# Joint Scheduling and Throughput Maximization in Self-backhauled Millimeter
Wave Cellular Networks

Chao Fang1, Charitha Madapatha1, Behrooz Makki2, and Tommy Svensson1  1
Department of Electrical Engineering, Chalmers University of Technology,
Gothenburg, Sweden
2 Ericsson Research, Ericsson AB, Gothenburg, Sweden
Email: 1 {fchao, charitha<EMAIL_ADDRESS>2
<EMAIL_ADDRESS>
###### Abstract

Integrated access and backhaul (IAB) networks have the potential to provide
high data rate in both access and backhaul networks by sharing the same
spectrum. Due to the dense deployment of small base stations (SBSs), IAB
networks connect SBSs to the core network in a wireless manner without the
deployment of high-cost optical fiber. As large spectrum is available in
mmWave bands and high data rate is achieved by using directional beamforming,
the access and backhaul links can be integrated in the same frequency band
while satisfying quality-of-service constraints. In this work, we optimize the
scheduling of access and backhaul links such that the minimum throughput of
the access links is maximized based on the revised simplex method. By
considering a probability based line-of-sight (LOS) and non-line-of-sight
(NLOS) path loss model and the antenna array gains, we compare the achievable
minimum access throughput of the IAB network with the network with only macro
base stations, and study the effect of the network topology and antenna
parameters on the achievable minimum throughput. Simulation results show that,
for a broad range of parameter settings, the implementation of IABs improves
the access minimum achievable throughput.

##  I Introduction

The evolution of wireless communication technology has led to advances in many
areas. Today, 5G has become a key enabler for applications such as virtual
reality, vehicle-to-everything communications, massive machine-type
communications, etc. The growth in the demand of data rate, reliability and
the number of connections require 5G to support high-speed, low-error-rate and
low-latency communications with massive connectivity. Compared to 4G Long-
term-evolution (LTE), 5G development includes many novel designs, in terms of
radio architecture, carrier frequency and network deployment, among which the
three most important technologies are millimeter-wave (mmWave) communications,
massive multiple-input and multiple-output (MIMO) and network densification
[1, 2, 3].

The large amount of spectrum available at mmWave frequency bands makes it
possible to achieve a data rate as high as giga-bit-per-second. However, due
to the high attenuation of mmWave signals and sensitivity to blockage, the
range of mmWave communication is limited to a few hundred meters and the
signal strength varies greatly depending on the line-of-sight (LOS) or non-
line-of-sight (NLOS) conditions [4, 5]. The drawbacks of mmWave communications
are expected to be reduced through beamforming with massive MIMO antenna
arrays and network densification. Due to the small wavelength of mmWave
signals, large antenna arrays have a small footprint, which makes beamforming
well suited for high-capacity transmission [6, 7]. Also, high density
deployments of small cells can greatly improve the NLOS situations and reduce
the path loss [4, 8, 9, 10].

With the dense deployments of small cells, a key challenge is to provide a
large amount of backhaul data into the core network. Traditional solutions
with fiber connections between small cells and a macro cell may become
infeasible due to the high cost, long installation time and low flexibility.
With this background and also motivated by the fact that the backhauling to
small cells needs to support standardazied NLOS connections, integrated access
and backhaul (IAB) networking has recently received considerable attention
[11, 12, 13, 14]. With IAB, the objective is to provide flexible wireless
backhauling using 3GPP new radio (NR) technology in international mobile
telecommunications (IMT) bands, and provide not only backhaul, but also the
existing cellular services in the same node and via the same hardware. Such a
network not only increases the flexibility but also may reduce the
implementation cost and time-to-market.

