,
only _probabilistic content placement_ can be obtained through using the tool
of stochastic geometry [35], that is, the probability of a certain file should
be cached in the access points. In [31], optimal caching placement was
obtained through a greedy algorithm to minimize the average bit error rate
(BER) in a macro cell with many cache-enabled helpers and each helper can
cache only one file. The user selects one helper with the highest
instantaneous received signal to noise ratio (SNR) among the helpers which
cache the requested file. If none of the helpers cache the requested file, the
user will fetch the file from the BS. In [32], cache-enabled BSs are connected
to a central controller via backhaul links. The aim was to minimize the
average download delay. Similar to [31], the user selects the BS with the
highest SNR in the candidate BSs caching the requested files. In [31] and
[32], the authors only considered small scale Rayleigh fading by assuming that
the user has the same large scale fading at any location, which is
unpractical. In addition, they focused on single-objective optimization
without considering the fronthaul/backhaul usage.

Inspired by the aforementioned researches, in this paper, outage probability
is used to reflect the wireless transmission performance, and fronthaul usage
is used to reflect the transmission latency and power consumption etc. Outage
probability and fronthaul usage are jointly considered when designing the
caching strategy, which leverages the tradeoff between caching the same
content files to obtain lower outage probability or caching different content
files to reduce fronthaul usage. Considering the distances from each RRH to
the user are different in a Cloud-RAN environment, a more practical fading
channel model which includes both large and small scale fading is adopted.

##  III System Model

It is assumed that there are $N$ cache-enabled RRHs in a circular cell with
radius $R$, and the set of RRH cluster is denoted as
$\mathcal{N}=\\{1,2,\cdots,N\\}$. The file library with a total of $L$ content
files is denoted as $\mathcal{F}=\\{F_{1},F_{2},\cdots,F_{L}\\}$, where
$F_{l}$ is the $l$-th ranked file in terms of popularity, i.e., $F_{1}$ is the
most popular content file. The popularity distribution of the files follows
the Zipf’s law [36], and the request probability of the $l$-th ranked content
file is

$P_{l}=\dfrac{l^{-\beta}}{\sum\nolimits_{n=1}^{L}{n^{-\beta}}}~{},$ (1)

where $\beta\in\left[0,+\infty\right)$ is the skewness factor. The popularity
is uniformly distributed over content files when $\beta=0~{}(P_{l}=1/L,\forall
l)$ and becomes more skewed towards the most popular files as $\beta$ grows,
while large popularity skewness is usually observed in wireless applications.

For simplicity, it is assumed that all content files have the same size, and
the file size is normalized to 1. Even though the file size will not be equal
in practice, each file can be segmented into equal-sized chunks for placement
and delivery [19, 37]. Considering the BBU pool can be equipped with
sufficient storage space, it is assumed that all the $L$ content files are
cached in the BBU pool 111Generally speaking, the backhaul connecting the BBU
pool and the core network will have larger transmission bandwidth than the
fronthaul, so only the fronthaul usage reduction is considered in this paper.
In practice, the BBU pool can not cache all the content files originated in
the Internet, however, if the requested file is not cached in the BBU pool, it
can be fetched from the core network through using backhaul, then it is the
same as the file is already cached in the BBU pool as we only focus on the
fronthaul usage rather than backhaul usage.. Some of the content files can be
further cached in the RRHs in order to improve the system’s performance, and a
file can be cached in one or more RRHs depending on the caching strategy. The
$n$-th RRH can cache $M_{n}$ files, and generally
$\sum\nolimits_{n=1}^{N}M_{n}<L$. That is, the total caching storage space in
all the RRHs is smaller than that in the BBU pool. The caching placement of
the content files in the RRHs can be denoted by a binary placement matrix
$\mathbf{A}^{L\times N}$, with the $(l,n)$-th entry

$a_{l,n}=\left\\{\begin{array}[]{cl}1,&\text{the $n$-th RRH caches the $l$-th
file}\\\ 0,&\text{otherwise}\\\ \end{array}\right.$ (2)

indicating whether the $l$-th content file is cached in the $n$-th RRH, and
$\sum_{l=1}^{L}a_{l,n}=M_{n},~{}\forall n$.

Single user scenario is considered in this paper. However, the proposed
algorithms can be applied in practical multiuser systems with orthogonal
multiple access technique such as OFDMA system, in which each user is
allocated with different subcarriers and there is no interference [38, 39,
40]. It is assumed that the user can only request for one file at one time,
and all the RRHs caching the requested file will serve the user. If none of
the RRHs caches the requested file, the file will be transferred to all the
RRHs from the BBU pool through fronthauls, and then to the user from all the
RRHs through wireless channels. The service RRH set for the user with respect
to (w.r.t.) the $l$-th file is denoted as

$\Phi_{l}=\left\\{\begin{array}[]{cl}\\{n|a_{l,n}=1,n\in\mathcal{N}\\}&,~{}\exists
n$ such that $a_{l,n}=1\\\ \mathcal{N}&,~{}\text{$a_{l,n}=0$ for $\forall
n$}\\\ \end{array}\right.,$ (3)

with cardinality $|\Phi_{l}|\in\\{1,2,\cdots,N\\}$,
($l\in\\{1,2,\cdots,L\\}$). The system model and file delivery scheme are
illustrated in Figure1. For example, when the user requests for the $l_{1}$-th
file which is not cached in any of the RRHs, the file will be transferred from
the BBU pool to all the RRHs through fronthauls and then transmitted to the
user. Then the user’s service RRH set is $\Phi_{l_{1}}=\\{1,2,3,4\\}$. When
the user requests for the $l_{2}$-th file which is already cached in RRH 2 and
RRH 3 via caching placement, the service RRH set is $\Phi_{l_{2}}=\\{2,3\\}$.

Figure 1: System model and file delivery scheme. Red dashed and green solid
lines represent the file fetching routes when user requests for the $l_{1}$-th
and $l_{2}$-th content file, respectively.

The wireless channel is assumed to be block-fading, i.e., the channel’s gain
is kept as constant within the duration of a block, and different blocks
experience independent and identically distributed (i.i.d.) fading. When being
requested, a file would be transmitted through different blocks of the
wireless channel. Assuming that both the RRH and the user’s device are
equipped with single antenna, the user’s received signal from the service RRH
set when requesting for the $l$-th file can be expressed as

$y=\sum\limits_{n\in\Phi_{l}}\sqrt{p_{T}Kd_{n}^{-\alpha}}h_{n}s+noise,$ (4)

where $p_{T}$ is the transmit power of each RRH, $K$ is a constant depending
on the antenna characteristics and the average channel attenuation, $d_{n}$ is
the distance between the $n$-th RRH and the user, $\alpha$ is the path loss
exponent, $h_{n}\sim\mathcal{CN}(0,1)$ represents complex Gaussian small scale
fading, $s$ represents the transmitted symbol and
$\mathbb{E}\left[|s|^{2}\right]=1$, and $noise$ denotes complex additive white
Gaussian noise (AWGN) with zero mean and variance $\sigma^{2}$.

The main modeling parameters and notations are summarized in Table I.

TABLE I: Modeling Parameters and Notations Symbol |  Definition
---|---
$N$ |  Number of RRHs
$L$ |  Number of content files
$P_{l}$ |  Request probability of the $l$-th file
$\beta$ |  Skewness factor of the Zipf’s distribution
$M_{n}$ |  The number of files that the $n$-th RRH can cache
$\mathbf{A}^{L\times N}$ |  Caching placement matrix
$a_{l,n}$ |  Binary variable indicating whether the $l$-th file is
|  cached in the $n$-th RRH, $(l,n)$-th entry of $\mathbf{A}^{L\times N}$
$\Phi_{l}$ |  The service RRH set for the user w.r.t. the $l$-th file
$p_{T}$ |  Transmit power of each RRH
$d_{n}$ |  Distance between the $n$-th RRH and the user

##  IV Problem Formulation and Analysis

###  IV-A Problem Formulation

Define the normalized fronthaul usage w.r.t. the $l$-th file as

$T_{l}(\mathbf{A})=\prod_{n=1}^{N}\left(1-a_{l,n}\right)=\left\\{\begin{array}[]{cl}1,&\text{$a_{l,n}=0$
for $\forall n$}\\\ 0,&\text{$\exists n$ such that $a_{l,n}=1$}\\\
\end{array}\right.,$ (5)

which indicates that if there is at least one copy of the requested file
cached in the RRHs, there will be no fronthaul usage, i.e., $T_{l}=0$, while
if the requested file is not cached in any of the RRHs, there will be
fronthaul usage, i.e., $T_{l}=1$. Note that $T_{l}$ does not depend on the
user’s location.

The caching strategy should be designed according to the long-term statistics
over the user’s locations and content file requests. The joint optimization
problem can be formulated through a weighted sum of the objectives [41],

$\displaystyle\min$ $\displaystyle
f_{obj}(\mathbf{A})=\eta\underbrace{\sum\limits_{l=1}^{L}P_{l}\mathbb{E}_{x_{0}}\left[{P}_{out}^{(l)}(x_{0})\right]}_{\textnormal{cell
average outage
probability}}+(1-\eta)\underbrace{\sum\limits_{l=1}^{L}{P}_{l}T_{l}}_{\textnormal{fronthaul
usage}},$ (6c) s.t. $\displaystyle\sum\limits_{l=1}^{L}a_{l,n}=M_{n},$
$\displaystyle a_{l,n}\in\\{0,1\\}.$

where $\eta\in[0,1]$ is a weighting factor to balance the tradeoff between
outage probability and fronthaul usage, $\mathbb{E}_{x_{0}}$ denotes
expectation in terms of the user’s location $x_{0}$, $P_{out}^{(l)}(x_{0})$ is
the outage probability when the user requests for the $l$-th file at location
$x_{0}$. Constraint (6c) describes the caching limit of each RRH, and
constraint (6c) indicates the joint optimization as a 0-1 integer problem.

