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1

Trafﬁc-Aware Coordinated Beamforming for

mmWave Backhauling of 5G Dense Networks

Marios Gatzianas, Member, IEEE, George Kalfas, Member, IEEE, Agapi Mesodiakaki, Member, IEEE,

Christos Vagionas, and Nikos Pleros, Senior Member, IEEE

Abstract—We study the problem of downlink Coordinated

Beamforming for dense ﬁxed-wireless millimeter wave (mmWave)

networks under a Centralized/Cloud Radio Access Network (C-

RAN) setting. To compensate for the increased mmWave path

loss, we consider directional transmissions via antenna array

analog beamforming, which leads to a discrete set of available

beams. We apply the Lyapunov optimization framework to

propose a throughput-optimal policy which explicitly accounts

for stochastic trafﬁc and channel ﬂuctuations and dynamically

performs joint Base Station-to-user association and analog beam

selection. Our model makes minimal assumptions and considers

realistic mmWave antenna radiation patterns, while it can be

easily extended to include additional MAC and PHY layer

controls. Since this ﬂexibility comes at the cost of high computa-

tional complexity, we also propose two heuristic policies offering

reduced complexity. Their performance is compared against a

baseline Round-Robin (RR)-based policy, while a theoretical

worst-case performance bound is derived. Extensive simulations

show the optimal Lyapunov policy achieving a stable throughput

increase of more than 2X per user compared to the RR policy,

while the proposed heuristics incur only a 20-30% performance

penalty with respect to the optimal Lyapunov policy with 60X-

900X computational time savings.

Index Terms—Dense 5G networks, mmWave, coordinated ana-

log beamforming, Lyapunov-based optimization.

I. INTRODUCTION

THE combination of huge-bandwidth mmWave bands,

massive MIMO processing, cell densiﬁcation, and new

architectures (e.g., C-RAN) is expected to play a key role

towards meeting the demanding 5G performance requirements.

However, these enabling technologies do not come without

challenges. Speciﬁcally, mmWave propagation faces much

higher path loss and blockage compared to sub-6 GHz bands

which can, fortunately, be mitigated using high gain antenna

arrays to create directional beams. At the same time, massive

MIMO can achieve very high spatial multiplexing within a

single cell but requires wireless channel estimation that is

much easier to perform in TDD (as opposed to the widely

deployed FDD) systems and suffers from potential Inter-Cell

Interference (ICI) from adjacent cells. ICI is a prominent

Part of this work was presented in the 2019 European Conference on

Networks and Communication (EuCNC) [1]. This work was supported in part

by the H2020 through the project 5G-COMPLETE under Grant GA 871900

and in part by the H2020 through the project 5G-PHOS under Grant GA

761989.

The authors are with the Department of Informatics, Aristotle Uni-

versity of Thessaloniki, 54124 Thessaloniki, Greece and also with the

Center for Interdisciplinary Research and Innovation (CIRI-AUTH), Aris-

totle University of Thessaloniki, 57001 Thessaloniki, Greece (e-mail:

mgkatzia@csd.auth.gr, gkalfas@csd.auth.gr, amesodia@csd.auth.gr, chva-

gion@csd.auth.gr, npleros@csd.auth.gr).

limiting factor for cell densiﬁcation as well, unless careful

multi-cell coordination is implemented [2], [3].

ICI mitigation was performed in 4G networks via Coor-

dinated Multipoint (CoMP) techniques [4], expected to also

play a key role in 5G, where multiple Base Stations (BSs)

collaborate to either (we next consider downlink only) 1)

transmit the same data to a given user on the same net-

work resource, thereby creating a distributed virtual MIMO

system that turns “interference” into useful transmission, or

2) transmit to their intended users so as not to cause high

interference to each other. In both cases, this leads to increased

SINR and throughput. The ﬁrst approach, known as Joint

Processing CoMP (CoMP-JP), requires tight BS synchroniza-

tion, accurate channel knowledge and high backhaul capacity

for exchanging user data among BSs [5], while the second

one, known as Coordinated Beamforming CoMP (CoMP-CB),

has lower backhaul requirements, as only channel information

needs to be exchanged among BSs. Although JP outperforms

CB when the aforementioned JP requirements are met, it

entails signiﬁcant processing complexity [5] and the literature

indicates that JP gains degrade when these requirements are

not met. Hence, CB is being considered as a low complexity

alternative to JP [6].

