#  RAN Slicing in Multi-MVNO Environment under Dynamic Channel Conditions

Darshan A. Ravi , Vijay K. Shah, Chengzhang Li, Tom Hou, and Jeffrey H. Reed
Bradley Department of ECE, Virginia Tech, Blacksburg, VA, 24061
Emails:{darshan19, vijays, licz17, hou<EMAIL_ADDRESS>This work has been
partially supported by DARPA HR0011-19-C-0096.

###### Abstract

With the increasing diversity in the requirement of wireless services with
guaranteed quality of service(QoS), radio access network(RAN) slicing becomes
an important aspect in implementation of next generation wireless systems(5G).
RAN slicing involves division of network resources into many logical segments
where each segment has specific QoS and can serve users of mobile virtual
network operator(MVNO) with these requirements. This allows the Network
Operator(NO) to provide service to multiple MVNOs each with different service
requirements. Efficient allocation of the available resources to slices
becomes vital in determining number of users and therefore, number of MVNOs
that a NO can support. In this work, we study the problem of Modulation and
Coding Scheme(MCS) aware RAN slicing(MaRS) in the context of a wireless system
having MVNOs which have users with minimum data rate requirement. Channel
Quality Indicator(CQI) report sent from each user in the network determines
the MCS selected, which in turn determines the achievable data rate. But the
channel conditions might not remain the same for the entire duration of user
being served. For this reason, we consider the channel conditions to be
dynamic where the choice of MCS level varies at each time instant. We model
the MaRS problem as a Non-Linear Programming problem and show that it is NP-
Hard. Next, we propose a solution based on greedy algorithm paradigm. We then
develop an upper performance bound for this problem and finally evaluate the
performance of proposed solution by comparing against the upper bound under
various channel and network configurations.

###### Index Terms:

5G and Beyond Networks, RAN Slicing, Dynamic Channel Conditions, Performance
Bound

##  I Introduction

With the advent of Internet of Things (IoT), number of devices accessing the
internet has been increasing exponentially. Ericsson has estimated that about
5 billion IoT devices will be connected to the internet and about 2.6 billion
5G subscriptions by the end of 2025 [1]. Efficient utilization of available
spectrum resources becomes vital to accommodate this growth. Adding to this
requirement, is the complexity of users having varied QoS requirements.

To address this complexity, Radio Access Network (RAN) slicing technology has
been widely adopted by several industrial communities [2, 3]. With the help of
RAN slicing, operators can perform service customization, isolation and multi-
tenancy support on a common physical network infrastructure by enabling
logical as well as physical separation of network resources [4]. This multi-
tenancy support enables network operators (NO) to support multiple mobile
virtual network operators (MVNOs) in the form of a slice. The Third-Generation
Partnership Project (3GPP) has identified network slicing as one of the key
technologies to achieve varied performance requirements — such as high
throughput, low high security goals in 5G networks [5].

One of the key features of RAN slicing is that - MVNOs are assigned slices
that are independent from one another [6]. That is, the allocation of the
radio resources is up to the NO who can allocate them at will, based on the
QoS requirement while ensuring complete isolation between slices. The NO we
consider is based on Software Defined - Radio Access Network (SD-RAN)
controller architecture comprising of a Slice Manager and MVNO specific
scheduler. The NO architecture is formally introduced in Section III.

Once the QoS requirements for each MVNO is collected by NO, the core problem
lies in allocation of scarce spectrum resources such that each MVNO’s QoS
requirement is met for all its users. We consider spectrum resources as
resource blocks (RB). This is a difficult problem because over provisioning of
RB for a user, will result in wastage and under provisioning might not meet
the QoS requirements. Therefore the design of efficient slicing algorithm to
meet each MVNO user’s requirement is key for optimal usage of RB. Also, from a
business standpoint, optimal usage of RB which will result in increased number
of users served in a time slot and thereby increased number of MVNOs supported
by fixed number of RB is of great interest.

One of the factors which influence the slicing decision, is the channel
condition experienced by the RB during its path towards the users. In order to
convey the channel information, each user in the network sends a CQI report
back to the NO. Often in real world scenarios, the channel conditions do not
remain the same. They keep varying with respect to time and frequency. In
order to take into account of this dynamic channel condition, the users send
the CQI report in regular intervals with its periodicity determined by the NO
and in between this interval, the channel conditions are assumed to remain
same [7]. In order to remain close to reality, we consider dynamic channel
conditions in our work.

We illustrate the problem of RAN Slicing under dynamic channel conditions by
considering minimum data rate per time slot for each user as a specification
by MVNOs. Calculation of data rate for a user at a given time depends on MCS
level chosen for the user by the NO at that time. Choice of MCS level in turn
depends on the CQI report sent from the users of MVNO. Now, the problem we are
addressing in this paper is, how do we create a channel conditions aware slice
for each MVNO such that, maximum number of MVNO user’s minimum data rate
requirement is met.

Even though RAN resource allocation issue has been studied extensively in the
recent past [8],[6],[9], the problem of resource allocation to MVNOs under
dynamic channel conditions is relatively new. This is discussed more in
Section II. Design of efficient resource allocation/slicing enforcement
algorithm is not trivial and is met with unique challenges:

  * •

Users maximization: Meeting the minimum data rate requirement for maximum
number of MVNO users in the slice time slot. This can be achieved by choosing
the optimal number of RB and the MCS level for each user.

  * •

Orthogonality:  Each RB should be allocated to only one user across all MVNOs
at a given time slot to avoid interference [10],[11],[12].

  * •

Support advanced 5G technologies: The RB allocation should also facilitate in
implementation of advanced 5G technologies such as CoMP and MIMO [9].

The aim of this work is to design, analyze and validate MCS aware RAN
Slicing(MaRS) algorithm that take into consideration the challenges mentioned
above. To summarize, this work makes the following contributions:

  * •

We formulate the MCS aware RAN Slicing (MaRS) problem as a Non-Linear
Programming Optimization problem in Section V using the model developed in
Section III. We will also prove the NP-Hardness of the MaRS problem.

  * •

We propose a solution for this problem using the greedy algorithm paradigm in
Section VI.

  * •

We develop an upper performance bound for the MaRS algorithm in Section VII.

  * •

We provide an implementation of the proposed solution and carry out an
exhaustive evaluation in Section VIII.

##  II Related Works

There has been significant work to address the problem of RAN slicing,
especially in the recent past. There have been many excellent surveys on this
topic [4, 10, 13, 14, 12]. The authors in these surveys provide a
comprehensive information regarding the work being done on this topic.
Additionally, a book has been published on the topic of RAN Slicing where many
slicing algorithms have been proposed [15]. Specifically [13] covers the
advancements in RAN slicing which is based on the SDN architecture. The
architecture considered in our work loosely follows the work covered in [13].

In the recent works, the RAN slicing problem [16, 17, 18] has been dealt by
designing solutions using various theoretical means optimization [19, 20],
game theory [21]. There has also been many advancements where several machine
learning approaches have been used to address the RAN Slicing problem –
{Reinforcement Learning [22], Deep Learning [23, 24, 25]}. These machine
learning approaches are not suitable for deployment due to their huge data
requirements for training and the time it requires to do so. Moreover accurate
predictions of the channel conditions are required to make the slicing
algorithm effective.

One of the key limitation of these works is that it does not show the actual
deployment of RAN slices on top of a physical network. Although the authors in
the paper [9] discuss RAN slicing policies and enforcement problem by
considering fine grained control of resources, it falls short when we bring in
dynamic channel conditions. Moreover, the problem formulation considers slice
as allocation of certain percentage of resource blocks from a given pool
without considering the underlying requirement for these slices.

