# URLLC and eMBB Coexistence in MIMO Non-orthogonal Multiple Access Systems

Qimei Chen, Jiajia Wang, and Hao Jiang Q. Chen, J. Wang, and H. Jiang are with
the School of Electronic Information, Wuhan University, Wuhan, China. e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Enhanced mobile broadband (eMBB) and ultra-reliable and low-latency
communications (URLLC) are two major expected services in the fifth-generation
mobile communication systems (5G). Specifically, eMBB applications support
extremely high data rate communications, while URLLC services aim to provide
stringent latency with high reliability communications. Due to their
differentiated quality-of-service (QoS) requirements, the spectrum sharing
between URLLC and eMBB services becomes a challenging scheduling issue. In
this paper, we aim to investigate the URLLC and eMBB co-scheduling/coexistence
problem under a puncturing technique in multiple-input multiple-output (MIMO)
non-orthogonal multiple access (NOMA) systems. The objective function is
formulated to maximize the data rate of eMBB users while satisfying the
latency requirements of URLLC users through joint user selection and power
allocation scheduling. To solve this problem, we first introduce an eMBB user
clustering mechanism to balance the system performance and computational
complexity. Thereafter, we decompose the original problem into two
subproblems, namely the scheduling problem of user selection and power
allocation. We introduce a Gale-Shapley (GS) theory to solve with the user
selection problem, and a successive convex approximation (SCA) and a
difference of convex (D.C.) programming to deal with the power allocation
problem. Finally, an iterative algorithm is utilized to find the global
solution with low computational complexity. Numerical results show the
effectiveness of the proposed algorithms, and also verify the proposed
approach outperforms other baseline methods.

###### Index Terms:

eMBB, URLLC, coexistence, MIMO-NOMA, user selection, power allocation.

##  I Introduction

The fifth-generation (5G) wireless communication classifies the various
service requirements into three types, namely enhanced Mobile Broad Band
(eMBB), massive Machine Type Communications (mMTC), and Ultra Reliable Low
Latency Communications (URLLC) [1, 2]. Precisely, eMBB users require extremely
high data rates, URLLC users concern high reliability and remarkably low
latency, and mMTC users focus on a bulk of connections. Generally, the eMBB
users would occupy the vast majority of wireless resources, whereas the URLLC
users occur spontaneously. The most toilless way to deal with this matter is
to reserve dedicated resources for URLLC users. Intuitively, this approach may
lead to the wireless resources under-utilized. Therefore, it is important to
investigate the URLLC and eMBB coexistence issue on the same wireless
resources.

3GPP has introduced a superposition/puncturing and the short transmission time
interval (short-TTI) technologies [3, 4]. The main idea of these two
technologies is to divide each transmission slot ($1$ ms) into several mini
slots (0.125 ms), and then allow arrival URLLC users occupy the ongoing eMBB
users. The transmission power of these interrupted eMBB users at specific mini
slots would be set as zero as long as any URLLC user arrives. Meanwhile,
academia and industries have also paid much attention on this issue.
Specifically, the authors in [5] propose a slicing architecture for orthogonal
URLLC and eMBB coexistence, where URLLC slices remain unchanged before each
transmission period. An optimal joint scheduling problem for URLLC and eMBB
traffic is proposed in [6], where linear, convex, and threshold based rate
loss models are introduced. The authors in [7] propose a URLLC and eMBB
coexistence mechanism on a cellular vehicle-to-everything (C-V2X) framework.
In [8], the authors propose a null space based spatial preemptive scheduling
for URLLC and eMBB traffic in a dense multi user 5G network. Furthermore, the
authors in [9] model the URLLC and eMBB coexistence problem as a deep
reinforcement learning issue. Nevertheless, it is still an open issue to
efficiently coexist URLLC and eMBB users.

On the other aspect, non-orthogonal multiple access (NOMA) has recently
emerged as a promising access technology, due to its potential in achieving
massive connections, user fairness, and high spectral efficiency [10]. Several
works have already leveraged the NOMA technology into the URLLC and eMBB
coexistence system. In [11], the authors propose a NOMA access based multi-
cell C-RAN architecture, where performance tradeoffs between URLLC and eMBB
users have been investigated. Moreover, the authors in [12] consider the
coexistence between URLLC and eMBB users in an analog C-RAN architecture,
where uplink NOMA transmission have been studied. Furthermore, the authors in
[13] propose a NOMA based Fog-Radio architecture for URLLC and eMBB
coexistence, which can schedule eMBB traffic centrally at the cloud and deal
with URLLC traffic at the edge nodes. However, the above URLLC and eMBB
coexistence systems have not considered the multiple-input-multiple-output
(MIMO)-NOMA access, which is a more attractive technology for further spectrum
efficiency enhancement through increasing the degree of spatial freedom.
However, the development of MIMO-NOMA technology also confronts several
challenges, such as the beamforming management and power allocation. In [14],
the authors propose a power allocation scheme to maximize the data rate of
MIMO-NOMA system. In addition, the authors in [15] compare the performance of
MIMO-NOMA with MIMO orthogonal multiple access (MIMO-OMA), and present the
impressive gain achieved by MIMO-NOMA. Moreover, the authors in [16, 17]
propose their MIMO-NOMA architectures based on a clustering technology. To the
best of our knowledge, there has no work leveraging MIMO-NOMA into the URLLC
and eMBB coexistence issue yet. Since the MIMO-NOMA based system has a
relatively high spatial freedom degrees and can connect more users, it
provides more access opportunities for the arrival URLLC users as well as
minimizes the performance loss of the connected eMBB users. Specifically, each
arrival URLLC users can select a more suitable access position from the large
number of connected eMBB users. Meanwhile, the connected eMBB users can also
provide more eMBB users with worse performance to be selected.

Based on the above analyses, in this paper, we introduce an efficient URLLC
and eMBB coexistence mechanism in a downlink MIMO-NOMA system. Firstly, we
introduce a dynamic user clustering mechanism before each transmission period.
A maximal-minimum clustering algorithm is investigated to ensure the tradeoff
between system performance and computational complexity. Thereafter, we allow
the arrived URLLC users to puncture the following mini slot immediately by
interrupting eMBB transmissions in related clusters. The URLLC and eMBB
coexistence issue is formulated as a joint user selection and power allocation
scheduling problem, which is a NP-hard mixed integer nonlinear programming
(MINLP). To deal with the optimization problem, we decompose it into two
subproblems, namely punctured eMBB user selection and power allocation. We
formulate the punctured user selection subproblem as a Gale-Shapley (GS)
matching process, which provides an adaptive and low-complexity framework.
Alternatively, the power allocation problem is simplified with a successive
convex approximation (SCA) and a difference of convex (D.C.) programming. The
suboptimal solution of the whole problem is found through iteratively solving
the above two subproblems.

The main contributions of this paper are summarized as follows.

  1. 1.

We propose an efficient URLLC and eMBB coexistence mechanism under a
puncturing technology in MIMO-NOMA systems, which provides more flexibility
for the access of arrival URLLC users and is more friendly to the existing
eMBB users. In detail, any arrival URLLC users would replace the transmission
of parts eMBB users in different spatial directions for the following mini
slot.

  2. 2.

We formulate the objective function to maximize the eMBB data rates while
ensuring the URLLC latency requirements. A joint user selection and power
allocation scheme is proposed to improve the spectrum efficiency.

  3. 3.

An eMBB user clustering mechanism in MIMO-NOMA systems is introduced to not
only ensure a stable eMBB user structure before transmission, but also strike
a balance between system performance and computational complexity.

  4. 4.

The MINLP optimization problem is decomposed into two subproblems: punctured
eMBB user selection and power allocation. We adopt a GS matching algorithm to
solve the user selection problem, and a SCA-DC algorithm to deal with the
power allocation problem.

The rest of this paper is organized as follows. Section II presents the system
model. Section III is the problem formulation. Section IV and Section V
propose the proposed user clustering and joint scheduling problems,
respectively. Simulation results are presented in Section VI. Finally, a
conclusion is summarized in Section VII.

Notations: The superscripts $\bm{A}^{T}$ and $\bm{A}^{H}$ denote its transpose
and conjugate transpose, respectively. Bold lowercase and uppercase letters
represent column vectors and matrices, respectively. $\mathbb{C}^{N\times M}$
denotes the set of all $N\times M$ matrices with complex entries.
$\mathbb{R}^{N\times M}$ denotes the set of all $N\times M$ matrices with real
entries. $\left|\cdot\right|$ denotes the absolute values of a complex scalar
or the cardinality of a set. $\|\cdot\|$ represents the vector norm.

##  II System Model

In this work, we consider a URLLC and eMBB coexistence scenario under a
downlink MIMO-NOMA based network. There are $K$ eMBB devices and several URLLC
devices within the coverage of a multi-antenna base station (BS), as shown in
Fig.1. The BS equips with $B$ antennas, and each eMBB or URLLC user equips
with $N$ antennas. Each time slot (transmission period of the framework) is
divided into $Q$ equivalent mini slots, denoted as
$\mathcal{Q}\triangleq\\{1,\cdots,Q\\}$, where $i\in\mathcal{Q}$ represents
the $i$-th mini slot.

Under the MIMO-NOMA system, $K$ eMBB users are grouped into $M$ clusters
before each transmission period, denoted as
${\mathcal{M}}=\left\\{{1,2,\ldots,M}\right\\}$, where the number of clusters
equals to the number of BS antennas, i.e., $M=B$. Furthermore, we denote
${\mathcal{L}}=\left\\{{{\mathcal{L}_{1}},{\mathcal{L}_{2}},\ldots,{\mathcal{L}_{M}}}\right\\}$,
where ${\mathcal{L}_{m}}\in{\mathcal{L}}$ represents the user set in the
$m$-th cluster. Cluster ${\mathcal{L}_{m}}$ contains
${L_{m}=\left|{\mathcal{L}_{m}}\right|}$ users. Here, we ensure each cluster
contains at least two users, i.e., ${\left|{\mathcal{L}_{m}}\right|}\geq 2$.

Figure 1: An illustration of the URLLC and eMBB coexistence in MIMO-NOMA
systems.

###  II-A MIMO-NOMA Models

Based on [18], the considered multi-user MIMO-NOMA scenario can be decomposed
into multiple separate single-antenna NOMA channels, to which conventional
NOMA technology can be applied directly by reasonable power allocation and
user clustering. The downlink MIMO-NOMA transmission is implemented by
superimposing the signals destined for end users at the same cluster. As a
result, the transmit signal vector
$\tilde{\mathbf{s}}^{(i)}\in\mathbb{C}^{M\times 1}$ can be written as

$\tilde{\mathbf{s}}^{(i)}=\left[{\\!{\begin{array}[]{*{20}{c}}{\sqrt{{\alpha_{1,1}^{(i)}}}{s_{1,1}^{(i)}}+\cdots+{\sqrt{{\alpha_{1,{L_{1}}}^{(i)}}}{s_{1,{L_{1}}}^{(i)}}}}\\\
\vdots\\\
{\sqrt{{\alpha_{M,1}^{(i)}}}{s_{M,1}^{(i)}}+\cdots+{\sqrt{{\alpha_{M,{L_{M}}}^{(i)}}}}}{s_{M,{L_{M}}}^{(i)}}\\\
\end{array}}\\!}\right]\buildrel\Delta\over{=}\left[{\\!{\begin{array}[]{*{20}{c}}{{{\tilde{\mathbf{s}}}_{1}^{(i)}}}\\\
\vdots\\\ {\tilde{\mathbf{{s}}}_{M}^{(i)}}\end{array}}\\!}\right]\\!,\qquad$
(1)

where $s_{m,k}^{(i)}$ and $\alpha_{m,k}^{(i)}$ are the transmitted information
and the NOMA power allocation coefficient of the $k$-th user in the $m$-th
cluster at mini slot $i$, respectively.