The use of mmWave presents unique challenges to CoMP,

as the inherently directional mmWave transmissions and the

fewer and weaker multipath components of the mmWave chan-

nel create a less diffuse interference environment that requires

careful BS coordination to reap the CoMP beneﬁts. In fact,

contrary to the initial belief that mmWave communications

operate in the noise-limited regime [7], later theoretical work

and simulations have shown that sufﬁciently dense networks

(e.g., cell size<100 m) actually operate in the interference-

limited regime [8].

The last remark motivates the need for enhanced CoMP

schemes in dense mmWave networks to be studied in this

paper. Although low complexity of such schemes is an at-

tractive property, this requirement can be slightly relaxed

in a C-RAN setting [9], [10] where signal processing and

coordination is performed in a centralized BaseBand Unit

(BBU) pool and computational resources are, in general,

readily available or can be easily added (if necessary) at rela-

tively low cost. The BBU pool allows for centralized resource

allocation and seamless effective BS coordination with respect

to (w.r.t.) a non-CRAN setting, while it offers high-speed

(e.g., 10/100G Ethernet) inter-BBU connectivity and relies

upon low hardware-complexity Remote Radio Heads (RRHs)

for RF processing and Over-the-Air transmission/reception.

2

The ensuing increased back/front-haul trafﬁc can be mitigated

by analog transport techniques [11].

Even though both CoMP-JP and CoMP-CB are facili-

tated by a C-RAN architecture, the inherent directionality

of mmWave transmissions implies that proper spatial beam

selection is a crucial component in both cases; hence, a

CoMP-CB solution to this component will provide useful

insights, and a stepping stone for the more involved CoMP-

JP solution. Especially for mmWave systems employing a

single RF chain per transmitter/receiver, CoMP-JP presents

additional implementation challenges, since the receiver must

employ “multi-pronged” directional patterns having multiple

lobes (within the same pattern) in order to properly “capture”

the highly-time synchronized transmission of the same symbol

by multiple transmission points. Such patterns may be difﬁcult

to produce with common phase shifter hardware in current

mmWave systems, which makes CoMP-CB a more viable

solution. As a result, this paper focuses on CoMP-CB and

leaves the extension to a CoMP-JP setting as future work.

A. Related work and contribution

Since the original proposition of CoMP for sub-6 GHz

cellular networks, the effects of imperfect channel acquisition,

limited backhaul capacity and lack of synchronization on

CoMP performance have been extensively studied (see [12],

[13] and references thereof). The literature on mmWave CoMP

is also extensive, with many works focusing on analytical

expressions for the probability of coverage and sum rate,

using stochastic geometry tools and speciﬁc models for the BS

and user locations [14]. To this end, various assumptions are

made regarding Rayleigh fading, probability of LOS/NLOS

paths and treating the direction of arrival of interfering signals

as a random variable [15], [16], while ideal (i.e., step-wise

constant) sectorized radiation patterns are also assumed, which

may be hard to generate with actual mmWave antennas.

Although very valuable, these works are based on assumptions

which are difﬁcult to hold in real networks.

In [17], analog mmWave beamforming is employed to

maximize the number of active links subject to SINR con-

straints in each active link and an iterative constant-factor

approximation algorithm is proposed along with upper/lower

performance bounds. The problem of joint beam selection,

user association and power control in mmWave hybrid beam-

forming systems is studied in [18], which pursues proportional

fairness subject to minimum rate constraints and solves the

joint problem by separately considering the individual user

association and beam selection/power control problems in an

alternating manner. Since none of the previous works considers

stochastic arrivals, any optimal policies proposed therein are

not necessarily optimal in the stochastic arrivals case, as they

cannot opportunistically exploit queue variations to prioritize

users.