One of the work which closely focuses on addressing RB allocation problem is
[11]. The authors propose a RB partitioning algorithm which focuses on
allocating RB to every MVNO by simultaneously maximizing the percentage of
satisﬁed MVNOs while allocating the minimum amount of RB. However, the problem
in [11] does not take into account the dynamic channel conditions.

Our work can be closely compared to [8]. The authors of [8] address the
problem of RAN slicing by considering dynamic channel conditions in a SD-RAN
based architecture. One of the key architectural differences between our work
and [8] is the flexibility offered to the MVNO in the SD-RAN architecture. In
[8], the authors consider individual slice managers for each slice but a
common scheduler for all the users. This provides very less flexibility for
MVNOs. In our work, we consider independent scheduler for all MVNOs. This
allows MVNO, the option of choosing its users for scheduling at each time
interval. Section III discusses this in detail.

In [26], the authors address the RAN Slicing problem for multiplexing eMBB and
URLLC slices. Although the paper [26], considers the MCS selection in the
design of the slicing algorithm, it again falls short in providing MVNO the
flexibility in scheduling as the architecture considered in completely
different.

The papers [27, 28] present a framework for LTE virtualization. The authors
propose an architecture for virtualizing the LTE base stations (called eNodeB
in LTE architecture) with the objective of having different operators sharing
the same physical resources. The solution is based on a hypervisor (as in CPU
virtualization), which hosts different virtual nodes, allocates the resources
and is responsible of the spectrum sharing and data multiplexing. In [29], the
framework from [27, 28] is used to present a algorithm for scheduling physical
RB for the virtual nodes. The main idea of this algorithm is that if a eNodeB
is overloaded and a neighbor eNodeB has available resources, a user is
selected to be migrated to the unloaded eNodeB. Although the concept of
centralized control is similar to our work, the problem statement is
completely different. In our work, we are addressing the problem of RAN
slicing in a multi-MVNO environment as opposed to resource sharing.

In summary, our work addresses the shortcomings of these papers by providing
more flexibility to the MVNOs, developing efficient slicing algorithm with
dynamic channel conditions and carrying out a thorough validation.

##  III System Model

We consider a NO administering a single $5G$ RAN base station $B$ and set of
$\mathcal{M}=\\{1,2...,$M$\\}$ MVNOs as depicted in Fig. 1.The NO serves the
MVNOs by creating virtual RAN slices built on top of underlying physical
network $B$. We split NO functionally into Slice Manager and MVNO scheduler.
This architecture lies inline with 5G RAN concepts, where the management and
orchestration is implemented as a Software Defined Network (SDN). We adopt the
architecture principle similar to [8], [30], and include additional features
to aid the proposed slicing procedure.

Once the NO collects the minimum data rate slice request from all MVNOs, it
creates an instance of MVNO scheduler for each MVNO in the network.We define
$\Lambda_{m}^{i}$ as the minimum data rate requirement for each user $i$ of
MVNO $\forall i\in m,m\in\mathcal{M},\forall m$. MVNO Scheduler for all
$m\in\mathcal{M}$ provides a scheduling order of users belonging to $m$,
$\mathcal{U}_{m}$, to the Slice Manager. The Slice Manager, which has the CQI
information for each user in the network, dynamically assigns the resources on
$B$ to each MVNO slice based on this scheduling order sent by the MVNO.
Advantage of this architecture is that it leaves the choice of scheduler
implementation, up to the MVNO. Each MVNO may employ unique scheduling
algorithm.

Since the BS follows $5G$ cellular technology, spectrum resources are
organized as grids of RB, that span across both time and frequency domains
[31]. Each RB represents the minimum spatio-temporal scheduling unit.
Considering $N_{RB}$ and $T$ as the number of available subcarriers and
temporal slot respectively, the set of available RB is $|R_{b}|=N_{RB}\times
T$ in the physical RAN network for a certain bandwidth.

Implication of time slot T. Theoretically, the time slot $T$ can range from 1
TTI($t$) to $1000$’s of TTIs, depending upon how dynamically the slice manager
wishes to operate resource slicing policy. Under realistic consideration and
in lieu with next-generation O-RAN architecture [32], it is expected that the
slicing manager will either reside in non-real-time Ran Intelligent
Controller(RIC) or near-real-time RIC, which are respectively, in order of
$>1$s and ($10-1000$ ms) time scales [33]. Thus, in our work, we consider that
$T$ will be a large value, in range of several milliseconds. Further, we
consider the user’s minimum data rate requirement is defined per time slot
$T$.

Dynamic Channel Conditions. We consider the channel conditions to be dynamic
in nature and may vary in frequency and time, but remain consistent within the
time slot $T$. This is similar to aperiodic csi reporting [7]. Depending upon
the channel condition obtained from CQI reports for users of the MVNOs being
served, the Slice Manager determines a suitable MCS for transmission depending
on each MVNO user’s minimum data rate requirement, out of $29$ MCS levels as
per 5G 3GPP specification [31]. Let $\mathcal{C}$ denote the set of available
MCS, i.e., $\mathcal{C}=\\{0,1,\dots,28\\}$. The MCS determines how much
information (in bits) is modulated and coded in each RB by the BS. The higher
the MCS is, the higher the modulation and coding rate is. That means, the
maximum amount of information that can be transmitted on one RB also depends
on the channel conditions. If the channel condition is poor and the NO uses a
high MCS, then the information carried in the RB will not be successfully
received and decoded. Therefore, the achievable data rate by an RB depends on
both MCS level chosen by the NO as well as the channel condition for this RB.

Let $q_{u^{i}_{m}}^{r,t}$ denotes the maximum MCS that can be used for a
certain RB $r$ to serve a user $u^{i}_{m}\in\mathcal{U}_{m}$ such that the
information carried in RB can be successfully received by the user at TTI
$t\in T$.

$1\leq q_{u^{i}_{m}}^{r,t}\leq|C|$

Let $v^{c,t}$ denote the modulation and coding rate for a resource block under
MCS $c\in\mathcal{C}$ and $d_{u^{i}_{m}}^{r,c,t}$ denote the maximum
achievable data rate by RB $r$ for the user $u^{i}_{m}$ under MCS
$c\in\mathcal{C}$ at time $t\in T$. If $c\leq q_{u^{i}_{m}}^{r,t}$, the
transmission would be successful and the achievable data rate is $v^{c,t}$.
Otherwise, the transmission would be unsuccessful and the data would be lost.
That is,

$d_{u^{i}_{m}}^{r,c,t}=\begin{cases}v^{c,t},\hskip 5.69054pt$if$\hskip
5.69054ptc\leq q_{u^{i}_{m}}^{r,t}\\\ 0\hskip 5.69054pt$otherwise$\end{cases}$
(1)

Figure 1: The SD-RAN slicing architecture.

Finally, the NO also imposes a restriction on maximum throughput allowed per
MVNO slice, namely $\overline{\Lambda_{m}}$, depending on channel conditions
or business requirements. This restriction prevents any individual MVNO slice
from overloading the network. An intuitive way of selecting $\Lambda_{m}^{i}$
may be from pure business perspective, i.e, whichever MVNO pays the most will
get higher throughput. However, in general the choice of maximum throughput in
a multi-MVNO, limited resources environment introduces a new problem that is
out of the scope for this work.

