# Cross Layer Resource Allocation in H-CRAN with Spectrum and Energy
Cooperation

Nazanin Moosavi  Mahnaz Sinaie  Paeiz Azmi  Pin-Hsun Lin and Eduard Jorswieck
• Nazanin Moosavi and Paeiz Azmi are with the Department of Electrical
Engineering, Tarbiat Modarres University, Tehran, Iran, (e.mail :
{n_moosavi,pazmi}@modares.ac.ir)• Mahnaz Sinaie is with the VTT Technical
Research Centre of Finland, Espoo, Finland,(e.mail mahnaz.sinaie@vtt.fi)• Pin-
Hsun Lin and Eduard Jorswieck are with the TU Braunschweig , Germany,
Braunschweig,(e.mail {lin,jorswieck}@ifn.ing.tu-bs.de)• The work of Mahnaz
Sinaie was supported by Iran National Science Foundation (INSF) under grant
96007791.• The work of Paeiz Azmi was supported by Iran National Science
Foundation (INSF) under grant 91059145.• The work of Pin-Hsun Lin was
supported in part by DFG LI 2886/2-1 and Fast Secure.• The work of E.
Jorswieck was partly supported by the German Research Foundation (DFG) under
grant JO 801/24-1.

###### Abstract

5G and beyond wireless networks are the upcoming evolution for the current
cellular networks to provide the essential requirement of future demands such
as high data rate, low energy consumption, and low latency to provide seamless
communication for the emerging applications. Heterogeneous cloud radio access
network (H-CRAN) is envisioned as a new trend of 5G that uses the advantages
of heterogeneous and cloud radio access networks to enhance both the spectral
and energy efficiency. In this paper, building on the notion of effective
capacity (EC), we propose a framework in non-orthogonal multiple access
(NOMA)-based H-CRAN to meet these demands simultaneously. Our proposed
approach is to maximize the effective energy efficiency (EEE) while
considering spectrum and power cooperation between macro base station (MBS)
and radio remote heads (RRHs). To solve the formulated problem and to make it
more tractable, we transform the original problem into an equivalent
subtractive form via Dinkelbach algorithm. Afterwards, the combinational
framework of distributed stable matching and successive convex algorithm (SCA)
is then adopted to obtain the solution of the equivalent problem. Hereby, we
propose an efficient resource allocation scheme to maximize energy efficiency
while maintaining the delay quality of service (QoS) requirements for the all
users. The simulation results show that the proposed algorithm can provide a
non-trivial trade-off between delay and energy efficiency in NOMA H-CRAN
systems in terms of EC and EEE and the spectrum and power cooperation improves
EEE of the proposed network. Moreover, our proposed solution complexity is
much lower than the optimal solution and it suffers a very limited gap
compared to the optimal method.

###### Index Terms:

Effective Capacity, QoS, Effective Energy Efficiency, Spectrum-Power Trading,
Non-Orthogonal Multiple Access (NOMA), Stable Matching

##  1 Introduction

The tremendous growth of wireless network services with a massive number of
users incurs an urgency for designing energy efficient communication systems
while guaranteeing quality of service (QoS) [1]. Compared with the 4G
communication networks, the 5G networks are required to achieve 1,000 times
higher system capacity, at least 100 times higher energy efficiency, 50 times
improvement in latency, and 100 times higher in connectivity density [2]. As
such, new techniques such as non-orthogonal multiple access (NOMA) and
spectrum and energy cooperation in new network architectures such as
heterogeneous cloud radio access networks (H-CRANs) have been envisioned as
promising solutions to enable efficient 5G communication networks [3], [4].

H-CRANs are such networks that can take the full advantage of both
heterogeneous networks and cloud radio access networks simultaneously [5]. The
H-CRAN architecture consists of a baseband unit (BBU) pool, fiber links, one
macro base station (MBS), and some remote radio heads (RRHs). In this network,
the centralized signal processing is performed in the BBU pool instead of
distributed base stations (BSs) in the heterogeneous networks. The control
plane and data plane are decoupled and the delivery of control and broadcast
signalling is shifted from RRHs to MBS. Hence, signaling overheads is reduced
and the cooperation between different RRHs is permitted because of the
centralized signal processing.

Another promising approach to improve the energy efficiency (EE), spectral
efficiency (SE), massive connectivity, and transmission latency is using new
multiple access transmission technique such as NOMA [3],[6]. In comparison
with the conventional orthogonal multiple access (OMA) schemes, NOMA schemes
use non-orthogonal resources to provide services for multiple users with the
main advantage to increase fairness between them, while increasing the user’s
receiver complexity and multiple access interference. In particular,
successive interference cancellation (SIC) is adjusted at receivers to
mitigate the mutual interference caused by the non-orthogonal transmission. In
order to get the full advantages of both H-CRAN architecture and NOMA
technique simultaneously, we consider NOMA based H-CRAN systems.

Another approach to improve system performance and EE in H-CRAN system is to
apply spectrum and energy cooperation [4] -[7]. In [4], there was a
cooperation strategy in small cell networks where the small cell base station
serves the macro cell users while it can use some bandwidth of the macro
cell[4]. Moreover, in [8], a joint energy and spectrum cooperation scheme
between different cellular systems was considered in order to decrease their
operational costs. In[9], energy efficient resource allocation for
heterogeneous cognitive radio networks with femtocells were studied, where a
cognitive BS maximizes its energy efficiency by allocating the spectrum
borrowed from the primary networks to the femtocells. In [7], the authors
considered spectrum sharing in a green communication system to reduce energy
consumption as much as possible with guaranteed users QoS. Hence, we will
apply the spectrum and energy cooperation in our NOMA based H-CRAN system to
enhance energy efficiency.

As it was mentioned, one of the main goals for 5G network design is to provide
very short end-to-end communication delay. In communication networks with
delay sensitive applications, optimizing solely the physical layer is not
sufficient to guarantee the QoS requirements due to the random variations of
wireless channel conditions. As an efficient alternative, one should consider
the delay requirement of each user statistically [1, 10, 11, 12]. Hence, the
notion of effective capacity was proposed in [13] which relates the Shannon
capacity to delay QoS requirements by introducing the data link layer
characteristics into the communication theory. There are some references
considering the statistical delay in CRAN architecture using effective
capacity notion. For example, in [14], the authors developed an effective
content RRH clustering and caching scheme to reduce the interference and
offload the traffic of the backhaul and fronthaul. In [15], the authors
considered effective capacity of the cluster content cashing to study the
impacts on the energy efficiency and delay. In [16], the authors studied the
distributed wireless MIMO links and minimized the BS usages while considering
heterogeneous statistical delay QoS constraints. In [17], the authors
establish a Markovian model to derive a closed-form expression of effective
capacity in device to device (D2D) communication in terms of QoS and the
transmission rate. In the context of EC in NOMA based networks, in [18], the
power control policy is applied to meet the delay QoS constraints in a NOMA
based network using the partial effective capacity. The authors in [19] obtain
the EC of NOMA based virtualized wireless network. In [20], the achievable EC
under the per-user statistical delay QoS requirements, for a downlink NOMA
based network is obtained. However, the analysis of the statistical delay and
energy efficiency simultaneously for the cross layer resource allocation in
the NOMA based H-CRAN networks have been missed in the literature, which,
given the discussion above, appears an important setting to investigate.

In this work, we obtain the EC and EEE definition of the NOMA based H-CRAN
systems while considering spectrum and energy cooperation between MBS and
RRHs. Afterward, we formulate the corresponding resource allocation problems
to allocate power, RRHs, and bandwidth to achieve delay aware energy efficient
framework for the proposed H-CRAN scheme. The objective function is EEE which
is the ratio of the effective capacity to the total power consumption where
the total power consumption has three parts e.g. fiber link power consumption,
active RRH power consumption, and the circuit power consumption. The
considered EEE optimization problem is a non-convex, NP-hard, intractable, and
fractional programming problem. Therefore, we apply Dinkelbach algorithm with
stable matching framework and successive convex approximation (SCA) to
allocate the resources efficiently. The simulation results demonstrate that
the spectrum and energy cooperation improves EEE compared to the other
networks. Moreover, EC is a decreasing function of delay and there is still a
trade off between delay and effective energy efficiency. In addition, our
proposed approach has lower complexity than the optimal solution with a
negligible gap with the optimal results.

The remainder of this paper is organized as follows. Section II describes our
system model and problem statement. In Section III, the proposed solution to
the problem is provided. The performance of the proposed algorithm and our
system model through different numerical specifications are examined in
Section IV. Finally, Section V concludes this paper.

TABLE I:  List of variables Variables |  Definitions
---|---
$K$ |  Number of users connected to the RRHs
$Z$ |  Number of users connected to MBS
$B$ |  Bandwidth of each RRHs
$W_{MC}$ |  Total bandwidth of MBS
$W_{z}$ |  Allocated bandwidth to the $z$-th MBS user
$M$ |  Number of RRHs
$\theta$ |  Delay QoS exponent
$\mathbb{E\\{.\\}}$ |  Expectation
$T_{f}$ |  Whole time slot duration
$g_{k,m}$ |  Channel gain between the $k$-th user and the $m$-th RRH
$p_{z}$ |  Allocated power from MBS to the $z$-th user
$p_{k,m}$ |  Allocated power from the $m$-th RRH to the $k$-th user
$x_{z,m}$ |  MBS user selection strategy by RRHs
$q_{k,m,z}$ |  Allocated power by the $m$-th RRH to the $k$-th user on the bandwidth $W_{z}$
$b_{m,z}$ |  Allocated bandwidth from $W_{z}$ to the users of the $m$-th RRH
$h_{k,m,z}$ |  Channel gain between the $k$-th user and the $m$-th RRH on the bandwidth $W_{z}$
$w_{m,z}$ |  Allocated bandwidth from $W_{z}$ of the $m$-th RRH to the $z$-th MBS user
$Q_{m,z}$ |  Allocated power to the $z$-th MBS user by the $m$-th RRH
$H_{m,z}$ |  Channel power between the $z$-th MBS user and the $m$-th RRH
$\zeta_{m}$ |  Reciprocal of drain efficiency of the $m$-th RRH
$P^{m}_{max}$ |  Maximum available power of the $m$-th RRH
$E_{c}$ |  Effective capacity
$\eta_{EEE}$ |  Effective energy efficiency
$P_{c,m}$ |  Circuit power consumption of the $m$-th RRH
$P_{m,f}$ |  Power consumption the $m$-th fronthaul fiber link
$R^{\text{ave-max}}_{m}$ |  Average rate of the $m$-th fronthaul link
$R^{\text{ave-max}}_{z}$ |  Average required rate of the $z$-th MBS user

##  2 System Model

We study a two-tier downlink transmission in a H-CRAN network consisting of
one MBS serving $Z$ users and $M$ RRHs serving $K$ users, as shown in Fig.1.
Let $\mathcal{M}=\\{1,2,...,M\\}$ denote the set of RRHs,
$\mathcal{Z}=\\{1,2,...,Z\\}$ denote set of MBS users and
$\mathcal{K}=\\{1,2,...,K\\}$ denote set of users supported by RRHs. The MBS
bandwidth is $W_{MC}$ and the available total bandwidth for all RRHs is $W$
which is equally divided into $M$ sections with a dedicated bandwidth $B$ for
each RRH. We further consider that all the RRHs encode their received data
using NOMA technique and then transmit to the users in the downlink. We study
the practical situation where there is spectrum and energy cooperation between
RRHs and MBS. Channels between the users and the RRHs are assumed to be flat
fading. Table I summarizes the parameters and symbols used in the system
model.