The BS precodes the signal vector with an identity precoding matrix
$\textbf{P}^{(i)}\in\mathbb{C}^{M\times M}$[16]. Hence, the received signal of
the $k$-th user over the $m$-th cluster in mini slot $i$ can be expressed as

${\textbf{y}_{m,k}^{(i)}}={\textbf{H}_{m,k}^{(i)}}\textbf{P}^{(i)}\tilde{\mathbf{s}}^{(i)}+{\textbf{n}_{m,k}^{(i)}},~{}\forall
m,k\in{\mathcal{L}_{m}},i,$ (2)

where $\textbf{H}_{m,k}^{(i)}\in\mathbb{C}^{N\times M}$ is the channel gain
matrix from the BS to the $k$-th user in the $m$-th cluster, and
${\textbf{n}_{m,k}^{(i)}}$ is additive white Gaussian noise (AWGN) vector. For
simplicity, the channel gains between the BS and users are assumed to be
independent and identically complex Gaussian distributed111Note that there has
no relationship between the BS/user’s channel gain distribution and the
channel spatial correlation.. Moreover, the channel state is invariant during
each mini slot and time-varying across different mini slots.

To further improve the spectral efficiency, a detection vector
$\textbf{v}_{m,k}^{(i)}\in\mathbb{C}^{N\times 1}$ has been applied by each
user. Therefore, the detection results of the $k$-th user in the $m$-th
cluster can be formulated as

${\textbf{v}_{m,k}^{(i)~{}H}}{{\textbf{y}_{m,k}^{(i)}}}={\textbf{v}_{m,k}^{(i)~{}H}}{\textbf{H}_{m,k}^{(i)}}\textbf{P}^{(i)}\tilde{\mathbf{s}}^{(i)}+{\textbf{v}_{m,k}^{(i)~{}H}}{\textbf{n}_{m,k}},\\\
\forall m,k\in{\mathcal{L}_{m}},i.$ (3)

Denote ${{\mathbf{p}}_{m}^{(i)}}$ as the $m$-th column of $\textbf{P}^{(i)}$,
(3) can be rewritten as

$\begin{split}{\textbf{v}_{m,k}^{(i)~{}H}}{{\mathbf{y}}_{m,k}^{(i)}}=\sum\limits_{f=1,f\neq
m}^{M}{{\textbf{v}_{m,k}^{(i)~{}H}}{{\mathbf{H}}_{m,k}^{(i)}}{{\mathbf{p}}_{f}^{(i)}}{{\tilde{\mathbf{s}}}_{f}^{(i)}}}+{\textbf{v}_{m,k}^{(i)~{}H}}{\textbf{n}_{m,k}^{(i)}}+\\\
{\textbf{v}_{m,k}^{(i)~{}H}}{{\mathbf{H}}_{m,k}^{(i)}}{{\mathbf{p}}_{m}^{(i)}}\bigg{(}{{\sqrt{{\alpha_{m,1}^{(i)}}}{s_{m,1}^{(i)}}+\cdots}}{{+\sqrt{{\alpha_{m,{L_{m}}}^{(i)}}}{s_{m,{L_{m}}}^{(i)}}}}\bigg{)},\\\
~{}\forall m,k\in{\mathcal{L}_{m}},i.\end{split}$ (4)

To remove the inter-cluster interference, the precoding and detection matrices
should satisfy the constraints of

${{\mathbf{v}_{{l,k}}^{(i)~{}H}}}\underbrace{\left[{\mathbf{h}_{1,lk}^{(i)}\cdots\mathbf{h}_{l-1,lk}^{(i)}\;\mathbf{h}_{l+1,lk}^{(i)}\cdots\mathbf{h}_{M,lk}^{(i)}}\right]}_{\tilde{\mathbf{H}}_{l,k}^{(i)}}=0,\\\
\forall l\in\mathcal{M},k\in\mathcal{L}_{\ell},i,$ (5)

where ${{\mathbf{h}}_{j,lk}^{(i)}}$ refers to the $j$-th column of
${\textbf{H}_{l,k}^{(i)}}$,
$\tilde{\mathbf{H}}_{l,k}^{(i)}\in\mathbb{C}^{N\times(M-1)}$ is a submatrix of
$\mathbf{H}_{l,k}^{(i)}$ by removing the $l$-th column.

Consequently, $\mathbf{v}_{l,k}^{(i)}$ can be obtained from the null space of
$\tilde{\mathbf{H}}_{l,k}^{(i)}$, i.e.,
$\mathbf{v}_{l,k}^{(i)}=\mathbf{U}_{l,k}^{(i)}\mathbf{z}_{l,k}^{(i)}$, where
$\mathbf{U}_{l,k}^{(i)}$ contains all the left singular vectors of
$\tilde{\mathbf{H}}_{l,k}^{(i)}$ corresponding to zero singular values, and
$\mathbf{z}_{l,k}^{(i)}\in\mathbb{C}^{(N-M+1)\times 1}$ is a normalized
vector. According to [16] and [17], $\mathbf{z}_{l,k}^{(i)}$ can be obtained
by using the maximal ratio combining (MRC) approach, which is given by

$\mathbf{z}_{l,k}^{(i)}=\frac{\mathbf{U}_{l,k}^{(i)~{}\\!\\!H}\mathbf{h}_{l,lk}^{(i)}}{\|\mathbf{U}_{l,k}^{(i)~{}\\!\\!H}\mathbf{h}_{l,lk}^{(i)}\|},~{}\forall
l\in\mathcal{M},k\in\mathcal{L}_{\ell},i.$ (6)

To reduce system overheads, we allow the BS transmits signals without
processing them that can avoid the channel state information (CSI) feedback.
However, to implement the successive interference cancellation (SIC)
effectively, the BS needs to obtain the information of the channel gain order.
We utilize the coefficients of
${\left|{{{\textbf{v}_{m,k}^{(i)~{}H}}{{\mathbf{H}}_{m,k}^{(i)}}{{\mathbf{p}}_{m}^{(i)}}}}\right|^{2}}$
to describe the channel conditions, which can be generally ordered as

${\left|{{{\textbf{v}_{m,1}^{(i)~{}H}}{{\mathbf{H}}_{m,1}^{(i)}}{{\mathbf{p}}_{m}^{(i)}}}}\right|^{2}}\geq\cdots\geq{\left|{{{\mathbf{v}}_{m,{L_{m}}}^{(i)~{}H}{{\mathbf{H}}_{m,{L_{m}}}^{(i)}}{{\mathbf{p}}_{m}^{(i)}}}}\right|^{2}},~{}\forall
m,i.$ (7)

Note that, we would sort the effective channel gain at the beginning of each
mini slot.

According to the SIC, user $k$ in the $m$-th cluster needs to decode the
messages of the users with poorer channel conditions in the same cluster
$s_{m,j}^{(i)},~{}k\leq j\leq{{L_{m}}}$ before detecting its own information
$s_{m,k}^{(i)}$. The signal $s_{m,j}^{(i)}$ can be detected at the $k$-th user
with the following SINR as

$SINR_{m,k,j}^{(i)}\\!=\\!\frac{{{{\left|{{{\mathbf{v}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\alpha_{m,j}^{(i)}}}}{{\sum\nolimits_{l=1}^{j-1}{{{\left|{{{\mathbf{v}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\alpha_{m,l}^{(i)}}+{{\|{{{{\mathbf{v}}_{m,k}^{(i)}}}}\|}^{2}}\frac{1}{\rho}}}},\\\
\forall
m,k\in{\mathcal{L}_{m}},j\in\left\\{{k,k+1,\ldots,{L_{m}}}\right\\},i,$ (8)

where $\rho$ denotes the transmit signal-to-noise-ratio (SNR).

Therefore, the effective SINR for user $k$ in the $m$-cluster at mini slot $i$
can be expressed as [10]

${SINR_{m,k}^{(i)}}=\frac{{{{\left|{{{\mathbf{v}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\alpha_{m,k}^{(i)}}}}{{\sum\nolimits_{l=1}^{k-1}{{{\left|{{{\mathbf{v}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\alpha_{m,l}^{(i)}}+{{\|{{{{\mathbf{v}}_{m,k}^{(i)}}}}\|}^{2}}\frac{1}{\rho}}}},\\\
\forall m,k\in{\mathcal{L}_{m}},i.$ (9)

###  II-B URLLC and eMBB Transmission Models

To ensure the strict latency requirement, any arrival URLLC user is allowed to
puncture the next mini slot immediately by interrupting an eMBB user’s
transmission in a specific cluster. Meanwhile, the BS allocates zero
transmission power to the punctured eMBB user [6]. After the puncturing, the
interrupted eMBB user would reoccupy the mini slot again. Denote
$\mathcal{S}^{(i)}_{~{}\\!u}$ as the set of arrival URLLC users in mini slot
$i$, which would interrupt the transmission of
$\left|{\mathcal{S}^{(i)}_{~{}\\!u}}\right|$ eMBB users. Let
$\mathcal{S}^{(i)}_{~{}\\!u,m}$ be the set of URLLC users puncturing in
cluster $m$ at mini slot $i$. Then, we have
${\mathcal{S}^{(i)}_{~{}\\!u}}=\mathcal{S}^{(i)}_{~{}\\!u,1}\cup\mathcal{S}^{(i)}_{~{}\\!u,2}\cup\cdots\cup\mathcal{S}^{(i)}_{~{}\\!u,M}$.
Denote $\mathcal{S}^{(i)}_{~{}\\!e,m}$ as the set of remaining eMBB users for
cluster $m$ at mini slot $i$, and $\mathcal{S}^{(i)}_{~{}\\!e}$ as the set of
all the remaining eMBB users at mini slot $i$, where
$\left|{\mathcal{S}^{(i)}_{~{}\\!e}}\right|=K-{\left|{\mathcal{S}^{(i)}_{~{}\\!u}}\right|}$
and
$\left|{\mathcal{S}^{(i)}_{~{}\\!e,m}}\right|={L_{m}}-{\left|{\mathcal{S}^{(i)}_{~{}\\!u,m}}\right|}$.
Let $x_{m,k,n}^{(i)}$ be a binary variable to indicate whether eMBB user $k$
in $m$-th cluster is punctured by the URLLC user $n$. $x_{m,k,n}^{(i)}=1$
represents eMBB user $k$ in the $m$-th cluster is punctured by URLLC user $n$,
and $x_{m,k,n}^{(i)}=0$ otherwise.

We further define $\beta_{m,n}^{(i)}$ and $\textbf{w}_{m,n}^{(i)}$ as the
power allocation coefficient and detection vector of the $n$-th URLLC user in
the $m$-th cluster at mini slot $i$, respectively. After puncturing, the
effective SINR for URLLC user $n\in\mathcal{S}^{(i)}_{~{}\\!u,m}$ in the
$m$-cluster at mini slot $i$ can be rewritten as

$SINR_{u,m,n}^{(i)}=\frac{{{{\left|{{{\mathbf{w}}_{m,n}^{(i)~{}H}{{\mathbf{h}}_{m,mn}^{(i)}}}}\right|}^{2}}{\beta_{m,n}^{(i)}}}}{I_{1}^{u}+I_{2}^{u}+\\!{{\|{{{{\mathbf{w}}_{m,n}^{(i)}}}}\|}^{2}}\frac{1}{\rho}},~{}\forall
m,n\in\mathcal{S}^{(i)}_{~{}\\!u,m},i,$ (10)

where

$I_{1}^{u}=\sum\limits_{l=1}^{j-1}(1-\\!\sum\limits_{n^{{}^{\prime}}\in\mathcal{S}^{(i)}_{~{}\\!\\!u,m}}\\!x_{m,l,n^{{}^{\prime}}}^{(i)}){{{\left|{{{\mathbf{w}}_{m,n}^{(i)~{}H}{{\mathbf{h}}_{m,mn}^{(i)}}}}\right|}^{2}}{\alpha_{m,l}^{(i)}}},$
(11)

and

$I_{2}^{u}=\sum\limits_{l=1}^{j-1}\sum\limits_{n^{{}^{\prime}}\in\mathcal{S}^{(i)}_{~{}\\!\\!u,m}}\\!{x_{m,l,n^{{}^{\prime}}}^{(i)}}{{{\left|{{{\mathbf{w}}_{m,n}^{(i)~{}H}{{\mathbf{h}}_{m,mn}^{(i)}}}}\right|}^{2}}{\beta_{m,l}^{(i)}}}.$
(12)

Since URLLC packets are typically quite short, the achievable rate cannot be
accurately captured by Shannon’s capacity. Derived by [21], the data rate of
URLLC user $n$ in $m$-th cluster at mini slot $i$ falls into the finite
blocklength channel coding regime, which can be formulated as

$\begin{split}r_{m,n}^{(i)}\leq{\log_{2}}(1+{SINR}_{u,m,n}^{(i)})-\sqrt{\frac{C_{m,n}^{(i)}}{b_{m,n}^{(i)}}}Q^{-1}(\epsilon)\log
e,\\\ \forall m,n\in\mathcal{S}^{(i)}_{~{}\\!u,m},i,\end{split}$ (13)

where $Q^{-1}(\cdot)$ is the inverse of the Gaussian Q-function, $\epsilon>0$
is the transmission error probability, $b_{m,n}^{(i)}$ is the length of
codeword block in symbols, and ${C_{m,n}^{(i)}}$ is the channel dispersion
with

$C_{m,n}^{(i)}=1-\frac{1}{(1+{SINR}_{u,m,n}^{(i)})^{2}},~{}\forall
m,n\in\mathcal{S}^{(i)}_{~{}\\!u,m},i.$ (14)

According to [22, 23], the upper bound of (13) can be achieved through a
suitable encoding/decoding couple. Therefore, we set the upper bound for
$r_{u,m,n}^{(i)}$ in the following paper.