##  IV SD-RAN Workflow

Before we proceed with designing of the Slicing Algorithm, it is important to
understand how different components in the SD-RAN architecture interact with
each other to serve users of MVNO. In this section, we present the workflow
for our SD-RAN architecture in Fig 1.

The NO communicates with several components before it assigns resources to a
specific user of a MVNO. After a MVNO requests services from a NO, the task of
allocating resources can be broadly divided into 4 steps.

####  Step 1: Acquiring RAN information

After a MVNO submits a request for a service to the Slice Manager of the NO,
the Slice Manager acquires the RAN information. This RAN information contains
the number of users in the network for the MVNO, the channel conditions
experienced by each user and the available resources in terms of RB in the
network to serve the MVNO. In Fig. 2, the base station and the users in the
network is represented by a single RAN block.

####  Step 2: Acquiring scheduling information

An instance of the scheduler is created in the NO for each MVNO that requests
a service. It is up to the MVNO on how the scheduling algorithm is
implemented. For example, a particular MVNO may use round robin and other MVNO
might opt for priority scheduling. Its one of the novelty in our work where we
provide the MVNO, the flexibility of choosing the scheduling algorithm. In
order to make a slicing decision, the Slice Manager interacts with the
instance of the MVNO scheduler to acquire the scheduling list which is a list
of users and its unique minimum data rate requirement that is generated
through MVNO specific scheduling algorithm.

####  Step 3: Making Slicing Decision

After Step 2, the Slice Manager has all the required information to make a
slicing decision. It has the list of users that it needs to serve with its
minimum data rate requirement, their channel conditions and the available
resources in the network to serve them. Now, the Slice Manager limits the
number of users that can be served for a MVNO by imposing a upper bound of
maximum throughput allowed per slice. Using the MCS-aware Ran Slicing
Algorithm, the slice manager makes a Slicing Decision by assigning resources
to the users of the multiple MVNOs across time $T$.

####  Step 4: Enforcing Slicing decision

After the Slice Manager makes the Slicing Decision for time slot $T$, it is
conveyed to the MVNO scheduler and enforced on RAN. The MVNO scheduler can use
this Slicing Decision as an input to generate the scheduling list for the next
time slot $T$.

Figure 2: SD-RAN slicing architecture workflow diagram.

##  V Problem Formulation

In this section, we formulate the MCS-aware RAN Slicing (MaRS) problem as a
Non Linear Optimization problem. The problem aims at determining the optimal
set of resource blocks to be allocated to each MVNO $m\in\mathcal{M}$ in time
slot $T$, such that maximum number of users can be served across MVNOs in $T$,
by considering – (1) MVNO’s minimum bit rate requirement is met for each of
its user, (2) Each MVNO scheduler’s unique user scheduling order is ensured,
and (3) The total throughput per slice does not exceed the maximum allowable
throughput set by the NO for that MVNO slice.

Notation. Let set $\mathcal{U}_{m}=\\{u^{1}_{m},\dots u^{j}_{m},\dots
u^{|U_{m}|}_{k}\\}$ denote the scheduling order of all users belonging to MVNO
$m\in\mathcal{M}$.

Decision variables. Let $u^{i}_{m}$ denotes whether a user $i$ belonging to
MVNO $m$ can be served by the Slice Manager. Let $x^{r,i,t}_{m}$ denote the
whether a certain RB $r\in R$ is allocated to any user $u^{i}_{m}$ in MVNO $m$
at TTI $t\in T$. Let $y^{c,t}_{i,m}$ denote whether a MCS level $c$ is chosen
by a user $u^{i}_{m}$ at TTI $t$.

$\displaystyle P1:\underset{x^{r,i,t}_{m},y_{i,m}^{c,t}}{\max}\sum_{m\in
M}\sum_{u^{i}_{m}\in\mathcal{U}_{m}}u^{i}_{m}$ (2)
$\displaystyle\sum_{u^{i}_{m}\in U_{m}}x^{r,i,t}_{m}\leq 1,\forall r,t$ (3)
$\displaystyle c\times y^{c,t}_{i,m}\leq c^{r,i,t}_{max}x^{r,i,t}_{m},\forall
t,i,m,r,c$ (4) $\displaystyle\sum_{c}y^{c,t}_{i,m}\leq 1,\forall m,i,t$ (5)
$\displaystyle u^{i}_{m}\geq u^{j}_{m},\forall i<j,\forall m$ (6)
$\displaystyle\sum_{t\in T}\sum_{r\in
R}x^{r,i,t}_{m}y^{c,t}_{i,m}d_{u^{i}_{m}}^{r,c,t}\geq
u^{i}_{m}\Lambda_{m}^{i},\forall m,i$ (7) $\displaystyle\sum_{t\in
T}\sum_{r\in R}\sum_{u^{i}_{m}\in
U_{m}}x^{r,i,t}_{m}y^{c,t}_{i,m}d_{u^{i}_{m}}^{r,c,t}\leq\overline{\Lambda_{m}},\forall
m$ (8) $\displaystyle x^{r,i,t}_{m},y^{c,t}_{i,m},u^{i}_{m}\in\\{0,1\\}$ (9)

The maximization problem given in (2) targets to accommodate maximum number of
users to satisfy the constraints. Constraint (3) indicates that a resource
block can be allocated to one UE at any given time. Equation (4) indicates
that MCS chosen for a user cannot be greater than maximum MCS supported by any
resource block r at that time. Moreover, equation (5), ensures that a single
MCS level is chosen for a user at time $t$. Constraint (6) ensures that
scheduling order determined by MVNO scheduler is maintained in allocating
resources. Equation (7) meets the minimum data rate requirement for each user
belonging to a MVNO. Equation (8) indicated the maximum data rate achieved by
the resources allocated to MVNO is under the maximum allowable throughput for
MVNO.

######  Theorem 1.

The MaRS problem is NP-Hard.

###### Proof.

In order to prove the NP Hardness, consider the optimization problem defined
in Equation (2) for a single MVNO and for a single time slot $t\in T$.
Therefore, we drop the $m$ and $t$ notation. Further, we consider channel
condition is the same across all base stations (and RBs), then the MCS level
for all the RBs will be same, $c\in\mathcal{C}$. This affects Equations (7)
and (8). Therefore, we can rewrite the optimization problem and the
constraints as follow,

$\displaystyle P2:\underset{x^{r,i}}{\max}\sum_{u^{i}\in\mathcal{U}}u^{i}$
(10) $\displaystyle\sum_{r\in R}x^{r,i}\geq u^{i}$ (11)
$\displaystyle\sum_{r\in R}x^{r,i}\leq\overline{\Lambda_{m}}$ (12)
$\displaystyle x^{r,i},u^{i}\in\\{0,1\\}$ (13)

Notice that P2 is a maximum coverage problem, which is a classic NP-Hard
problem [34]. Since MaRS problem can be modeled as a maximum coverage problem,
MaRS problem is also a NP-Hard problem. ∎