The fronthaul capacity for each RRH is practically limited, in general it can
only receive the data for a selected set of users from the BBU pool, and then
forward them to the selected users in the NOMA-based downlink transmission. As
a result, each RRH $m$ transmits only on the corresponding set of users whose
data are received from the BBU pool.

Figure 1: H-CRAN system with one macro base station (MBS), some RRHs, MBS
users and the other users.

Hence, on the system model assumptions, the received signal at the $k$-th user
is given by 0pt

$\mathbf{y}_{k}=\mathbf{x}_{k}\mathbf{g}_{k}+\\!\sum_{k^{\prime}\neq{k}}\\!\\!\mathbf{x}_{k^{\prime}}\mathbf{g}_{k}+\mathbf{z},$
(1)

where
$\mathbf{x}_{k}\\!\\!\\!=\\!\\!\\![x_{k,1},...,x_{k,M}]\\!=\\![\sqrt{p_{k,1}},...,\sqrt{p_{k,M}}]\\!\\!$
is the transmitted signal to the $k$-th user, $p_{k,m}$ is the allocated power
from the $m$-th RRH to the $k$-th user,
$\mathbf{g}_{k}=[g_{k,1},g_{k,2},...,g_{k,M}]$ is the channel coefficient and
$\mathbf{z}$ denotes white Gaussian noise having zero mean and variance
$\sigma^{2}$.

In order to decode signals, in the NOMA-based system, the SIC process is
implemented at each user receiver. To get the desired signal, each SIC
receiver first decodes the dominant interference which are sufficiently
stronger than the receiver’s desired signal and then subtracts them from the
superposed signal[21]. Hence, the SINR condition is satisfied in the following
Lemma.
_Lemma 1:_ The SINR condition $|g_{k^{\prime},m}|^{2}\leq|g_{k,m}|^{2}$ is
satisfied automatically in our considered NOMA system.

###### Proof:

The proof is provided in Appendix A. ∎

The SINR of the $k$-th user which is supported by the $m$-th RRH after
performing SIC is as follow

$\gamma_{k,m}=\frac{p_{k,m}{|g_{k,m}|}^{2}}{\sum_{\begin{subarray}{c}\footnotesize{k^{\prime}\neq{k}}\\\
{|g_{k^{\prime}\\!,m}|}^{2}\leq{|g_{k,m}|}^{2}\end{subarray}}p_{k^{\prime},m}|g_{k,m}|^{2}+\footnotesize{BN_{0}}}.$
(2)

Therefore, the achievable data rate is
$r_{k,m}=BT_{f}\log_{2}(1+\gamma_{k,m})$ where $T_{f}$ is time frame duration.
Moreover, it should be noticed that each user can only be connected to one RRH
and cooperation between RRHs due to the central signal processing is
restricted. The exclusive RRH allocation strategy can be written as

$\displaystyle p_{k,m}p_{k,m^{\prime}}=0,\forall m\neq
m^{\prime},m,m^{\prime}\in\mathcal{M}.$ (3)

This statement ensures that each user can be at most connected to one RRH.
Then, if $p_{k,m}\neq 0$ for $m$-th RRH then $p_{k,m^{\prime}}=0$ for any RRH
$m\neq m^{\prime}$ [22].

Remark 1: It is worth noting that H-CRAN network decouples the control and
data plane which results in the significant signaling overhead saving in the
radio connection between RRHs and BBU [23]. It is because instead of
interaction between BBU pool and each RRH for interference management, the BBU
pool is interfaced with the MBS for inter-tier interference coordination and
totally the MBS is configured with the entire communication functionalities
from physical to network layers, and therefore the delivery of control and
broadcast signalling is shifted from RRHs to MBS, which decreases the capacity
and time delay constraints on the fronthaul. Hence, the signaling overhead and
complexity between BBU, RRHs and users are negligible and we do consider such
overheads neither in the analytical nor the simulation analyses.

###  2.1 Spectrum and Energy Cooperation

In our system model, we consider a spectrum and energy cooperation scenario
which consists of MBS and some RRHs. Each RRH can provide higher data rate by
getting more bandwidth from MBS and the MBS aims at offloading the data
traffic of some of its cell edge users to the RRH in order to decrease its own
energy consumption with the spectrum and energy cooperation. By considering
that, each MBS user can be served by only one RRH, the user selection strategy
is denoted by $x_{z,m}$ as follows:

$x_{z,m}=\left\\{\begin{array}[]{ll}1&\text{If MBS user $z$ is served by RRH
$m$}\\\ 0&\text{otherwise.}\\\ \end{array}\right.$ (4)

The $m$-th RRH which is selected to serve the $z$-th MBS user with a dedicated
bandwidth $W_{z}$, allocates $w_{m,z}$ bandwidth to that MBS user. Hence, the
achievable data rate of $z$-th user over $m$-th RRH is
$r_{m,z}=w_{m,z}T_{f}\log_{2}(1+\frac{Q_{m,z}|H_{m,z}|^{2}}{w_{m,z}N_{0}})$
where $Q_{m,z}$ is the allocated power to the $z$-th MBS user by the $m$-th
RRH, and $H_{m,z}$ is the channel gain between the $z$-th MBS user and the
$m$-th RRH on the bandwidth $w_{m,z}$. Then $b_{m,z}$ which is the remaining
bandwidth of $W_{z}$, is assigned to transmit data to the users connected to
the $m$-th RRH. Consequently, the achievable data rate of the $k$-th user
connected to the $m$-th RRH on the assigned bandwidth $b_{m,z}$, is expressed
as 0pt

$\displaystyle r_{k,m,z}=b_{m,z}T_{f}\log_{2}(1+\nu_{k,m,z}),$ (5)

where
$\nu_{k,m,z}\\!=\\!\frac{q_{k,m,z}|h_{k,m,z}|^{2}}{b_{m,z}N_{0}+I_{k,m,z}}$.
In this equation, $I_{k,m,z}$ is
$\sum_{\begin{subarray}{c}k^{\prime}\neq{k}\\\
|h_{k^{\prime},m,z}|^{2}\leq|h_{k,m,z}|^{2}\end{subarray}}\\!\\!\\!q_{k^{\prime},m,z}|h_{k,m,z}|^{2}$
,$h_{k,m,z}$ denotes the channel gain between the $k$-th user and the $m$-th
RRH on the bandwidth $b_{m,z}$ and $q_{k,m,z}$ is the corresponding transmit
power respectively.

It is worth to stress that based on _Lemma 1_ , the SINR condition for
successive decoding is satisfied automatically in our proposed scheme and also
for $r_{k,m,z}$. Hence, the achievable total data rate of the $k$-th user by
considering spectrum and energy cooperation is given as

$r_{k_{t}}=\sum_{m=1}^{M}r_{k,m}+\sum_{z=1}^{Z}\sum_{m=1}^{M}x_{z,m}r_{k,m,z}.$
(6)

###  2.2 Effective Capacity

The theory of link-layer effective capacity was introduced in [24] to take
into account the communication delay, by incorporating the users’ delay-QoS
requirements into the Shannon capacity. The effective capacity determines the
maximum arrival rate that the channel can support with the given delay
exponent $\theta$. It captures the delay QoS constraints including link layer
characteristics such as the queue distribution and the buffer overflow
probability [1]. Assuming independent and identically distributed (i.i.d.)
block fading, this leads to express the effective capacity as [13] and [25]

$E_{c}=-\frac{1}{\theta}\ln\left(\mathbb{E}_{\gamma}\\{e^{-\theta
r}\\}\right),$ (7)

where $\mathbb{E}$ is the statistical expectation concerning the so-called
service rate $r=\log_{2}\left(1+\text{SINR}\right)$, with SINR which is
modeled as a random variable due to the fading channel $\gamma$. A larger
$\theta$ enforces more stringent delay-QoS constraints[26], whereas if
$\theta=0$ no delay QoS is enforced, and the effective capacity can be shown
to simplify the usual Shannon capacity. Thus, the effective capacity can be
seen to be a generalization of the Shannon capacity, which jointly accounts
for both the communication reliability and delay. Instead, the Shannon
capacity considers only the reliability aspect, without explicitly accounting
for the communication delay.

Moreover, effective capacity maximization is clearly technically more
challenging than maximizing the Shannon capacity, mainly due to the
expectation operator in (7) and the more involved functional form. Hence,
Plugging (6) into (7) yields the effective capacity of $k$-th user in
$\mathcal{K}$, namely

$\displaystyle
E^{k}_{c}\\!\\!=\\!\\!\\!\sum_{m=1}^{M}{\frac{-1}{\theta_{k}}}\log\mathbb{E}_{g}\\{e^{-\theta_{k}BT_{f}\log_{2}(1+\gamma_{k,m})}\\}$
$\displaystyle\hskip 15.00002pt\\!\\!+\\!\\!\
\sum_{z=1}^{Z}\\!\sum_{m=1}^{M}\frac{-x_{z,m}\\!}{\theta_{k}}\log\mathbb{E}_{h}\\!\\{e^{-\theta_{k}b_{m,z}T_{f}\\!\log_{2}(1+\nu_{k,m,z})\\!}\\!\\},$
(8)

where $\mathbb{E}_{g}$ and $\mathbb{E}_{h}$ give the statistical average of
inner arguments with respect to channel gains $g$ and $h$ respectively.