In this paper, we study the minimum achievable access throughput of multi-hop
IAB networks. Considering a probability based LOS and NLOS path loss model, we
utilize the revised simplex method to derive a proper scheduling and resource
allocation scheme maximizing the minimum access rate. Moreover, we evaluate
the effect of different parameters such as blockage, antenna gain and network
density on the system performance, and compare the results with those achieved
in the cases with only macro base stations (MBSs). The simulation results show
that, despite of the additional resources needed for backahuling, IAB networks
with the optimal scheduling and resource allocation can provide a higher
achievable minimum throughput than that of the macro-only networks for a broad
range of MBS transmit powers, especially in dense blockage environments. Also,
we show that the achievable minimum throughput increases with network
desification and refined antenna parameters.

##  II System model

We consider a multi-hop downlink mmWave IAB network with $M$ MBSs and $R$
small base stations (SBSs). The BSs serve a total of $K$ users in the access
network. The MBSs are connected to the core networks via non-IAB backhauling,
e.g., fiber, while the SBSs are connected to the MBSs and receive backhauling
data wirelessly. We use graphs to represent the network topology where the BSs
or users that receive data from BS $m$ are treated as the child nodes of BS
$m$, and the maximum number of child nodes per BS is denoted by $C$. The
network expands from the MBSs in a tree structure such that there are $R+K$
links in the network. An example network with $M=1,R=2,C=2$ is shown in Fig.
1. The SBS nodes receive backhauling data from the MBS and each SBS maintains
a number of a ccess links in order to serve the user nodes.

Figure 1: A wireless backhaul and access network example.

The BSs and users are assumed to be equipped with antenna arrays and use
directional beamforming for array gain. We assume that the array gain is
$M_{t}$ for all angles in the main lobe of the transmission antenna arrays and
$m_{t}$ otherwise. Similarly, the $M_{r}$ and $m_{r}$ are the antenna gain for
receivers. The beamwidth of the main lobe in the transmitter and receiver are
denoted as $\theta_{t}$ and $\theta_{r}$, respectively. We assume that for the
desired link, the maximum array gain is always achieved while the array gain
is a random variable $D_{j,k}$ for any interfering link between nodes $j$ and
$k$. We assume that angles of departure (AOD) and angles of arrival (AOA) are
uniformly distributed in $(0,2\pi]$. Hence, the array gain of interfering
links is decided depending on if the AOD and AOA falls within the main lobe of
the transmitter and receiver.

Each link can be LOS or NLOS according to the blockage situations, we use a
random LOS model to calculate the path loss of the links. The path loss
between node $m$ and $k$ is given by

$\displaystyle
l_{m,k}=\mathbb{I}(p_{L}(d))\mathrm{PL}^{-1}_{\textrm{LOS}}+\left(1-\mathbb{I}(p_{L}(d))\right)\mathrm{PL}^{-1}_{\textrm{NLOS}},$
(1)

where $d$ is the distance between node $m$ and $k$, $\mathbb{I}(p_{L}(d))$ is
a Bernoulli random variable with LOS probability $p_{L}(d)=e^{-\beta d}$, and
$\beta$ models the blockage density [4]. $\mathrm{PL}_{\textrm{LOS}}$ and
$\mathrm{PL}_{\textrm{NLOS}}$ are the path loss for LOS and NLOS links,
respectively. We use the close-in free space reference distance model given by

$\displaystyle\mathrm{PL}[\mathrm{dB}](d)=20\log_{10}\left(\frac{4\pi}{\lambda}\right)+10\alpha_{\mathrm{LOS}/\mathrm{NLOS}}\log_{10}(d)$
$\displaystyle+X_{\sigma_{\mathrm{LOS}/\mathrm{NLOS}}},d\geq 1~{}\mathrm{m},$
(2)

where $\alpha_{\mathrm{LOS}/\mathrm{NLOS}}$ is the path loss exponent for
LOS/NLOS links and $X_{\sigma_{\mathrm{LOS}/\mathrm{NLOS}}}$ is the log-normal
distributed shadowing parameter with standard deviation
$\sigma_{\mathrm{LOS}/\mathrm{NLOS}}$. In addition to the path loss, we assume
independent Nakagami fading on each link and the signal-to-interference-plus-
noise ratio (SINR) for a link between nodes $m$ and $k$ is given by