TABLE I: Notation Table Symbol |  Definition
---|---
$\mathcal{M}$ |  A set of MVNOs requesting slices from NO.
$\Lambda_{m}^{i}$ |  Minimum data rate req. for user $i$ in MVNO $m\in\mathcal{M}$.
$\overline{\Lambda_{m}}$ |  Maximum allowable throughput for a slice $m\in\mathcal{M}$.
$\mathcal{U}_{m}$ |  Scheduling List for MVNO $m\in\mathcal{M}$
$\mathcal{C}$ |  A set of possible MCS values as per 3GPP specifications.
$u_{m}^{i}$ |  Represents a UE $i$ belonging to MVNO $m\in\mathcal{M}$.
$v^{c,t}$ |  The modulation and coding rate for an RB
|  under MCS $c\in\mathcal{C}$ at time $t\in T$.
$q_{u^{i}_{m}}^{r,t}$ |  The maximum MCS that can be used for a certain RB $r$
|  to user belonging to MVNO $m\in\mathcal{M}$ at time $t\in T$.
$d_{u^{i}_{m}}^{r,c,t}$ |  The maximum achievable data rate by RB $r$
|  for a UE $u_{m}^{i}$ under MCS $c\in\mathcal{C}$ at time $t\in T$.
$c^{r,i,t}_{max}$ |  Maximum mcs that can be selected for a RB $r$
|  for user $i$ at time $t\in T$.
$\mathcal{L}^{T}$ |  Slicing List - List of users to be scheduled across MVNOs
|  at time $T$.
$C_{max}$ |  Maximum MCS that can be selected for a user at any
|  TTI $t\in T$.
$h^{t}$ |  Maximum achievable data rate for a user at each tti $t$.
$\tilde{c}$ |  MCS used to achieve maximum data rate at each TTI $h^{t}$.
$A^{t}$ |  List to hold maximum data rate for each user for
|  every TTI $t\in T$.
$R_{tot}$ |  Total available RB in the network.
$\hat{R}$ |  RB that have been already allocated in $T$.
$R^{\prime}$ |  RB that contribute to achieve maximum data rate $h^{t}$ at each
|  TTI $t\in T$.
$c^{\prime}$ |  MCS value for $R^{\prime}$ that achieve the maximum data rate $h^{t}$.
$R*$ |  Total RBs used to meet data rate requirements all the users
|  in time slot $T$.
$C*$ |  MCS used for all the RB in $R*$.
$U$ |  Users served in time slot $T$.

##  VI MCS Aware RAN Slicing Algorithm

In this section, we develop the MCS aware RAN slicing algorithm based on a
greedy paradigm.

###  VI-A Key Intuitions behind the Proposed Algorithm

The design of the MaRS algorithm is based on the following key intuitions.

Figure 3: An example for MCS selection.

Intuition 1. From MaRS problem objective function (See Eq. 2), it is obvious
that we need to maximize the number of users that can be served in $T$ for
each MVNO $m\in\mathcal{M}$. We consider the minimum data rate requirement to
be per time slot $T$ and we say that a user $u_{m}^{i}$ is served only if it
is allocated sufficient resource blocks such that its minimum data rate
$\Lambda^{i}_{m}$ is met in $T$. Based on this observation, we should minimize
the number of resource blocks utilized to serve each user.

Intuition 2. We sort the users across MVNOs in increasing order based on their
minimum data rate requirement $\Lambda^{i}_{m}$. We call it as Slicing List
$\mathcal{L}^{T}$. Even though, each user can have its own minimum data rate
requirement, we must follow the scheduling order defined by the MVNO (equation
(6)). That is, for some MVNO $m$ if the scheduling order is
${u^{1}_{m},u^{2}_{m}}$, we must always allocate resources to $u^{1}_{m}$
first even if $\Lambda_{m}^{1}>\Lambda_{m}^{2}$. This ensures that in a case
of insufficient resource blocks to support all users in $T$, the user which is
first in the scheduling order gets higher priority than other users. However,
we do not maintain any scheduling order across MVNOs. That is, for any MVNOs
$m,n$, the Slicing List can be
$\mathcal{L}^{T}=\\{\Lambda_{m}^{1},\Lambda_{m}^{2},\Lambda_{n}^{1}\\}$ if
$\Lambda_{m}^{2}<\Lambda_{n}^{1}$.

Intuition 3. To incorporate the channel conditions in the slicing decision, we
must consider the effect of MCS selection on RB. In Fig. 3, we use an example
to show the dependency between MCS selection and number of RB. Suppose a user
is requesting a data rate of 8 from a Base Station which has 9 RB. The channel
conditions for each of the resource block is denoted in their respective grid
position. If the BS chooses MCS 3 for transmission, 3 RB are required to meet
the user’s data rate requirement as $3\times 3=9$. If BS choose MCS 4 for
transmission, the user’s data rate requirement can be met by just 2 RB as
$2\times 4=8$. Therefore, choosing the higher MCS reduces the RB utilization
to meet the data rate requirement. From the previous ideas, we know that we
must use least amount of resources to serve users to maximize the number of
users served. This implies, we must choose the maximum MCS for each user at
any given time.

Intuition 4. The slicing decision is an iterative approach where in, we
allocate the subset of unallocated resource blocks based on its MCS level to a
user of MVNO at each iteration. The slicing decision is controlled by two main
factors, the minimum data rate requirement for each user $\Lambda^{i}_{m}$ and
the maximum allowable throughput decided by the NO for each MVNO,
$\overline{\Lambda_{m}}$. Eventually, the algorithm exits when all the users
have been served or when all RB are allocated.

###  VI-B Algorithm Details

In this section, we discuss on how we utilize the MCS levels on the RB in
making the slicing decisions.

Recall that the first step in our algorithm is the generation of the slicing
list $\mathcal{L}^{T}$. This depends on:

  * •

the minimum data rate requirement for each user $\Lambda^{i}_{m},\forall
u^{i}_{m},m$.

  * •

the scheduling list sent by each MVNO $\mathcal{U}_{m},\forall m$.

Using these information, the Slice Manager develops $\mathcal{L}^{T}$ which is
valid for time slot $T$ by two-stage sorting, as shown in Alg. 1.

Algorithm 1 Slicing List

1:Collect scheduling order and minimum data rate requirement $\Lambda_{m}^{i}$
for each user.

2:Generate a tuple for each user which contains MVNO id, scheduling order,
minimum data rate $<m,i,\Lambda_{m}^{i}>,\forall i,m$.

3:Add all users to the list $\mathcal{L}^{T}=[<m,i,\Lambda_{m}^{i}>],\forall
i,m$.

4:Sort $\mathcal{L}^{T}$ based on $\Lambda_{m}^{i}$.

5:Sort $\mathcal{L}^{T}$ based on $i$.

6:return $\mathcal{L}^{T}$

With the Slicing List $\mathcal{L}^{T}$ as the input, we present the MCS-aware
RAN Slicing algorithm in Alg. 2. The algorithm outputs the least number of
resource blocks and their MCS level in the time slot $T$ such that each user’s
minimum data rate requirement is met.

As discussed in the previous section, the algorithm uses an iterative approach
wherein at each iteration, it serves a user according to $\mathcal{L}^{T}$.
This algorithm consists of two key steps.

(i) Step 1. Finding the optimal number of resource blocks and their MCS level
which maximizes the achievable data rate at each TTI $t\in T$ \- This is
addressed by iterating over the MCS values that a user can support, followed
by iterating over each TTI. Remember, the achievable data rate at each TTI, is
directly related to the MCS level chosen for its resource blocks. Therefore,
in our algorithm, we iterate over each TTI, starting with the maximum MCS
$C_{max}$, first to calculate the data rate. We keep track of the maximum
achievable data rate by updating $h^{t}$ after each iteration of MCS
$\tilde{c}$.