Different values of $\eta$ will lead to different balances between outage
probability and fronthaul usage. Given $\eta$, the caching strategy can be
determined through solving the optimization problem in (6). In practice,
$\eta$ is chosen by the decision maker (e.g., RAN’s operator) according to the
system’s long-term statistics of outage probability and fronthaul usage. For
example, when the fronthauls’ average payload is heavy, a small value of
$\eta$ should be chosen to reduce the fronthaul usage, and the price is to
increase the outage probability. On the other hand, when the cell average
outage probability is high, a large value of $\eta$ should be chosen to reduce
the outage probability, so that the price is to increase the fronthaul usage.

###  IV-B Outage Probability Analysis

When the user requests for the $l$-th file at location $x_{0}$, the SNR of the
received signal is given by

$\gamma_{l}(x_{0})=\sum_{n\in\Phi_{l}}\frac{p_{T}}{\sigma^{2}}Kd_{n}^{-\alpha}|h_{n}|^{2}=\sum_{n\in\Phi_{l}}\gamma_{0}S_{n}|h_{n}|^{2}=\sum_{n\in\Phi_{l}}\gamma_{n},$
(7)

where $\gamma_{0}=\frac{p_{T}}{\sigma^{2}}$ is SNR at the transmitter of each
RRH, $S_{n}=Kd_{n}^{-\alpha}$ is the large scale fading, and
$\gamma_{n}=\gamma_{0}S_{n}|h_{n}|^{2}$ represents the received SNR from the
$n$-th RRH. For a specific file, without ambiguity, we omit the subscript of
file index $l$ and the user’s location $x_{0}$ in the following analysis.

In the service RRH set $\Phi$ with cardinality $|\Phi|$, the RRHs with the
same distance to the user are grouped together. Assuming there are
$I~{}(I\leq|\Phi|)$ groups, the number of RRHs in the $i$-th group is denoted
by $J_{i}$, and $\sum_{i=1}^{I}J_{i}=|\Phi|$. The distance between the user
and the RRH in the $i$-th group is denoted by
$d_{i}~{}(i\in\\{1,2,3,\cdots,I\\})$. Letting
$\lambda_{i}=\frac{1}{\gamma_{0}Kd_{i}^{-\alpha}}$, the probability density
function (PDF) of the received SNR can be obtained as

$f_{\gamma}(\gamma)=\sum_{i=1}^{I}\sum_{j=1}^{J_{i}}\dfrac{\lambda_{i}^{j}A_{ij}}{(j-1)!}\gamma^{j-1}e^{-\lambda_{i}\gamma},$
(8)

and the cumulative distribution function (CDF) is given by

$\displaystyle F_{\gamma}(\gamma)$
$\displaystyle=\sum_{i=1}^{I}\sum_{j=1}^{J_{i}}\dfrac{\lambda_{i}^{j-1}A_{ij}}{(j-1)!}$
(9)
$\displaystyle~{}\cdot\left[\dfrac{(j-1)!}{\lambda_{i}^{j-1}}-\left(e^{-\lambda_{i}\gamma}\sum_{k=0}^{j-1}\dfrac{(j-1)!}{(j-1-k)!\lambda_{i}^{k}}\gamma^{j-1-k}\right)\right],$

where

$\begin{aligned}
&A_{ij}=\dfrac{(-\lambda_{i})^{J_{i}-j}}{(J_{i}-j)!}\dfrac{d^{J_{i}-j}}{ds^{J_{i}-j}}\left[M_{\gamma}(s)\left(1-\dfrac{1}{\lambda_{i}}\cdot
s\right)^{J_{i}}\right]\Bigg{|}_{s=\lambda_{i}}\end{aligned},$ (10)

and

$M_{\gamma}(s)=\prod\nolimits_{n\in\Phi}\frac{1}{1-\gamma_{0}S_{n}\cdot
s}\;\;.$ (11)

The derivations of (8) and (9) are given in Appendix A.

When the distance between any service RRH and the user is distinct, i.e.,
$d_{n}\neq d_{m},\forall n\neq m\in\Phi,$ (8) and (9) are written as

$\displaystyle f_{\gamma}(\gamma)$ $\displaystyle=$
$\displaystyle\sum_{n\in\Phi}\dfrac{1}{\gamma_{0}S_{n}}\left(\prod_{\begin{subarray}{c}m\in\Phi\\\
m\neq
n\end{subarray}}\dfrac{S_{n}}{S_{n}-S_{m}}\right)\exp\left({-\dfrac{\gamma}{\gamma_{0}S_{n}}}\right)$
(12)

and

$\displaystyle
F_{\gamma}(\gamma)=\sum_{n\in\Phi}\left(\prod_{\begin{subarray}{c}m\in\Phi\\\
m\neq
n\end{subarray}}\dfrac{S_{n}}{S_{n}-S_{m}}\right)\left[1-\exp\left({-\dfrac{\gamma}{\gamma_{0}S_{n}}}\right)\right],$
(13)

respectively.

The accuracy of the derived CDF of (9) (written as (13) in special case) is
illustrated in Fig. 2 through three scenarios. Assuming there are 6 service
RRHs for the user, and the distances between the service RRHs and the user are
denoted by a vector $\mathbf{D}$. The three different scenarios are (1)
scenario 1: $\mathbf{D_{1}}=[0.8R,0.8R,0.8R,0.8R,0.8R,0.8R]$, ($R$ is the cell
radius), i.e., all the RRHs are with the same distance to the user; (2)
scenario 2: $\mathbf{D_{2}}=[0.6R,0.7R,0.7R,0.8R,0.8R,0.8R]$, i.e., some of
the RRHs have same distance with the user; (3) scenario 3:
$\mathbf{D_{3}}=[0.5R,0.6R,0.7R,0.8R,0.9R,1.0R]$, i.e., all the RRHs are with
different distances to the user. It can be seen from Fig. 2 that the
analytical results match the simulation results, which demonstrates the
accuracy of the derived expression of (9).

Figure 2: CDF of the user’s received SNR at a fixed location.

The outage probability according to a certain SNR threshold $\gamma_{th}$ is

$P_{out}(\gamma_{th})=F_{\gamma}(\gamma_{th}).$ (14)

It is difficult to find a closed form solution of the cell average outage
probability w.r.t. the $l$-th file, i.e.,
$\mathbb{E}_{x_{0}}[P_{out}^{(l)}(x_{0})]$. However, we can use the composite
Simpson’s integration in forms of polar coordinates, where the user’s location
is denoted by $(\rho,\theta)$ and $x_{0}=\rho e^{j\theta}$.

$\displaystyle\mathbb{E}_{x_{0}}\left[{P}^{(l)}_{out}(x_{0})\right]$ (15)
$\displaystyle=$
$\displaystyle\int_{0}^{2\pi}\int_{0}^{R}{P}^{(l)}_{out}(\rho,\theta)f_{x_{0}}(\rho,\theta)\rho
d\rho d\theta$ $\displaystyle\approx$ $\displaystyle\dfrac{\Delta h\Delta
k}{9}\sum_{u=0}^{U}\sum_{v=0}^{V}w_{u,v}\rho_{u}{P}^{(l)}_{out}(\rho_{u},\theta_{v})f_{x_{0}}(\rho_{u},\theta_{v}),$

where $R$ is the cell radius, even integers $U$ and $V$ are chosen such that
$\Delta h=R/U$ and $\Delta k=2\pi/V$ meeting the requirement of calculation
accuracy, $\rho_{u}=u\Delta h$, $\theta_{v}=v\Delta k$,
$f_{x_{0}}(\rho,\theta)$ is the probability density function of the user’s
location, which is $1/\pi R^{2}$ when the user’s location is uniformly
distributed in the cell, and $\\{w_{u,v}\\}$ are constant coefficients (please
refer to [42] and Chapter 4 in [43]).

Substituting (5), (9), (14) and (15) into (6c), the optimization problem is
formulated as a function of the caching placement matrix $\mathbf{A}^{L\times
N}=\\{a_{l,n}\\}$. However, the problem is a 0-1 integer nonlinear problem,
and it is difficult to obtain a closed form solution. The following section
will focus on how to solve this problem.

##  V Caching Placement Scheme

In this section, two efficient approaches are proposed to solve the joint
optimization problem: one is GA-based approach and the other is mode selection
approach.

###  V-A Genetic Algorithm Based Approach

Genetic algorithm is inherently suitable for solving optimization problems
with binary variables [44]. The algorithm structure is shown in Fig. 3.
Firstly, $N_{p}$ candidate caching placement matrices are generated, known as
the initial population (with population size $N_{p}$), and each matrix is
called an individual. Then the objective value of each individual is evaluated
through (6c). $N_{e}$ individuals with best objective values are chosen as
elites and passed into next generation (children of current generation
population) directly. The rest of the next generation population are generated
through crossover and mutation operations. The crossover function operates on
two individuals (known as parents) and generates a crossover child, and the
mutation function operates on a single individual and generates a mutation
child. The number of individuals generated through crossover and mutation
operations are denoted as $N_{c}$ and $N_{m}$, respectively, where
$N_{e}+N_{c}+N_{m}=N_{p}$, and the crossover fraction is defined as
$f_{c}=\frac{N_{c}}{N_{c}+N_{m}}$. The selection function selects $2N_{c}$ and
$N_{m}$ individuals from the current generation for the crossover and mutation
function, respectively, where some individuals will be selected more than
once. Stochastic uniform sampling selection [45] is adopted, and individuals
with lower objective values in current generations will have a higher
probability to generate offsprings. Repeat the evaluation-selection-generation
procedures until termination criterion is reached. Finally, the best
individual in the current population is chosen as the output of the algorithm.
The initial population, crossover function and mutation function of the
proposed GA approach are described as follows.