The work in [19] formulates a downlink CoMP-JP OFDMA

problem for stationary users with stochastic packet arrivals

and proposes a Lyapunov-based throughput-optimal policy.

Though applying same techniques as in our work, [19] uses a

more abstract interference-based packet reception model, rely-

ing on a priori knowledge of probability of correct reception

(at the data-link layer) and assumes ﬁxed (as opposed to our

dynamic) BS-to-user association. Lyapunov drift minimization

is also employed in [20], which considers a heterogeneous

macro/small-cell C-RAN and formulates a downlink joint

admission control, user association and OFDMA Resource

Block/power allocation problem subject to average power

constraints, albeit in a non-CoMP setting. The work in [21]

studies CoMP-CB with digital beamforming in a C-RAN

setting and employs the Lyapunov framework to minimize

average RRH transmission power subject to network stability.

In the resulting joint beamforming and RRH-to-user associa-

tion problem, a heuristic association scheme is used, leading

to a beamforming only problem with an outage constraint,

which is approximately solved via semi-deﬁnite relaxation.

Since [19]–[21] study a non-mmWave setting, they do not

consider directional transmissions or analog beamforming.

This is crucial as mmWave analog beamforming is mainly

performed via quantized phase shifters [22], for cost and hard-

ware complexity reasons. Hence, the available beam control

set is discrete rather than continuous (as is typical for digital

beamforming), which implies that many of the relaxation or

approximation techniques proposed for digital beamforming

cannot be carried over to analog, invalidating any performance

guarantees offered by them.

Being the closest to our work, [23] studies the joint user as-

sociation and beam selection problem in hybrid analog/digital

beamforming with a predetermined codebook for the analog

part. Using the Lyapunov framework, [23] ﬁrst formulates

a weighted sum rate maximization problem and proposes a

greedy algorithm without any theoretical performance guaran-

tees. It then applies a simpliﬁed threshold-based interference

model to create a link conﬂict graph and recasts the problem

into a maximum weight independent set graph problem for

which an approximation greedy algorithm with a guaranteed

performance bound is proposed. Finally, as a heuristic solution

to the original problem, a new problem of maximizing the

number of scheduled links subject to SINR constraints is

examined and another greedy approximation algorithm with

a guaranteed performance bound is proposed. Though very

elegant, the ﬁrst bound depends on the conﬂict graph structure,

which may be difﬁcult to predict, and the second one is

derived for idealized channel and antenna gain (i.e., step-

wise constant) models. In realistic scenarios, where antenna

side lobes may be present, both bounds may be too loose.

Unfortunately, due to the high computational complexity of

the optimal algorithm, [23] evaluates the performance of the

proposed greedy algorithms and only compares among them

and not w.r.t. the optimal one.

The differences of our paper with the above works and our

contributions are as follows:

• We study a mmWave CoMP-CB problem with purely

analog beamforming for 5G back/mid-haul and create a model

with minimal assumptions regarding interference that also

accounts for realistic (i.e., not step-wise constant), instead

of idealized, antenna beams in addition to stochastic trafﬁc

arrivals and channel ﬂuctuations. Our treatment of analog, as

opposed to hybrid, beamforming is motivated by its lower

cost, lower power consumption, lower hardware complexity

3

and required feedback overhead (especially regarding channel

estimation), as well as by recent literature [24], [25], which

even proposes outdoors usage of analog-beamforming-based

802.11ad systems [26]. At the same time, the design of an

analog beamformer is an important building block of the

hybrid beamforming solution, since the latter employs a single

analog beamformer for all subcarriers in OFDM-based systems

[27].

• Using the well-known Lyapunov optimization framework,

we formulate a joint user association and beam selection prob-

lem and propose a throughput-optimal policy which requires a

scheduling problem to be solved at each slot. Due to the high

computational complexity of this policy, we also propose two

suboptimal heuristic policies, namely Serve-Longest-Queues

(SLQ) and Limited-Beam-Set (LBS), with better scalability

w.r.t. the number of BSs and users. Note that the decision

of user association is dynamically performed by the proposed

policies and is not a priori ﬁxed.