$\displaystyle\textrm{SINR}_{m,k}=\frac{M_{r}M_{t}g_{m,k}l_{m,k}}{\sigma^{2}+\sum_{j,j\neq
m}D_{j,k}g_{j,k}l_{j,k}},$ (3)

where the Nakagami fading variable $g_{j,k}$ follows the gamma distribution
$p_{g}(x)=q^{q}\frac{x^{(q-1)}}{\Gamma(q)}\exp(-qx)$ with factor
$q=1/N_{\text{LOS/NLOS}}$, $N_{\text{LOS}}$ and $N_{\text{NLOS}}$ model the
fluctuations of the signal strength due to multipath scattering for LOS and
NLOS links, respectively, and $\sigma^{2}$ is the noise power. Therefore,
assuming Gaussian signaling and for unit bandwidth, the link capacity is given
by $c_{m,k}=\log_{2}(1+\textrm{SINR}_{m,k})$ between link $m$ and $k$. The
users are assumed to be associated to the BS with maximum received power
averaged over fading. Therefore, BS $m$ serves user $k$ if
$l_{m,k}=\max_{i=1}^{M+R}l_{i,k}$.

We assume that the BSs can only transmit or receive, i.e., not simultaneously,
in a given time slot and the BSs transmit to their serving users and BSs in a
time-division multiple access (TDMA) manner. Hence, only one of the links
connected to a BS can be active and the active links can not share a common BS
at a given time. In order to find the optimal scheduling that maximizes the
user throughput, we divide each frame into a number of time slots with
variable length. In each time slot, a certain set of links are active.
Therefore, the optimal scheduling problem is equivalent to finding the optimal
time resource allocation to every possible set of activation links. We denote
the duration of a time slot as $t_{i}$. By defining a unit time frame, we have
$\sum_{i=1}^{N}t_{i}=1$ where $N$ is the total number of time slots in a unit
frame.

##  III Minimum throughput maximization

In this section, we formulate the optimization problem which maximizes the
minimum throughput of each access link subject to the time resource
constraint. Such an optimization guarantees better fairness in the network, at
the cost of losing the peak access rate for the users with good link quality.

Let $S$ be a schedule where in each time slot a certain set of links are
active. The throughput of a link between nodes $m$ and $k$ is given by
$\gamma_{m,k}=\sum_{i=1}^{N}t_{i}c_{m,k}$. If the link $m\rightarrow k$ is not
active in time slot $t_{i}$, we assume $c_{m,k}=0$. Hence, the total
throughput delivered to a node $k$ is given by

$\displaystyle r_{k}$ $\displaystyle=$ $\displaystyle\sum_{m\in
B_{k}}\gamma_{m,k}-\sum_{n\in A_{k}}\gamma_{k,n},$ (4) $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}t_{i}\left(\sum_{m\in B_{k}}c_{m,k}-\sum_{n\in
A_{k}}c_{k,n}\right)$ (5)

where $B_{k}$ is the set of IAB nodes that provide backhaul data to node
(either in access or backhaul) $k$ and $A_{k}$ is the set of nodes to which
node $k$ transmits data. Denoting $\mathbf{t}=[t_{1},\cdots,t_{N}]$ and
$\mathbf{C}=[\mathbf{c}_{1},\cdots,\mathbf{c}_{i},\cdots,\mathbf{c}_{N}]$ as
the capacity matrix where the elements in $\mathbf{c}_{i}$ are the aggregated
rates of each node based on a given link activation pattern in time slot
$t_{i}$, i.e., $\mathbf{c}_{i}[k]=\sum_{m\in B_{k}}c_{m,k}-\sum_{n\in
A_{k}}c_{k,n},\forall k$.