####  V-A1 Initial Population

The initial population is created as a set of $\\{\mathbf{A}^{L\times N}\\}$.
For each column in each individual, $M_{n}$ out of the first $L^{\prime}$
entries (i.e., $\\{a_{1,n},a_{2,n},\cdots,a_{L^{\prime},n}\\}$) are set to be
one randomly, and all the remaining entries are set to be zero, where

$L^{\prime}=\sum\nolimits_{n=1}^{N}M_{n}<L$ (16)

is based on the fact that the total different files with higher popularity can
be cached in the RRHs are $\\{F_{l}|l=1,2,\cdots,L^{\prime}\\}$. There is no
benefit to cache files $\\{F_{l}|l>L^{\prime}\\}$ with lower popularity.

####  V-A2 Crossover Function

The crossover function generates a child $\mathbf{A}_{c}$ from parents
$\mathbf{A}_{1}$ and $\mathbf{A}_{2}$. A two-point crossover function is used,
which is described in Algorithm 1, in which steps 1 to 1 are heuristic
operations to meet constraint (6c).

1 Get parent $\mathbf{A}_{1}=\\{a_{l,n}^{(1)}\\}$ and
$\mathbf{A}_{2}=\\{a_{l,n}^{(2)}\\}$ from selection function, initialize their
child $\mathbf{A}_{c}=\\{a_{l,n}^{(c)}\\}=\mathbf{0}^{L\times N}$.

2 for  _$n=1,2,\cdots,N$_ do

3 Generate random integers $l_{1},l_{2}\in[1,L^{\prime}]$, $l_{1}\neq l_{2}$

4 if  _$l_{1} <l_{2}$_ then

5 Replace $a_{l,n}^{(1)},~{}l=\\{l_{1},l_{1}+1,\cdots,l_{2}\\}$ of
$\mathbf{A}_{1}$ with $a_{l,n}^{(2)},~{}l=\\{l_{1},l_{1}+1,\cdots,l_{2}\\}$ of
$\mathbf{A}_{2}$, and then set $a^{(c)}_{l,n}=a^{(1)}_{l,n},~{}\forall
l\in\\{1,2,\cdots,L\\}$.

6 else

7 Replace $a_{l,n}^{(2)},~{}l=\\{l_{2},l_{2}+1,\cdots,l_{1}\\}$ of
$\mathbf{A}_{2}$ with $a_{l,n}^{(1)},~{}l=\\{l_{2},l_{2}+1,\cdots,l_{1}\\}$ of
$\mathbf{A}_{1}$, and then set $a^{(c)}_{l,n}=a^{(2)}_{l,n},~{}\forall
l\in\\{1,2,\cdots,L\\}$.

8 end if

9 while  _$\sum\nolimits_{l=1}^{L}a^{(c)}_{l,n} >M_{n}$ _ do

10 Set nonzero $a^{(c)}_{l,n}$ to 0 in descending order of $l$.

11 end while

12 while  _$\sum\nolimits_{l=1}^{L}a^{(c)}_{l,n} <M_{n}$ _ do

13 Set zero $a^{(c)}_{l,n}$ to 1 in ascending order of $l$.

14 end while

15

16 end for

Algorithm 1 Crossover function

####  V-A3 Mutation Function

The mutation function operates on a single individual and generates its
mutation child. For each column of the individual, one of the first
$L^{\prime}$ entries is randomly selected and the value is set to be the
opposite ($0$ to $1$ and vice versa), then steps 1 to 1 described in Algorithm
1 are executed to meet constraint (6c). The mutation operation reduces the
probability that the algorithm converges to local minimums.

If $N_{g}$ generations are evaluated, there is a total of $N_{p}N_{g}$
calculations of the objective values. In order to further reduce the
computational complexity of caching strategy, a mode selection approach is
proposed in next subsection.

Figure 3: Genetic algorithm structure.

###  V-B Mode Selection Approach

There are two particular caching placement schemes: one is the most popular
content (MPC) caching, and the other one is the largest content diversity
(LCD) caching [32, 46, 47]. In MPC, each RRH caches the most popular files,
i.e., the $n$-th RRH caches $\\{F_{l}|l=1,2,\cdots,M_{n}\\}$, which will have
low outage probability while high fronthaul usage. In the LCD scheme, a total
of $L^{\prime}=\sum\nolimits_{n=1}^{N}M_{n}~{}(<L)$ different most popular
content files are cached in the RRHs, which can have lowest fronthaul usage
while relatively high outage probability. If the LCD scheme is adopted in
Cloud-RAN, the impact of locations of caching content files on the cell
average outage probability needs to be considered. Assuming the locations of
the user are uniformly distributed in the cell, caching the most popular files
in the RRH nearest to the cell center will achieve better outage probability
performance, which is similar to the RRH placement problem [48]. Therefore,
for Cloud-RAN, we improve the LCD scheme and propose a location-based LCD (LB-
LCD) scheme which is described in Algorithm 2.

1 Sort the RRH set as
$\mathcal{N}_{s}=\\{n_{i}|i=1,2,\cdots,N,\;\;D_{n_{1}}\leq
D_{n_{2}}\leq\cdots\leq D_{n_{N}}\\}$, where $D_{n_{i}}$ denotes the distance
between the $n_{i}$-th RRH and the cell center.

2 Fill the cache of the RRH set $\mathcal{N}_{s}$ in sequence from $n_{1}$ to
$n_{N}$ with content files
$\\{F_{l}|l=1,2,\cdots,\sum\nolimits_{n=1}^{N}M_{n}\\}$ in ascending order of
$l$.

Algorithm 2 Proposed LB-LCD caching strategy

For example, there are 3 RRHs $\\{1,2,3\\}$, and each RRH can cache 3 files,
so that all the RRHs can cache 9 different content files. The distance between
RRH $i$ and the cell center is $D_{i}$, assuming $D_{1}<D_{2}<D_{3}$. The LB-
LCD caching strategy is illustrated in Table II.

TABLE II: Example of The LB-LCD Caching Strategy RRH | 1 | 2 | 3
---|---|---|---
| Files
---
cached
| $F_{1}$
---
$F_{2}$
$F_{3}$
| $F_{4}$
---
$F_{5}$
$F_{6}$
| $F_{7}$
---
$F_{8}$
$F_{9}$

######  Proposition 1.

The objective value of both the MPC scheme and the LCD scheme is linear with
$\eta$. When $\eta$ is small, i.e., minimization of the fronthaul usage is
weighted more, the LCD scheme is superior to the MPC scheme. When $\eta$ is
large, i.e., minimization of the cell average outage probability is weighted
higher, the MPC scheme is superior to the LCD scheme. There exists a crossover
point of the two schemes, the weighting factor of the crossover point is

$\begin{array}[]{c}\eta_{0}=\dfrac{1}{1+\dfrac{\sum\nolimits_{l=1}^{L}P_{l}\mathbb{E}_{x_{0}}\left[P_{out,MPC}^{(l)}(x_{0})-P_{out,LCD}^{(l)}(x_{0})\right]}{\sum\nolimits_{l=1}^{L}P_{l}\left(T_{l,LCD}-T_{l,MPC}\right)}}\end{array}.$
(17)

When $M_{n}=M,~{}\forall n$, $\eta_{0}$ can be further expressed as

$\begin{array}[]{c}\eta_{0}=\dfrac{1}{1+\dfrac{\sum\nolimits_{l=1}^{NM}P_{l}\mathbb{E}_{x_{0}}\left[P_{out,LCD}^{(l)}(x_{0})-P_{out,MPC}^{(l)}(x_{0})\right]}{\sum\nolimits_{l=M+1}^{NM}P_{l}}}\end{array}.$
(18)

###### Proof.

Please refer to Appendix B. ∎

Based on proposition 1, we propose a mode selection caching strategy. The RAN
can make a decision of the tradeoff according to the statistics of cell
average outage probability and fronthaul usage in the cell, and a tradeoff
weighting factor $\eta$ is chosen. When $\eta\leq\eta_{0}$, select the LB-LCD
caching scheme, while when $\eta>\eta_{0}$, select the MPC caching scheme.

###  V-C Computational Complexity Analysis

The number of objective function calculations w.r.t. a certain value of $\eta$
is evaluated to measure the complexities of the exhaustive search method, the
proposed GA approach and the proposed mode selection approach. The complexity
of exhaustive search is $\prod_{n=1}^{N}\binom{L}{M_{n}}$. When
$M_{n}=M,~{}\forall n$, it is clear that the complexity of exhaustive search
is exponential w.r.t. the number of RRHs, i.e., $\binom{L}{M}^{N}$. The
complexity of the proposed GA is $N_{p}N_{g}$, where $N_{p}$ and $N_{g}$ are
the population size and the number of generations evaluated, respectively.
$N_{g}$ is determined by the convergence behavior of the GA. While the
complexity of the proposed mode selection scheme is only 2. The reason is
that, once the value of $\eta_{0}$ is solved from (17), the RAN can choose a
mode between MPC and LCD based on whether $\eta>\eta_{0}$, and 2 objective
function calculations are involved in solving the equation. Further more, once
$\eta_{0}$ is obtained, caching schemes for all values of $\eta$ are obtained.
The computational complexities of the three approaches are summarized in Table
III.