To satisfy the latency requirements, URLLC users should satisfy [5]

$\frac{F_{m,n}^{(i)}}{Br_{m,n}^{(i)}}\leq D_{m,n}^{(i),\max},~{}\forall
m,n\in\mathcal{S}^{(i)}_{~{}\\!u,m},i,$ (15)

where $B$ is the bandwidth, $D_{m,n}^{(i),\max}$ and $F_{m,n}^{(i)}$ are the
maximum tolerant delay and the packet length of URLLC user $n$ in the $m$-th
cluster at mini slot $i$, respectively.

Similarly, the effective SINR of eMBB user $k\in\mathcal{S}^{(i)}_{~{}\\!e,m}$
in the $m$-th cluster at mini slot $i$ can be expressed as

$\displaystyle
SINR_{e,m,k}^{(i)}=\frac{{{{\left|{{{\mathbf{v}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\alpha_{m,k}^{(i)}}}}{{{I_{1}^{e}+I_{2}^{e}+\|{{{{\mathbf{v}}_{m,k}^{(i)}}}}\|}^{2}}\frac{1}{\rho}},~{}\forall
m,k\in\mathcal{S}^{(i)}_{~{}\\!e,m},i,$ (16)

where

$\displaystyle
I_{1}^{e}=\sum\limits_{l=1}^{j-1}(1-\\!\sum\limits_{n\in\mathcal{S}^{(i)}_{~{}\\!\\!u,m}}\\!x_{m,l,n}^{(i)}){{{\left|{{{\mathbf{v}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\alpha_{m,l}^{(i)}}},$
(17)

and

$\displaystyle
I_{2}^{e}=\sum\limits_{l=1}^{j-1}\sum\limits_{n\in\mathcal{S}^{(i)}_{~{}\\!\\!u,m}}\\!{x_{m,l,n}^{(i)}}{{{\left|{{{\mathbf{v}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\beta_{m,l}^{(i)}}}.$
(18)

Thereafter, the data rate of eMBB user $k$ in the $m$-th cluster at mini slot
$i$ can be formulated as

$\displaystyle{R_{m,k}^{(i)}}\\!=\\!{\log_{2}}\left({1\\!+\\!SINR_{e,m,k}^{(i)}}\right),~{}\forall
m,k\in\mathcal{S}^{(i)}_{~{}\\!e,m},i.$ (19)

To ensure the data rate requirements, eMBB users should satisfy

${R_{m,k}^{(i)}}\geq{R_{m,k}^{\min}},~{}\forall
m,k\in\mathcal{S}^{(i)}_{~{}\\!e,m},i,$ (20)

where ${R_{m,k}^{\min}}$ is the minimum throughput constraint of eMBB user $k$
in the $m$-th cluster.

##  III Problem Formulation

In this section, we first formulate the user clustering problem. Thereafter,
we introduce the joint user selection and power allocation problem.

###  III-A User Clustering

To ensure the tradeoff between system performance and computational
complexity, the user clustering problem is formulated to maximize the SINR of
the worst user among all the eMBB users, which is a double maximize minimum
problem of

$\displaystyle\mathcal{P}_{0}:\quad\max\limits_{\bm{\Lambda}}\min\limits_{m}{{{{\max\limits_{{\bm{\alpha}}_{m}^{(i)}}\min\limits_{k,j}\left({{SINR_{m,k,j}^{(i)}}}\right)}}}},$
(21) $\displaystyle\quad\,\mathrm{subject}~{}\mathrm{to}$
$\displaystyle\qquad\quad\log_{2}\left(1+SINR_{m,k,j}^{(i)}\right)\geq{R_{m,j}^{\min}},$
$\displaystyle\qquad\quad\qquad\forall
m,k\in{\mathcal{L}_{m}},j\in\left\\{{k,k+1,\ldots,{L_{m}}}\right\\},$ (21a)
$\displaystyle\qquad\quad{\sum\limits_{k=1}^{L_{m}}{{\alpha_{m,k}^{(i)}}}\leq\frac{L_{m}}{K},}~{}\forall
m,$ (21b) $\displaystyle\qquad\quad{\alpha_{m,k}^{(i)}}\geq 0,~{}\forall
m,k\in{\mathcal{L}_{m}},$ (21c)

where $\bm{\Lambda}$ is defined as all possible clustering combination sets of
all eMBB users,
${\bm{\alpha}}_{m}^{(i)}=\\{\alpha_{m,1}^{(i)},...,\alpha_{m,L_{m}}^{(i)}\\},\forall
m$ is the power allocation coefficient vector for the $m$-th cluster.
Constraints (21a) are the minimum data rate requirements. Constraints
(21b)-(21c) are the total power limitations in different clusters.

In detail, the inner maximize minimum problem
$\max\limits_{{\bm{\alpha}}_{m}^{(i)}}\min\limits_{k,j}\left({{SINR_{m,k,j}^{(i)}}}\right)$
is formulated to maximize the worst SINR in the $m$-th cluster at a given user
clustering structure through power assignment. Thereafter, the problem
$\min\limits_{m}{{{{\max\limits_{{\bm{\alpha}}_{m}^{(i)}}\min\limits_{k,j}\left({{SINR_{m,k,j}^{(i)}}}\right)}}}}$
means to select the smallest SINR from all the $M$ clusters. Finally, the
outermost maximize problem
$\max\limits_{\bm{\Lambda}}\min\limits_{m}{{{{\max\limits_{{\bm{\alpha}}_{m}^{(i)}}\min\limits_{k,j}\left({{SINR_{m,k,j}^{(i)}}}\right)}}}}$
denotes to get the maximum value of the worst SINR among all clusters through
traversing all user clustering combinations.

###  III-B Optimization Problem

The goal of this work is to maximize the data rate of eMBB users during all
the transmission period, while satisfying the QoS requirements of URLLC and
eMBB users. Therefore, the optimal scheduling problem can be formulated as

$\displaystyle\mathcal{P}_{1}:~{}\max\limits_{\left\\{{x_{m,k,n}^{(i)},{\bm{\alpha_{m}^{(i)}}},\bm{\beta_{m}^{(i)}}}\right\\}}\sum\limits_{i=1}^{Q}\sum\limits_{m=1}^{M}\sum\limits_{k=1}^{{L_{m}}}\log_{2}\left({{1+SINR_{e,m,k}^{(i)}}}\right),$
(22) $\displaystyle\mathrm{subject}~{}\mathrm{to}$
$\displaystyle\qquad\quad\frac{F_{m,n}^{(i)}}{Br_{m,n}^{(i)}}\leq
D_{m,n}^{(i),\max},~{}\forall m,n\in\mathcal{S}^{(i)}_{~{}\\!u,m},i,$ (22a)
$\displaystyle\qquad\quad{R_{m,k}^{(i)}}\geq{R_{m,k}^{\min}},~{}\forall
m,k\in\mathcal{S}^{(i)}_{~{}\\!e,m},i,$ (22b) $\displaystyle\qquad\quad
x_{m,k,n}^{(i)}\in\left\\{{{0,1}}\right\\},~{}\forall
m,k\in\mathcal{L}_{m},n\in\mathcal{S}^{(i)}_{~{}\\!u,m},i,$ (22c)
$\displaystyle\qquad\quad\sum\limits_{m\in{\mathcal{M}}}\sum\limits_{k\in\mathcal{L}_{m}}x_{m,k,n}^{(i)}=1,~{}\forall
n\in\mathcal{S}^{(i)}_{~{}\\!u},i,$ (22d)
$\displaystyle\qquad\quad\sum\limits_{n\in\mathcal{S}^{(i)}_{~{}\\!u}}x_{m,k,n}^{(i)}\leq
1,~{}\forall m,k\in\mathcal{L}_{m},i,$ (22e)
$\displaystyle\qquad\quad{\alpha_{m,k}^{(i)}}>0,~{}\forall
m,k\in{S}_{~{}\\!e,m}^{(i)},i,$ (22f)
$\displaystyle\qquad\quad{\beta_{m,n}^{(i)}}>0,~{}\forall
m,n\in\mathcal{S}_{~{}\\!u,m}^{(i)},i,$ (22g)
$\displaystyle\qquad\quad\sum\limits_{k\in\mathcal{S}^{(i)}_{~{}\\!\\!e,m}}\\!\\!{\alpha_{m,k}^{(i)}}\\!+\\!\sum\limits_{n\in\mathcal{S}^{(i)}_{~{}\\!\\!u,m}}\\!\\!{\beta_{m,n}^{(i)}}\leq\frac{{{L_{m}}}}{K},~{}\forall
m,i,$ (22h)

where
${\bm{\alpha}}_{m}^{(i)}=\left\\{{{\alpha_{m,1}^{(i)}},\cdots,{\alpha_{m,{\left|{\mathcal{S}^{(i)}_{~{}\\!\\!e,m}}\right|}}^{(i)}}}\right\\}$
and
${\bm{\beta}}_{m}^{(i)}=\left\\{{{\beta_{m,1}^{(i)}},\cdots,{\beta_{m,{\left|{\mathcal{S}^{(i)}_{~{}\\!\\!u,m}}\right|}}^{(i)}}}\right\\}$
are the power allocation coefficient vector of URLLC and eMBB users in the
$m$-th cluster, respectively. Constraints (22c)-(22e) ensure each URLLC user
occupies one eMBB user and each eMBB user is punctured by at most one URLLC
user. Constraints (22f)-(22h) denote the transmit power limitation in each
cluster.

Note that the computational complexity of the NOMA technology is based on the
decoding and implementation complexities of the SIC, which increases with the
number of users. Based on [10], the SIC complexity without user clustering is
${\rm\mathcal{O}}\left({{{{K}}}^{3}}\right)$. However, with user clustering,
the computational complexity of SIC reduces to
${\rm\mathcal{O}}\left({\sum\nolimits_{m=1}^{M}{\left(L_{m}\right)}^{3}}\right)$,
where $K\\!=\\!\sum\nolimits_{m=1}^{M}{L_{m}}$.

##  IV User Clustering Mechanism

In this section, a user clustering mechanism would be proposed to solve
$\mathcal{P}_{0}$.

Recalling (8), the objective user clustering function $\mathcal{P}_{0}$ is
obviously a non-convex problem. To solve this problem, we split the objective
function (21) into two steps.

  * •

Step 1: Under a given eMBB user clustering combination set
$\varphi\in\bm{\Lambda}$, $\mathcal{P}_{0}$ can be reformulated as

$\displaystyle\mathcal{P}_{2}:\quad\max\limits_{{\bm{\alpha}}_{m}^{(i)}}\min\limits_{k,j}\left({{\bm{SINR_{m}}}\left({{\bm{\alpha}}_{m}^{(i)}}\right)}\right),$
(23)

subject to (21a), (21b) and (21c).