(ii) Step 2. Greedily allocate the resource blocks for each user such that
their requirement is met. Once we have the list containing the maximum
achievable data rate and the corresponding resource blocks with the MCS value
for each TTI $A^{t}$, we now allocate the resources to the user in $T$. Our
key idea is to minimize the number of resource blocks for each user which will
subsequently help us in maximizing served users in $T$. As discussed in
previous section, in order to minimize the RB utilization, we need to choose
the higher MCS. Following this idea, we follow a greedy approach where in we
choose the resource blocks with maximum MCS first in $A^{t}$ to meet the
minimum data rate requirement for each user. This enables us to choose the
least amount of resource blocks and corresponding MCS at each TTI $t\in T$,
such that each user’s requirement $\Lambda_{m}^{i}$ is met for time $T$.

Algorithm 2 MCS-aware RAN Slicing Algorithm

Input: Slicing list $\mathcal{L}^{T}$, $\Lambda^{i}_{m}$ minimum bit rate
requirement for each user belonging to mvno $m$, $v^{c}$ achievable bit rate
with MCS level $c\in C$, maximum allowable throughput for a MVNO
$\overline{\Lambda_{m}},\forall m$, the maximum mcs that can be supported by a
resource block at TTI $t$, $q^{r,t}$.

Output: Set of allocated RBs $R^{*}$ and MCS level $C^{*}$, for each user in
$\mathcal{L}^{T}$.

1:Initialize $R^{*}=\phi$ and $C^{*}=\phi$

2:Initialize already allocated RBs, $\hat{R}=\phi$

3:Total RBs, $R_{tot}$

4:for each user, $u_{m}^{i}$ in $\mathcal{L}^{T}$ do

5: current data rate for each user, $d_{m}^{i}=\phi$

6: current data rate for each MVNO, $d_{m}=\phi$

7: if $d_{m}\geq\lambda_{m}$ then

8: break

9: for each MCS, $c=C_{max},\dots,1$ do

10: list to hold each TTI information, $A^{t}=\phi$

11: if $\hat{R}\cap R_{tot}=\phi$ then

12: return “No solution”

13: for $t=0,\dots,T$ do

14: maximum data rate at tti t $h^{t}=\phi$

15: for $\tilde{c}=C_{max},\dots,c$ do

16: candidate RB for MCS $\tilde{c}$, $R_{can}=\phi$

17: for $r\in R_{tot}$ do

18: if $r\cap\hat{R}=\phi$ and $q^{r,t}\geq\tilde{c}$ then

19: $R_{can}=R_{can}\cup r$

20: if $R_{can}\times v^{c}>h^{t}$ then

21: $h^{t}=R_{can}\times v^{c}$

22: $R^{\prime}=R_{can},c^{\prime}=\tilde{c}$

23: add tuple $\delta^{t}_{i}=<c^{\prime},R^{\prime},h^{t}>$ to the list
$A^{t}$

24: sort $A^{t}$ based on decreasing order of $c^{\prime}$

25: for each tuple $\delta_{i}^{t}$ in $A^{t}$ do

26: $h_{u}=h_{u}+\delta_{i}^{t}[h^{t}]$

27: $R_{u}=R_{u}\cup\delta_{i}^{t}[R^{\prime}]$

28: $c_{u}=c_{u}\cup\delta_{i}^{t}[c^{\prime}]$

29: if $h_{u}\geq\Lambda_{m}^{i}$ then

30: $R^{*}=R^{*}\cup R_{u}$, $\hat{R}=\hat{R}\cup R^{*}$

31: $C^{*}=C^{*}\cup c_{u}$

32: $U=U+1$

33: break

34:return $U$, $R^{*}$ and $C^{*}$

###  VI-C Time Complexity

We will now discuss the complexity of MaRS algorithm. To compute the maximum
data rate for a user at each TTI, the time complexity is $O(|C||R_{tot}|)$. We
need to compute this for the each TTI $t\in T$. Therefore, the total time
complexity to compute maximum data rate for a user in time $T$ is,
$O(|T||C||R_{tot}|)$. After that, we sort maximum data rate achieved across
TTIs. The sorting operation in represented in Line 24 which has the complexity
of $O(|T||logT|)$. Now, we iterate over each element in the sorted list to
meet the data rate requirement, $O(|T|)$. Then MaRS algorithm allocates the
optimal resources $R_{u}$ and chooses its MCS $c_{u}$ for each user across TTI
if possible for the current MCS $c$. Therefore the total complexity for each
iteration of $c$ is
$O(|T||C||R_{tot}|)+O(|T||logT|)+O(|T|)=O(|T||C||R_{tot}|)$. Since there are
$|C|$ possible values of $c$, the complexity is $O(|T||C|^{2}|R_{tot}|)$. Now
MaRS algorithm calculates this for every user in $|L^{T}|$. Therefore, the
total time complexity of MaRS algorithm is $O(|L^{T}||T||C|^{2}|R_{tot}|)$.

######  Theorem 2.

If there exists a feasible solution for any given user in $\mathcal{L}^{T}$,
Algorithm 2 will find it.

###### Proof.

As discussed in the previous section, for each user, Algorithm 2, calculates
the maximum achievable data rate for each TTI $t\in T$. The algorithm goes
over every possible combination of the resource blocks and MCS to determine
this data rate. Once the algorithm generates $A^{t}$, the maximum achievable
data rate for a user at each TTI, it proceeds with RB allocation. Now, as long
as the sum of the data rates in $A^{t}$ is greater than minimum user data rate
requirement, the algorithm provides a solution. That is, given a finite set of
unallocated RB and a user $u^{i}_{m}$ with minimum data rate requirement of
$\Lambda_{m}^{i}$,

$\displaystyle\sum_{\forall\delta_{i}^{t}\in
A^{t}}{\delta_{i}^{t}[h^{t}]}\geq\Lambda_{m}^{i}$ (14)

the algorithm will find sub set of RB and corresponding MCS across $T$ which
meet $\Lambda_{m}^{i}$ as long as equation (14) is met. ∎

##  VII Performance Bound

As proven in Theorem 1, the MaRS problem is NP-hard and it is not feasible to
find a polynomial-time optimal solution. Therefore, it is vital to develop an
upper bound for the objective function defined in Exp. (2). This upper bound
can be used as a benchmark to measure the performance of the scheduling
algorithm that we presented in section V.

When $R$, the maximum number of resource blocks in time $T$ is given, our
problem aims to find a subset of $R$ for each user. Choice of this subset
depends on $c$, the MCS selected for them. Note that, if we want to maximize
Exp. (2), we need to find a subset which contains least amount of RBs.

Since we want to find an upper bound for objective function in Exp. (2) , let
us consider a fictitious scenario of excellent channel condition for every
user in time $T$. Therefore, every resource block $r\in R$, can support the
maximum MCS value that a user can support during its allocation. That is,

$\displaystyle q_{u^{i}_{m}}^{\max}=\max_{r,t}\ q_{u^{i}_{m}}^{r,t}.$ (15)

We further consider that the MVNO scheduler always uses the maximum MCS for
each user. That is,

$\displaystyle d_{u^{i}_{m}}^{r,c,t}=q_{u^{i}_{m}}^{\max}$ (16)

We then proceed with the allocation of the resource blocks for each user
following the slicing list, $\mathcal{L}^{t}$. In this fictitious scenario,
the data rate achievable for each user in time $T$, is directly proportional
to the number of resource blocks allocated to it at time $T$. Therefore, we
can re-write constraints (7), (8) as,

$\displaystyle\sum_{t\in T}\sum_{r\in R}x^{r,i,t}_{m}\times
d_{u^{i}_{m}}^{r,c,t}\geq u^{i}_{m}\Lambda_{m}^{i},\forall m,i$ (17)
$\displaystyle\sum_{t\in T}\sum_{r\in R}\sum_{u^{i}_{m}\in
U_{m}}x^{r,i,t}_{m}\times
d_{u^{i}_{m}}^{r,c,t}\leq\overline{\Lambda_{m}},\forall m$ (18)

We can see that, the criterion to meet each user’s minimum data rate
requirement completely depends on number of resource blocks allocated to it.
Since we assume the maximum MCS level for each resource block, any allocation
of the resource block with MCS $d_{u^{i}_{m}}^{r,c,t}$, for a user to meet
$\Lambda_{m}^{i}$ would use least amount of resource blocks. Therefore, if we
use the least amount of resource blocks for every user, we can find the
maximum users that can be supported by the given set of resource blocks $R$
for time slot $T$.