TABLE III: Computational Complexity Scheme | Objective function calculations
---|---
Exhaustive search |  $\prod\nolimits_{n=1}^{N}\binom{L}{M_{n}}$
Proposed GA-based approach |  $N_{p}N_{g}$
Proposed mode selection approach |  2

##  VI Numerical Results

In this section, the performances of the proposed two caching strategies are
investigated through some representative numerical results. Firstly, the
accuracy of the cell average outage probability and fronthaul usage analysis
are verified by evaluating two typical caching schemes, i.e., the MPC and the
LB-LCD schemes. Then the effectiveness of the proposed GA approach is verified
by comparing its performance with exhaustive search, where the Pareto optimal
solutions [41] of the joint optimization problem are presented. In the
proposed GA approach, placement matrices of the MPC and the LB-LCD schemes are
added into the initial population to further improve the performance. Finally,
performances of different caching strategies are compared and the convergence
behavior of the proposed GA is presented.

The MATLAB software is used for the Monte-Carlo simulations and numerical
calculations. Throughout the simulation, it is assumed that each RRH has the
same cache size $M_{n}=M$. The transmit power of each RRH is
$p_{T}=\frac{P}{N}$, where $P$ is the total transmit power in the cell and
$\frac{P}{\sigma^{2}}=23~{}\text{dB}$. The constant $K$ in (4) is chosen such
that the received power attenuates 20 dB when the distance between the RRH and
the user is $R$ [49]. In such setting, the outage probability does not depend
on the absolute value of $R$, that is, $R$ can be regarded as the normalized
radius. The main simulation parameters are summarized in Table IV.

TABLE IV: Simulation Parameters Parameter | Value
---|---
Path loss exponent $\alpha$ | $3$
$P/\sigma^{2}$ |  $23$ dB
SNR threshold $\gamma_{th}$ |  $3$ dB
User location distribution | uniform
$U$ and $V$ in Simpson’s integration | $6,6$
Population size $N_{p}$ in GA | $50$
Selection function | stochastic universal sampling
Number of elites $N_{e}$ | $10$
Crossover fraction $f_{c}$ | $0.85$

###  VI-A MPC and LB-LCD Caching Placements

Cell average outage probability
($\sum\nolimits_{l=1}^{L}P_{l}\mathbb{E}_{x_{0}}[P_{out}^{(l)}(x_{0})]$) and
average fronthaul usage ($\sum\nolimits_{l=1}^{L}P_{l}T_{l}$) of the MPC and
LB-LCD schemes are shown in Fig. 4 and Fig. 5, respectively. There are $L=50$
files, $N=7$ RRHs with one RRH located at the cell center and the other 6 RRHs
evenly distributed on the circle with radius ${2R}/{3}$ [23, 50], and each RRH
can cache $M=5$ files. Both the simulation and numerical results are shown in
this subsection. In the Monte-Carlo simulations, there are $10^{4}$
realizations of the user’s different locations and content file requests.

Cell average outage probability with different SNR threshold $\gamma_{th}$ and
popularity skewness factor $\beta$ is illustrated in Fig. 4. It can be seen
that the outage probability of both the MPC and the LB-LCD schemes increases
with the increase of $\gamma_{th}$, and the outage probability of the MPC
scheme is lower than that of the LB-LCD scheme. The MPC curves of different
values of $\beta$ coincide. The reason is as follows: according to the file
delivery scheme and the MPC caching strategy, no matter whether the requested
file is cached in the RRHs or not, the file will be transmitted to the user
from all the RRHs, thus the cell average outage probability w.r.t. any $l$-th
file is the same, denoting as
$\mathbb{E}_{x_{0}}[P_{out}^{(l)}(x_{0})]=P_{cell,out}$, then the cell average
outage probability expected on all the file requests is

$\sum\limits_{l=1}^{L}P_{l}\mathbb{E}_{x_{0}}[P_{out}^{(l)}(x_{0})]=P_{cell,out}\cdot\sum\limits_{l=1}^{L}P_{l}=P_{cell,out},$
(19)

which is not related to $\beta$, i.e., no matter how the popularity is
distributed over the files, the cell average outage probability is kept as a
constant.

Figure 4: Cell average outage probability. $L=50,M=5,N=7$.

For the LB-LCD scheme, cell average outage probability reaches the minimum
value when $\beta=0$, and it increases as $\beta$ increases and approaches the
maximum value when $\beta$ is large enough, e.g., $\beta=2,2.5,3$. The reason
is explained as follows. According to the file delivery scheme and the LB-LCD
caching strategy, if the requested file is not cached in any of the RRHs, the
file will be fetched from the BBU pool, and then transmitted to the user from
all the RRHs. The outage probability will then achieve the minimum value due
to wireless diversity. While if the requested file is cached in the RRH (only
cached in one of the RRHs), the file will be transmitted to the user from only
one RRH, and the outage probability will be relatively higher. When $\beta=0$,
$P_{l}=1/L,\forall l$, i.e., the request probability is the same for all the
content files, which means that the cell average outage probability depends
evenly on the outage probability of each file, and the outage probability of
the files which are not cached in the RRHs is lower than that of the files
cached in the RRHs. As $\beta$ increases, the more skewness of the popularity
will toward the first few files with high ranks, i.e., the cell average outage
probability depends more on these files, and there is a higher probability
that there is only one copy for each of these files cached in one certain RRH,
and the corresponding outage probability is high, so the outage probability
increases as $\beta$ increases and the curve with $\beta=0$ is the lower
bound. Note that

$\sum_{l=1}^{5}P_{l}=\left\\{\begin{aligned} 0.90,&~{}\beta=2.0\\\
0.96,&~{}\beta=2.5\\\ 0.99,&~{}\beta=3.0\end{aligned}\right.$ (20)

which means when $\beta$ is large enough ($\beta\geqslant 2.0$), the cell
average outage probability depends mainly on the first 5 most popular files.
These 5 files are cached in the RRH located at the cell center, and the cell
average outage probability w.r.t any one of the 5 files is the same, so the
cell average outage probability is nearly the same for different values of
$\beta~{}(\geqslant 2.0)$, which approaches the maximum value.

Fig. 5 shows the fronthaul usage of the two caching schemes. Because the
fronthaul usage is independent of $\gamma_{th}$, the curve versus different
values of the skewness factor $\beta$ is evaluated. The LB-LCD scheme has
lower fronthaul usage than the MPC scheme, which is because that the LB-LCD
scheme caches a total of $MN=5\times 7=35$ different files in the RRHs while
the MPC scheme caches only $M=5$ different files. The average fronthaul usage
of both the MPC and LB-LCD scheme decreases with the increase of $\beta$,
which is due to the same reason that as $\beta$ increases, the popularity
becomes more skewed towards the first few files with higher ranks, and there
is a higher probability that these few files are cached in the RRHs. As shown
in (20), when $\beta=3$, the fronthaul usage almost depends on the first 5
popular files, since they are all cached in the RRHs under both the MPC and
the LB-LCD caching strategies, the fronthaul usages of both schemes approach
zero.

It is seen from Fig. 4 and Fig. 5 that the analytical results are highly
consistent with the simulation results. Therefore, analytical results will be
used instead of time-consuming simulations in the following evaluations.

Figure 5: Cell average fronthaul usage. $L=50,M=5,N=7$.

###  VI-B Tradeoffs between Cell Average Outage Probability and Fronthaul
Usage

Tradeoffs between cell average outage probability and fronthaul usage obtained
by exhaustive search and the proposed GA-based approach are shown in this
subsection. There are three RRHs, and the polar coordinates of which are
$\left(\frac{R}{4},0\right)$, $\left(\frac{R}{3},\frac{2\pi}{3}\right)$, and
$\left(\frac{R}{2},\frac{4\pi}{3}\right)$, respectively. There are $L=9$
content files, and the popularity skewness factor $\beta=1.5$.

Figure 6: Cell average outage probability and fronthaul usage tradeoff region.
$L=9,M=2,N=3,\beta=1.5$. Each red point corresponds to a caching placement,
and the 5 points emphasized by small blue circles are the Pareto optimal
solutions of the joint optimization problem.

Fig. 6 is focused on the scenario when the cache size $M$ is equal to $2$. All
tradeoffs between cell average outage probability and fronthaul usage are
given by exhaustive search. Since the popularities of the content files
$\\{P_{l}\\}$, the fronthaul usage $\\{T_{l}\\}$ and the outage probability
$\\{P^{(l)}_{out}\\}$ are all discontinuous values w.r.t. integer $l$, the
cell average outage probability and average fronthaul usage region of all
caching placements is a set of discrete points as shown in the figure, where
each red point corresponds to a caching placement. The 5 points emphasized by
small blue circles are the Pareto optimal solutions (nondominated set [41]) of
the joint optimization problem, i.e., there is no other point dominating with
the Pareto optimal solutions in terms of both the cell average outage
probability and fronthaul usage.

The cell average outage probability is minimized when the files cached in each
RRH are the same, and the popularity of these cached files will have an impact
on the average fronthaul usage. The corresponding points of these caching
placements lie on the line segment $AD$, i.e., line segment $AD$ represents
the lower bound of the cell average outage probability. The MPC scheme
represented by point $A$ achieves the minimum fronthaul usage among these
caching placements. The reason is that the MPC scheme caches the most popular
files which can reduce the fronthaul usage to a minimum value among these
caching placements.

The fronthaul usage is minimized when all RRHs cache different files with
higher popularity ranks, and the cache locations of these files will have an
impact on the cell average outage probability. The corresponding points of
these caching placements lie on the line segment $BC$, i.e., line segment $BC$
represents the lower bound of the average fronthaul usage. The LB-LCD scheme
represented by point $B$ achieves the minimum cell average outage probability
among these caching placements. The reason is that the LB-LCD scheme caches
the files with higher ranks in the RRHs near to the cell center, which has the
minimum cell average outage probability among these caching placements.