Here, ${\bm{SINR_{m}}}$ denotes all possible values of $SINR_{m,k,j}^{(i)}$,
$\forall k,j$ in the $m$-th cluster under a given power allocation coefficient
vector ${\bm{\alpha}}_{m}^{(i)}$.

However, $\mathcal{P}_{2}$ is still non-convex, which leads us to find an
equivalent conversions to make it tractable. Therefore, we define
${\mathbb{G}_{\Theta}}=\left\\{{{{\bm{SINR}}_{\mathbf{m}}\left({{\bm{\alpha}}_{m}^{(i)}}\right)\geq{\Theta}}}\right\\}$
as the set of ${\bm{\alpha}}_{m}^{(i)}$ when the objective function is larger
than $\Theta$, where $\Theta$ is a real number. Hence, $\Theta$ is equivalent
to the minimum value of ${\bm{SINR_{m}}}$, i.e., $SINR_{\min}^{m}=\Theta$.
Therefore, we can equally transform $\mathcal{P}_{2}$ as [17, 24]

$\displaystyle\mathcal{P}_{3}:\quad\quad\quad\quad\mathrm{Find}\quad\quad{{\bm{\alpha}}_{m}^{(i)}},$
(24) $\displaystyle\quad\,\mathrm{subject}~{}\mathrm{to}$
$\displaystyle\qquad\quad{\left|{{{\mathbf{v}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|^{2}}{\alpha_{m,j}^{(i)}}\geq\Theta
D,$ (24a) $\displaystyle\qquad\quad\qquad\forall
m,k\in{\mathcal{L}_{m}},j\in\left\\{{k,k+1,\ldots,{L_{m}}}\right\\},$
$\displaystyle\qquad\quad{\left|{{{\mathbf{v}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|^{2}}{\alpha_{m,j}^{(i)}}\geq
D\left(2^{{R_{m,j}^{min}}}-1\right),$ (24b)
$\displaystyle\qquad\quad\qquad\forall
m,k\in{\mathcal{L}_{m}},j\in\left\\{{k,k+1,\ldots,{L_{m}}}\right\\},$
$\displaystyle\qquad\quad{\sum\limits_{k=1}^{L_{m}}{{\alpha_{m,k}^{(i)}}}\leq\frac{L_{m}}{K},}~{}\forall
m,$ (24c) $\displaystyle\qquad\quad{\alpha_{m,k}^{(i)}}\geq 0,~{}\forall
m,k\in{\mathcal{L}_{m}},$ (24d)

where
$D={{\sum\nolimits_{l=1}^{j-1}{{{\left|{{{\mathbf{v}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\alpha_{m,l}^{(i)}}+{{\|{{{{\mathbf{v}}_{m,k}^{(i)}}}}\|}^{2}}\frac{1}{\rho}}}}$.

We utilize a bisection searching method to find the optimal $\Theta$. Under
given $\Theta$, $\mathcal{P}_{3}$ becomes a linear programming (LP) problem,
which can be solved by standard LP tools to obtain the corresponding power
allocation coefficient vector ${\bm{\alpha}}_{m}^{(i)}$.

  * •

Step 2: To find the optimal user clustering mechanism, we would traverse all
the user clustering combinations. Here, we adopt the exhaustive searching
method.

The detailed process is shown in Algorithm 1. $SINR_{\min}^{\varphi}$ in
Algorithm 1 denotes the smallest SINR of all the $M$ clusters under a given
eMBB user clustering combination $\varphi\in\bm{\Lambda}$. The computational
complexity of step 1 and step 2 are ${\rm\mathcal{O}}\left({2^{K}}\right)$ and
${\rm\mathcal{O}}\left({\frac{K!}{\prod_{m=1}^{M}{L_{m}}!}}\right)$,
respectively. Therefore, Algorithm 1 has the complexity of
${\rm\mathcal{O}}\left({2^{K}\frac{(K)!}{\prod_{m=1}^{M}{L_{m}}!}}\right)$.

Algorithm 1 Algorithm for user clustering.

1: Initialize the tolerance $\delta$ and the ${\small{SINR}}_{\min}=0$.

2: For each user combination $\varphi\in\bm{\Lambda}$ do

3: Initialize the $SINR_{\min}^{\varphi}=0$ when the user group is $\Lambda$.

4: For each cluster $m\in{\mathcal{M}}$ do

5: Initialize the lower bound of $\bm{SINR}_{m}$ as $\Theta_{LB}$, the upper
bound of $\bm{SINR}_{m}$ as $\Theta_{UB}$.

6: While ${|\Theta_{UB}}-{\Theta_{LB}|}\geq\delta$ do

7: Set $\Theta=\left({\Theta_{UB}}+{\Theta_{LB}}\right)/2$.

8: Obtain ${\bm{\alpha}}_{m}$ by solving the linear problem $\mathcal{P}_{3}$.

9: If $\mathcal{P}_{3}$ is feasible then

10: Set $SINR_{\min}^{m}=\Theta$ and $\Theta_{LB}=\Theta$.

11: else

12: Set $\Theta_{UB}=\Theta$.

13: end If

14: end While

15: Update
$SINR_{\min}^{\varphi}\\!=\\!{\max}\\!\left\\{\\!SINR_{\min}^{\varphi},SINR_{\min}^{m}\\!\right\\}$.

16: end For

17: If $SINR_{\min}^{\varphi}>SINR_{\min}$ then

18: Set $SINR_{\min}=SINR_{\min}^{\varphi}$ and ${\varphi}^{*}=\varphi$.

19: end if

20: end For

21: Output: $SINR_{\min}$ and the optimal user clustering combination
${\varphi}^{*}$.

##  V Joint URLLC and eMBB Scheduling

In this section, we aim to find the optimal joint URLLC and eMBB scheduling
scheme.

Intuitively, $\mathcal{P}_{1}$ is a non-convex optimization problem. According
to (16) and (22), the objective function relies on URLLC-eMBB punctured user
selection and power assignment. Therefore, it follows two key problems: 1)
which eMBB users to be punctured when URLLC users arrived? 2) how to assign
transmission power to the users in each cluster?

As we can observed from (22), the objective function is a coupled problem.
Considering the computational complexity, we decouple $\mathcal{P}_{1}$ into
two subproblems, namely, the punctured user selection problem and the multi-
cluster power allocation problem. Thereafter, we propose an iterative
algorithm, where the two subproblems are performed in an iterative way to
obtain a solution for $\mathcal{P}_{1}$. In this work, the punctured user
selection problem can be formulated as a matching game. Under a punctured user
selection scheme, the power allocation problem can be transformed as a convex
problem by means of SCA and D.C. programming.

###  V-A The Power Allocation Problem

Under a given punctured user selection scheme, the effective SINR for URLLC
and eMBB users in the $m$-cluster at mini slot $i$ can be simplified as

$\begin{split}SINR_{m,k}^{e/u,(i)}={{{{\left|{{{\mathbf{m}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\gamma_{m,k}^{(i)}}}\Big{/}{\Big{(}{{\|{{{{\mathbf{m}}_{m,k}^{(i)}}}}\|}^{2}}\frac{1}{\rho}}}+\\\
\\!\sum\limits_{u\in\mathfrak{U_{1}}}{{{\left|{{{\mathbf{m}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\beta_{m,u}^{(i)}}}+\sum\limits_{e\in\mathfrak{E_{1}}}{{\left|{{{\mathbf{m}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\alpha_{m,e}^{(i)}}\\!\Big{)},\\\
\forall
m,k\in\\{{\mathcal{S}^{(i)}_{~{}\\!e,m}}\cup{\mathcal{S}^{(i)}_{~{}\\!u,m}}\\},i,\end{split}$
(25)

where ${\gamma_{m,k}^{(i)}}$ is the unified expression of
${\alpha_{m,k}^{(i)}}$ and ${\beta_{m,k}^{(i)}}$, ${\mathbf{m}}_{m,k}^{(i)}$
is the unified expression of ${\mathbf{v}}_{m,k}^{(i)}$ and
${\mathbf{w}}_{m,k}^{(i)}$, the user sets $\mathfrak{S_{1}}$ and
$\mathfrak{U_{1}}$ are respectively defined as

$\mathfrak{S_{1}}={\left\\{{{l\in{\mathcal{S}^{(i)}_{~{}\\!e,m}}\Big{|}|{{\mathbf{m}}_{m,l}^{(i)~{}H}{{\mathbf{h}}_{m,ml}^{(i)}}}|>|{{\mathbf{m}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}|}}\right\\}},$
(26)

and

$\mathfrak{U_{1}}={\left\\{{{u\in{\mathcal{S}^{(i)}_{~{}\\!u,m}}\Big{|}|{{\mathbf{m}}_{m,u}^{(i)~{}H}{{\mathbf{h}}_{m,mu}^{(i)}}}|>|{{\mathbf{m}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}|}}\right\\}}.$
(27)

After puncturing, URLLC and eMBB users in cluster $m$ need to be rearranged
based on the effective channel gains in a decreasing order for the
implementation of SIC decoding. For simplicity, we use the sorted order for
subscripts in each formula below. Hence, (25) can be rewritten as

$\begin{split}SINR_{m,k}^{e/u,(i)}=\frac{{{{\left|{{{\mathbf{m}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\gamma_{m,k}^{(i)}}}}{{\\!\sum\limits_{l=1}^{k-1}{{{\left|{{{\mathbf{m}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\gamma_{m,l}^{(i)}}\\!+\\!{{\|{{{{\mathbf{m}}_{m,k}^{(i)}}}}\|}^{2}}\frac{1}{\rho}}}},\\\
\forall
m,k\in\\{{\mathcal{S}^{(i)}_{~{}\\!e,m}}\cup{\mathcal{S}^{(i)}_{~{}\\!u,m}}\\},i.\end{split}$
(28)

The subscripts of ${\mathbf{m}}_{m,k}^{(i)}$, ${\mathbf{h}}_{m,mk}^{(i)}$, and
${\gamma_{m,k}^{(i)}}$ are all in order of decreasing
${\left|{{{\mathbf{m}}_{m,k,i}^{H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}$.
Thus, the data rate for URLLC users ${r_{m,k}^{(i)}},\forall
k\in\mathcal{S}^{(i)}_{~{}\\!u,m}$ and eMBB users ${R_{m,k}^{(i)}},\forall
k\in\mathcal{S}^{(i)}_{~{}\\!e,m}$ can be further converted based on (28).

Taking into account the similarity of the problems among different mini slots,
$\mathcal{P}_{1}$ can be rewritten as

$\displaystyle\mathcal{P}_{4}:\quad\max\limits_{\bm{\gamma}^{(i)}}\sum\limits_{m=1}^{M}\sum\limits_{k\in\mathcal{S}^{(i)}_{~{}\\!\\!e,m}}\log_{2}\left({1+SINR_{m,k}^{e/u,(i)}}\right),$
(29)
$\displaystyle\quad\,\mathrm{subject}~{}\mathrm{to}~{}\eqref{20a},\eqref{20b},$
$\displaystyle\qquad\quad{\gamma_{m,k}^{(i)}}>0,~{}\forall
m,k\in\\{{\mathcal{S}_{~{}\\!e,m}^{(i)}}\cup{\mathcal{S}_{~{}\\!u,m}^{(i)}}\\},i,$
(29a)
$\displaystyle\qquad\quad\sum\nolimits_{k\in\\{{\mathcal{S}_{~{}\\!\\!e,m}^{(i)}}\cup{\mathcal{S}_{~{}\\!\\!u,m}^{(i)}}\\}}\\!\\!{\gamma_{m,k}^{(i)}}\leq\frac{{L_{m}}}{K},~{}\forall
m,i,$ (29b)

where $\bm{\gamma}^{(i)}\in\mathbb{R}^{K\times 1}$ is the power allocation
coefficient collections for the URLLC and eMBB users.