###  2.3 power consumption model

Here, we turn our attention to the derivation of the power consumption $P_{T}$
in the considered scenario. By definition, the effective capacity considers a
time frame which embraces multiple channel realizations, due to the
expectation in (7). Accordingly, the consumed power must be evaluated over the
same time frame. Our power consumption model consists of two parts, dynamic
and static power consumption. Dynamic power consumption includes both the
transmit power consumption for MBS users and other users connected to RRHs as
follow:

$\displaystyle
P_{d}=\mathbb{E}_{g}\left\\{\sum_{m=1}^{M}\sum_{k=1}^{k}\zeta_{m}p_{k,m}\right\\}$
$\displaystyle\hskip
11.99998pt+\mathbb{E}_{h}\left\\{\sum_{z=1}^{Z}\sum_{m=1}^{M}\sum_{k=1}^{K}\zeta_{m}x_{z,m}q_{k,m,z}\right\\}$
$\displaystyle\hskip
11.99998pt+\mathbb{E}_{H}\left\\{\sum_{z=1}^{Z}\sum_{m=1}^{M}\zeta_{m}x_{z,m}Q_{z,m}\right\\},$
(9)

where $\zeta_{m}$ is the reciprocal of drain efficiency of the $m$-th RRH and
the value of it depends on the specific type of the RRH, and $\mathbb{E}_{H}$
gives the statistical average of included arguments with respect to channel
gains $H$. The static power consumption consists of fiber power consumption,
and the power consumed in the hardware blocks of the network nodes. Hence, the
total power consumption is given as

$\displaystyle
P_{T}=\sum_{m=1}^{M}p_{c,m}+\sum_{m=1}^{M}P_{m,f}+\mathbb{E}_{g}\left\\{\sum_{m=1}^{M}\sum_{k=1}^{k}\zeta_{m}p_{k,m}\right\\}$
$\displaystyle\hskip
11.99998pt+\mathbb{E}_{h}\left\\{\sum_{z=1}^{Z}\sum_{m=1}^{M}\sum_{k=1}^{K}\zeta_{m}x_{z,m}q_{k,m,z}\right\\}$
$\displaystyle\hskip
11.99998pt+\mathbb{E}_{H}\left\\{\sum_{z=1}^{Z}\sum_{m=1}^{M}\zeta_{m}x_{z,m}Q_{z,m}\right\\},$
(10)

wherein $p_{c,m}$ is the circuit power consumption of the $m$-th RRH, and
$P_{m,f}$ is the power consumption of fiber links.

###  2.4 Problem formulation

Our main goal is to maximize the system bit-per-Joule effective energy
efficiency while guaranteeing throughput, energy efficiency and delay-QoS
requirement simultaneously. It is defined as the ratio

$\displaystyle\eta_{EEE}=\frac{E_{c}}{P_{T}},$ (11)

In this context, the radio resource allocation problem to be tackled is the
maximization of the effective energy efficiency via jointly optimizing
transmit powers, and bandwidth, $\\{p_{k,m}$, $q_{k,m,z}$, $Q_{m,z}$,
$b_{m,z}$, $\forall k\in\mathcal{K},m\in\mathcal{M},z\in\mathcal{Z}$}, subject
to QoS requirements, total power constraint and RRH and MBS user selection
constraints. Formally, The system EEE maximization problem is given as

$\mathcal{P}_{1}:\max_{\begin{subarray}{c}\\!\left\\{p_{k,m}\right\\},\\!\left\\{\\!q_{k,m,z}\\!\right\\}\\\
\\!\\{\\!Q_{m,z}\\},\\{b_{m,z}\\},\\{x_{z,m}\\}\end{subarray}}\eta_{EEE}=\frac{\sum_{k=1}^{K}E^{k}_{c}}{P_{T}}\\\
$ (12)

subject to:

${\begin{cases}\text{C1:}~{}E^{k}_{c}\geq R^{\text{ave-max}}_{m},~{}\forall
k,m,\\\
\text{C2:}~{}\mathbb{E}_{H}\\!\\{\\!w_{z,m}T_{f}\log_{2}\\!(1+\frac{Q_{m,z}H_{m,z}}{w_{m,z}N_{0}}\\!)\\!\\!\\}\\!\geq\\!x_{z,m}R^{\text{ave-
max}}_{z}\\!,~{}\forall z\\\
\text{C3:}~{}\sum_{m=1}^{M}b_{m,z}+w_{m,z}\leq\sum_{m=1}^{M}x_{z,m}W_{z},~{}\forall
z\\\
\text{C4:}~{}\mathbb{E}_{g}\\!\\!\left\\{\\!\sum_{k=1}^{K}\\!p_{k,m}\\!\right\\}+\mathbb{E}_{H}\\!\left\\{\sum_{z=1}^{Z}x_{z,m}Q_{m,z}\\!\right\\}\\\
+\mathbb{E}_{h}\\!\\!\left\\{\\!\sum_{z=1}^{Z}\\!\sum_{k=1}^{K}\\!x_{z,m}q_{k,m,z}\\!\right\\}\leq
P^{m}_{\text{max}},~{}\forall m\\\ \text{C5:}~{}p_{k,m}\geq 0,~{}\forall
k,m,\\\ \text{C6:}~{}b_{m,z},q_{k,m,z}\geq 0,~{}\forall m,z,\\\
\text{C7:}~{}Q_{m,z},w_{m,z}\geq 0,~{}\forall m,z,\\\
\text{C8:}~{}p_{k,m}p_{k,m^{\prime}}=0,~{}\forall
m,m^{\prime}\in\mathcal{M},k\in\mathcal{M},\\\
\text{C9:}~{}x_{z,m}\in\\{0,1\\},~{}\forall z,m,\\\
\text{C10:}~{}\sum_{z=1}^{Z}x_{z,m}\leq 1,~{}\forall m,\\\
\text{C11:}~{}\sum_{m=1}^{M}x_{z,m}\leq 1,~{}\forall z,\\\ \end{cases}}$

where C1 is to guarantee the statistical delay bound of the $k$-th user, C2
guarantees the average required rate for the $z$-th MBS user, C3 ensures that
the bandwidth allocated to RRH users and the $z$-th MBS user does not exceed
the available bandwidth, $W_{z}$ that has been licensed to the $z$-th MBS
user. C4 ensures that the average transmit power by the $m$-th RRH is non-
negative and below the maximum feasible average transmit power,
$P^{m}_{\text{max}}$, C5, C6, and C7 ensure that all allocated powers and
bandwidth are non-negative, C8 indicates whether the $k$-th user is supported
by the $m$-th RRH, and guarantees that each user is not connected to more than
one RRH. C9 indicates whether to serve the $z$-th MBS user or not and C10 and
C11 is to ensure that each RRH can just served one MBS user and each MBS user
is not served by more than one RRH.

The EEE maximization problem is non-concave mixed integer fractional
programming and therefore direct use of standard convex optimization tool is
not possible. To tackle this burden, we applied stable matching framework, SCA
and Dinkelbach algorithm. Stable matching is one of the well known approach
which is suited to analysis of mixed-integer assignment problem. It is
implemented inside Dinkelbach algorithm to obtain the MBS user assignment.
Moreover, SCA is also considered to deal with the non-concavity of the
numerator. Consequently, Dinkelbach algorithm solves the optimization problem
with polynomial complexity when the numerator and denominator are respectively
concave and convex functions over a convex set which is shown to converge with
super-linear convergence rate[27]. The proposed algorithm is applicable from
the point of the complexity and it guarantees the convergence to the
suboptimal solution. In the simulation results, we compare the suboptimal
approach with the optimal approach which is branch and bound algorithm
demonstrating that our proposed suboptimal solution has a negligible gap with
the optimal solution.

##  3 Proposed solution

After formulating the resource allocation problem in Section 2.4, this section
is devoted to the development of the corresponding radio resource allocation
algorithms. The problem (12) is non-convex due to the integer variable
$\\{x_{z,m}\\}_{z,m}$ and has a fractional form. Therefore, there is no
generally efficient method to solve this problem and the complexity grows
exponentially.

At first, we turn our attention to the constraints C2 and C3 of problem
$\mathcal{P}_{1}$. There is a discussion about network with a spectrum and
power cooperation as how much power should RRH use to serve an MBS user
against how much bandwidth the MBS user can give to the RRH. Since it is
always useful for RRH not only to seek as much as bandwidth but also consuming
as less as power in the spectrum and energy cooperation with the MBS, we can
consider equality at the optimal solution so the constraint C2 and C3 can be
satisfied in equality and change to 0pt

$\displaystyle
C2:~{}\mathbb{E}_{H}\\!\\{\\!w_{z,m}T_{f}\log_{2}\\!(1+\frac{Q_{m,z}H_{m,z}}{w_{m,z}N_{0}}\\!)\\}=x_{z,m}R^{\text{ave-
max}}_{z}\\!,~{}\forall z$ (13) $\displaystyle
C3:~{}b_{m,z}+w_{m,z}=x_{z,m}W_{z},~{}\forall z,m,$ (14)

Hence, we have
$Q_{m,z}=\mathbb{E}_{H}\\{\frac{w_{m,z}N_{0}}{H_{m,z}}(2^{\frac{R_{z}^{\text{ave-
max}}}{T_{f}w_{m,z}}}-1)\\},~{}\forall z$. Moreover, since
$b_{m,z}=W_{z}-w_{m,z}$, the optimization variable $b_{m,z}$ is replaced by
$w_{m,z}$ for simplicity. Then, we have
$\eta_{EEE}=\frac{\sum_{k=1}^{K}\widehat{E}^{k}_{c}}{\widehat{P}_{T}}$ where
$\widehat{E}_{c}$ and $\widehat{P}_{T}$ are defined as follows