Figure 7: Cell average outage probability and fronthaul usage tradeoff.
$L=9,M=\\{1,2,3\\},N=3,\beta=1.5$. Unless otherwise specified in the figure,
$\eta$ is evaluated from 0 to 1 with a step 0.1 in the proposed GA-based
approach, i.e., $\eta=\\{0.1,0.2,\cdots,1\\}$.

Fig. 7 shows the Pareto optimal tradeoffs between the cell average outage
probability and the fronthaul usage with different cache size $M$. The results
obtained through the proposed GA approach are almost the same as exhaustive
search, which means that the proposed GA approach can achieve near-optimal
performance. The minimum cell average outage probability is achieved at point
$A_{1}$ when $M=1$, $A_{2}$ when $M=2$, and $A_{3}$ when $M=3$, respectively.
The minimum cell average outage probability represented by the three points
are the same, and the corresponding caching placements of the three points are
the MPC scheme. The reason is that according to the file delivery scheme and
the MPC caching placement, all the RRHs will serve the user no matter how many
files the RRHs can cache. It is also seen that the corresponding fronthaul
usage of the three points decrease as $M$ increases, which is obvious because
larger cache size can cache more files thus the fronthaul usage can be
reduced.

On the other hand, the minimum fronthaul usage is achieved at point $B_{1}$
when $M=1$, $B_{2}$ when $M=2$, and $B_{3}$ when $M=3$, respectively. The
corresponding caching placements of the three points are the LB-LCD scheme.
Obviously, the corresponding fronthaul usage of the three points decreases as
$M$ increases. The fronthaul usage is zero at point $B_{3}$ when $M=3$, the
reason is that all the RRHs can cache a total of $MN=3\times 3=9$ files, which
is equal to the number of files in the file library, i.e., all the files are
cached in the RRHs. The corresponding cell average outage probability of the
three points increases as $M$ increases. The reason is that according to the
file delivery scheme and the LB-LCD caching strategy, more different files can
be cached in the RRHs as $M$ increases, however, there is only one copy of
each file and the outage probability w.r.t. these cached files will be higher,
i.e., more different files cached in the RRHs, higher the cell average outage
probability is.

Note that as the cache size $M$ increases, the GA-based approach should
evaluate more values of $\eta$ in order to obtain all the Pareto optimal
solutions of the joint optimization problem. For example, when $M=3$,
additional values of $\eta=0.15$ and $\eta=0.45$ are evaluated to obtain the
Pareto optimal solutions represented by point $C$ and $D$.

Table V shows all the optimal caching placements obtained by the proposed GA-
based approach when $M=2$. For illustration, we use a $M\times N$ matrix to
represent the caching placement, with the $(m,n)$-th entry
$b_{m,n}\in\\{1,2,3,\cdots,L\\}$ denotes the file index cached in the $m$-th
cache space of the $n$-th RRH. From (18), $\eta_{0}=0.3312$. It can be seen
that the LB-LCD scheme is the optimal placement when $\eta=0,0.1<\eta_{0}$,
while the MPC scheme is the optimal solution when $\eta=0.6\sim 1.0>\eta_{0}$,
and some files are duplicately cached in the RRHs when $\eta=0.2\sim 0.5$.

TABLE V: Optimal Caching Strategy Obtained by The Proposed GA $\begin{array}[]{c}\eta=0\\\ f_{obj}=0.0689\\\ \left[\begin{array}[]{ccc}1&3&5\\\ 2&4&6\end{array}\right]\end{array}$ | $\begin{array}[]{c}\eta=0.1\\\ f_{obj}=0.1186\\\ \left[\begin{array}[]{ccc}1&3&5\\\ 2&4&6\end{array}\right]\end{array}$ | $\begin{array}[]{c}\eta=0.2\\\ f_{obj}=0.1651\\\ \left[\begin{array}[]{ccc}1&1&4\\\ 2&3&5\end{array}\right]\end{array}$
---|---|---
$\begin{array}[]{c}\eta=0.3\\\ f_{obj}=0.1938\\\ \left[\begin{array}[]{ccc}1&1&1\\\ 2&3&4\end{array}\right]\end{array}$ | $\begin{array}[]{c}\eta=0.4\\\ f_{obj}=0.2087\\\ \left[\begin{array}[]{ccc}1&1&1\\\ 2&3&4\end{array}\right]\end{array}$ | $\begin{array}[]{c}\eta=0.5\\\ f_{obj}=0.2144\\\ \left[\begin{array}[]{ccc}1&1&1\\\ 3&2&2\end{array}\right]\end{array}$
$\begin{array}[]{c}\eta=0.6\\\ f_{obj}=0.2077\\\ \left[\begin{array}[]{ccc}1&1&1\\\ 2&2&2\end{array}\right]\end{array}$ | $\begin{array}[]{c}\eta=0.7\\\ f_{obj}=0.1905\\\ \left[\begin{array}[]{ccc}1&1&1\\\ 2&2&2\end{array}\right]\end{array}$ | $\begin{array}[]{c}\eta=0.8\\\ f_{obj}=0.1733\\\ \left[\begin{array}[]{ccc}1&1&1\\\ 2&2&2\end{array}\right]\end{array}$
$\begin{array}[]{c}\eta=0.9\\\ f_{obj}=0.1561\\\ \left[\begin{array}[]{ccc}1&1&1\\\ 2&2&2\end{array}\right]\end{array}$ | $\begin{array}[]{c}\eta=1.0\\\ f_{obj}=0.1390\\\ \left[\begin{array}[]{ccc}1&1&1\\\ 2&2&2\end{array}\right]\end{array}$ | $\begin{array}[]{l}L=9\\\ M=2\\\ N=3\\\ \beta=1.5\end{array}$

According to the above evaluations, the MPC and LB-LCD caching schemes are two
special solutions of the joint optimization problem when $\eta=1$ and
$\eta=0$, respectively. The former can achieve the lowest cell average outage
probability while the latter can achieve the minimum fronthaul usage. The
proposed GA-based approach can achieve different tradeoffs between the cell
average outage probability and fronthaul usage according to different
weighting factors, which can achieve better performance than the MPC and LB-
LCD schemes.

###  VI-C Performances of the GA-based Approach and Mode Selection Approach

The performances of the proposed GA-based approach and the mode selection
approach are analyzed in this subsection. Besides the MPC and the LB-LCD
caching schemes, two other widely used caching strategies are evaluated for
comparison, one is _random caching_ , where each RRH caches the content files
independently and randomly regardless of the files’ popularity distribution,
the other one is _probabilistic caching_ , where each RRH caches the files
independently and randomly according to the files’ popularity distribution,
i.e., high-ranked files have higher probability to be cached [23, 51]. There
are $L=50$ files 222Alougth there is a huge amount of content files in
practice, they can be classified into different categories [46], and the
number of files in each category (or subcategory) is relatively limited, so
the proposed algorithms can be performed on each category, the number of files
evaluated in the simulation will not lose meaningful insights of the tradeoff
caching optimization., $N=7$ RRHs with one RRH located at the cell center and
the other 6 RRHs evenly distributed on the circle with radius ${2R}/{3}$, and
$\beta=1.5$.

Figure 8: Objective function value versus weighting factor $\eta$.
$L=50,M=5,N=7$.

Fig. 8 shows the objective value of different caching strategies with $M=5$.
It can be seen from the figure that as the weighting factor $\eta$ increases,
i.e., more focus on minimization of outage probability, the objective value of
MPC decreases linearly, while the objective value of the LB-LCD scheme
increases linearly. The horizontal coordinate of the crossover point of the
MPC and LB-LCD scheme ($\eta_{0}$) approaches zero as the popularity skewness
factor $\beta$ increases. This is because that when $\beta$ increases, the
requesting probability $P_{l}$ of the first few popular files increase
significantly, then $\sum\nolimits_{l=M+1}^{NM}P_{l}\rightarrow 0$ in (18),
thus $\eta_{0}\rightarrow 0$. That is, as $\beta$ increases, the MPC scheme
will dominate with most values of $\eta$. This can also be explained as
follows. When $\beta$ increases, the average fronthaul usage will depend more
and more on the few files with higher ranks. These files can be cached in the
RRHs under both of the MPC and the LB-LCD schemes, thus the MPC and the LB-LCD
schemes are equivalent in terms of fronthaul usage, while the MPC can achieve
lower outage probability. Therefore the MPC scheme is superior to the LB-LCD
scheme. The crossover point $\eta_{0}=0.23$ when $\beta=1.5$ calculated
through (18) exactly matches the simulation results. The above mentioned
results are consistent with Proposition 1.

The random caching strategy has a relative poor performance for all values of
$\eta$, which is because the files cached in the RRHs are selected randomly,
there is neither a high probability to cache the same file for reducing the
outage probability nor to cache different high-ranked files for reducing the
fronthaul usage. While the probabilistic caching strategy can achieve better
performance than the proposed mode selection approach in the middle range of
$\eta$, e.g., for $\eta=0.2\sim 0.4$, where both the cell average outage
probability minimization and the fronthaul usage reduction are treated
approximate equally. Which is because that, in probabilistic caching, each RRH
will cache the high-ranked files with a higher probability, so there is a high
probability for different RRHs to cache the same high-ranked files, which can
reduce the cell average outage probability, and meanwhile the inherent
randomness in the placement makes it possible to cache different files to
reduce the fronthaul usage. It is also seen that the proposed GA-based
approach can achieve better performance than the other caching strategies, for
instance, the objective function value of the proposed GA algorithm is 18.25%
lower than a typical probabilistic caching scheme when $\eta=0.4$, and this
improvement goes up to 87.9% when $\eta=1$, the average improvement over all
values of $\eta$ is 47.5%.