Nevertheless, it is still hard to tackle $\mathcal{P}_{4}$ due to the non-
convexity of the objective function as well as constraints (22a) and (22b). In
what follows, we aim to solve $\mathcal{P}_{4}$ based on D.C. programming and
SCA method.

####  V-A1  D.C. programming for the objective function

The objective function (29) in $\mathcal{P}_{4}$ can be equivalently
transformed into a canonical form with the help of D.C. programming as [25]

$\underset{\bm{\gamma}^{(i)}}{\mathop{\mathrm{minimize}}\nolimits}\;\;{G_{1}\left({{\bm{\gamma}^{(i)}}}\right)-{G_{2}}\left({{\bm{\gamma}^{(i)}}}\right)},$
(30)

where $G_{1}\left({\bm{\gamma}}^{(i)}\right)$ and
$G_{2}\left({\bm{\gamma}}^{(i)}\right)$ are respectively given by

${G_{1}}\left({\bm{\gamma}}^{(i)}\right)=-\sum\limits_{m=1}^{M}{\sum\limits_{k\in{\mathcal{S}^{(i)}_{~{}\\!\\!e,m}}}{\log_{2}\left({D_{1}^{m,k}\left({\bm{\gamma}}^{(i)}\right)}\right)}},$
(31)

and

${G_{2}}\left({\bm{\gamma}}^{(i)}\right)=-\sum\limits_{m=1}^{M}{\sum\limits_{k\in{\mathcal{S}^{(i)}_{~{}\\!\\!e,m}}}{\log_{2}\left({D_{2}^{m,k}\left({\bm{\gamma}}^{(i)}\right)}\right)}},$
(32)

where $D_{1}^{m,k}\left({\bm{\gamma}}^{(i)}\right)$ and
$D_{2}^{m,k}\left({\bm{\gamma}}^{(i)}\right)$ are respectively given by

$D_{1}^{m,k}\left({\bm{\gamma}}^{(i)}\right)=\\!\sum\limits_{l=1}^{k}{{\left|{{{\mathbf{m}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\gamma_{m,l}^{(i)}}\\!+\\!{{\|{{{{\mathbf{m}}_{m,k}^{(i)}}}}\|}^{2}}\frac{1}{\rho},$
(33)

and

$D_{2}^{m,k}\left({\bm{\gamma}}^{(i)}\right)={{\\!\sum\limits_{l=1}^{k-1}{{{\left|{{{\mathbf{m}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\gamma_{m,l}^{(i)}}\\!+\\!{{\|{{{{\mathbf{m}}_{m,k}^{(i)}}}}\|}^{2}}\frac{1}{\rho}}}}.$
(34)

Note that $G_{1}\left({\bm{\gamma}}^{(i)}\right)$ and
$G_{2}\left({\bm{\gamma}}^{(i)}\right)$ are different convex functions with
respect to ${\bm{\gamma}}^{(i)}$. For any feasible solution
${\bm{\gamma}}^{(i),p}$ in the $p$-th iteration, we can obtain a lower bound
for $G_{2}\left({\bm{\gamma}}^{(i)}\right)$, which is given by

$G_{2}\left({\bm{\gamma}}^{(i)}\right)\\!\geq\\!G_{2}\left({\bm{\gamma}}^{(i),p}\right)\\!+\\!{\nabla_{{\bm{\gamma}}^{(i)}}}{G_{2}}{\left({\bm{\gamma}}^{(i),p}\right)^{\mathrm{T}}}\left({{\bm{\gamma}}^{(i)}\\!-\\!{\bm{\gamma}}^{(i),p}}\right),$
(35)

where
${\nabla_{{\bm{\gamma}}^{(i)}}}{G_{2}}\left({\bm{\gamma}}^{(i),p}\right)=\left\\{{{\frac{{\partial{G_{2}}\left({\bm{\gamma}}^{(i)}\right)}}{{\partial{\gamma_{m,k}^{(i)}}}}\left|{{{}_{{{\bm{\gamma}}^{(i),p}}}}}\right.}}\right\\}\in\mathbb{R}^{{K}\times
1}$ denotes the gradient of ${G_{2}}\left({\cdot}\right)$ with respect to
${\bm{\gamma}}^{(i)}$ and

$\frac{\partial{G_{2}\left({\bm{\gamma}}^{(i)}\right)}}{{\partial{\gamma_{m,k}^{(i)}}}}|{{{}_{{{\bm{\gamma}}^{(i),p}}}}}=-\frac{1}{\ln(2)}\sum\limits_{k^{\prime}\in\mathfrak{R_{2}}}{\frac{{{\left|{{{\mathbf{m}}_{m,k^{\prime}}^{(i)~{}H}{{\mathbf{h}}_{m,mk^{\prime}}^{(i)}}}}\right|}^{2}}}{D_{2}^{m,k^{\prime}}\left({\bm{\gamma}}^{(i),p}\right)}},$
(36)

where

$\mathfrak{R_{2}}\\!=\\!\left\\{\\!{{k^{\prime}\\!\in\\!{\mathcal{S}^{(i)}_{~{}\\!u,m}}\\!\cup\\!{\mathcal{S}^{(i)}_{~{}\\!e,m}}\Big{|}|{{\mathbf{m}}_{m,k^{\prime}}^{(i)~{}H}{{\mathbf{h}}_{m,mk^{\prime}}^{(i)}}}|\\!<\\!|{{\mathbf{m}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}|}}\right\\}.$
(37)

Based on (35), we can obtain an upper bound for the objective function (30) as

${G_{1}\\!\left({\bm{\gamma}}^{(i)}\right)\\!-\\!{G_{2}}\\!\left({\bm{\gamma}}^{(i)}\right)}\leq
G_{1}\\!\left({\bm{\gamma}}^{(i)}\right)\\!-\\!{G_{2}}\\!\left({\bm{\gamma}}^{(i),p}\right)\\!\\\
-\\!{\nabla_{{\bm{\gamma}}^{(i)}}}{G_{2}}\\!{\left({\bm{\gamma}}^{(i),p}\right)^{\mathrm{T}}}\left({{\bm{\gamma}}^{(i)}\\!-\\!{\bm{\gamma}}^{(i),p}}\right).$
(38)

Therefore, the objective function (30) is transferred to minimize the upper
bound of (38), which is a convex problem and can be solved by standard convex
tools.

####  V-A2  SCA for constraints (22a) and (22b)

Substituting (13) and (19), (22a) and (22b) can be respectively rewritten as

$\frac{F_{m,k}^{(i)}}{BD_{m,k}^{(i),\max}}\leq\log_{2}(1+SINR_{m,k}^{e/u,(i)})-\\\
\sqrt{C_{m,k}^{(i)}}\frac{Q^{-1}(\epsilon)\log
e}{\sqrt{b_{m,k}^{(i)}}},~{}\forall m,k\in\mathcal{S}_{~{}\\!u,m}^{(i)},i,$
(39)

and

${R_{m,k}^{\min}}\leq\log_{2}\left({1+SINR_{m,k}^{e/u,(i)}}\right),~{}\forall
m,k\in\mathcal{S}_{~{}\\!e,m}^{(i)},i,$ (40)

where $\sqrt{C_{m,k}^{(i)}}$ and
$\log_{2}\left({1+SINR_{m,k}^{e/u,(i)}}\right)$ can be respectively expressed
as

$\sqrt{C_{m,k}^{(i)}}={\Big{(}{{{\left|{{{\mathbf{m}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\gamma_{m,k}^{(i)}}}\Big{)}\\!^{1\\!/\\!2}\Big{(}{\\!2{{{\|{{{{\mathbf{m}}_{m,k}^{(i)}}}}\|}^{2}}\frac{1}{\rho}}}}\\\
{+\\!{{\left|{{{\mathbf{m}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\gamma_{m,k}^{(i)}}+\\!2\\!\sum\limits_{l=1}^{k-1}{{\left|{{{\mathbf{m}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\gamma_{m,l}^{(i)}}}\Big{)}\\!^{1\\!/\\!2}\\!\Big{/}\\\
\Big{(}{2\\!\sum\limits_{l=1}^{k-1}{{{\left|{{{\mathbf{m}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\gamma_{m,l}^{(i)}}\\!+\\!{{{\left|{{{\mathbf{m}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\gamma_{m,k}^{(i)}}}}}\Big{)},$
(41)

and

$\begin{split}&\log_{2}\left({1+SINR_{m,k}^{e/u,(i)}}\right)\\\
&=\log_{2}\left({{\\!\sum\limits_{l=1}^{k}{{{\left|{{{\mathbf{m}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\gamma_{m,l}^{(i)}}\\!+\\!{{\|{{{{\mathbf{m}}_{m,k}^{(i)}}}}\|}^{2}}\frac{1}{\rho}}}}\right)\\\
&~{}~{}~{}-\log_{2}\left({{\\!\sum\limits_{l=1}^{k-1}{{{\left|{{{\mathbf{m}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\gamma_{m,l}^{(i)}}\\!+\\!{{\|{{{{\mathbf{m}}_{m,k}^{(i)}}}}\|}^{2}}\frac{1}{\rho}}}}\right).\end{split}$
(42)

Intuitively, (40) is concave. By analyzing (14), we can verify that
$\sqrt{C_{m,k}^{(i)}}$ is concave with respect to $1+SINR_{m,k}^{e/u,(i)}>1$.
On this basis, we can employ the SCA mechanism to deal with the non-convex
constraints (22a) and (22b) [5]. Specifically, SCA performs a locally tight
approximation of the original problem at each iteration to produce a tight
convex constrain sets. According to [5], the approximation functions for both
(22a) and (22b) should hold the following conditions: 1) they are continuous,
consistent, and convex; 2) they are upper-bounds of the original functions.

In what follows, we would apply SCA to (39) and (40) to obtain their upper
bounds. Let’s define

$H_{1}^{m,k}({\bm{\gamma}^{(i)}})=\log_{2}\left({{\\!\sum\limits_{l=1}^{k-1}{{{\left|{{{\mathbf{m}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\gamma_{m,l}^{(i)}}\\!+\\!{{\|{{{{\mathbf{m}}_{m,k}^{(i)}}}}\|}^{2}}\frac{1}{\rho}}}}\right),\\\
\forall m,k\in\mathcal{S}_{~{}\\!u,m}^{(i)},i.$ (43)

Applying SCA to $H_{1}^{m,k}(\bm{\gamma}^{(i)})$, we have

$H_{1}^{m,k}\left(\bm{\gamma}^{(i)}\right)\\!\leq\\!H_{1}^{m,k}\left(\bm{\gamma}^{(i),p}\right)\\!+\\\
\\!{\nabla_{\bm{\gamma}^{(i)}}}{H}_{1}^{m,k}\\!{\left(\bm{\gamma}^{(i),p}\right)\\!^{\mathrm{T}}}\left(\bm{\gamma}^{(i)}-\bm{\gamma}^{(i),p}\right),$
(44)

where we denote the gradient of $H_{1}^{m,k}\left(\cdot\right)$ with respect
to ${\bm{\gamma}^{(i)}}$ as
${\nabla_{\bm{\gamma}^{(i)}}}{H}_{1}^{m,k}\left({{\bm{\gamma}}^{(i),p}}\right)\\!=\\!\left\\{{{\frac{{\partial{H}_{1}^{m,k}\left({{\bm{\gamma}^{(i)}}}\right)}}{{\partial{\gamma_{m^{\prime},k^{\prime}}^{(i)}}}}\left|{{{}_{{{\bm{\gamma}^{(i),p}}}}}}\right.}}\right\\}_{k^{\prime}\in\\{{\mathcal{S}_{~{}\\!\\!e,m^{\prime}}^{(i)}}\cup{\mathcal{S}_{~{}\\!\\!u,m^{\prime}}^{(i)}}\\}}^{m^{\prime}\in\mathcal{M}}$
and
$\frac{\partial{H_{1}^{m,k}\left({{\bm{\gamma}^{(i)}}}\right)}}{{\partial{\gamma_{m^{\prime},k^{\prime}}^{(i)}}}}|{{{}_{{{\bm{\gamma}^{(i),p}}}}}}$
is defined in (45).