In this section, we developed a very intuitive based upper performance bound
for the MaRS algorithm. In the following section, we perform simulations of
this upper bound and compare the performance of MaRS algorithm with it.

##  VIII Performance Evaluation

In this section, we assess the performance of the MaRS algorithm proposed in
Section VI. We evaluate MaRS algorithm in terms of its ability to achieve our
objective function of maximizing the users served by varying various 5G
network parameters. We use the upper bound developed in section VII as the
benchmark for this purpose.

#### Network Setting

We simulate an 5G NR base station deployed in a certain environment serving
$\mathcal{N}$ number of users. This BS and user deployment can be modeled
using any standard approaches such as hexagonal, square lattice or stochastic
geometry based Poisson Point process [35]. We consider this BS to be operating
as frequency division duplexing(FDD) system with a channel bandwidth of 20MHz,
which is divided into 1200 subcarriers organized into $R_{tot}=$100 RB. while
considering sub-carrier spacing of 15Khz. Each PRB represents the minimum
scheduling unit and consists of 12 subcarriers and 14 symbols.

For each user in the network, the expected channel condition (in terms of MCS)
is randomly chosen. For each MCS $c$, the modulation and the coding rate
$v^{c,t}$ is obtained from [36].

Configurable Parameters. There are many configurable system parameters such as
the time slot $T$, number of MVNO $M$, users of each MVNO $u^{i}_{m}$, minimum
data rate for each user $\lambda_{m}^{i}$, the maximum throughput allocated
per MVNO slice by NO $\Lambda_{m}$. We evaluate the performance of MaRS
algorithm under various combination of these settings.

The number of users served in the time slot $T$ would be the sole performance
metric for all our simulation settings.

#### Results

In this section, we evaluate the proposed algorithm against the upper bound
against varying parameters.

Varying Channel Propagation. We first evaluate the performance of MaRS
algorithm under channels with varying LOS signal strength. We assume Rician
fading channel with no frequency and time correlation.

Figure 4: MaRS algorithm Performance under different Rician factors.

Fig. 4 compares the performance of MaRS algorithm against the upper bound for
different Rician factor $K$. The configuration used for the experiments is
listed in Table. II.

Time Slot, $T$ | 5
---|---
Number of MVNOs, $M$ | 2
Number of users per MVNO, $u_{m}^{i}$ | 10
Minimum Throughput required per user, $\Lambda_{m}^{i}$ | 50 Mb/Slot
Maximum allowed throughput per MVNO slice, $\overline{\Lambda_{m}}$ | 500 Mb/slot
TABLE II: Simulation Parameters - varying channel propagation.

Under this configuration, we can see that MaRS algorithm can achieve near-
optimal performance. In particular, when the Rician factor $K=0$ (i,e.
Rayleigh fading), 4 and 8, the number of users served by the MaRS algorithm is
within 10% of the respective upper bound. For $K=8$, the performance of MaRS
algorithm is as good as the upper bound. This can be attributed towards the
higher availability of resources that can be allocated to fewer number of
users.

Varying Time Slot $T$. We now evaluate MaRS algorithm by varying the slice
time slot. Increasing the time slot $T$, increases the available resources to
meet slice requirements per $T$. Therefore, for this experiment, we increase
the minimum data rate requirement for each user while also increasing the
maximum available throughput. We consider Rayleigh fading to model the channel
and generate MCS values for each user.

$T$ | 20,50,100
---|---
$M$ | 2
$u_{m}^{i}$ | 10
$\Lambda_{m}^{i}$ | 100 Mb/Slot
$\overline{\Lambda_{m}}$ | 5 Gb/slot
TABLE III: Simulation Parameters - varying slice slot time $T$. Figure 5: MaRS
algorithm performance under different Time Slot $T$.

Fig. 5 shows the performance of the MaRS algorithm in comparison with the
upper bound. Clearly increasing the number of available resources, increases
the performance of the MaRS algorithm. This is evident in Fig 5 for $T=100$
TTIs, where MaRS algorithm catches up with the upper bound in terms of number
of users served across MVNOs.

Varying other simulation parameters. We now vary other system parameters to
evaluate MaRS algorithm performance. We understand the behaviour of MaRS
algorithm by considering three scenarios of the network configurations as
shown in Table IV for our simulations. We assume Rayleigh fading channels for
all the simulations.

| Scenario 1 | Scenario 2 | Scenario 3
---|---|---|---
$T$ | 50 | 50 | 50
$M$ | 3 | 2 | 3
$u_{m}^{i}$ | 15 | 10 | 5
$\Lambda_{m}^{i}$ | 100 Mb/Slot | 100 Mb/Slot | 50 Mb/Slot
$\overline{\Lambda_{m}}$ | 5 Gb/slot | 5 Gb/slot | 5 Gb/slot
TABLE IV: Simulation Parameters - 3 network scenarios. Figure 6: MaRS
algorithm performance comparison for 3 scenarios. Figure 7: MaRS algorithm
performance comparison for 3 scenarios with random data rate for each user.

In Fig. 6, Scenario 1 represents a network scenario where there are many users
with high minimum data rate requirement and few resources to allocate them.
Here, we can see that MaRS algorithm is within 5% of the upper bound. In
Scenario 2, we decrease the load on the base station by reducing the number of
MVNOs and users. Even in this case, we can see MaRS algorithm achieves near-
optimal performance. Finally in Scenario 3, where the number of resource
blocks are plenty, we see that MaRS algorithm performs as well as the upper
bound. Further, we also tried varying the data rate requirement for each user
in the network under these 3 scenarios(Fig 7).We choose a random data rate for
each user between 10 Mb/Slot to 150 Mb/slot, the results obtained is similar
to the previous case where the data rate is fixed.

Fast changing channel. Until now, we have considered time correlation for each
user in the network where the channel conditions remains constant for each
user in time slot $T$. We now consider a network scenario where the channel
conditions for each user changes at each TTI. We still assume Rayleigh fading
channels with no frequency correlation. Table V shows the settings used for
this evaluation.

$T$ | 20,50,100
---|---
$M$ | 2
$u_{m}^{i}$ | 30
$\Lambda_{m}^{i}$ | 10 Mb/Slot
$\overline{\Lambda_{m}}$ | 250 Mb/slot
TABLE V: Simulation Parameters - fast changing channel. Figure 8: MaRS
algorithm performance for fast changing channels.

Fig 8, represents the obtained results. We can see that MaRS algorithm
performance is with in 5% of the upper bound. As mentioned in the earlier
section, we have developed the MaRS algorithm and evaluated its performance
for near real-time and non-real-time configuration of the Ran Intelligent
Controller (RIC) in O-RAN architecture. By demonstrating that MaRS algorithm’s
performance is near optimal, we can say MaRS algorithm is a viable option for
deployment for non-real-time and near-real-time RIC.