$\displaystyle\widehat{E}_{c}\\!\\!=\\!\\!\\!\sum_{k=1}^{K}\sum_{m=1}^{M}{\frac{-1}{\theta_{k}}}\log\mathbb{E}_{g}\left\\{e^{-\theta_{k}BT_{f}\log_{2}(1+\gamma_{k,m})}\right\\}$
$\displaystyle\hskip
11.00008pt+\sum_{k=1}^{K}\sum_{z=1}^{Z}\\!\sum_{m=1}^{M}\frac{-x_{z,m}\\!}{\theta_{k}}\log\mathbb{E}_{h}\\!\left\\{e^{-\theta_{k}(W_{z}-w_{m,z})T_{f}\\!\log_{2}(1+\nu_{k,m,z})\\!}\\!\right\\},$
(15)
$\displaystyle\\!\widehat{P}_{T}=\\!\\!\sum_{m=1}^{M}p_{c,m}+\mathbb{E}_{g}\left\\{\sum_{m=1}^{M}\sum_{k=1}^{k}\zeta_{m}p_{k,m}\right\\}+\sum_{m=1}^{M}P_{m,f}$
$\displaystyle\hskip
11.99998pt+\mathbb{E}_{H}\left\\{\sum_{z=1}^{Z}\sum_{m=1}^{M}\zeta_{m}x_{z,m}\frac{w_{m,z}N_{0}}{H_{m,z}}(2^{\frac{R_{z}^{\text{ave-
max}}}{T_{f}w_{m,z}}}-1)\right\\}$ $\displaystyle\hskip
11.99998pt+\mathbb{E}_{h}\left\\{\sum_{z=1}^{Z}\sum_{m=1}^{M}\sum_{k=1}^{K}\zeta_{m}x_{z,m}q_{k,m,z}\right\\}.$
(16)

Accordingly, we have the following problem instead of problem
$\mathcal{P}_{1}$ which will be tackled in the following subsections. 0pt

$\displaystyle\mathcal{P}_{2}:\max_{\begin{subarray}{c}\\!\left\\{p_{k,m}\\!\right\\}_{k,m},\\!\left\\{\\!q_{k,m,z}\\!\right\\}_{k,m,z}\
,\\!\left\\{\\!w_{m,z},x_{z,m}\\!\right\\}_{m,z}\end{subarray}}\eta_{EEE}$
(17)

subject to

${\begin{cases}\text{C1:}~{}E^{k}_{c}\geq R^{\text{ave-max}}_{m},~{}\forall
m,k,\\\
\text{C4:}~{}\mathbb{E}_{g}\\!\\!\\{\\!\sum_{k=1}^{K}\\!p_{k,m}\\!\\}+\mathbb{E}_{H}\\!\\{\sum_{z=1}^{Z}x_{z,m}Q_{m,z}\\!\\}\\\
+\mathbb{E}_{h}\\{\\!\sum_{z=1}^{Z}\\!\sum_{k=1}^{K}x_{z,m}\\!q_{k,m,z}\\!\\}\leq
P^{m}_{\text{max}},~{}\forall m,\\\ \text{C5:}~{}p_{k,m}\geq 0,~{}\forall
m,k,\\\ \text{C6:}~{}q_{k,m,z}\geq 0,~{}\forall m,z,\\\
\text{C7:}~{}w_{m,z}\geq 0,~{}\forall m,z,\\\
\text{C8:}~{}p_{k,m}p_{k,m^{\prime}}=0,~{}\forall m,m^{\prime},k.\end{cases}}$

Remark 2: It is worth noting that it is proved in [4] that C2 and C3 are
satisfied in the equality. It is due to the fact that by using spectrum and
energy cooperation although we focus on improving the effective energy
efficiency of RRHs, the EEE of the MBS will also improve. Because MBS desires
to offload its users with poor channel condition which it needs more power to
be served. Therefore, the EEE of the MBS will obviously increase via
offloading. Furthermore, at the optimal point the most energy efficient
strategy for each RRH is to search as much as bandwidth while consuming less
power. Hence, in order to maximize the achievable bandwidth, the constraint C3
should reach its maximum value which is $W_{z}$ and therefore constraint C3
changes to equality. In addition, each RRH desires to consume less power,
which means the $z$-th MBS user rate should be in its minimum value and it
happens when it reaches its average rate value $R^{\text{ave-max}}_{z}$, so
the constraint C2 changes to equality.

###  3.1 Dinkelbach Algorithm

The next step is to tackle the fractional programming and non-convexity of the
constraints. The optimization problem $\mathcal{P}_{2}$ is a non-linear
fractional problem, which can be solved with the fractional programming theory
[28]. Based on the fractional programming, the problem is transformed into an
equivalent subtractive form via the parametric approach. An auxiliary function
$:\mathbf{q}\in\mathbb{R}\longrightarrow F(\mathbf{q})$ is introduced and the
following problem must be solved.

$\mathcal{P}_{3}:F(\mathbf{q})=\text{max}_{\left\\{p_{k,m},w_{m,z},q_{k,m,z},x_{z,m}\right\\}}\widehat{E}_{c}-\mathbf{q}\widehat{P}_{T},$
(18)

where $\mathbf{q}$ is a non-negative parameter. Hence, the optimal solutions
of $\mathcal{P}_{2}$ and $\mathcal{P}_{3}$ are related as
$\mathbf{q}^{*}=\eta^{*}_{EEE}$. Hence, we can conclude that, a pair of power
and bandwidth allocation strategy is a global solution of $\eta_{EEE}$ if and
only if $F(\mathbf{q}^{*})=0$, with $\mathbf{q}^{*}$ being the maximum value
of $\eta_{EEE}$, i.e, $\mathbf{q}^{*}=\eta^{*}_{EEE}$. Hence, the maximization
of $\eta_{EEE}$ is the same as finding the zeros of the auxiliary function
$F(\mathbf{q})$. This can be accomplished by the well-known Dinkelbach
algorithm which is an iterative approach where a sequence of the equivalent
subtractive form (18) is solved at each iteration of this algorithm to update
the auxiliary variable $\mathbf{q}$. It can be concluded that solving
$\mathcal{P}_{2}$ is resort to obtaining $\mathbf{q}$ with
$F(\mathbf{q}^{*})=0$.

Although the original problem $\mathcal{P}_{2}$ can be transformed into
$\mathcal{P}_{3}$ with an equivalent solution, $\mathcal{P}_{3}$ is obvious a
non-convex problem due to the non-integer variable and existence of co-channel
interference terms in rate formulation. Hence, to make $\mathcal{P}_{3}$
tractable, a combinational framework of stable matching and SCA are introduced
in the following sections. It should be stressed that Dinkelbach’s algorithm
is guaranteed to converge to the global solution of the corresponding
fractional problem, provided one is able to globally solve the inner problem
in (18), regardless of any concavity / convexity property of the numerator and
denominator of the fractional objective to maximize.

###  3.2 Power and Bandwidth Allocation Algorithm

Due to the non-convexity of problem 18, a logarithmic approximation method
based on SCA is adapted with any given $\mathbf{q}$ and $\\{x_{z,m}\\}_{z,m}$.
Let us define $\Psi_{m}$ which is a set of MBS users supported by each RRH,
i.e.,
$\Psi_{m}=\\{z|x_{z,m}=1,z\in\mathcal{Z}\\}=\\{\Psi_{1,m},...,\Psi_{Z,m}\\}$.
By considering a given $\Psi_{m}$, problem $\mathcal{P}_{3}$ is just a joint
bandwidth and power allocation problem but it still has non-convexity in the
rate formulation.
The main idea of SCA method is to approximate a non-convex problem into a
series of solvable problems, to obtain the near optimal solution satisfying
KKT conditions of the original problem. It can be demonstrated analytically
that the SCA approach has a convergence to a local optimal point[29]. We
should apply the following lower bound
$\log_{2}{(1+z)}\leq\alpha\log_{2}{z}+\beta$, where
$\alpha=\frac{\bar{z}}{1+\bar{z}}$, and
$\beta=\log_{2}(1+\bar{z})-\frac{\bar{z}}{1+\bar{z}}\log_{2}\bar{z}$ in which
$\bar{z}\in[0,\infty)$ is a positive real-valued number. As a consequence, the
rate of the $k$-th user which is connected to the $m$-th RRH on bandwidth $B$
and $b_{m,z}$ are respectively transformed to the following equations.

$\hat{r}_{k_{t}}\geq\alpha_{k,m}\log_{2}\gamma_{k,m}+\beta_{k,m},\\\ $ (19)
$\hat{r}_{k,m,z}\geq\kappa_{k,m,z}\log_{2}\widehat{\nu}_{k,m,z}+\xi_{k,m,z},\\\
$ (20)

where $\\{\alpha_{k,m}$, $\beta_{k,m}$, $\kappa_{k,m,z}$, $\xi_{k,m,z}\\}$ are
the approximation constants computed for some $\bar{z}=\gamma_{k,m}$ for (19)
and $\bar{z}=\nu_{k,m,z}$ for (20). By considering the transformation
$\bar{p}=\ln{p}$ and $\bar{q}=\ln{q}$, $\bar{E}^{k}_{c}$ and $\bar{P}_{T}$ are
given as

$\displaystyle\bar{E}^{k}_{c}=\sum_{m=1}^{M}\frac{-1}{\theta_{k}}\log\mathbb{E}_{g}\left\\{e^{\\!-\theta_{k}BT_{f}(\alpha_{k,m}\log_{2}(\bar{\gamma}_{k,m})+\beta_{k,m}\\!}\right\\}$
$\displaystyle\\!\\!+\\!\\!\sum_{z=1}^{Z}\\!\\!\sum_{m=1}^{M}\\!\frac{-1}{\theta_{k}}\\!\\!\log\\!\mathbb{E}_{h}\\!\left\\{e^{\\!\\!-\theta_{k}\\!\left(W_{z}\\!-w_{m,z}\\!\right)T_{f}(\kappa_{k,m,z}\\!\\!\log_{2}\\!\left(\bar{\nu}_{k,m,z}\\!\right)+\xi_{k,m,z})}\\!\\!\\!\right\\},$

where

$\displaystyle\bar{\gamma}_{k,m}=\\!\frac{e^{\bar{p}_{k,m}}\mid
g_{k,m}\mid^{2}}{\sum_{\begin{subarray}{c}k^{\prime}\neq{k}\\\ \mid
g_{k^{\prime},m}\mid^{2}\leq\mid
g_{k,m}\mid^{2}\end{subarray}}e^{\bar{p}_{k^{\prime},m}}\mid
g_{k,m}\mid^{2}+BN_{0}},$ (22)
$\displaystyle\bar{\nu}_{k,m,z}\\!\\!=\\!\\!\frac{e^{\bar{q}_{k,m,z}}\mid
h_{k,m,z}\mid^{2}}{\sum_{\begin{subarray}{c}k^{\prime}\neq{k}\\\ \mid
h_{k^{\prime},m,z}\mid^{2}\leq\mid
h_{k,m,z}\mid^{2}\end{subarray}}\\!\\!\\!\\!e^{\bar{q}_{k^{\prime},m,z}}\\!\mid
h_{k,m,z}\mid^{2}\\!\\!+\\!\\!\left(\\!W_{z}\\!\\!-\\!w_{m,z}\\!\right)\\!N_{0}}.$
(23)