$\displaystyle\frac{\partial{H_{1}^{m,k}\left({{\bm{\gamma}^{(i)}}}\right)}}{{\partial{\gamma_{m^{\prime},k^{\prime}}^{(i)}}}}|{{{}_{{{\bm{\gamma}^{(i),p}}}}}}=\left\\{\begin{array}[]{rcl}\frac{{{{\left|{{{\mathbf{m}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}}}{\ln(2)\left({{\\!\sum\limits_{l=1}^{k-1}{{{\left|{{{\mathbf{m}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\gamma_{m^{\prime},l}^{(i),p}}\\!+\\!{{\|{{{{\mathbf{m}}_{m,k}^{(i)}}}}\|}^{2}}\frac{1}{\rho}}}}\right)},&m^{\prime}=m,k^{\prime}=1,2,\cdots
k-1,\\\ 0,&\mathrm{otherwise}.\end{array}\right.$ (45)
$\displaystyle\frac{\partial{H_{2}^{m,k}\left({{\bm{\gamma}^{(i)}}}\right)}}{{\partial{\gamma_{m^{\prime},k^{\prime}}^{(i)}}}}|{{{}_{{{\bm{\gamma}^{(i),p}}}}}}=\left\\{\begin{array}[]{rcl}\frac{S_{1}{{{\left|{{{\mathbf{m}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}(S_{3}-2S_{2}^{2})}}{S_{2}S_{3}^{2}},&m^{\prime}=m,k^{\prime}=1,2,\cdots
k-1,\\\
\frac{{{\left|{{{\mathbf{m}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}S_{2}}{2S_{1}S_{3}}-\frac{S_{1}{{{\left|{{{\mathbf{m}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}(S_{3}-2S_{2}^{2})}}{2S_{2}S_{3}},&m^{\prime}=m,k^{\prime}=k,\\\
0,&\mathrm{otherwise}.\end{array}\right.$ (47)

Similarly, applying SCA to ${H_{2}^{m,k}}=\sqrt{C_{m,k}^{(i)}},\forall
m,k\in\mathcal{S}_{~{}\\!u,m}^{(i)},i$, we have

$H_{2}^{m,k}\\!\left({{\bm{\gamma}^{(i)}}}\right)\\!\leq\\!H_{2}^{m,k}\left({{\bm{\gamma}^{(i),p}}}\right)\\!+\\\
\\!{\nabla_{\bm{\gamma}^{(i)}}}{H}_{2}^{m,k}\\!{\left({{\bm{\gamma}^{(i),p}}}\right)\\!^{\mathrm{T}}}\\!\left({{\bm{\gamma}^{(i)}}\\!-\\!{\bm{\gamma}^{(i),p}}}\right),$
(46)

where we denote the gradient of ${H_{2}^{m,k}}\left({\cdot}\right)$ with
respect to ${\bm{\gamma}^{(i)}}$ as
${\nabla_{\bm{\gamma}^{(i)}}}{H}_{2}^{m,k}\left({{\bm{\gamma}^{(i),p}}}\right)=\left\\{{{\frac{{\partial{H}_{1}^{m,k}\left({{\bm{\gamma}^{(i)}}}\right)}}{{\partial{\gamma_{m^{\prime},k^{\prime}}^{(i)}}}}\\!\left|{{{}_{{{\bm{\gamma}^{(i),p}}}}}}\right.}}\right\\}_{k^{\prime}\in\\{{\mathcal{S}_{~{}\\!\\!e,m^{\prime}}^{(i)}}\cup{\mathcal{S}_{~{}\\!\\!u,m^{\prime}}^{(i)}}\\}}^{m^{\prime}\in\mathcal{M}}$
and
$\frac{\partial{H_{2}^{m,k}\left({{\bm{\gamma}^{(i)}}}\right)}}{{\partial{\gamma_{m^{\prime},k^{\prime}}^{(i)}}}}\\!|\\!{{{}_{{{\bm{\gamma}^{(i),p}}}}}}$
is defined in (47). Here, ${S_{1}}$, ${S_{2}}$, and ${S_{3}}$ are respectively
given by

${S_{1}}=\Big{(}{{{\left|{{{\mathbf{m}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\gamma_{m,k}^{(i),p}}}\Big{)}\\!^{1\\!/\\!2},$
(48)
${S_{2}}={\Big{(}{\sum\limits_{l=1}^{k-1}{{\left|{{{\mathbf{m}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\gamma_{m,l}^{(i),p}}\\!+\\!2{{{\|{{{{\mathbf{m}}_{m,k}^{(i)}}}}\|}^{2}}\frac{1}{\rho}}}}\\\
+{{{{\left|{{{\mathbf{m}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\gamma_{m,k}^{(i),p}}}}\Big{)}\\!^{1\\!/\\!2},$
(49)
${S_{3}}={2\\!\sum\limits_{l=1}^{k-1}{{{\left|{{{\mathbf{m}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\gamma_{m,l}^{(i),p}}\\!+\\!{{{\left|\\!{{{\mathbf{m}}_{m,k}^{(i)~{}H}\\!{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\gamma_{m,k}^{(i),p}}}}}.$
(50)

Therefore, constraints (22a) at the $p$-th iteration can be approximated as

$\frac{F_{m,k}^{(i)}}{BD_{m,k}^{(i),\max}}\leq\log_{2}\left({{\\!\sum\limits_{l=1}^{k}{{{\left|{{{\mathbf{m}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\gamma_{m,l}^{(i)}}\\!+\\!{{\|{{{{\mathbf{m}}_{m,k}^{(i)}}}}\|}^{2}}\frac{1}{\rho}}}}\right)\\\
-{\frac{Q^{-1}(\epsilon)\log
e}{\sqrt{b_{m,k}^{(i)}}}\Big{(}{H_{2}^{m,k}\left({\bm{\gamma}}^{(i),p}\right)+{\nabla_{{\bm{\gamma}}^{(i)}}}{H}_{2}^{m,k}{\left({\bm{\gamma}}^{(i),p}\right)^{\mathrm{T}}}\cdot}}\\\
\left({{\bm{\gamma}}^{(i)}-{\bm{\gamma}}^{(i),p}}\right)\Big{)}-{\nabla_{{\bm{\gamma}}^{(i)}}}{H}_{1}^{m,k}{\left({\bm{\gamma}}^{(i),p}\right)^{\mathrm{T}}}\left({{\bm{\gamma}}^{(i)}-{\bm{\gamma}}^{(i),p}}\right)\\\
\quad\quad\quad\quad\quad\quad\quad\quad-
H_{1}^{m,k}\left({\bm{\gamma}}^{(i),p}\right),~{}\forall
m,k\in\mathcal{S}_{~{}\\!u,m}^{(i)},i,$ (51)

and constraints (22b) at the $p$-th iteration can be approximated as

${R_{m,k}^{\min}}\leq\log_{2}\left({{\\!\sum\limits_{l=1}^{k}{{{\left|{{{\mathbf{m}}_{m,k}^{(i)~{}H}{{\mathbf{h}}_{m,mk}^{(i)}}}}\right|}^{2}}{\gamma_{m,l}^{(i)}}\\!+\\!{{\|{{{{\mathbf{m}}_{m,k}^{(i)}}}}\|}^{2}}\frac{1}{\rho}}}}\right)\\\
-H_{1}^{m,k}\left({{\bm{\gamma}^{(i),p}}}\right)-{\nabla_{\bm{\gamma}^{i}}}{H}_{1}^{m,k}{\left({{\bm{\gamma}^{(i),p}}}\right)^{\mathrm{T}}}\left({{\bm{\gamma}^{(i)}}-{\bm{\gamma}^{(i),p}}}\right),\\\
\forall m,k\in\mathcal{S}_{~{}\\!e,m}^{(i)},i.$ (52)

In this way, we can transform $\mathcal{P}_{4}$ as

$\mathcal{P}_{5}:\quad\underset{\bm{\gamma}^{(i)}}{\mathop{\mathrm{minimize}}\nolimits}\;\;G_{1}\left({{\bm{\gamma}^{(i)}}}\right)-{G_{2}}\left({{\bm{\gamma}^{(i),p}}}\right)\\\
-{\nabla_{\bm{\gamma}^{(i)}}}{G_{2}}{\left({{\bm{\gamma}^{(i),p}}}\right)^{\mathrm{T}}}\left({{\bm{\gamma}^{(i)}}-{\bm{\gamma}^{(i),p}}}\right),$
(53)

subject to (29a), (29b), (51), and (52).

Obviously, $\mathcal{P}_{5}$ is a convex problem. An iteration algorithm is
developed to solve $\mathcal{P}_{5}$, which is shown in Algorithm 2.
Specifically, in the $p$-th iteration, the updated solution
${\bm{\gamma}}^{(i),p}$ is obtained by solving $\mathcal{P}_{5}$ with
${\bm{\gamma}}^{(i),p-1}$. The algorithm will terminate when the change of
optimal objective function value becomes smaller than a given convergence
tolerance, i.e.,
$|W^{p}\left({{\bm{\gamma}^{(i)}}}\right)-W^{p-1}\left({{\bm{\gamma}^{(i)}}}\right)|\leq\epsilon$.
Here, $\epsilon>0$ is a small constant, and

$W^{p}\left({{\bm{\gamma}^{(i)}}}\right)=G_{1}\\!\left({{\bm{\gamma}^{(i)}}}\right)\\!-\\!{G_{2}}\\!\left({{\bm{\gamma}^{(i),p}}}\right)\\!-\\\
\\!{\nabla_{\bm{\gamma}^{(i)}}}{G_{2}}\\!{\left({{\bm{\gamma}^{(i),p}}}\right)^{\mathrm{T}}}\left({{\bm{\gamma}^{(i)}}\\!-\\!{\bm{\gamma}^{(i),p}}}\right).$
(54)

Algorithm 2 Joint URLLC and eMBB scheduling algorithm.

1: Initialize the convergence tolerance $\epsilon$.

2: Set the mini slot index $i=1$.

3: Do

4: Define $\mathcal{S}^{(i)}_{~{}\\!u}$ as the set of arrival URLLC users at
mini slot $i$.

5: The selection of the punctured eMBB users for arriving URLLC users is
determined by Algorithm 3.

6: Set the iteration index $p=1$.

7: Initialize the feasible solution ${\bm{\gamma}}^{(i),1}$.

8: While
$|W^{p}\left({{\bm{\gamma}^{(i)}}}\right)-W^{p-1}\left({{\bm{\gamma}^{(i)}}}\right)|\geq\epsilon$
do

9: Solving problem $\mathcal{P}_{5}$ under given ${\bm{\gamma}}^{(i),p-1}$ to
obtain the power allocation ${\bm{\gamma}}^{(i),p}$.

10: Set $p=p+1$.

11: end While

12: Return the solution ${\bm{\gamma}}^{(i),*}={\bm{\gamma}}^{(i),p}$.

13: Set $i=i+1$.

14: Until $i=Q+1$.

Since $\mathcal{P}_{5}$ is a convex formulation with linear matrix inequality
(LMI) and second order cone (SOC) constraints, the computational complexity of
Algorithm 2 consists of two parts [26]:

  1. 1.

_Iteration Complexity_ : The number of iterations required to reach
$|W^{p}\left({{\bm{\gamma}^{(i)}}}\right)-W^{p-1}\left({{\bm{\gamma}^{(i)}}}\right)|<\epsilon$
is on the order of $\sqrt{\beta}$, where $\beta=\sum_{j=1}^{J}k^{L}_{j}+2m$,
$J$ is the number of LMI constraints, $k^{L}_{j}$ means the dimension of the
$j$-th LMI constraint, and $m$ denotes the number of SOC constraints. Here, we
can find that $J=K+M$, $k^{L}_{j}=1,\forall j$, and $m=K$, i.e.,
$\sqrt{\beta}=\sqrt{3K+M}$.

  2. 2.