RB Utilization.  Finally, we evaluate the performance of MaRS algorithm in
terms of number of RB utilized to serve the users across all MVNOs in the
networks. We say a user is served when its minimum data rate is met at time
slot $T$. We measure the number of RB utilized to serve users in 3 scenarios
presented earlier under different MCS selection criterion. MCS selection
criterion:

  * •

Maximum MCS:  We assume that each RB in $T$ for a user can support the maximum
MCS.

  * •

Average MCS:  We calculate the average MCS level for a user across $T$ and
assume that each RB in $T$ supports this average value.

  * •

Lowest MCS:  We calculate the lowest MCS level for a user across $T$ and
assume that each RB in $T$ can only support the lowest value.

Figure 9: Comparison of MaRS algorithm against static allocation algorithms
for RB utilization.

Fig 9 shows the obtained results. It is evident that the Maximum MCS selection
criteria uses the least amount of resource blocks to serve users. This is
understandable as we assume the best channel conditions for all RBs. But,
there may be significant re-transmissions which would increase latency.
However, The performance of MaRS algorithm out performs Average MCS and Lowest
MCS selection criteria. There is a significant decrease in the number of
resource blocks used to serve the users using MaRS algorithm when compared to
these criteria. Therefore, using MaRS algorithm we can serve more users in a
time slot $T$ than using Lowest MCS and Average MCS static algorithms.

##  IX CONCLUSIONS

In this paper, we investigated the problem of RAN slicing in multi-MVNO
environment with varied users having minimum data rate requirement as a
specification for the users. First, we discussed the SD-RAN architecture and
discussed its operation flow. Then, we formulated the MCS-aware RAN Slicing
(MaRSP) problem as optimization problem with an objective function to increase
the number of supported users at each time slot $T$. We proved that MaRSP
problem is NP-Hard. Next, we developed the novel MCS-aware RAN Slicing (MaRS)
algorithm where we maximize the data rate for each user at each TTI and assign
resources to it based on a greedy paradigm. We also showed that the MaRS
algorithm has a polynomial time complexity. Following that, we developed a
upper performance bound for the MaRS algorithm by considering no frequency and
time correlation. Finally, we carry out thorough evaluation of the MaRS
algorithm under various network and channel scenarios. Results conclude that
the proposed slicing algorithm achieves near-optimal performance when compared
with the upper bound. Through various simulation settings, we have also shown
that MaRS algorithm is easily scalable. In compliance with the O-RAN
architecture, we have seen through results that MaRS algorithm can be applied
to non-real-time and near-real-time RIC deployments. Using RB utilization as a
metric, we have compared the performance of MaRS algorithm with other static
allocation algorithms. We see that MaRS algorithm outperforms many static
allocation algorithms by using least amount of resource blocks to serve
minimum data rate requirement for each user.

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According to [W87, Chap. 3] the preminimal operator is closable and its
closure

(B.4) ${\mathsf{T}}_{0}:=\overline{{\mathsf{T}}_{0}^{\prime\prime}}$

is the minimal symmetric operator in $L^{2}({\mathcal{I}};{\mathbb{C}}^{2})$

Let us define the Lagrange brackets for absolutely continuous functions
$f,g\colon{\mathcal{I}}\rightarrow{\mathbb{C}}^{2}$ by

(B.5) $[f,g]_{x}:=f_{2}(x)\overline{g_{1}(x)}-f_{1}(x)\overline{g_{2}(x)}.$

Using the concept of the Lagrange bracket we can provide the Green’s identity
for ${\mathsf{T}}$.

######  Proposition B.1 (Green’s formula, [W03, Satz 15.3]).

Let the maximal operator ${\mathsf{T}}$ be as in (B.2). Let the Lagrange
bracket $[f,g]_{x}$ be as in (B.5). Then for any
$f,g\in\mathrm{dom}\,{\mathsf{T}}$ there exist the limits

$[f,g]_{0}:=\lim_{x\rightarrow
0^{+}}[f,g]_{x},\qquad[f,g]_{b}:=\lim_{x\rightarrow b^{-}}[f,g]_{x},$

and it holds that

$\langle{\mathsf{T}}f,g\rangle-\langle
f,{\mathsf{T}}g\rangle=[f,g]_{b}-[f,g]_{0}.$

The domain of the closed symmetric operator ${\mathsf{T}}_{0}$ can be
explicitly characterised.

######  Proposition B.2.

[W87, Thm. 3.11] For $z\in{\mathbb{C}}\setminus{\mathbb{R}}$ we set
${\mathcal{N}}_{z}:=\ker({\mathsf{T}}-z)$, then

$\begin{split}\mathrm{dom}\,{\mathsf{T}}_{0}&=\left\\{f\in\mathrm{dom}\,{\mathsf{T}}\colon[g,f]_{0}=[g,f]_{b}=0\,\,\forall\,g\in\mathrm{dom}\,{\mathsf{T}}\right\\}\\\
&=\left\\{f\in\mathrm{dom}\,{\mathsf{T}}\colon[g,f]_{0}=[g,f]_{b}=0\,\,\forall\,g\in{\mathcal{N}}_{z}+{\mathcal{N}}_{\overline{z}}\right\\}.\end{split}$

The endpoints of the interval ${\mathcal{I}}$ admit some classifications.
First, one can classify whether the endpoints are regular or singular.

######  Definition B.3.

Let the differential $\tau$ be as in (B.1). Then $\tau$ is called _regular_ at
$x=0$ if there exists $c\in{\mathcal{I}}$ so that $\|q(\cdot)\|$ is integrable
on $[0,c]$. The expression $\tau$ is called _regular_ at $x=b$ if $b<\infty$
and there exists $c\in{\mathcal{I}}$ so that $\|q(\cdot)\|$ is integrable on
$[c,b]$. If $\tau$ is not regular, respectively, at $x=0$ and/or at $x=b$,
then it is called _singular_ at $x=0$ and/or $x=b$, respectively.

The functions in $\mathrm{dom}\,{\mathsf{T}}$ have certain regularity in the
neighbourhood of a regular endpoint. This is clarified in the next
proposition.

######  Proposition B.4.

[W03, Satz 15.4] Let the maximal operator ${\mathsf{T}}$ be as in (B.2).
Assume that $\tau$ is regular at $x=b$. Then the following hold.

  * (i)

For any $f\in\mathrm{dom}\,{\mathsf{T}}$ there exists a finite limit
$f(b):=\lim_{x\rightarrow b^{-}}f(x)$

  * (ii)

The mapping $\mathrm{dom}\,{\mathsf{T}}\ni f\mapsto f(b)\in{\mathbb{C}}^{2}$
is surjective.

Analogous statements hold if $\tau$ is regular at $x=0$.

The connection between the maximal operator and the minimal and preminimal
operators is revealed in the next proposition.

######  Proposition B.5.

Let the maximal operator ${\mathsf{T}}$, the preminimal operator
${\mathsf{T}}_{0}^{\prime\prime}$ and the minimal operator ${\mathsf{T}}_{0}$
be as in (B.2), (B.3), and (B.4), respectively. Then

$({\mathsf{T}}_{0}^{\prime\prime})^{*}={\mathsf{T}}_{0}^{*}={\mathsf{T}}.$

Next we define the concepts of being right or left in
$L^{2}({\mathcal{I}};{\mathbb{C}}^{2})$.