$\displaystyle\bar{P}_{T}=\sum_{m=1}^{M}p_{c,m}+\mathbb{E}_{g}\left\\{\sum_{m=1}^{M}\sum_{k=1}^{k}\zeta_{m}e^{\bar{p}_{k,m}}\right\\}+\sum_{m=1}^{M}P_{m,f}$
$\displaystyle\hskip
11.99998pt+\mathbb{E}_{H}\left\\{\sum_{z=1}^{Z}\sum_{m=1}^{M}\zeta_{m}\frac{w_{m,z}N_{0}}{H_{m,z}}(2^{\frac{R_{z}^{\text{ave-
max}}}{T_{f}w_{m,z}}}-1)\right\\}$ $\displaystyle\hskip
11.99998pt+\mathbb{E}_{h}\left\\{\sum_{z=1}^{Z}\sum_{m=1}^{M}\sum_{k=1}^{K}\zeta_{m}e^{\bar{q}_{k,m,z}}\right\\}.$
(24)

In order to be compatible with SCA method, the equality in constraint C8 in
problem $\mathcal{P}_{2}$ should be replaced with an inequality. Therefore,
the constraint C8 is replaced by

$e^{\bar{p}_{k,m}}e^{\bar{p}_{k,m^{\prime}}}\leq\varepsilon,~{}\forall
m,m^{\prime}\in\mathcal{M},k\in\mathcal{M},$ (25)

where $\varepsilon$ is a small positive number. Consequently problem
$\mathcal{P}_{3}$ is equivalent to the following problem. 0pt

$\displaystyle\mathcal{P}_{3}:\max_{\begin{subarray}{c}\\!\left\\{\bar{p}_{k,m}\\!\right\\}_{k,m},\\!\left\\{\\!\bar{q}_{k,m,z}\\!\right\\}_{k,m,z}\
,\\!\left\\{\\!w_{m,z}\\!\right\\}_{m,z}\end{subarray}}\bar{E}_{c}-\mathbf{q}\bar{P}_{T}$
(26)

subject to

${\begin{cases}\text{C1:}~{}\bar{E}^{k}_{c}\geq R^{\text{ave-
max}}_{m},~{}\forall m\in\Psi_{z},k,\\\
\text{C4:}~{}\mathbb{E}_{g}\\!\\!\left\\{\\!\sum_{k=1}^{K}\\!e^{\bar{p}_{k,m}}\\!\right\\}+\mathbb{E}_{H}\\!\left\\{\sum_{z=1}^{Z}Q_{m,z}\\!\right\\}\\\
+\mathbb{E}_{h}\\!\\!\left\\{\\!\sum_{z=1}^{Z}\\!\sum_{k=1}^{K}\\!e^{\bar{q}_{k,m,z}}\\!\right\\}\leq
P^{m}_{\text{max}},~{}\forall m\in\Psi_{z},\\\
\text{C5:}~{}e^{\bar{p}_{k,m}}\geq 0,~{}\forall m\in\Psi_{z},k,\\\
\text{C6:}~{}e^{\bar{q}_{k,m,z}}\geq 0,~{}\forall m,z\in\Psi_{m},\\\
\text{C7:}~{}w_{m,z}\geq 0,~{}\forall m,z\in\Psi_{m},\\\
\text{C8:}~{}e^{\bar{p}_{k,m}}e^{\bar{p}_{k,m^{\prime}}}\leq\varepsilon,~{}\forall
m,m^{\prime},k.\end{cases}}$

Presently, the equivalent subtractive problem (26) which should be solved at
each iteration of Dinkelbach algorithm, is jointly concave with respect to the
powers and bandwidth and the duality approach can be applied to find the
optimal solutions. The Lagrangian function of the proposed problem is given in
(3.2) which is in the top of the next page.

$\displaystyle
L(\bar{p}_{k,m},\bar{q}_{k,m,z},w_{m,z},\mu_{k,m},\omega_{m},\varpi_{k,m,m^{\prime}})=\bar{E}_{c}-\mathbf{q}\bar{P_{T}}+\sum_{m=1}^{M}\sum_{k=1}^{K}\frac{\mu_{k,m}}{\theta_{k}}\Big{(}\log\mathbb{E}_{g}\left\\{e^{-\theta_{k}BT_{f}\\!\left(\alpha_{k,m}\log_{2}(\bar{\gamma}_{k,m})+\beta_{k,m}\\!\right)}\\!\right\\}$
$\displaystyle+\sum_{z\in\Psi_{m}}\\!\log\\!\mathbb{E}_{h}\\!\left\\{e^{-\theta_{k}(W_{z}-w_{m,z}\\!)T_{f}\\!\left(\kappa_{k,m,z}\log_{2}\\!\left(\bar{\nu}_{k,m,z}\\!\right)+\xi_{k,m,z}\\!\right)}\\!\right\\}-\\!R^{\text{ave-
max}}_{m}\Big{)}-\\!\\!\sum_{m\in\Psi_{z}}\\!\omega_{m}\Big{(}\mathbb{E}_{\gamma}\Big{\\{}\sum_{k=1}^{K}\\!e^{\bar{p}_{k,m}\\!\\!}\Big{\\}}\\!+\mathbb{E}_{h}\Big{\\{}\sum_{z=1}^{Z}\\!e^{\bar{q}_{k,m,z}}\Big{\\}}$
$\displaystyle+\\!\mathbb{E}_{H}\Big{\\{}\sum_{z=1}^{Z}\frac{w_{m,z}N_{0}}{H_{m,z}}(2^{\frac{R_{z}^{\text{ave-
max}}}{T_{f}w_{m,z}}}-1)\Big{\\}}-P^{m}_{\text{max}}\Big{)}\\!+\\!\sum_{\begin{subarray}{c}m^{\prime}=1\\\
m^{\prime}\neq m\end{subarray}}^{M}\sum_{\begin{subarray}{c}m=1\\\
\end{subarray}}^{M}\sum_{k=1}^{K}\varpi_{k,m,m^{\prime}}e^{\bar{p}_{k,m}\bar{p}_{k,m^{\prime}}}.$
(27)

In this equation, $\\{\mu_{k,m},\omega_{m},\varpi_{k,m,m^{\prime}}\\}$ are the
non-negative Lagrange multipliers associated to the above constraints
respectively. Thus, the dual problem is given as,

$\small\min_{\begin{subarray}{c}\mu_{k,m},\omega_{m}\\\
\varpi_{k,m,m^{\prime}}\\!\end{subarray}}\\!\max_{\begin{subarray}{c}\bar{p}_{k,m},\bar{q}_{k,m,z}\\\
w_{m,z}\end{subarray}}L\left(\bar{p}_{k,m},\bar{q}_{k,m,z},w_{m,z},\mu_{k,m},\omega_{m},\varpi_{k,m,m^{\prime}}\right).$
(28)

By applying the Lagrange dual decomposition, Problem (28) is decomposed into a
slave problem, which solves the Lagrange function with respect to the transmit
powers and bandwidth, for fixed Lagrange multipliers and a master problem
updates the Lagrange multipliers using the solution of the slave problem [30].
The slave and master problems are solved iteratively, and the process
converges to the optimal power and bandwidth allocation for each set of MBS
user selection strategy $\Psi$. The slave problem is decomposed into two
subproblems to find bandwidth and powers iteratively. The closed form of the
power allocation strategies are obtained in _Lemma 2_.

_Lemma 2:_ Consider Problem (28). For the fixed Lagrange multipliers, given
$\Psi$ and bandwidth $w_{m,z}$, the optimal power allocation for $p$ and $q$
are obtained as

$\displaystyle\\!\\!p_{k,m}=\exp[{\frac{\log{\Theta_{k,m}}+\sigma_{k}(\alpha_{k,m}\\!\log{\Gamma_{k,m}}+\beta_{k,m})}{-1-\sigma_{k}\alpha_{k,m}}}],$
(29)

where  0pt

$\displaystyle\Theta_{k,m}=\frac{\delta\ln{2}(\mathbf{q}\zeta_{m}+\omega_{m}+\varpi_{k,m,m^{\prime}}p_{k,m^{\prime}})}{BT_{f}\alpha_{k,m}(1+\mu_{k,m})},$
(30)

0pt

$\displaystyle\Gamma_{k,m}=\frac{g_{k,m}}{\sum_{i=1,i\neq{k}}^{K}p_{i,m}g_{k,m}+BN_{0}},$
(31)

with
$\delta=\mathbb{E}_{g}\\{e^{-\theta_{k}BT_{f}(\alpha_{k,m}log(\bar{\gamma}_{k,m})+\beta_{k,m})}\\}$
and $\sigma_{k}=\theta_{k}BT_{f}$.

$\displaystyle
q_{k,m,z}=\exp[{\frac{\log{\Upsilon_{k,m,z}}+\varsigma_{k}(\xi_{k,m,z}\\!\log{\Lambda_{k,m,z}}+\kappa_{k,m,z})}{-1-\varsigma_{k}\xi_{k,m,z}}}],$
(32)

where 0pt

$\displaystyle\Lambda_{k,m,z}=\frac{h_{k,m}}{\sum_{i=1,i\neq{k}}^{K}q_{i,m,z}h_{k,m}+\left(W_{z}-w_{m,z}\right)N_{0}},$
(33)

0pt

$\displaystyle\Upsilon_{k,m,z}=\frac{\lambda\ln{2}\left(\omega_{m}-\mathbf{q}\zeta_{m}\right)}{(W_{z}-w_{m,z})T_{f}\xi_{k,m,z}(1+\mu_{k,m})},$
(34)

with
$\lambda=\mathbb{E}_{h}\\{e^{-\theta_{k}(W_{z}-w_{m,z})T_{f}(\xi_{k,m,z}log(\bar{\vartheta}_{k,m,z})+\kappa_{k,m,z})}\\}$
and $\varsigma_{k}=\theta_{k}\left(W_{z}-w_{m,z}\right)T_{f}$.