• We provide a theoretical worst-case performance bound

for LBS based on ofﬂine knowledge of network topology and

employed radiation patterns. Unfortunately, this bound can be

loose (i.e., pessimistic) so that we also perform an extensive

simulation-driven performance study, where the optimal policy

is actually simulated to provide a yardstick against which the

heuristic policies are evaluated. We also employ a simpler

baseline Round-Robin-based (RR) policy to determine whether

the optimal policy and its accompanying heuristics are “worth

the extra effort” performance-wise. Simulation results show

the optimal policy achieving an average of 2X throughput

increase per user compared to RR, while the heuristic policies

perform only 20-30% worse than the optimal policy with huge

computational savings.

The proposed Lyapunov-based policy can also be readily

modiﬁed to account for additional factors (such as power

control) and can be integrated into a protocol stack, where an

external MAC layer algorithm determines the user schedul-

ing and the Lyapunov framework determines the PHY layer

controls, such as beam selection. Although we do not aim

at full compliance with current 5G standardization activities,

all presented policies can, in principle, be implemented at a

Central or Distributed Unit (CU, DU), as long as it meets

the requirements set forth by the policies in term of available

computational resources.

The rest of this paper is structured as follows: Section II

contains the system model and problem deﬁnition, while

Section III presents the proposed policies. A theoretical per-

formance bound for policy LBS is derived in Section IV and a

framework for comparing different policies performance-wise

is described. The policies are evaluated via simulations and

the results are presented in Section V, while Section VI con-

cludes the paper. Notation-wise, vector quantities are denoted

with boldface aand sets with calligraphic A. Expectation is

denoted as E[·], set cardinality as |·|, Cartesian set product

as ×and equality by deﬁnition as M

=. Finally, for any integer

N > 0, we denote [N]M

={1,2, . . . , N }.

BBU Pool/Cntrl

PON

Splitter

Nla=9

Nro=3

Fig. 1. Dense network with Nro rooftop and Nla lamppost antennas

interconnected via a PON topology with the BBU pool, where the centralized

controller (Cntrl) is located. Rooftop antennas can generate multiple radiation

patterns (aka. beams) for transmission via analog beamforming.

II. SY ST EM M OD EL

We consider a dense area network consisting of Nro rooftop

and Nla lamppost antennas, as depicted in Fig. 1. All antennas

are stationary and their numbers and positions are assumed to

be known a priori to an operator-owned centralized Controller

(hereafter abbreviated as Cntrl) in the BBU pool, which

coordinates the entire network. We focus on the downlink

rooftop-to-lamppost communication only and the more typical

case Nro ≤Nla, although the proposed policies can naturally

handle the case Nro > Nla as well.

Each rooftop antenna also has a ﬁber connection to

BBU/Cntrl, possibly via a Passive Optical Network (PON).

The PON is included only to showcase the compatibility of

the examined model with current deployments, as the proposed

model is agnostic to the employed transport network topology.

The lampposts are assumed to host additional equipment

(e.g., eNBs, gNBs or Small Cell BSs) to provide wireless

access to the “real” users served by them. Any Radio Access

Technology can be employed between the lampposts and their

actually served users, as the subsequent analysis does not

depend on this. However, depending on the type of lamppost

access network equipment and employed functional split,

the rooftop-to-lamppost link can be considered as part of a

back/mid/front-haul transport network segment with differ-

ent throughput and latency requirements (i.e., higher layer

functional splits typically have less stringent requirements).

Since fronthaul trafﬁc typically has a Constant Bit Rate proﬁle

with very tight latency requirements and the policies to be

proposed do not offer deterministic latency guarantees, the

examined setting in this paper mainly refers to a backhaul (or

even midhaul) transport scenario with more relaxed latency

requirements.