In general, assume node $k$ is required to achieve a minimum throughput
$\theta_{k}$, our objective is to maximize the minimum throughput of the nodes
in the access network subject to the time resource constraint. The problem can
be formulated as

$\displaystyle\mathcal{P}:\mathrm{max}~{}~{}~{}\theta$ (6)
$\displaystyle\mathrm{s.t.}~{}~{}~{}\mathbf{C}\mathbf{t}\geq\theta\mathbf{w}_{\theta}$
(7) $\displaystyle\mathbf{1}^{T}\mathbf{t}=1$ (8)
$\displaystyle\mathbf{t}\geq\mathbf{0},$ (9)

where $\theta\mathbf{w}_{\theta}$ is the minimum throughput of all nodes and
$\mathbf{w}_{\theta}[k]$ is the weight for node $k$ in order to achieve
different minimum throughput among nodes and provide larger throughput for
users with higher data demand. Here we are interested in maximizing the
minimum throughput to the users. To that end, for BS nodes we can simply set
the weights to be 0 so that the BSs are treated as gateways to provide
throughput to the users.

Problem $\mathcal{P}$ is difficult to solve, because, in each time slot
$t_{i}$, there exists a number of feasible link activation patterns, and thus
$\mathbf{c}_{k}$ varies with different link activation patterns. In order to
find the optimal schedule in which $\mathbf{C}\mathbf{t}$ gives the maximum
throughput of each users, we propose an iterative algorithm based on the
revised simplex algorithm[15] and maximum matching theory [16, 17]. We first
create an initial feasible schedule and $\mathbf{C}$, then iteratively update
the schedule and $\mathbf{C}$ until there is little or no improvement to the
throughput.

First, we convert the linear program $\mathcal{P}$ to its canonical form by
introducing surplus variables. The equivalent problem can be written by

$\displaystyle\mathcal{P}_{0}:\mathrm{min}~{}~{}~{}\mathbf{f}^{T}\mathbf{x}$
(10) $\displaystyle\mathrm{s.t.}~{}~{}~{}\mathbf{U}\mathbf{x}=\mathbf{g}$ (11)
$\displaystyle\mathbf{x}\geq\mathbf{0},$ (12)

where

$\displaystyle\mathbf{U}=\begin{bmatrix}\mathbf{C}&\mathbf{-\mathbf{w}_{\theta}}&-\mathbf{I}\\\
\mathbf{1}^{T}&0&\mathbf{0}^{T}\end{bmatrix},$ (13)

$\mathbf{f}^{T}=\begin{bmatrix}\mathbf{0}^{T}&-1&\mathbf{0}^{T}\end{bmatrix},\mathbf{x}^{T}=\begin{bmatrix}\mathbf{t}^{T}&\theta&\mathbf{s}^{T}\end{bmatrix},\mathbf{g}^{T}=\begin{bmatrix}\mathbf{0}^{T}&1\end{bmatrix}$
and $\mathbf{s}$ are the surplus variables.

In order to solve $\mathcal{P}_{0}$, conditioning on $N=R+K$, we first find an
initial schedule in which only one of the links is active in each time slot.
Since the network consists of $R+K$ links, in each time slot $t_{i}$, a
different link is active and $\mathbf{c}_{i},\forall i$ is computed according
to $\mathbf{c}_{i}[k]=\sum_{m\in B_{k}}c_{m,k}-\sum_{n\in
A_{k}}c_{k,n},\forall k$. Based on the initial $\mathbf{C}$, $\mathcal{P}_{0}$
can be solved via the simplex method and we obtain the optimal basis $B$.
Next, we iteratively update the schedule and $\mathcal{C}$ based on revised
simplex method and the maximum matching theory [15].