######  Definition B.6.

Let $f\colon{\mathcal{I}}\rightarrow{\mathbb{C}}^{2}$ be measurable. One says
that $f$ lies "left in $L^{2}({\mathcal{I}};{\mathbb{C}}^{2})$" if there
exists $c\in{\mathcal{I}}$ such that $f|_{(0,c)}\in
L^{2}((0,c);{\mathbb{C}}^{2})$. One says that $f$ lies "right in
$L^{2}({\mathcal{I}};{\mathbb{C}}^{2})$" if there exists $c\in{\mathcal{I}}$
such that $f|_{(c,b)}\in L^{2}((c,b);{\mathbb{C}}^{2})$

The solutions of the differential equation $(\tau-z)f=0$ obey a dichotomy with
respect to the property stated in Definition B.6.

######  Proposition B.7 (Weyl’s alternative).

For any Dirac differential expression $\tau$ as in (B.1) on the interval
${\mathcal{I}}$ one of the following two alternatives holds.

  * (i)

For all $z\in{\mathbb{C}}$ every solution of $(\tau-z)f=0$ lies left in
$L^{2}({\mathcal{I}};{\mathbb{C}}^{2})$.

  * (ii)

For all $z\in{\mathbb{C}}$ there is at least one solution of $(\tau-z)f=0$
that does not lie left in $L^{2}({\mathcal{I}};{\mathbb{C}}^{2})$. In this
case there exists for any $z\in{\mathbb{C}}\setminus{\mathbb{R}}$ exactly one
solution of $(\tau-z)f=0$ (up to a constant factor) that lies "left in
$L^{2}({\mathcal{I}};{\mathbb{C}}^{2})$".

The same alternative holds for “right in
$L^{2}({\mathcal{I}};{\mathbb{C}}^{2})$”.

The Weyl’s alternative stated above yields yet another classification of the
endpoints of the interval ${\mathcal{I}}$ with respect to the differential
expression $\tau$.

######  Definition B.8.

If the first alternative in Propoosition B.7 holds, then one says that $\tau$
is limit-circle at $x=0$ (respectively, at $x=b$). If the second alternative
in Proposition B.7 holds, the one says that $\tau$ is limit-point at $x=0$
(respectively, at $x=b$).

The limit-point/limit-circle classification provides extra properties of
functions in $\mathrm{dom}\,{\mathsf{T}}$ at the endpoints and allows to
compute the deficiency indices of the closed symmetric operator
${\mathsf{T}}_{0}$.

######  Proposition B.9.

Let $\tau$ be the Dirac differential expression as in (B.1) on the interval
${\mathcal{I}}$. Let the maximal operator ${\mathsf{T}}$ and the minimal
operator ${\mathsf{T}}_{0}$ be as in (B.2) and (B.4), respectively.

  * (i)

If $\tau$ is limit-point at $x=0$ (respectively, at $x=b$) then $[f,g]_{0}=0$
(respectively, $[f,g]_{b}=0$) for all $f,g\in\mathrm{dom}\,{\mathsf{T}}$.

  * (ii)

The deficiency indices of ${\mathsf{T}}_{0}$ are

    * -

$(2,2)$ if $\tau$ is limit-circle at $x=0$ and $x=b$.

    * -

$(1,1)$ if $\tau$ is limit-circle at either $x=0$ or at $x=b$ and limit-point
at the other endpoint of ${\mathcal{I}}$.

    * -

$(0,0)$ if $\tau$ is limit-point at both endpoints of ${\mathcal{I}}$.

##  Appendix C Numerical results

In Remark 2.4 we conjectured that Theorem 2.3 is in fact valid for all
$\omega\in(0,\pi)\setminus\\{\pi/2\\}$; in this appendix we provide a
numerical evidence to support this claim.

The key step in the proof of Theorem 2.3 for all
$\omega\in(0,\pi)\setminus\\{\pi/2\\}$ would be showing the property

(C.1)
$\sigma({\mathsf{T}}_{k})\cap\left[-\frac{1}{2},\frac{1}{2}\right]=\emptyset\quad\text{
for all }k\in\mathbb{Z}\text{ and
}\omega\in(0,\pi)\setminus\left\\{\frac{\pi}{2}\right\\}.$

In our analytical results we are able to show it only for
$\omega\in(0,\pi/2)$: this poses a limitation in the statement of Theorem 2.3.

In fact, (C.1) holds for $k\in\mathbb{Z}\setminus\\{-1,0\\}$ and all
$\omega\in(0,\pi)\setminus\\{\pi/2\\}$, since it is true that
$\lambda\in\sigma({\mathsf{T}}_{k})={\mathcal{Z}}_{k}\cup(-{\mathcal{Z}}_{k})\implies|\lambda|\geq(\sqrt{3}+1)/2\approx
1.37$, see Remark 4.8. It remains to consider the cases $k=-1,0$ and
$\omega\in(\pi/2,\pi)$. We remark that
$\sigma({\mathsf{T}}_{k})={\mathcal{Z}}_{k}\cup(-{\mathcal{Z}}_{k})$ is a
discrete set and it is expressed by the solutions of the transcendental
equation in (5.8): thanks to [DLMF, Eq. (14.9.5)], it is true that
${\mathcal{Z}}_{-1}=-{\mathcal{Z}}_{0}$, so it is enough to treat the case
$k=0$ and $\omega\in(\pi/2,\pi)$. We provide here a plot where we draw the set
${\mathcal{Z}}_{0}$ for all $\omega\in(0,\pi)\setminus\\{\frac{\pi}{2}\\}$ and
understand if (C.1) holds true.

In Figure C.1, the set ${\mathcal{Z}}_{0}$ is plotted with a black line for
$\omega\in(0,\pi)\setminus\\{\frac{\pi}{2}\\}$. For $\omega<\frac{\pi}{2}$,
Proposition 4.7 implies the spectral bound
$\lambda\in{\mathcal{Z}}_{0}\implies|\lambda|\geq\frac{\pi}{4\omega}+\frac{1}{2}$:
the curves $\lambda=\pm\left(\frac{\pi}{4\omega}+\frac{1}{2}\right)$ are
plotted with dotted green curves. Finally, the lines $\lambda=1/2$ and
$\lambda=-1/2$ are plotted with dashed red lines.

It is evident from Figure C.1 that
${\mathcal{Z}}_{0}\cap[-1/2,1/2]=\emptyset$, so Theorem 2.3 should be true for
all $\omega\in(0,\pi)\setminus\\{\pi/2\\}$.

Figure C.1. The set ${\mathcal{Z}}_{0}$ is plotted with black lines; the
curves $\lambda=\pm(\frac{\pi}{4\omega}+\frac{1}{2})$ are plotted with dotted
green curves; the lines $\lambda=\pm 1/2$ are plotted with dashed red lines.

######  Remark C.1.

In Proposition 4.7, for $\omega\in(0,\pi/2)$ the bound
$|\lambda|\geq\frac{\pi}{4\omega}+\frac{1}{2}$ on the eigenvalues
$\lambda\in\sigma({\mathsf{T}}_{k})={\mathcal{Z}}_{k}\cup(-{\mathcal{Z}}_{k})$
is provided uniformly in $k\in\mathbb{Z}$. Figure C.1 shows that such a bound
is not sharp for $k=0$, and consequently neither for $k=-1$ since
${\mathcal{Z}}_{-1}=-{\mathcal{Z}}_{0}$. For
$k\in\mathbb{Z}\setminus\\{-1,0\\}$, Remark 4.8 gives the better bound
$|\lambda|\geq(\sqrt{3}+1)/2\approx 1.37$, so we conclude that the bound given
by Proposition 4.7 is not sharp.

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