From the point of view of the rooftop-to-lamppost link

(which is the paper’s focus), the lampposts can be considered

as trafﬁc aggregators of all “real” users in their coverage and

are, hence, treated as Nla “effective” users (hereafter referred

to as users) w.r.t. the rooftop antennas. Since we focus on a

backhaul scenario and the lampposts are not UEs themselves,

the lampposts are not required to use the PHY of current

5G standards (also note that, due to the single RF chain at

rooftops/lampposts, SU/MU-MIMO is not applicable).

The rooftop-to-lamppost link operates in the mmWave

bands; hence, both ends may use antenna arrays of discrete

4

(a) ς= 60◦. (b) ς=−60◦. (c) ς= 120◦. (d) ς=−120◦.

-20

-10

0

10

20

150

180 0

30

60

90

120

Pattern #5

(e) ς= 180◦.

Fig. 2. Gain azimuth radiation patterns of a 5-microstrip-element rooftop linear array for different phase progression (denoted as ς) values between adjacent

radiating elements. Magnitude units are in dBi (see [28] for details regarding the modeling used to construct these patterns).

radiating elements. Due to the larger real estate available in

the rooftop antenna (compared to a lamppost), each rooftop

antenna is equipped with specialized hardware (e.g., phase-

shifters, attenuators, power couplers) that can modify the

signal fed into each radiating element, thus creating a phased

array that can generate N(pat)

ro different radiation patterns via

analog beamforming. Note that, depending on relative place-

ment, different beams from different rooftops may spatially

interfere with each other, as indicated by the dotted oval in

Fig. 1. For illustration purposes, an example of N(pat)

ro = 5

azimuth radiation patterns for each rooftop antenna is shown

in Fig. 2 for a linear array of 5 circular microstrip elements at

60 GHz. The parameters of each individual microstrip element

(i.e., radius of 1.5 mm and height separation of 0.09993 mm

from the ground plane), the element separation (i.e., 0.625λat

60 GHz) and respective radiation patterns have been computed

by Matlab’s Antenna Toolbox.

Due to the limited real estate in lampposts and general trend

towards simplifying Small Cell BS hardware, lamppost anten-

nas typically have reduced capabilities (compared to rooftops),

in terms of the number of radiating elements, beamforming

capabilities and gain and can only employ N(pat)

la different

radiation patterns, where N(pat)

la < N(pat)

ro . For sufﬁciently

dense networks, it may even be possible to use omnidirectional

beams at the lampposts.

We consider a discrete-time system where slot tcorresponds

to the continuous time interval [tT, tT +T), where Tis the

slot duration. At the end of each slot t, an amount Al(t)

of exogenous trafﬁc (in bits) arrives for each user l, with

Al(t)being independent (for different users l) processes with

mean E[Al(t)] = λl.Cntrl maintains a group of First-In First-

Out (FIFO) queues Ql, where Qlstores the bits intended for

lamppost l(actually, intended for the users served by lamppost

l). The queue size for user lat the beginning of slot tis

denoted as Ql(t).

We hereafter denote with k∈[Nro]and l∈[Nla]the

indices for the rooftop and lamppost antennas, respectively. At

each slot, all wireless transmissions from rooftops to lampposts

occur over the same frequency band and, hence, interfere

with each other to an extent determined by the antennas’

physical proximity, channel propagation characteristics and the

speciﬁc radiation patterns used. Also, at each slot, a rooftop

transmission can carry trafﬁc for exactly one user/lamppost.

The notation (k→l)is hereafter used to state that rooftop

kserves trafﬁc for lamppost l. For simplicity, all rooftops

are assumed to transmit at the same power Pt; however,

power control is naturally supported by our model and can

be incorporated by enlarging the available control set, at the

cost of increased computational complexity. The quantitative

performance beneﬁt of this additional control is an interesting

topic of future work.

A. Notation and deﬁnitions

Deﬁnition 1: A set Sof rooftop-to-lamppost assignments

(i.e., S ⊆ {(k→l) : k∈[Nr o], l ∈[Nla ]}) is valid iff

the following properties hold: P1) there do not exist distinct

k, k0∈[Nr o]such that (k→l)∈ S and (k0→l)∈ S for

some l∈[Nla], and P2) if it holds Nro ≤Nla, then, for

each k∈[Nro], there exists exactly one l∈[Nla ]such that

(k→l)∈ S.