We then update $\mathbf{C}$ according to

$\displaystyle
c_{m,k}=\begin{cases}c_{m,k}(p_{m}-p_{k})~{}~{}\text{if}~{}\text{nodes $m,k$
are SBSs or users}~{}\\\
c_{m,k}p_{m}~{}~{}~{}~{}\text{otherwise,}\par\end{cases}$ (14)

where $p_{m}$ is the $m$-th element in $\mathbf{p}=\mathbf{f}_{B}^{T}B^{-1}$
and $\mathbf{f}_{B}$ are the coefficients in $\mathbf{f}^{T}$ corresponding to
the initial basis. According to maximum matching with $c_{m,k}$ being the
weights of each link, we obtain a new schedule where in each time slot, an
updated set of links are active. We then compute

$\displaystyle\eta_{1}=-\sum_{m}c_{m,k}-p_{R+K+1},~{}~{}~{}\text{for all MBS
nodes $m$ }$ (15) $\displaystyle\eta_{2}=-1+\sum^{R+K}_{i=1}p_{i}$ (16)
$\displaystyle\eta_{3}=\min_{1\leq i\leq R+K}p_{i}$ (17)
$\displaystyle\eta=\min(\eta_{1},\eta_{2},\eta_{3}).$ (18)

If $\eta\geq 0$, then the global optimal $\theta$ is found, otherwise $B$ is
updated by solving $\mathcal{P_{0}}$ with updated $\mathbf{C}$. With each
iteration, we compute a new $\mathbf{C}$ with a new schedule and the
throughput are improved, the stopping criterion applies if there is no further
improvement of the throughput. Problem $\mathcal{P}_{0}$ are guaranteed to
converge in polynomial time [18].

It is worth noting that the algorithm assume that there is always enough data
at each BS to transmit to other SBSs or users. This may not be true in the
initial few time slots, however, by assuming a data buffer in each BS, the
maximum throughput is achieved on average.

##  IV Simulation Results

In this section, we present the simulation results based on the proposed
minimum throughput optimization algorithm and show that the achievable minimum
throughput of IAB networks with the optimal scheduling and resource allocation
can outperform that of macro-only networks for a broad range of network
parameters. In addition, we analyze the effect of the network, blockage and
the array gain on the achievable minimum throughput.

Figure 2: An example realization of the locations of the BSs and users.
‘$M=1,2,4$’ denotes the locations of the MBSs when the number of MBSs is
$1,2,4$, respectively. Here, $K=10$ and $R=4$.

The locations of the MBSs are predefined and the locations of SBSs and users
are randomly generated in a 2D plane, one example realization of the network
is shown in Fig. 2. We assume the maximum node degree $C=2$, i.e., each BS can
provide backhauling to a maximum of 2 BSs, and the backhaul network expands
from the MBSs. The mmWave network is assumed to operate at 28 GHz with
bandwidth 100 MHz per link. The transmit power of SBSs is
$P_{\mathrm{SBS}}=30$ dBm and the transmit power of MBSs is
$P_{\mathrm{MBS}}=40$ dBm unless otherwise stated. The LOS and NLOS path loss
exponents and the LOS and NLOS log-normal shadowing standard deviations are
$\sigma_{\mathrm{LOS}}=3.6$ dB, $\sigma_{\mathrm{NLOS}}=9.7$ dB, respectively.
The Nakagami fading scaling parameters are $N_{\text{LOS}}=3$ and
$N_{\text{NLOS}}=2$. The LOS probability parameter is $\beta=0.01$ unless
otherwise stated. The backhaul links are assumed LOS while the access links
and the interfering links follow the random LOS/NLOS path loss model. The
array gains are $M_{t}=M_{r}=10$ dB and $m_{t}=m_{r}=-10$ dB unless otherwise
stated and the main lobe beamwidths are $\theta_{t}=30^{\circ}$ and
$\theta_{r}=90^{\circ}$. The simulation results are averaged over $10^{5}$
channel realizations.