_Per-Iteration Complexity Cost_ : In each iteration, the computation cost is
dominated by: (a) the formation of the coefficient matrix, which can be
formulated as
$C_{form}=n\sum_{j=1}^{J}(k_{j}^{L})^{3}+n^{2}\sum_{j=1}^{J}(k_{j}^{L})^{2}+n\sum_{j=1}^{m}(k_{j}^{S})^{2}$,
where $n$ is the total number of optimization parameters, $k^{S}_{j}$ is the
dimension of the $j$-th SOC constraint; (b) the factorization of the
coefficient matrix, which is given by $C_{fact}=n^{3}$. For Algorithm 2, we
can find that $n=K$ and $k^{S}_{j}=N+1,\forall j$. Hence, the per-iteration
complexity cost is $C_{form}+C_{fact}=K(K+M)+K^{2}(K+M)+K^{2}(N+1)^{2}+K^{3}$.

By combining the above two parts, the computational complexity of Algorithm 2
is on the order of $\sqrt{\beta}\left(C_{form}+C_{fact}\right)$, which is
derived as
$\mathcal{O}\left(\sqrt{3K+M}\left(2K^{3}+K^{2}\left(N^{2}+2N+M+2\right)+KM\right)\right)$.
Moreover, the convergence of the proposed algorithm is proved in Appendix A.

###  V-B The Punctured User Selection Problem

In this subsection, we investigate the punctured user selection problem. The
user selection is represented by a binary variable $x_{m,k,n}^{(i)}$, where
$x_{m,k,n}^{(i)}=1$ represents eMBB user $k$ in the $m$-th cluster is
punctured by URLLC user $n$, and $x_{m,k,n}^{(i)}=0$ otherwise. Matching
theory [27] provides a framework to tackle the above problem through pairing
URLLC and eMBB players in two distinct sets based on each player’s individual
preference. Similar with [7], we apply the Gale-Shapely (GS) algorithm with
incomplete preference list here to match URLLC and eMBB pairs.

The GS algorithm can be used to find a stable one-to-one matching between men
and women, which has been respectively regarded as URLLC and eMBB users in our
work. The basic idea of GS algorithm is that one gender have a sequence of
“proposals” to the other sex (i.e. women). When a woman receives a proposal,
she rejects if she already holds a better proposal, and otherwise agrees to
hold it for consideration. In our matching model, the punctured user selection
problem occurs in each mini slot. At mini slot $i$ if BS allocates resources
to URLLC $u_{n}\in\mathcal{S}^{(i)}_{~{}\\!u}$ that is already hold by eMBB
user $e_{mk}\in{\mathcal{L}}$, then URLLC user $u_{n}$ is said to be matched
with eMBB user $e_{mk}$ and form a matching pair $(u_{n},e_{mk})$. Therefore,
in this work, the matching is an assignment of URLLC users in
$\mathcal{S}^{(i)}_{~{}\\!u}$ to eMBB users ${\mathcal{L}}$, which can be
defined as follows [28].

######  Definition 1.

A matching $\Omega$ between $\mathcal{S}^{(i)}_{~{}\\!u}$ and ${\mathcal{L}}$
is a mapping from the set $\mathcal{S}^{(i)}_{~{}\\!u}\cup{\mathcal{L}}$ to
the set of all subsets of $\mathcal{S}^{(i)}_{~{}\\!u}\cup{\mathcal{L}}$ such
that for $\forall u_{n}\in\mathcal{S}^{(i)}_{~{}\\!u}$ and $\forall
e_{mk}\in{\mathcal{L}}$: (i) $\Omega(u_{n})\in{\mathcal{L}}$ and
$\Omega(e_{mk})\in\mathcal{S}^{(i)}_{~{}\\!u}$, (ii)
$\left|{\Omega(u_{n})}\right|=1$, (iii) $\left|{\Omega(e_{mk})}\right|\leq 1$.

Definition 1 illustrates that an URLLC user $u_{n}$ must be matched with one
eMBB user from ${\mathcal{L}}$, while one eMBB user $e_{mk}$ can only be
matched at most one URLLC user from $\mathcal{S}^{(i)}_{~{}\\!u}$ at any mini
slot.

The matching is performed based on the players’ preference lists. The
preference list for eMBB users is defined based on a proposed concept of
$only$ $one$ $URLLC$ $rate$ $(OOUR)$ of a certain URLLC user $u_{n}$. $OOUR$
is the throughput if the BS only accept one URLLC user $u_{n}$, which
punctures the eMBB user $k$ in the $m$-th cluster $e_{mk}$ and reject all
other URLLC users at mini slot $i$, denoted as $R_{n}^{mk}$. $R_{n}^{mk}$ can
be obtained by solving problem $\mathcal{P}_{5}$ under a given matching
matrix. Thus, we define the preference relationship for eMBB user $e_{mk}$
between URLLC user $u_{n}$ and $u_{n^{\prime}}$ in Definition 2.

######  Definition 2.

At mini slot $i$, eMBB user $e_{mk}$ prefers to choose URLLC user $u_{n}$ than
URLLC user $u_{n^{\prime}}$, if $R_{n}^{mk}>R_{n^{\prime}}^{mk}$, denoted by
$u_{n}\succ_{e_{mk}}{u_{n^{\prime}}}$, for
$u_{n}\in\mathcal{S}^{(i)}_{~{}\\!u}$,
$u_{n}^{\prime}\in\mathcal{S}^{(i)}_{~{}\\!u}$, $e_{mk}\in\mathcal{L}$.

Algorithm 3 The GS algorithm for URLLC and eMBB users.

1: Set up eMBB users’ preference lists as $\mathcal{PL}_{mk}^{e},\forall
e_{mk}\in{\mathcal{L}}$ based on Definition 2.

2: Set up URLLC users’ preference lists as $\mathcal{PL}_{n}^{u},\forall
u_{n}\in\mathcal{S}^{(i)}_{~{}\\!u}$ based on Definition 3 and Definition 4.

3: Define $\mathcal{M}_{c}=\mathcal{S}^{(i)}_{~{}\\!u}$ as the set of
unmatched URLLC users at mini slot $i$.

4: While $\mathcal{M}_{c}$ is nonempty do

5: URLLC user $u_{n}$ selects the eMBB user $e_{mk}$ that locates first in its
preference list $\mathcal{PL}_{n}^{u}$.

6: If $e_{mk}$ has received a proposal $u_{n^{\prime}}$ in its preference
list, then

7: If $e_{mk}$ prefers $u_{n^{\prime}}$, then

8: $e_{mk}$ holds $u_{n^{\prime}}$ and rejects $u_{n}$.

9: Remove $u_{n^{\prime}}$ from $\mathcal{M}_{c}$.

10: Add $u_{n}$ into $\mathcal{M}_{c}$.

11: else

12: $e_{mk}$ rejects $u_{n^{\prime}}$ and accept $u_{n}$.

13: Remove $u_{n}$ from $\mathcal{M}_{c}$.

14: Add $u_{n}^{\prime}$ into $\mathcal{M}_{c}$.

15: end If

16: else

17: $e_{mk}$ accept $u_{n}$.

18: Remove $u_{n}$ from $\mathcal{M}_{c}$.

19: end If

20: End while

When considering URLLC users’ preference list, we similarly take $R_{n}^{mk}$
as an evaluation indicator. To satisfy the QoS requirements of both URLLC and
eMBB users, we introduce $\overline{x}_{mk}^{(i)},\forall m,k,i$ as another
measure indicator to represent the number of mini slots that eMBB user
$e_{mk}$ punctured by URLLC users from mini slot $1$ to mini slot $(i-1)$.
Thereafter, we define the preference relationship between URLLC user $u_{n}$
to eMBB user $e_{mk}$ and $e_{m^{\prime}k^{\prime}}$ in Definition 3 and
Definition 4.

######  Definition 3.

At mini slot $i$, URLLC user $u_{n}$ prefers to replace eMBB user $e_{mk}$
than eMBB user $e_{m^{\prime}k^{\prime}}$, if
$\overline{x}_{mk}^{(i)}<\overline{x}_{m^{\prime}k^{\prime}}^{(i)}$, denoted
by $e_{mk}\succ_{u_{n}}{e_{m^{\prime}k^{\prime}}}$, for
$u_{n}\in\mathcal{S}^{(i)}_{~{}\\!u}$, $e_{mk}\in\mathcal{L}$,
$e_{m^{\prime}k^{\prime}}\in\mathcal{L}$.

######  Definition 4.

At mini slot $i$, URLLC user $u_{n}$ prefers to replace eMBB user $e_{mk}$
than eMBB user $e_{m^{\prime}k^{\prime}}$, if
$\overline{x}_{mk}^{(i)}=\overline{x}_{m^{\prime}k^{\prime}}^{(i)}$ and
$R_{n}^{mk}>R_{n}^{m^{\prime}k^{\prime}}$, denoted by
$e_{mk}\succ_{u_{n}}{e_{m^{\prime}k^{\prime}}}$, for
$u_{n}\in\mathcal{S}^{(i)}_{~{}\\!u}$, $e_{mk}\in\mathcal{L}$,
$e_{m^{\prime}k^{\prime}}\in\mathcal{L}$.

The preference list of eMBB user $e_{mk}\in\mathcal{L}$ over URLLC users
$u_{n}\in\mathcal{S}^{(i)}_{~{}\\!u}$, denoted by $\mathcal{PL}_{mk}^{e}$, is
ranked by Definition 2 in a descending order. Similarly URLLC user $u_{n}$’s
preference list $\mathcal{PL}_{n}^{c}$ over $e_{mk}$ is ranked by Definition 3
and Definition 4 in a descending order. Now we define stable matching in
Definition 5.

######  Definition 5.

A matching $\Omega$ is stable, if there exists no blocking pair
$(u_{n},e_{mk})$, such that $e_{mk}\succ_{u_{n}}{\Omega(u_{n})}$ and
$u_{n}\succ_{e_{mk}}{\Omega(e_{mk})}$, where ${\Omega(u_{n})}$ represents
URLLC user $u_{n}$’s partner in $\Omega$ and ${\Omega(e_{mk})}$ represents
eMBB user $e_{mk}$’s partner in $\Omega$.

Algorithm 3 presents how to use the GS algorithm to find a stable matching
between URLLC and eMBB users, which has the complexity of
${\rm\mathcal{O}}\left({\left|{{\mathcal{S}^{(i)}_{~{}\\!u}}}\right|}^{2}\right)$.

As shown in Algorithm 3, the exchange operations occur only if the utilities
of URLLC and eMBB users are strictly improved. After searching all the
possible swaps, the matching stage terminates, which means there has no swap
operation to further improve the utilities of URLLC and eMBB users. Therefore,
Algorithm 3 is also stable.

##  VI Simulation Results

Figure 2: Performance comparison under different punctured schemes.

In this section, we would evaluate the performance of the proposed URLLC and
eMBB coexistence network. We consider $9$ eMBB users and $3$ clusters in the
system. For simplicity, we let $4$, $3$, and $2$ eMBB users respectively exist
in each cluster at the beginning of the transmission period. In each
transmission period, there are $8$ mini slots. Furthermore, we let either the
BS, the eMBB user, or the URLLC user equips $M=3$ antennas. In addition, in
this work, we consider rayleigh fading channels. Each URLLC user’s packet
length $F_{m,n}^{(i)}$ and blocklength $b_{m,n}^{(i)}$ are set as $500$ bytes
and $168$, respectively [5]. The reliability requirements of URLLC users are
set as $99.999\%$. The tolerance $\delta$ and $\epsilon$ are set as the same,
i.e., $10^{-6}$. We assume the QoS requirements of eMBB users are the same,
which is set as $R_{m,k}^{\min}=1$ bit/s/Hz. The delay requirements of URLLC
users are the same, which is set as $D_{m,n}^{(i),\max}=0.7$ ms. In addition,
the transmit SNR $\rho$ is set as $30$ dB. For simplicity, we further assume
there has at most one URLLC user arrives at a mini slot.