###### Proof:

See Appendix B.∎

It can be seen that the power allocations in (29) and (32) are different from
the classical water-filling solution and it depends on the delay QoS
requirements, channel gain and bandwidth. To obtain the optimal bandwidth
allocation for the given power allocation schemes, we set the derivative of
the Lagrangian function with respect to $w_{m,z}$ to zero as follows,

$\displaystyle\frac{\partial{L}}{\partial{w_{m,z}}}\\!=\sum_{k=1}^{K}[T_{f}(1+\mu_{k,m})(\xi_{k,m,z}-\kappa_{k,m,z}{\log_{2}\nu_{k,m,z}})$
$\displaystyle\hskip
30.00005pt-\frac{T_{f}\xi_{k,m,z}}{N_{0}+I_{k,m,z}(W_{z}-w_{m,z})^{-1}}]-\frac{N_{0}(\omega_{m}+\mathbf{q}\zeta_{m})}{H_{m,z}}$
$\displaystyle\hskip 30.00005pt\times(1-2^{\frac{R^{\text{ave-
max}}_{z}}{w_{m,z}}}-\ln 2\frac{R^{\text{ave-
max}}_{z}}{w_{m,z}}2^{\frac{R^{\text{ave-max}}_{z}}{w_{m,z}}}=0.$ (35)

The form of (3.2) does not allow us to find an elegant closed-form analytic
solution for the bandwidth allocation therefore we resort to a numerical
search, such as Newton’s algorithm. Based on the above derivations, the
overall resource allocation algorithm is formulated by alternatively
optimizing the power allocation strategies according to (29) and (32) for
fixed bandwidth, and then optimizing the bandwidth for fixed powers, as
described above.
Next, we turn our attention to solve the master problem. Since the master
problem is always convex, the sub-gradient method updates the Lagrange
multipliers with guaranteed convergence and as it is mentioned in [31], the
gradient algorithm is more suitable for distributed implementation, where each
user may update its own dual variable autonomously. By applying the gradient
method, which leads to the following update formulas: 0pt

$\displaystyle\mu^{(t+1)}_{k,m}\\!\\!=\bigg{[}\mu^{(t)}_{k,m}+\\!\beta_{1}\\!\left(\sum_{m=1}^{M}\sum_{k=1}^{K}\bar{E}^{k}_{c}-\\!R^{ave-
max}_{m}\\!\right)\bigg{]}^{+},$
$\displaystyle\omega^{(t+1)}_{m}=\bigg{[}\omega^{(t)}_{m}-\beta_{2}\bigg{(}P^{m}_{\text{max}}-\sum_{m=1}^{M}p_{k,m}-\sum_{z=1}^{Z}\sum_{m\in\Psi_{z}}q_{k,m,z}$
$\displaystyle-\mathbb{E}_{H}\\!\left\\{\sum_{z=1}^{Z}\sum_{m\in\Psi_{z}}\frac{(W_{z}-w_{m,z})N_{0}}{H_{m,z}}(2^{\frac{R_{z}^{\text{ave-
max}}}{T_{f}\\!\left(W_{z}-w_{m,z}\right)}}-1)\\!\right\\}\\!\bigg{)}\\!\bigg{]}^{+}\\!\\!,$
(37)
$\varpi^{(t+1)}_{k,m,m^{\prime}}\\!\\!=\bigg{[}\\!\varpi^{(t)}_{k,m,m^{\prime}}+\\!\beta_{3}\sum_{\begin{subarray}{c}m=1\\\
m\neq
m^{\prime}\end{subarray}}^{M}\sum_{m^{\prime}=1}^{M}\sum_{k=1}^{K}\\!p_{k,m}p_{k,m^{\prime}}\bigg{]}^{+},\\\
$ (38)

where $t$ is the iteration index, $\beta_{1}$, $\beta_{2}$, and $\beta_{3}$
are positive step-sizes for the $t$-th iteration, (e.g., $\frac{1}{\sqrt{t}}$
from [30]) and $[.]^{+}$ denotes the projection onto the non-negative orthant.
Now, we turn our attention to obtain MBS user assignment by matching theory.

###  3.3 MBS User Assignment by Distributed Stable Matching

Consider problem $\mathcal{P}_{4}$ with the set of RRHs as
$\mathcal{M}=\\{1,...,M\\}$ and set of MBS users as
$\mathcal{Z}=\\{1,...,Z\\}$. Each RRH prefers to serve the $z$-th MBS user
which can get more bandwidth from it. At the same time, each MBS user should
be connected to the RRH which gives it more power. Clearly, this is a conflict
situation, since two RRHs might prefer the same MBS users. To obtain a
distributed implementation in the MBS user assignment problem, we would need
to implement a rejection mechanism which does not require information transfer
to a central point or the use of shared memory. In this context, we intend to
use stable matching which provides a framework to match RRHs and MBS users in
a stable way, where stable means that no RRH can gain by unilaterally changing
its assigned MBS user[28]. In order to make this general idea formally
precise, at first we should have some discussion about matching theory.
Matching occurs in every aspects of our lives. When there are non-divisible
goods and entities with different interests in these goods, there is a
corresponding matching market. Based on the local information, each entity has
preferences on the goods which is determined by utility function. Let us
define RRH as entities and MBS as goods, then the utility function of the
$m$-th RRH on the $z$-th MBS user is defined as $u_{z,m}$. Hence, the $z$-th
MBS user is acceptable to the $m$-th RRH if it leads to a larger utility value
than the other MBS users as $u_{z,m}>u_{z^{\prime},m},\forall z^{\prime}\in
Z\setminus\\{z\\}$. It should be emphasized that the MBS users also have a
preference relation, which, for the case at hand, is based on the utility
value $u_{z,m}$. If the same utility value determines both preference lists
(as in our case), an easier representation is to replace the preference
relations with the utility matrix $\mathbf{U}=\\{u_{z,m}\\}_{z,m}$. Finally
matching market is completely defined by the tuple
$(\mathcal{M},\mathcal{Z},\mathbf{U})$. The matching problem is stable
matching if no RRH/MBS user wants to change the current assignment.

Now, we turn our attention to define the utility function $u_{z,m}$. Due to
sum-based form of the total effective capacity with respect to $x_{z,m}$, the
objective function can be decoupled and the parts with $x_{z,m}$ can be
defined as utility function as follows,

$\displaystyle
u_{z,m}\\!\\!=\\!\\!\sum_{k=1}^{K}\\!\\!\left\\{\\!\\!\frac{-\log\\!\mathbb{E}_{h}\\!\left\\{e^{\\!\\!-\theta_{k}(W_{z}\\!-w_{m,z}\\!)T_{f}\left(\kappa_{k,m,z}\\!\\!\log_{2}\\!(\nu_{k,m,z}\\!)+\xi_{k,m,z}\\!\right)}\right\\}}{\theta_{k}}\\!\\!\\!\right\\}$
$\displaystyle-\mathbf{q}\left(\mathbb{E}_{h}\left\\{\zeta_{m}q_{k,m,z}\right\\}+\mathbb{E}_{H}\left\\{\zeta_{m}\frac{N_{0}w_{m,z}}{H_{m,z}}(2^{\frac{R_{z}^{\text{ave-
max}}}{T_{f}w_{m,z}}}-1)\right\\}\right).$ (39)

By considering stable matching perspectives, the RRH-MBS user assignment
problem is formulated by identifying the RRH and MBS user sets $\mathcal{M}$
and $\mathcal{Z}$ as the two groups to be matched, while the preference lists
are represented by the utility function (3.3). Specifically, considering the
utility matrix $\mathbf{U}$, sorting the entries of the elements in the $z$-th
row of $\mathbf{U}$ in the decreasing order yields the preference list of the
$z$-th MBS user, whereas sorting the elements of the $m$-th column of
$\mathbf{U}$ in decreasing order yields the preference list of the $k$-th RRH.
In general matching problems, each matching group have special matching matrix
which represents its preference list. In our case, the preferences of all RRHs
and MBS users are uniquely determined by a single matrix instead of two
matrices which is the special case of the general matching problems. Matching
problems in which all preference lists can be determined by a single matrix
enjoy stronger properties than general matching problems. With given utilities
$\mathbf{U}$ and $\mathbf{q}$, we define the function
$D:(\mathbf{U},\mathbf{q})\in\mathbb{R}^{M\times Z}\times\mathbb{R}\rightarrow
D(\mathbf{U},\mathbf{q})\in\mathcal{A}$ where $\mathcal{A}$ is the set of all
feasible assignment which maps a preference matrix $\mathbf{U}$ and a
parameter $\mathbf{q}$ to the corresponding stable matching
$D:(\mathbf{U},\mathbf{q})$. Consequently, the overall resource allocation
algorithm can be formally stated as in Algorithm 1.

Algorithm 1 Resource Allocation Algorithm

1: $q=0$

2: Solve
$F(\mathbf{q})=\max_{\mathcal{F}}\left(\bar{E}_{c}-\mathbf{q}\bar{P}_{T}\right)$.

while  _$F(\mathbf{q})\geq\epsilon$_ do

3: Compute $p_{k,m}$, $q_{k,m,z}$, $w_{m,z}$ and $\mathbf{U}$ based on Eq.

(29), (32), (3.2) and (3.3).

4: $\mathbf{x}=D(\mathbf{U},\mathbf{q})$

5: Compute $F(\mathbf{q})$ for given $\mathbf{q}$.

6: $\mathbf{q}=\frac{\bar{E}_{c}}{\bar{P}_{T}}$

end while

7:Return the optimal solutions $p_{k,m}$,$q_{k,m,z}$, $w_{m,z}$, and
$x_{z,m}$.

###  3.4 Complexity Analysis

The optimal solution of this problem to select the best MBS user is exhaustive
search algorithm. It is an algorithm to solve the discrete and combinatorial
optimization problem. It searches over all possible MBS users for each RRH and
it requires $\mathcal{O}{(2^{M}ZK)}$ operations, which is prohibitive for
large number of $M$. Greedy algorithm and matching theory are two promising
approaches having extremely low computational complexity, but weaker
optimality properties. The greedy algorithm which is proposed in [4] for MBS
selection, should be run outside of Dinklebach algorithm and it needs to solve
fractional programming at each iteration which leads to complexity
$\mathcal{O}{(M^{2}ZK)}$. However, the matching algorithms execute the
preference matrix inside the Dinkelbach algorithm and select the best matched
RRH and MBS users based on sorting the preference matrix. Hence, for the case
at hand, the stable matching is guaranteed to converge in $M$ steps with
complexity $\mathcal{O}(Mlog(M))$ and the overall complexity of Algorithm 1 is
obtained as $\mathcal{O}{(ZKMlog(M))}$ which is much lower than the Greedy
algorithm.