Figure 9: Cell average outage probability and fronthaul usage versus weighting
factor $\eta$. $L=50,M=5,N=7,\beta=1.5$.

Cell average outage probability and fronthaul usage of the proposed GA and the
proposed mode selection approach versus weighting factor are shown in Fig. 9.
Note that the mode selection scheme is actually the LB-LCD scheme when
$\eta\leq\eta_{0}$ and the MPC scheme when $\eta>\eta_{0}$, respectively. For
the proposed GA approach, the solution is exactly the LB-LCD scheme when
$\eta=0$, as $\eta$ increases, the cell average outage probability decreases
and the fronthaul usage increases, and they reach the lower and upper bounds
when $\eta\geqslant 0.6$, respectively, where the solution is the MPC scheme.
The proposed GA approach can adjust the caching placement according to
different weighting factors $\eta$ while the mode selection scheme only
chooses a caching placement between the MPC and the LB-LCD schemes based on
whether $\eta>\eta_{0}$, so the proposed GA approach can achieve better
performance than the mode selection scheme. However, the computational
complexity of the mode selection scheme is extremely low.

Figure 10: Objective function value versus weighting factor $\eta$.
$L=50,N=7,\beta=1.5$.

Fig. 10 shows the performance of the proposed GA and the mode selection scheme
with different cache size $M$. It can be seen from the figure that the mode
selection scheme can achieve near-optimal performance over a wide range of the
weighting factor $\eta$. The vertex of the mode selection scheme, i.e., the
crossover point of the MPC and the LB-LCD schemes moves toward the origin as
the cache size $M$ increases, i.e., the MPC scheme will dominate with most
values of $\eta$ as $M$ increases. The reason is explained as follows. When
$M$ increases, more content files can be cached in the RRHs. The fronthaul
usage depends mostly on the first few popular files cached in the RRHs, so the
fronthaul usage will tend to be the same between the two schemes as $M$
increases. The MPC scheme can achieve lower outage probability, further more,
the cell average outage probability of the LB-LCD scheme increases as $M$
increases, so the objective value of the MPC scheme will be much lower than
that of the LB-LCD scheme, and the MPC scheme is superior to the LB-LCD scheme
with most values of $\eta$.

Figure 11: Convergence behavior of the proposed GA approach.
$L=50,M=5,N=7,\beta=1.5$.

Fig. 11 shows the convergence behavior of the proposed GA approach. It can be
seen from the figure that the mean objective value of the population converges
within average 8 generations. The computational complexity is
$N_{g}N_{p}=8\times 50=400$. While the computational complexity of the
exhaustive search is $\binom{50}{5}^{7}=1.92\times 10^{44}$, which is not
feasible in practice. As stated earlier, the popularity of the content files
will remain the same for a relative long period, so the convergence behavior
of the caching placement algorithm is not time-critical. Unlike the delivery
stage, which needs to make an instant decision for coping with the dynamics of
mobile networking systems, e.g., the rapid change of channel state
information, it is not necessary for the caching strategy to make an instant
decision due to the slow change of the statistics data (e.g., the request
probabilities of the files). In addition, if parallel computing is adopted,
the population size of the proposed GA approach can be increased without
introducing additional execution time, while the converging speed of the GA
will be accelerated. Thus, the GA approach can perform well with a satisfying
converging speed in practice.

##  VII Conclusion

In this paper, we have investigated tradeoff caching strategy in Cloud-RAN for
future mobile communications. In order to jointly minimize the cell average
outage probability and fronthaul usage, the optimization problem is formulated
as a weighted sum of the two objectives, with weighting factor $\eta$ (and
$1-\eta$). Analytical expressions of cell average outage probability and
fronthaul usage have been presented and verified through simulations.
Performances of two particular caching strategies have been analyzed, namely
the MPC and the LB-LCD schemes. When the minimization of the cell average
outage probability is more focused on, the MPC scheme is superior to the LB-
LCD scheme, while the latter is superior to the former in the opposite
situation, i.e., where the reduction of average fronthaul usage is more
focused on. When the content files’ popularity skewness factor $\beta$ is
larger, or the cache size of each RRH increase, the MPC scheme will dominate
in a wide range of $\eta$. Two heuristic approaches have been proposed to
solve the joint optimization problem: one is the GA based approach which can
achieve nearly the same optimal performance of exhaustive search, while the
computational complexity is significantly reduced; the other is the mode
selection approach with extremely low computational complexity, which can
obtain near-optimal performance within a wide range of $\eta$. Compared with a
typical probabilistic caching scheme, the proposed GA approach can reduce the
objective function value by up to 45.7% on average and the proposed two mode
selection caching strategy can provide an average improvement of 36.9%. In
practice, the RAN can make a decision of the tradeoff according to the
system’s statistics of fronthaul traffic and outage probability, and then
adopt caching strategy through the proposed schemes.

##  Appendix A  Derivations of (8) and (9)

For a specific file $F_{l}$, the subscript of file index $l$ and the user’s
location $x_{0}$ are omitted without ambiguity. In (7),
$|h_{n}|^{2}\sim\chi^{2}(2)$, and the PDF is given by [52]

$f_{|h_{n}|^{2}}(x)=\exp(-x),~{}x>0.$ (A.1)

Then the PDF of $\gamma_{n}=\gamma_{0}S_{n}|h_{n}|^{2}$ is

$f_{\gamma_{n}}(\gamma)=\dfrac{1}{\gamma_{0}S_{n}}\exp\left({-\dfrac{\gamma}{\gamma_{0}S_{n}}}\right),~{}\gamma>0,~{}n\in\Phi.$
(A.2)

The moment generation function (MGF) [53] of the random variable $\gamma_{n}$
is

$\displaystyle M_{\gamma_{n}}(s)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}f_{\gamma_{n}}(\gamma)e^{s\gamma}d\gamma$ (A.3)
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\frac{1}{\gamma_{0}S_{n}}\exp\left(-\frac{\gamma}{\gamma_{0}S_{n}}\right)e^{s\gamma}d\gamma$
$\displaystyle=$ $\displaystyle\frac{1}{1-\gamma_{0}S_{n}\cdot{s}}~{},$

and the range of convergence (ROC) is
$\operatorname{Re}\left(s\right)<\frac{1}{\gamma_{0}S_{n}}$. Since the RRHs
are distributed at different locations, $\\{\gamma_{n},n\in\Phi\\}$ is
independent of each other, the MGF of received SNR
$\gamma=\sum_{n\in\Phi}\gamma_{n}$ is given by

$M_{\gamma}(s)=\prod_{n\in\Phi}M_{\gamma_{n}}(s)=\prod_{n\in\Phi}\frac{1}{1-\gamma_{0}S_{n}\cdot{s}}~{},\\\
$ (A.4)

and the ROC is
${\bigcap}_{n\in\Phi}\operatorname{Re}\left(s\right)<\frac{1}{\gamma_{0}S_{n}}$.

Since there are $I$ distinct distances $d_{1}\neq d_{2}\neq\cdots\neq
d_{i}\neq\cdots\neq d_{I}$ between the service RRHs and the user, and the
$i$-th distance has multiplicity of $J_{i}$, (A.4) can be rewritten as

$M_{\gamma}(s)=\dfrac{1}{\left(1-\dfrac{1}{\lambda_{1}}s\right)^{J_{1}}\left(1-\dfrac{1}{\lambda_{2}}s\right)^{J_{2}}\cdots\left(1-\dfrac{1}{\lambda_{I}}s\right)^{J_{I}}}~{},\\\
$ (A.5)

where
$\lambda_{i}=\frac{1}{\gamma_{0}Kd_{i}^{-\alpha}},~{}i\in\\{1,2,\cdots,I\\}$
is the $i$-th pole of multiplicity $J_{i}$ of $M_{\gamma}(s)$, using partial
fraction expansion, $M_{\gamma}(s)$ can be expressed as

$M_{\gamma}(s)=\sum_{i=1}^{I}\sum_{j=1}^{J_{i}}{\dfrac{A_{ij}}{\left(1-\dfrac{1}{\lambda_{i}}s\right)^{j}}}~{},$
(A.6)

where $\\{A_{ij}\\}$ are the undetermined coefficients. Multiplying
$(1-\frac{1}{\lambda_{i}}s)^{J_{i}}$ to both sides of (A.6), then calculating
the $(J_{i}-j)$-th order derivate for both sides and let $s=\lambda_{i}$, we
have

$\displaystyle\dfrac{d^{J_{i}-j}}{ds^{J_{i}-j}}\left[M_{\gamma}(s)\left(1-\dfrac{1}{\lambda_{i}}s\right)^{J_{i}}\right]\Bigg{|}_{s=\lambda_{i}}$
(A.7) $\displaystyle=$
$\displaystyle\dfrac{d^{J_{i}-j}}{ds^{J_{i}-j}}\left.\left[\sum_{i=1}^{I}\sum_{j=1}^{J_{i}}{\dfrac{A_{ij}}{\left(1-\dfrac{1}{\lambda_{i}}\cdot
s\right)^{j}}}\left(1-\dfrac{1}{\lambda_{i}}s\right)^{J_{i}}\right]\right|_{s=\lambda_{i}}$
$\displaystyle=$
$\displaystyle(J_{i}-j)!\left(-\dfrac{1}{\lambda_{i}}\right)^{J_{i}-j}A_{ij}~{}.$

Thus $A_{ij}$ is obtained as (10).

The PDF of $\gamma$ can be obtained by inversely transforming the MGF in
(A.6). Considering a general form of the PDF,

$f(\gamma)=\gamma^{n}e^{-a\gamma},\quad\gamma\geq 0,$ (A.8)

where $n\in\\{0\\}\cup\mathbb{Z}^{+}$, $a\in\mathbb{R}^{+}$. The MGF of
$f(\gamma)$ can be obtained by continuously using the method of integration by
parts.