Figure 3: Minimum throughput versus the MBS transmit power. The network
parameters for the IAB network are $M=1$, $R=2$, $K=10$, and
$P_{\mathrm{SBS}}=30$ dBm. The macro-only network consists of only $M=1$ MBS
serving $K=10$ users. $\beta$ controls the LOS probability with a larger
$\beta$ defining a denser blockage environment. Figure 4: MBS association
probability versus the MBS transmit power. The network parameters for the IAB
network are $M=1$, $K=10$, and $P_{\mathrm{SBS}}=30$ dBm. $\beta$ controls the
LOS probability with a larger $\beta$ defining a denser blockage environment.

Figure 3 compares the achievable minimum throughput between IAB and macro-only
networks for different MBS transmit power levels. It can be seen that the
achievable minimum throughput for both IAB and macro-only networks increases
with the macro BS power level, as a result of increased desired link power.
Although IAB networks require additional resources on the backhaul links, the
IAB network has a higher achievable minimum throughput than that of the macro-
only network for a broad range of MBS transmit powers, especially in denser
blockage environments. This is due to the reduced path loss and increased LOS
probability for access links. The achievable minimum throughput of the macro-
only network outperforms that of the IAB networks only when the MBS power
level is considerably larger than the SBS power level (approximately 20 times
for $\beta=0.001$ and 250 times for $\beta=0.01$), since larger MBS power
causes more interference to users associated with SBSs in IAB networks.

In Fig. 4, we show the probability that a user is associated to the MBS with
different MBS power levels and SBS density. As expected, the MBS association
probability increases with the MBS transmit power, LOS probability and
decreases with the number of SBSs, as it is more likely for a user to be
associated to BSs with shorter link distance and less path loss.

Figure 5: Minimum throughput versus the number of SBSs. The number of users is
$K=10$.

In Fig 5, we show the achievable minimum throughput for different numbers of
MBSs and SBSs. The achievable minimum throughput increases with the number of
SBSs $R$, as the network is further densified with SBSs and the links have
shorter link distance, lower path loss and higher probability for LOS
conditions on average. Also, we observe that the achievable minimum throughput
increases with the number of MBSs $M$. This is intuitive because the number of
backhauling hops decreases with $M$, since the SBSs are partitioned into IAB
sub-networks starting from the MBSs. Therefore, less time resources are spent
in backhauling links. It is worthwhile to mention that as the total number of
BSs approach or exceeds the number of users, there is a decrease in the
incremental throughput. This is because, according to the maximum received
power association rule, some BSs have no attached users and serve as relay
nodes or become inactive, when the number of SBSs increases.

Figure 6: Minimum throughput versus the main lobe array gain. The number of
users is $K=10$, the number of MBSs is $M=1$, and the number of SBSs is $R=2$.

The achievable minimum throughput for different transmit antenna main lobe
gain and beamwidth is shown in Fig. 6. Here, the results are presented for the
cases with the side lobe gains of the transmitters and the antenna parameters
of the receivers are fixed. It can be seen that larger minimum throughput can
be achieved by increasing the main lobe gain or decreasing the main lobe
beamwidth. Particularly, for the considered parameter settings, the minimum
access rate increases almost linearly with the transmit antenna main lobe
gain. Also, the relative throughput gain when increasing the main lobe gain
increases as the main lobe beamwidth decreases.

##  V  Acknowledgment

This work was supported in part by VINNOVA (Swedish Government Agency for
Innovation Systems) within the VINN Excellence Center ChaseOn.

##  VI Conclusion

In the paper, we proposed a minimum throughput maximization algorithm for
mmWave IAB networks. Our model considered the path loss difference due to
blockages and the antenna array gains depending on if AODs or AOAs are within
the main lobe. Based on the considered channel model, we maximized the minimum
throughput of each access link based on the revised simplex method. The
simulation results show that the achievable minimum throughput of IAB networks
outperforms that of the macro-only networks for a broad range of MBSs power
levels and IAB networks can provide better throughput in denser blockage
environments. Moreover, we showed that the achievable minimum throughput
increases when densifying the network with SBSs, reducing the number of
backhaul hops, increasing the transmit antenna main lobe gain or reducing the
main lobe beamwidth.

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