##  4  SIMULATION RESULTS

In this section, we illustrate the numerical results to verify the analytical
results of the proposed delay aware and effective energy efficient resource
allocation schemes in the NOMA based H-CRAN. The considered H-CRAN consists of
$1$ MBS, $2$ to $12$ RRHs for different scenarios, $2$ MBS user and up tp $14$
users. The MBS is located in the center of the cell area, while the RRHs, MBS
users and users are uniformly distributed. The bandwidth of each RRH is
$B=700$ kHz and the bandwidth of MBS is $W_{MC}=550$ kHz and the slot duration
is $1ms$. Without loss of generality, we assume that all MBS users have
identical parameters, i.e., the same amount of available bandwidth $W_{z}=200$
kHz and average required rate $R_{z}^{ave-max}=700$ Kbps and also the average
rate of the each fronthaul link is specifies as $R_{m}^{ave-max}=1000$ Kbps.
Channels between all users even the MBS users and RRH are flat fading with
Rayleigh distributions and have been modelled as
$\alpha^{2}|\frac{d_{0}}{d_{k}}|^{(2.5)}$, where $\alpha$ is a standard
complex circularly symmetric Gaussian random variable which models fading
effects and $|\frac{d_{0}}{d_{k}}|$ models the communication power path-loss
at $d_{0}=1m$ as the reference distance, and $d_{k}$ denotes the physical
distance between the transmitter and the $k$-th user. The noise power is
$-102dBm$. For each RRH, the reciprocal of the drain efficiency of the power
amplifier, circuit power consumption and power consumption of the fronthaul
fiber link are given by $\zeta=16\%$, $P_{c}=0.01W$ and $P_{f}=1W$.

We also stress that the proposed optimization problem contains statistical
expectations as a direct effect of the fact that the long-term performance
measure is required to define effective capacity. Since closed form
expressions are not available, numerical computations based on Monte-Carlo
simulation with Matlab are obtained.

###  4.1 Convergence of the Proposed Iterative Algorithm

Fig.2 shows the total effective capacity for different values of delay QoS
exponent for the exhaustive search and our proposed solution. It is obvious
that the effective capacity is decreasing function of delay QoS exponent. This
is due to the fact that there is a trade-off between achievable low delay and
large effective capacity. When the delay requirement is stricter, the
effective capacity decreases. Note that when $\theta$ goes to zero, the
effective capacity equals Shannon capacity. Moreover, it is observed that the
proposed suboptimal solution performs almost as well as the optimal solution
with a negligible gap.

The comparison between the achieved EEE for the exhaustive search and the
proposed algorithm for the different values of $P^{SC}_{max}$ is given in
Fig.3. It can be seen that the EEE is an increasing function of
$P^{SC}_{max}$. Here we observe that the same as Fig.2 the proposed suboptimal
solution is also performs almost near the optimal solution. Moreover, both
figures demonstrate that our proposed algorithm converges to the near optimal
solution. It should be noticed that the simulation results for the different
number of users and the RRHs will discuss in further figures with details.

Figure 2: Effective Capacity versus QoS exponent ($\theta$) for 3 users and 2
RRHs.  Figure 3: Effective Energy Efficiency versus $P_{max}$ for 3 users and
2 RRHs with $\theta=10^{-5}$

###  4.2 EEE Performances of the Proposed Solutions

In this section the EEE performance of our proposed algorithm is considered.
Fig.4 shows the variation of the effective energy efficiency obtained by the
resource allocations which maximize the effective energy efficiency and the
effective capacity versus the maximum average transmit power. In Fig.4, we
observe that by increasing the maximum available transmit power, the network
effective energy efficiency first increases and then saturates when the
resources are optimized for the effective energy efficiency maximization. This
is due to the fact that the effective energy efficiency is a unimodal function
with respect to the transmit powers which can be calculated from the 1st
derivative. Specifically when $P^{SC}_{max}$ is so large to get the peak,
increasing the transmit power is not useful as it would just decrease the
effective energy efficiency. Actually, when the resources are optimized for
effective capacity maximization, which is a monotonically increasing function
of the transmit power, we see that since the excess transmit power is used to
maximize the effective capacity, the attained effective energy efficiency
value decreases.

Moreover, Fig.4 shows that by considering the spectrum and energy cooperation,
the effective energy efficiency in comparison to the traditional architectures
is increased which demonstrates the effectiveness of the proposed solution.
Moreover, it is seen that there is a performance gap between the proposed
scheme and the scheme without spectrum and energy cooperation . As we can see
that the gap first increases and then reaches a constant. This is because when
the transmit power $P^{SC}_{max}$ is limited, RRH does not have sufficient
power to serve more MBS users. Hence, our spectrum and energy cooperation is
less likely to happen and it limits the improvement in performance. As
$P^{SC}_{max}$ increases, RRH has more power to serve its own connected users
and more freedom to get more bandwidth from the MBS users.

Similar conclusions can be drawn from Fig.5, which considers a similar
scenario as in Fig.4, with the effective energy efficiency metric. It can be
seen that the effective capacity increases and then saturates when the
resources are allocated for effective energy efficiency maximization and also
as we can see in the figure, by considering the spectrum and energy
cooperation we have about 20% gain.

Figure 4: Effective Energy Efficiency versus $P_{max}$ for 3 users and 2 RRHs.
Figure 5: Effective Capacity versus $P_{max}$ for 3 users and 2 RRHs.

###  4.3 Performance Comparisons

To be applicable in realistic scenarios, the impact of number of users and
number of RRHs on effective energy efficiency is considered in this part. In
Fig.6(a), the number of RRHs is fixed to 10 and the number of users changes
from 2 to 14. In Fig.6(b), the number of users is fixed to 10 and the number
of RRHs change from 1 to 12. It is shown that the effective energy efficiency
is an increasing function of the number of users and the number of RRHs. It is
obvious that by increasing the number of RRHs and users in the network, the
proposed spectrum and energy cooperation can have more impact and therefore
higher spectral and energy efficiency of the system happens.

(a) Effective Energy Efficiency versus the number of users, number of RRH is
10.

(b) Effective Energy Efficiency versus the number of RRHs, number of users is
10.

Figure 6: Impact of different number of users and different number of RRH on
EEE and $\theta=10^{-5}$

##  5 conclusion

In this paper, we have studied the downlink transmission in a NOMA based
H-CRAN with different statistical delay requirements. This has been achieved
by introducing and optimizing the EC and EEE metrics. In order to increase the
energy and spectral efficiency, the spectrum and energy cooperation is also
applied. We formulate an optimization problem to jointly allocate power,
bandwidth, and RRHs. Although the problem is fractional and non-convex, an
efficient solution with low complexity based on the Dinkelbach algorithm ,SCA
and stable matching is proposed to obtain the near optimal solution. Through
numerical simulations, it is shown that there is a trade off between delay and
EEE, and EC is a decreasing function of delay. Throughout the work, it has
been assumed that the there is no cooperation between different RRHs, which
force the users to connect to just one RRH. Relaxing this assumption appears a
relevant topic for the problem formulation as future work.

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##  Appendix A  Proof of Lemma 1

In our NOMA system, we assume that SIC receiver at the $k$-th user wants to
cancel the interference from the $k^{\prime}$-th user. The $k$-th user SIC
receiver can decode and remove the interference from the $k^{\prime}$-th user,
if the SINR of the $k^{\prime}$-th user, which its data stream is decoded at
the $k$-th user, is higher than its own SINR [32]. Therefore, mathematically
we have $\gamma_{k,m}(k^{\prime})>\gamma_{k^{\prime},m}(k^{\prime})$, where
$\gamma_{k,m}(k^{\prime})$ is the SINR of the $k^{\prime}$-th user at the
$k$-th user and $\gamma_{k^{\prime},m}(k^{\prime})$ is the SINR of the
$k^{\prime}$-th user. Therefore the SINR condition for the successive decoding
is given by the following equation, 0pt

$\displaystyle\\!\frac{p_{k^{\prime},m}|g_{k,m}|^{2}}{\mathcal{I}_{k,m}+BN_{0}}\geq\frac{p_{k^{\prime},m}|g_{k^{\prime},m}|^{2}}{\mathcal{I}_{k^{\prime},m}+BN_{0}}\\!,$
(40)

where $\mathcal{I}_{k,m}=\sum_{\tiny{\begin{subarray}{c}i\neq{k^{\prime}}\\\
|g_{k^{\prime},m}|^{2}\leq|g_{i,m}|^{2}\end{subarray}}}p_{i,m}|g_{k,m}|^{2}$
and
$\mathcal{I}_{k^{\prime},m}=\sum_{\tiny{\begin{subarray}{c}i\neq{k^{\prime}}\\\
|g_{k^{\prime},m}|^{2}\leq|g_{i,m}|^{2}\end{subarray}}}\\!p_{i,m}|g_{k^{\prime},m}|^{2}$,
then the equation (40) is equivalent to, 0pt

$\displaystyle
BN_{0}\\!\\!\left(|g_{k^{\prime},m}|^{2}-|g_{k,m}|^{2}\right)\\!\\!+\\!\\!|g_{k,m}|^{2}|g_{k^{\prime},m}|^{2}(\\!\\!\\!\\!\\!\\!\\!\sum_{\tiny{\begin{subarray}{c}i\neq{k^{\prime}}\\\
|g_{k^{\prime},m}|^{2}\leq|g_{i,m}|^{2}\end{subarray}}}\\!\\!\\!\\!\\!\\!p_{i,m}\\!)$
$\displaystyle-(\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\tiny{\begin{subarray}{c}i\neq{k^{\prime}}\\\
|g_{k^{\prime},m}|^{2}\leq|g_{i,m}|^{2}\end{subarray}}}\\!\\!\\!\\!\\!\\!p_{i,m})\leq
0=BN_{0}\left(|g_{k^{\prime},m}|^{2}-|g_{k,m}|^{2}\right)\leq 0,$

which results to $|g_{k^{\prime},m}|^{2}\leq|g_{k,m}|^{2}$.