$\displaystyle M(s)$ (A.9) $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\gamma^{n}e^{-a\gamma}e^{s\gamma}d\gamma$
$\displaystyle=$
$\displaystyle-\dfrac{1}{a-s}\int_{0}^{\infty}\gamma^{n}de^{-(a-s)\gamma}$
$\displaystyle=$
$\displaystyle-\dfrac{1}{a-s}\left(\gamma^{n}e^{-(a-s)\gamma}\Big{|}_{0}^{\infty}-n\int_{0}^{\infty}e^{-(a-s)\gamma}\gamma^{n-1}d\gamma\right)$
$\displaystyle\hskip 85.35826pt\vdots$ $\displaystyle=$
$\displaystyle\dfrac{n!}{(a-s)^{n+1}}~{},$

and the ROC is $\operatorname{Re}\left(s\right)<a$. Denote the pair of the PDF
and its corresponding MGF as

$f(\gamma)=\gamma^{n}e^{-a\gamma}\Longleftrightarrow
M(s)=\dfrac{n!}{(a-s)^{n+1}}~{}.$ (A.10)

The CDF can be calculated in the same manner,

$\displaystyle F(\gamma)$ $\displaystyle=\int_{0}^{\gamma}f(\gamma)d\gamma$
(A.11) $\displaystyle=\int_{0}^{\gamma}\gamma^{n}e^{-a\gamma}d\gamma$
$\displaystyle=\dfrac{1}{a}\left[\dfrac{n!}{a^{n}}-\left(e^{-a\gamma}\sum_{k=0}^{n}\dfrac{n!}{(n-k)!a^{k}}\gamma^{n-k}\right)\right]~{}.$

According to (A.6) and (A.10), the PDF of the received SNR is obtained, as
shown in (8). According to (A.11), the CDF of the received SNR is obtained as
shown in (9).

##  Appendix B  Proof of Proposition 1

Without loss of generality, it is assumed that $M_{n}=M,~{}\forall
n\in\mathcal{N},~{}|\mathcal{N}|>1$. According to (6c), it is obvious that the
objective functions of the MPC and LCD schemes $f_{obj}^{MPC}$ and
$f_{obj}^{LCD}$ are linearly (thus monotonic) continuous function of $\eta$ on
closed interval $[0,1]$.

When $\eta=0$, $f_{obj}=\sum\nolimits_{l=1}^{L}P_{l}T_{l}$. The objective
values of the two schemes are

$\displaystyle f_{obj}^{MPC}$ $\displaystyle=$
$\displaystyle\sum\limits_{l=1}^{M}P_{l}\underset{\begin{subarray}{c}\uparrow\\\
0\end{subarray}}{{T_{l}}}$ $\displaystyle+$
$\displaystyle\sum\limits_{l=M+1}^{L}P_{l}\underset{\begin{subarray}{c}\uparrow\\\
1\end{subarray}}{{T_{l}}}$ $\displaystyle=$
$\displaystyle\sum\limits_{l=M+1}^{L}P_{l}~{},$ (B.1) $\displaystyle
f_{obj}^{LCD}$ $\displaystyle=$
$\displaystyle\sum\limits_{l=1}^{NM}P_{l}\underset{\begin{subarray}{c}\uparrow\\\
0\end{subarray}}{{T_{l}}}$ $\displaystyle+$
$\displaystyle\sum\limits_{l=NM+1}^{L}P_{l}\underset{\begin{subarray}{c}\uparrow\\\
1\end{subarray}}{{T_{l}}}$ $\displaystyle=$
$\displaystyle\sum\limits_{l=NM+1}^{L}P_{l}~{}.$

Note that $\sum\nolimits_{l=1}^{L}P_{l}=1$ and $P_{1}\geqslant
P_{2}\geqslant\cdots\geqslant P_{L}$, where equality holds if and only if
$\beta=0$. Thus

$f_{obj}^{MPC}\Big{|}_{\eta=0}>f_{obj}^{LCD}\Big{|}_{\eta=0}~{}.$ (B.2)

When $\eta=1$,
$f_{obj}=\sum\limits_{l=1}^{L}P_{l}\mathbb{E}_{x_{0}}\left[P_{out}^{(l)}(x_{0})\right]$.
Denoting $\mathbb{E}_{x_{0}}\left[P_{out}^{(l)}(x_{0})\right]$ as
$P_{cell,out}(l)$, then

$\displaystyle f_{obj}^{MPC}=$
$\displaystyle\underbrace{\sum\limits_{l=1}^{NM}P_{l}P^{MPC}_{cell,out}(l)}_{C}$
$\displaystyle+$
$\displaystyle\underbrace{\sum\limits_{l=NM+1}^{L}P_{l}P^{MPC}_{cell,out}(l)}_{D}~{},$
(B.3) $\displaystyle f_{obj}^{LCD}=$
$\displaystyle\underbrace{\sum\limits_{l=1}^{NM}P_{l}P^{LCD}_{cell,out}(l)}_{E}$
$\displaystyle+$
$\displaystyle\underbrace{\sum\limits_{l=NM+1}^{L}P_{l}P^{LCD}_{cell,out}(l)}_{F}~{}.$

According to the wireless transmission strategy, $D=F$, where $D$ and $F$
correspond to the scenario that all the RRHs serve the user, while $E>C$
because $E$ denotes there is only one RRH serving the user, while $C$
corresponds to all the RRHs serving the user. Thus

$f_{obj}^{MPC}\Big{|}_{\eta=1}<f_{obj}^{LCD}\Big{|}_{\eta=1}~{}.$ (B.4)

According to (B.2), (B.4) and the linearity of $f^{MPC}_{obj}$ and
$f^{LCD}_{obj}$, there exists a crossover point $\eta_{0}\in[0,1]$ of the two
objective functions. When $\eta<\eta_{0}$, the LCD scheme is superior to the
MPC scheme, while when $\eta>\eta_{0}$, the MPC scheme is superior to the LCD
scheme.

Substituting $\\{a_{l,n}\\}$ of the MPC and LCD schemes into (6c),
respectively, a linear equation of $\eta$ is formulated, and the solution is
shown as in (17). Because $M=M_{n},~{}\forall n$, (17) can be further written
as (18).

The proof can be extended to the case that $M_{n}$ is different with $n$.