In Fig. 2, we present the punctured eMBB performance under different user
selection algorithms, i.e., random searching method, GS algorithm, Hungarian
algorithm, and exhaustive searching method. Due to the computational
complexity of the exhaustive searching method, we consider either the BS or
the eMBB user has $3$ antennas. Moreover, we assume there are $1$, $3$, $4$,
$2$, and $1$ URLLC users arriving in mini slot $1$, $2$, $4$, $6$, and $8$,
respectively. From this figure, the performance gap between GS algorithm and
exhaustive searching method would be less than $10\%$. Moreover, compared with
the random searching method, the GS algorithm has a significant performance
improvement. Since the number of punctured eMBB users increases with the
number of arrival URLLC users, the performance gaps among the proposed $3$
schemes are the most apparent in STS4. The performance of the Hungarian
algorithm is similar to the GS algorithm. However, the stability of the
Hungarian algorithm is much weaker than the GS algorithm. Therefore, the
proposed GS algorithm with acceptable performance and computational complexity
is suitable in this work.

Figure 3: Performance comparison under different QoS requirements.

Fig. 3 shows the effect of the URLLC and eMBB users’ QoS requirements on the
eMBB throughput performance in the whole transmission period. In this
simulation, we let two URLLC users arrive to the system. From this figure, we
can find that the total eMBB throughput increases with the URLLC delay
requirements. This is because the more tolerant of the delay requirements, the
lower power consumptions, and thus the BS can use the excessive power to
improve the data rate of eMBB users. On the other hand, under fixed URLLC
delay requirements, the performance of all eMBB users increase with the
increment of eMBB users’ data rate requirements.

Figure 4: Performance comparison among different number of punctured URLLC
users.

Fig. 4 presents the total eMBB users’ throughput and each eMBB user’s
throughput in a transmission period under different number of arrival URLLC
users. The MIMO-OMA scheme is selected as a benchmark. From this figure, the
MIMO-OMA scheme provides the highest per-user eMBB throughput, but the lowest
total eMBB throughput. This is because the MIMO-OMA scheme would connect fewer
users than the MIMO-NOMA scheme to eliminate interference, and the MIMO-NOMA
scheme has higher spectrum utilization. Furthermore, the total eMBB
performance decreases with the increased arrival URLLC users, since more URLLC
users would preempt more eMBB resources. Under the fixed number of arrival
URLLC users, the total eMBB performance increases with $K$ due to the provided
multiuser diversity gain.

Figure 5: Performance comparison among different power allocation schemes.

In Fig. 5, we compare the proposed MIMO-NOMA based adaptive power allocation
scheme with four baseline schemes, namely the equal power allocation scheme
and the fixed power allocation scheme in [29], the best channel condition
(BCC) power allocation scheme in [30], and the MIMO-OMA based adaptive power
allocation scheme in [31]. Note that the fixed power allocation scheme
distributes ${\frac{k}{\sum_{k\in\mathcal{L}_{m}}{k}}}\frac{L_{m}}{K}$ to the
$k$-th user in $m$-th cluster to due with the SIC. In the BCC power allocation
scheme, except for the user with the best channel condition, the achievable
transmission rates for all the other users in the same cluster are set to
satisfy the minimum QoS requirement. Here, we let $R_{m,k}^{\min}=0.2$
bit/s/Hz. From this figure, we can find that the proposed adaptive power
control scheme has the best performance. This is because the adaptive power
control scheme can adjust the power allocation scheme based on the channel
conditions and system parameters. As the SNR increases, the equal power
control scheme, the BCC power allocation scheme, and the proposed scheme
almost have the same performance, however, the fixed allocation scheme has a
clear performance decreasing. It is due to the equal power control scheme and
the BCC power allocation scheme force the user with the largest channel gain
to achieve the relatively high power compared to the proposed scheme, which
can improve the spectral efficiency. However, both the equal power control
scheme and the BCC power allocation scheme can not guarantee users’ fairness
well, which is shown in Fig. 6. The fixed power allocation scheme has a worse
performance than the equal power allocation scheme, since the fixed allocation
scheme allocates more power to the user with a weak channel gain, resulting in
a lower power utilization efficiency. Hence, it makes sense to select a proper
non-orthogonal power optimization scheme according to the service
characteristics in MIMO-NOMA systems. Moreover, it is also intuitively that
MIMO-NOMA can significantly outperform MIMO-OMA under reasonably power
allocation schemes.

Figure 6: User fairness index under different number of eMBB users.

Fig. 6 shows the user fairness under different number of eMBB users. We
compare the proposed adaptive power allocation scheme with the equal power
allocation scheme, the fixed power allocation scheme, and the BCC power
allocation scheme. According to [32][33], Jain’s fairness index is obtained by
$\left(\sum_{m\in\mathcal{M}}\sum_{k\in\mathcal{L}_{m}}R_{m,k}^{(i)}\right)^{2}/{\left|{\mathcal{S}^{(i)}_{~{}\\!e}}\right|}\left(\sum_{m\in\mathcal{M}}\sum_{k\in\mathcal{L}_{m}}{R_{m,k}^{(i)~{}2}}\right)$,
where $R_{m,k}^{(i)}$ is the achievable data rate of eMBB user $k$ in the
$m$-th cluster at mini slot $i$. With the increment of eMBB users, the
fairness index among them decreases due to the competition. The fixed power
allocation scheme achieves higher user fairness than the proposed adaptive
power allocation, since the proposed adaptive power allocation improve the
spectrum utilization at the cost of fairness. By combining Fig. 5 and Fig. 6,
we can conclude that the proposed adaptive power allocation scheme can
significantly improve the performance at an acceptable fairness level.

Figure 7: Performance comparison under different number of antennas and
transmission power.

Fig. 7 shows the system data rate under different transmission power and user
antennas. From this figure, we can observe that the total eMBB throughput in a
transmission period of our proposed system is improved with the increment of
transmit power. Moreover, the total data rate increases with the increment of
user antennas, since the dimension of the null space $\mathbf{U}_{l,k}^{(i)}$,
defined in (6), increases and the ability of detection vector to eliminate
interference improves.

Figure 8: Performance comparison among different URLLC and eMBB coexistence
mechanisms.

Fig. 8 illustrates the performance comparison among different URLLC and eMBB
coexistence mechanisms. The linear model for eMBB throughput loss under
superposition mechanism in [6] (LM-SM) and the OMA based spatial coexistence
mechanism in [7] are selected as two benchmarks. In the LM-SM scheme, we adopt
the superposition mechanism for URLLC scheduling and leverage the GS method
proposed in our paper to perform the user selection in the clustered MIMO-NOMA
system. Moreover, the rate loss of the superimposed eMBB users is calculated
through the linear model proposed in [6]. In the coexistence scheme in [7],
each eMBB user occupies one spatial resource. Any arriving URLLC user can
select an eMBB user to replace and occupy the entire spatial dimension under
the puncturing mechanism. From this figure,the proposed coexistence scheme
performs best. Moreover, the LM-SM scheme has better performance than the
coexistence scheme in [7]. The reason is that the resource utilization in [7]
is insufficient. Moreover, QoS requirements of both URLLC and eMBB users are
ignored in the user scheduling program in [7].

##  VII Conclusion

This paper proposes a cluster-based MIMO-NOMA technology to the URLLC and eMBB
coexistence network. Before transmission, we propose a dynamic eMBB user
clustering mechanism in MIMO-NOMA systems to strike a balance between the
system performance and the computational complexity. Thereafter, we adopt a
punctured scheduling scheme, which allows arriving URLLC users interrupt parts
of associated eMBB users. An eligible and optimal interrupted eMBB users can
be found through a GS matching algorithm with low computational complexity.
For further spectrum utility improvement, we propose an iterative power
allocation algorithm based on the SCA and D.C. programming. Our numerical
results demonstrate a notable performance improvement of the proposed
mechanism over the baseline methods in the presented indicators.

##  Appendix A Convergence proof of Algorithm 2

We first introduce the following stationary point definition according to
[34].

######  Definition 6.

Let $f:\mathcal{D}\rightarrow\mathbb{R}$ be a function where
$\mathcal{D}\subseteq\mathbb{R}^{m}$ is a convex set. The point $x$ is a
stationary point of $f(\cdot)$ if $f^{\prime}(x;d)\geq 0$ for all $d$ such
that $x+d\in\mathcal{D}$. Here, $d$ is the distance of a point from a set.

Thereafter, the conditions to setup the approximation functions to locally
approximate the non-convex constraints can be summarized as the following
assumption [5].

######  Assumption 1.

A function $u(x,y)$ is called as the _approximation function_ for the non-
convex function $f(x)$, when the following conditions satisfy:

$u(x,x)=f(x),\forall x,$ (55) $u(x,y)\geq f(x),\forall x,y,$ (56)
$\frac{\partial u(x,y)}{\partial x}|_{x=y}=\triangledown f(x)|_{x=y},\forall
x,$ (57) $u(x,y)~{}\mathrm{is~{}continuous~{}in}~{}(x,y),$ (58)
$u(x,y)~{}\mathrm{is~{}convex~{}in}~{}x.$ (59)

Define the non-convex function of $\mathcal{P}_{4}$ as
$f(\bm{\gamma}^{(i)})={G_{1}\left({{\bm{\gamma}^{(i)}}}\right)-{G_{2}}\left({{\bm{\gamma}^{(i)}}}\right)}$
and the approximation function of $\mathcal{P}_{5}$ as
$u(\bm{\gamma}^{(i)},\bm{\gamma}^{(i),p})=G_{1}\\!\left({{\bm{\gamma}^{(i)}}}\right)\\!-\\!{G_{2}}\\!\left({{\bm{\gamma}^{(i),p}}}\right)\\!-\\!{\nabla_{\bm{\gamma}^{(i)}}}{G_{2}}\\!{\left({{\bm{\gamma}^{(i),p}}}\right)^{\mathrm{T}}}\left({{\bm{\gamma}^{(i)}}\\!-\\!{\bm{\gamma}^{(i),p}}}\right)$.

We can observe the following series of relationships as

$f({\bm{\gamma}^{(i),p+1}})\overset{(a)}{\leq}u(\bm{\gamma}^{(i),p+1},\bm{\gamma}^{(i),p})\\\
\overset{(b)}{\leq}u({\bm{\gamma}^{(i),p}},\bm{\gamma}^{(i),p})\overset{(c)}{=}f({\bm{\gamma}^{(i),p}}),$
(60)

where $(a)$ is due to (56), $(b)$ follows from the optimality of
$\bm{\gamma}^{(i),p+1}=\arg\underset{\bm{\gamma}^{(i)}}{\min}~{}u({\bm{\gamma}^{(i)}},{\bm{\gamma}^{(i),p}})$,
and $(c)$ is due to (55).

A straightforward consequence of (60) is that the sequence of $P_{4}$ values
are non-increasing at each iteration, which can be formulated as

$f({\bm{\gamma}^{(i),0}})\geq f({\bm{\gamma}^{(i),1}})\geq
f({\bm{\gamma}^{(i),2}})\geq\cdots.$ (61)

Assume that there exists a subsequence
$\left\\{\bm{\gamma}^{(i),p_{j}}\right\\}$ converging to a limit point $z$.
Combing (55), (56), and (61), we can derive that

$u({\bm{\gamma}^{(i),p_{j+1}}},{\bm{\gamma}^{(i),p_{j+1}}})=f({\bm{\gamma}^{(i),p_{j+1}}})\leq
f({\bm{\gamma}^{(i),p_{j}+1}})\\\ \leq
u({\bm{\gamma}^{(i),p_{j}+1}},{\bm{\gamma}^{(i),p_{j}}})\leq
u({\bm{\gamma}^{(i)}},{\bm{\gamma}^{(i),p_{j}}}).$ (62)

Letting $j\rightarrow\infty$, we can obtain from (62) that

$u(z,z)\leq u({\bm{\gamma}^{(i)}},z),$ (63)

which implies that

$u^{\prime}({\bm{\gamma}^{(i)}},z;d)|_{{\bm{\gamma}^{(i)}}=z}\geq 0,~{}\forall
d\in\mathbb{R}^{m}.$ (64)

With the assumption of (57), we have

$f^{\prime}(z;d)\geq 0,~{}\forall d\in\mathbb{R}^{m},$ (65)

which implies that $z$ is a stationary point of $f(\cdot)$. It ends the proof.

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