##  Appendix B Proof of Lemma 2

In the following, we solve the optimal power allocation $p_{k,m}$, by assuming
$N$ fading states (or sub-channels) and introducing an additional index $n$
for each allocated power, namely, $p_{k,m,n}$. For notational ease, we
introduce the following definition:

$\displaystyle\mathcal{F}_{n}=e^{-\theta_{k}BT_{f}(\alpha_{k,m}log(\bar{\gamma}_{k,m,n})+\beta_{k,m})},$
(42)

0pt

$\displaystyle
L_{1}=\sum_{m=1}^{M}\sum_{k=1}^{K}(1+\mu_{k,m})\frac{-1}{\theta_{k}}\log\\!\left(\frac{1}{N}\sum_{n=1}^{N}\mathcal{F}_{n}\\!\right)\\!$
$\displaystyle\hskip
10.00002pt-\frac{1}{N}\sum_{n=1}^{N}{\sum_{m=1}^{M}\sum_{k=1}^{k}e^{\bar{p}_{k,m,n}}\left(\mathbf{q}\zeta_{m}-\omega_{m}\right)}$
$\displaystyle\hskip 10.00002pt+\sum_{\begin{subarray}{c}m=1\\\ m\neq
m^{\prime}\end{subarray}}^{M}\sum_{m^{\prime}=1}^{M}\sum_{k=1}^{K}\
\varpi_{k,m,m^{\prime}}e^{p_{k,m}p_{k,m^{\prime}}},$ (43)

where $L_{1}$ corresponds to the terms with $p_{k,m,n}$ in (3.2). Taking the
first derivative of $L_{1}$ with respect to $p_{k,m,n}$ is as follows: 0pt

$\displaystyle\frac{\partial{L_{1}}}{\partial{p_{k,m,n}}}=\frac{(1+\mu_{k,m})}{-\theta_{k}\bar{\delta}\ln{2}}\mathcal{F^{\prime}}_{n}-e^{\bar{p}_{k,m,n}}(\omega_{m}-\mathbf{q}\zeta_{m}$
$\displaystyle\hskip
37.00002pt+\varpi_{k,m,m^{\prime}}e^{p_{k,m^{\prime},n}}),$ (44)

where $\bar{\delta}=\frac{1}{N}\sum_{n=1}^{N}\mathcal{F}_{n}$. By (42) the
first derivative of $\mathcal{F^{\prime}}_{n},$ with respect to $p_{k,m,n}$ is
as follows,

$\displaystyle\mathcal{F^{\prime}}_{n}=-\theta_{k}BT_{f}\alpha_{k,m}\mathcal{F}_{n}.$
(45)

Then set $L_{1}$ to zero leads to a stationary condition: 0pt

$\displaystyle\frac{\partial{L_{1}}}{\partial{p_{k,m,n}}}=\frac{BT_{f}\alpha_{k,m}(1+\mu_{k,m})}{\bar{\delta}\ln{2}}\mathcal{F}_{n}-e^{\bar{p}_{k,m,n}}(\omega_{m}$
$\displaystyle\hskip
37.00002pt-\mathbf{q}\zeta_{m}+\varpi_{k,m,m^{\prime}}e^{p_{k,m^{\prime},n}})=0,$
(46)

which can be simply rewritten as

$\displaystyle\frac{\mathcal{F}_{n}}{e^{\bar{p}_{k,m,n}}}\\!=\\!\frac{(\omega_{m}-\mathbf{q}\zeta_{m}+\varpi_{k,m,m^{\prime}}e^{\bar{p}_{k,m^{\prime},n}})\bar{\delta}\ln{2}}{BT_{f}\alpha_{k,m}(1+\mu_{k,m})}.$
(47)

Then we can define

$\displaystyle
e^{-\sigma_{k}(\alpha_{k,m}\log{(e^{\bar{p}_{k,m,n}}\Gamma}_{k,m,n})+\beta_{k,m})}\\!=\Theta_{k,m,n}e^{\bar{p}_{k,m,n}},$
(48)

where $\sigma_{k}=\theta_{k}BT_{f}$ and $\Theta_{k,m,n}$ and $\Gamma_{k,m,n}$
are given as

$\displaystyle\Theta_{k,m,n}=\frac{\bar{\delta}\ln{2}(\omega_{m}-\mathbf{q}\zeta_{m}+\varpi_{k,m,m^{\prime}}p_{k,m^{\prime},n})}{BT_{f}\alpha_{k,m}(1+\mu_{k,m})},$
(49)
$\displaystyle\Gamma_{k,m,n}=\frac{|g_{k,m,n}|^{2}}{\sum_{i\neq{k}}^{K}e^{\bar{p}_{i,m,n}}|g_{k,m,n}|^{2}+BN_{0}},$
(50)

Finally, the optimal power is obtained as follows

$\displaystyle\\!\\!p_{k,m,n}\\!\\!=\\!\\!\exp[{\frac{\log{\Theta_{k,m,n}}+\sigma_{k}(\alpha_{k,m}\\!\log{\Gamma_{k,m,n}}+\beta_{k,m})}{-1-\sigma_{k}\alpha_{k,m}}}].$
(51)

Similar to [33], we let $N\rightarrow\infty$, which yields to

$\displaystyle\\!\\!p_{k,m}\\!\\!=\\!\\!\exp[{\frac{\log{\Theta_{k,m}}+\sigma_{k}(\alpha_{k,m}\\!\log{\Gamma_{k,m}}+\beta_{k,m})}{-1-\sigma_{k}\alpha_{k,m}}}],$
(52)

and $\bar{\delta}\rightarrow\delta$.
The proof to find $q_{k,m,z,n}$ follows similar steps as to find $p_{k,m,n}$.
Therefore we introduce the following definition in (53) 0pt

$\displaystyle\mathcal{G}_{n}=e^{-\theta_{k}(W_{z}-w_{m,z})T_{f}(\xi_{k,m,z}log(\bar{\vartheta}_{k,m,z,n})+\kappa_{k,m,z})},$
(53)

0pt

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!L_{2}=\sum_{z=1}^{Z}\sum_{m=1}^{M}\sum_{k=1}^{K}(1+\mu_{k,m})\frac{-1}{\theta_{k}}\log\\!\left(\frac{1}{N}\sum_{n=1}^{N}\mathcal{G}_{n}\\!\right)\\!$
$\displaystyle\\!\\!-\frac{1}{N}\sum_{n=1}^{N}{\sum_{m=1}^{M}\sum_{k=1}^{k}e^{\bar{q}_{k,m,z,n}}\left(\mathbf{q}\zeta_{m}-\omega_{m}\right)},$
(54)

where $L_{2}$ corresponds to the terms with $q_{k,m,z,n}$ in (3.2). Taking the
first derivative of $L_{2}$ with respect to $q_{k,m,z,n}$ is as follows 0pt

$\displaystyle\frac{\partial{L_{2}}}{\partial{q_{k,m,z,n}}}=\frac{-(1+\mu_{k,m})}{\theta_{k}\bar{\lambda}\ln{2}}\mathcal{G^{\prime}}_{n}-e^{\bar{q}_{k,m,z,n}}\left(\omega_{m}-\mathbf{q}\zeta_{m}\right),$
(55)

where $\bar{\lambda}=\frac{1}{N}\sum_{n=1}^{N}\mathcal{G}_{n}$. By (53) the
first derivative of $\mathcal{G^{\prime}}_{n},$ with respect to $q_{k,m,z,n}$
is as follows

$\displaystyle\mathcal{G^{\prime}}_{n}=-\theta_{k}(W_{z}-w_{m,z})T_{f}\xi_{k,m,z}\mathcal{G}_{n}.$
(56)

Then set $L_{2}$ to zero leads to a stationary condition 0pt

$\displaystyle\frac{\partial{L_{2}}}{\partial{q_{k,m,z,n}}}=\frac{(W_{z}-w_{m,z})T_{f}\xi_{k,m,z}(1+\mu_{k,m})}{\bar{\lambda}\ln{2}}\mathcal{G}_{n}-$
$\displaystyle\hskip
45.00006pte^{\bar{q}_{k,m,z,n}}(\omega_{m}-\mathbf{q}\zeta_{m})=0,$ (57)

which can be simply rewritten as

$\displaystyle\frac{\mathcal{G}_{n}}{e^{\bar{p}_{k,m,n}}}\\!=\\!\frac{(\omega_{m}-\mathbf{q}\zeta_{m})\bar{\lambda}\ln{2}}{(W_{z}-w_{m,z})T_{f}\xi_{k,m,z}(1+\mu_{k,m})}.$
(58)

Then we can define

$\displaystyle
e^{-\varsigma_{k}(\xi_{k,m,z}log(\bar{\vartheta}_{k,m,z})+\kappa_{k,m,z})}\\!=\Upsilon_{k,m,z,n}e^{\bar{q}_{k,m,z,n}},$
(59)

where $\varsigma_{k}=\theta_{k}\left(W_{z}-w_{m,z}\right)T_{f}$ and
$\Upsilon_{k,m,z,n}$ and $\Lambda_{k,m,z,n}$ are as follows

$\displaystyle\Upsilon_{k,m,z,n}=\frac{\bar{\lambda}\ln{2}\left(\omega_{m}-\mathbf{q}\zeta_{m}\right)}{(W_{z}-w_{m,z})T_{f}\xi_{k,m,z}(1+\mu_{k,m})},$
(60)
$\displaystyle\Lambda_{k,m,z,n}=\frac{|h_{k,m,z,n}|^{2}}{\sum_{i\neq{k}}^{K}q_{i,m,z,n}|h_{k,m,z,n}|^{2}+\left(W_{z}-w_{m,z}\right)N_{0}},$
(61)

We finally obtain

$\displaystyle\\!\\!q_{k,m,z,n}\\!\\!=$
$\displaystyle\exp[{\frac{\log{\Upsilon_{k,m,z,n}}+\varsigma_{k}\\!(\xi_{k,m,z}\\!\log{\Lambda_{k,m,z,n}}+\kappa_{k,m,z}\\!)}{-1-\varsigma_{k}\xi_{k,m,z}}}].$
(62)

Similar to [33], we let $N\rightarrow\infty$, which yields to (32).

$\displaystyle\\!\\!q_{k,m,z}\\!\\!=\exp[{\frac{\log{\Upsilon_{k,m,z}}+\varsigma_{k}\\!(\xi_{k,m,z}\\!\log{\Lambda_{k,m,z}}+\kappa_{k,m,z}\\!)}{-1-\varsigma_{k}\xi_{k,m,z}}}],$
(63)

and $\bar{\lambda}\rightarrow\lambda$.