## References

  * [1] Z. Ye, C. Pan, H. Zhu, and J. Wang, “Outage probability and fronthaul usage tradeoff caching strategy in Cloud-RAN,” in _2017 IEEE International Conference on Communications (ICC)_ , May 2017, pp. 1–6.
  * [2] S. Buzzi, C. L. I, T. E. Klein, H. V. Poor, C. Yang, and A. Zappone, “A survey of energy-efficient techniques for 5G networks and challenges ahead,” _IEEE Journal on Selected Areas in Communications_ , vol. 34, no. 4, pp. 697–709, April 2016.
  * [3] A. Gupta and R. K. Jha, “A survey of 5G network: architecture and emerging technologies,” _IEEE Access_ , vol. 3, pp. 1206–1232, 2015.
  * [4] O. Galinina, A. Pyattaev, S. Andreev, M. Dohler, and Y. Koucheryavy, “5G multi-RAT LTE-WiFi ultra-dense small cells: performance dynamics, architecture, and trends,” _IEEE Journal on Selected Areas in Communications_ , vol. 33, no. 6, pp. 1224–1240, June 2015.
  * [5] K. M. S. Huq, S. Mumtaz, J. Bachmatiuk, J. Rodriguez, X. Wang, and R. L. Aguiar, “Green HetNet CoMP: energy efficiency analysis and optimization,” _IEEE Transactions on Vehicular Technology_ , vol. 64, no. 10, pp. 4670–4683, October 2015.
  * [6] A. Checko, H. L. Christiansen, Y. Yan, L. Scolari, G. Kardaras, M. S. Berger, and L. Dittmann, “Cloud RAN for mobile networks —a technology overview,” _IEEE Communications Surveys & Tutorials_, vol. 17, no. 1, pp. 405–426, 2015.
  * [7] C. L. I, J. Huang, R. Duan, C. Cui, J. Jiang, and L. Li, “Recent progress on C-RAN centralization and cloudification,” _IEEE Access_ , vol. 2, pp. 1030–1039, 2014.
  * [8] H. Zhu, “Performance comparison between distributed antenna and microcellular systems,” _IEEE Journal on Selected Areas in Communications_ , vol. 29, no. 6, pp. 1151–1163, June 2011.
  * [9] J. Wang, H. Zhu, and N. J. Gomes, “Distributed antenna systems for mobile communications in high speed trains,” _IEEE Journal on Selected Areas in Communications_ , vol. 30, no. 4, pp. 675–683, May 2012.
  * [10] H. Zhu and J. Wang, “Radio resource allocation in multiuser distributed antenna systems,” _IEEE Journal on Selected Areas in Communications_ , vol. 31, no. 10, pp. 2058–2066, October 2013.
  * [11] C. Pan, H. Zhu, N. J. Gomes, and J. Wang, “Joint precoding and RRH selection for user-centric green MIMO C-RAN,” _IEEE Transactions on Wireless Communications_ , vol. 16, no. 5, pp. 2891–2906, May 2017.
  * [12] ——, “Joint user selection and energy minimization for ultra-dense multi-channel C-RAN with incomplete CSI,” _IEEE Journal on Selected Areas in Communications_ , vol. 35, no. 8, pp. 1809–1824, Aug 2017.
  * [13] S. C. Zhan and D. Niyato, “A coalition formation game for remote radio head cooperation in cloud radio access network,” _IEEE Transactions on Vehicular Technology_ , vol. 66, no. 2, pp. 1723–1738, Feb 2017.
  * [14] R. Yu, J. Ding, X. Huang, M. T. Zhou, S. Gjessing, and Y. Zhang, “Optimal resource sharing in 5G-enabled vehicular networks: A matrix game approach,” _IEEE Transactions on Vehicular Technology_ , vol. 65, no. 10, pp. 7844–7856, Oct 2016.
  * [15] M. Jaber, M. A. Imran, R. Tafazolli, and A. Tukmanov, “5G backhaul challenges and emerging research directions: a survey,” _IEEE Access_ , vol. 4, pp. 1743–1766, 2016.
  * [16] J. Liu, S. Xu, S. Zhou, and Z. Niu, “Redesigning fronthaul for next-generation networks: beyond baseband samples and point-to-point links,” _IEEE Wireless Communications_ , vol. 22, no. 5, pp. 90–97, October 2015.
  * [17] U. Siddique, H. Tabassum, E. Hossain, and D. I. Kim, “Wireless backhauling of 5G small cells: challenges and solution approaches,” _IEEE Wireless Communications_ , vol. 22, no. 5, pp. 22–31, October 2015.
  * [18] X. Wang, M. Chen, T. Taleb, A. Ksentini, and V. C. M. Leung, “Cache in the air: exploiting content caching and delivery techniques for 5G systems,” _IEEE Communications Magazine_ , vol. 52, no. 2, pp. 131–139, February 2014\.
  * [19] K. Poularakis, G. Iosifidis, V. Sourlas, and L. Tassiulas, “Exploiting caching and multicast for 5G wireless networks,” _IEEE Transactions on Wireless Communications_ , vol. 15, no. 4, pp. 2995–3007, April 2016.
  * [20] M. A. Maddah-Ali and U. Niesen, “Fundamental limits of caching,” _IEEE Transactions on Information Theory_ , vol. 60, no. 5, pp. 2856–2867, May 2014\.
  * [21] J. Duan, X. Lagrange, and F. Guilloud, “Performance analysis of several functional splits in C-RAN,” in _2016 IEEE 83rd Vehicular Technology Conference (VTC Spring)_ , May 2016, pp. 1–5.
  * [22] X. Peng, J. C. Shen, J. Zhang, and K. B. Letaief, “Joint data assignment and beamforming for backhaul limited caching networks,” in _2014 IEEE 25th Annual International Symposium on Personal, Indoor, and Mobile Radio Communication (PIMRC)_ , September 2014, pp. 1370–1374.
  * [23] M. Tao, E. Chen, H. Zhou, and W. Yu, “Content-centric sparse multicast beamforming for cache-enabled Cloud RAN,” _IEEE Transactions on Wireless Communications_ , vol. 15, no. 9, pp. 6118–6131, Sept. 2016.
  * [24] F. Pantisano, M. Bennis, W. Saad, and M. Debbah, “Match to cache: joint user association and backhaul allocation in cache-aware small cell networks,” in _2015 IEEE International Conference on Communications (ICC)_ , June 2015, pp. 3082–3087.
  * [25] H. Hsu and K. C. Chen, “A resource allocation perspective on caching to achieve low latency,” _IEEE Communications Letters_ , vol. 20, no. 1, pp. 145–148, January 2016.
  * [26] X. Li, X. Wang, S. Xiao, and V. C. M. Leung, “Delay performance analysis of cooperative cell caching in future mobile networks,” in _2015 IEEE International Conference on Communications (ICC)_ , June 2015, pp. 5652–5657.
  * [27] Y. Zhou, Z. Zhao, R. Li, H. Zhang, and Y. Louet, “Cooperation-based probabilistic caching strategy in clustered cellular networks,” _IEEE Communications Letters_ , vol. 21, no. 9, pp. 2029–2032, Sept 2017.
  * [28] J. Liao, K. K. Wong, M. R. A. Khandaker, and Z. Zheng, “Optimizing cache placement for heterogeneous small cell networks,” _IEEE Communications Letters_ , vol. 21, no. 1, pp. 120–123, Jan 2017.
  * [29] S. Wang, X. Zhang, K. Yang, L. Wang, and W. Wang, “Distributed edge caching scheme considering the tradeoff between the diversity and redundancy of cached content,” in _2015 IEEE/CIC International Conference on Communications in China (ICCC)_ , November 2015, pp. 1–5.
  * [30] Q. Li, W. Shi, X. Ge, and Z. Niu, “Cooperative edge caching in software-defined hyper-cellular networks,” _IEEE Journal on Selected Areas in Communications_ , vol. 35, no. 11, pp. 2596–2605, Nov 2017.
  * [31] J. Song, H. Song, and W. Choi, “Optimal caching placement of caching system with helpers,” in _2015 IEEE International Conference on Communications (ICC)_ , June 2015, pp. 1825–1830.
  * [32] X. Peng, J. C. Shen, J. Zhang, and K. B. Letaief, “Backhaul-aware caching placement for wireless networks,” in _2015 IEEE Global Communications Conference (GLOBECOM)_ , December 2015, pp. 1–6.
  * [33] Z. Yan, S. Chen, Y. Ou, and H. Liu, “Energy efficiency analysis of cache-enabled two-tier HetNets under different spectrum deployment strategies,” _IEEE Access_ , vol. 5, pp. 6791–6800, 2017.
  * [34] Y. Chen, M. Ding, J. Li, Z. Lin, G. Mao, and L. Hanzo, “Probabilistic small-cell caching: Performance analysis and optimization,” _IEEE Transactions on Vehicular Technology_ , vol. 66, no. 5, pp. 4341–4354, May 2017\.
  * [35] J. Wen, K. Huang, S. Yang, and V. O. K. Li, “Cache-enabled heterogeneous cellular networks: Optimal tier-level content placement,” _IEEE Transactions on Wireless Communications_ , vol. 16, no. 9, pp. 5939–5952, Sept 2017.
  * [36] L. Breslau, P. Cao, L. Fan, G. Phillips, and S. Shenker, “Web caching and Zipf-like distributions: evidence and implications,” in _Eighteenth Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM ’99)_ , vol. 1, March 1999, pp. 126–134.
  * [37] K. Shanmugam, N. Golrezaei, A. G. Dimakis, A. F. Molisch, and G. Caire, “Femtocaching: Wireless content delivery through distributed caching helpers,” _IEEE Transactions on Information Theory_ , vol. 59, no. 12, pp. 8402–8413, Dec 2013.
  * [38] H. Zhu and J. Wang, “Chunk-based resource allocation in OFDMA systems —part I: chunk allocation,” _IEEE Transactions on Communications_ , vol. 57, no. 9, pp. 2734–2744, September 2009.
  * [39] ——, “Chunk-based resource allocation in OFDMA systems —part II: joint chunk, power and bit allocation,” _IEEE Transactions on Communications_ , vol. 60, no. 2, pp. 499–509, February 2012.
  * [40] H. Zhu, “Radio resource allocation for OFDMA systems in high speed environments,” _IEEE Journal on Selected Areas in Communications_ , vol. 30, no. 4, pp. 748–759, May 2012.
  * [41] M. Ehrgott, _Multicriteria Optimazation_ , 2nd ed. Springer, 2005.
  * [42] J. Y. Wang, J. B. Wang, M. Chen, H. M. Chen, X. Dang, and H. Y. Li, “System outage probability analysis of uplink distributed antenna systems over a composite channel,” in _2011 IEEE 73rd Vehicular Technology Conference (VTC Spring)_ , May 2011, pp. 1–5.
  * [43] R. L. Burden and J. D. Faires, _Numerical Analysis_ , 9th ed. Brooks/Cole, Cengage Learning, 2011.
  * [44] M. Srinivas and L. M. Patnaik, “Genetic algorithms: a survey,” _Computer_ , vol. 27, no. 6, pp. 17–26, June 1994.
  * [45] M. Mitchell, _An Introduction to Genetic Algorithms_. Cambridge, MA, USA: MIT Press, 1998.
  * [46] H. Ahlehagh and S. Dey, “Video-aware scheduling and caching in the radio access network,” _IEEE/ACM Transactions on Networking_ , vol. 22, no. 5, pp. 1444–1462, Oct 2014.
  * [47] N. Golrezaei, P. Mansourifard, A. F. Molisch, and A. G. Dimakis, “Base-station assisted device-to-device communications for high-throughput wireless video networks,” _IEEE Transactions on Wireless Communications_ , vol. 13, no. 7, pp. 3665–3676, July 2014.
  * [48] E. Park, S. R. Lee, and I. Lee, “Antenna placement optimization for distributed antenna systems,” _IEEE Transactions on Wireless Communications_ , vol. 11, no. 7, pp. 2468–2477, July 2012.
  * [49] H. Kim, S. R. Lee, K. J. Lee, and I. Lee, “Transmission schemes based on sum rate analysis in distributed antenna systems,” _IEEE Transactions on Wireless Communications_ , vol. 11, no. 3, pp. 1201–1209, March 2012.
  * [50] J. Zhang and J. G. Andrews, “Distributed antenna systems with randomness,” _IEEE Transactions on Wireless Communications_ , vol. 7, no. 9, pp. 3636–3646, September 2008.
  * [51] A. Alameer and A. Sezgin, “Resource cost balancing with caching in C-RAN,” in _2017 IEEE Wireless Communications and Networking Conference (WCNC)_ , March 2017, pp. 1–6.
  * [52] J. G. Proakis and M. Salehi, _Digital Communications_ , 5th ed. London: McGraw-Hill, 2008.
  * [53] M. K. Simon and M.-S. Alouini, _Digital Communication over Fading Channels_. Hoboken: John Wiley & Sons, 2005.