P1 in Deﬁnition 1 forbids multiple rooftops to collaborate

towards serving the same lamppost, while P2 forces each

rooftop to serve a lamppost. Note that there exist pathological

topologies (e.g., all rooftops and lampposts being placed along

a straight line) where it would be advantageous for a rooftop

to not serve any lamppost at all (thus, violating the second

property) in order to avoid creating strong interference to

other transmissions and increase the sum throughput for the

current slot. However, this action would also increase the

latency for some lamppost that would otherwise be served

in the slot (i.e., if the second property were imposed) and

this might be unacceptable for transport networks with very

tight latency requirements, which motivates the need for the

second property. P1 does not constitute a modeling limitation

per se, since it is easy to account for intentionally idle rooftops

(i.e., switched off to counter interference) by enlarging the

control sets, as described in [28]. Clearly, more than one valid

assignment sets Sexist for given Nro, Nla;1we denote with

˚

Mthe class containing all valid assignments S.

Let Gro,Gla be the sets of radiation patterns/beams that can

be used by a single rooftop and lampost antenna, respectively.

Note that each element of Gro,Gla is a function that essentially

describes the antenna gain variation vs. azimuth (including

gain variation vs. elevation is a straightforward extension)

for each pattern, as illustrated in Fig. 2. We denote with

bk(resp. ˇ

bl) the pattern/beam index selected by rooftop k

(resp. lamppost l), where bk∈hN(pat)

ro iand ˇ

bl∈hN(pat)

la i.

We also denote with Gbk(φk)(resp. ˇ

Gˇ

bl(ˇ

φl)) the speciﬁc

antenna gain value, for the selected pattern index bk(resp. ˇ

bl),

at azimuth φk(resp. ˇ

φl) w.r.t. a local coordinate system collo-

cated with each rooftop (resp. lamppost), which is assumed to

1For example, for Nro = 2,Nla = 3, assignment sets {(1 →2),(2 →

3)},{(1 →3),(2 →2)}are both valid.

5

be known to Cntrl. Finally, we introduce shorthand notation

bM

= (bk:k∈[Nro]),ˇ

bM

=ˇ

bl:l∈[Nla]for the pattern

indices selected by all rooftops and lampposts, respectively,

and Gb

M

= (Gbk(·) : k∈[Nro]),ˇ

Gˇ

b

M

=ˇ

Gˇ

bl(·) : l∈[Nla]for

the selected pattern functions. Deﬁning Gnet

M

=GNro

ro × GNla

la

as the set of all possible patterns for the entire network, it

follows that (Gb,ˇ

Gˇ

b)∈ Gnet.

At each slot t,Cntrl must decide upon the following controls

for the entire network:

•the speciﬁc valid assignment S(t)∈˚

M, i.e., which

rooftop will serve which lamppost.

•the speciﬁc radiation pattern indices b(t),ˇ

b(t)and their

associated patterns (Gb(t),ˇ

Gˇ

b(t))∈ Gnet to be used by

all rooftops and lampposts.

The second control in the above list may, at ﬁrst, seem

superﬂuous, since a valid assignment S(t)suggests that, to

achieve reduced interference and increased system throughput,

the antenna patterns should be selected so that the respective

rooftop and lamppost antenna beams point towards each

other. However, such perfect alignment is not always possible,

due to the discrete nature of Gnet. Additionally, given the

geographically-constrained antenna placement in an urban

environment, it is still possible for two rooftop transmissions

to strongly interfere with each other, even when they are

perfectly aligned w.r.t. their assigned lampposts under S, as

shown in the oval box of Fig. 1. Hence, a greedy approach

targeting “perfect” alignment between each rooftop and its

served lamppost may not always be optimal, which motivates

the second control above.

In summary, at each slot t,Cntrl selects a control a(t)M

=

S(t),(Gb(t),ˇ

Gˇ

b(t))in the (a priori known to it) control set

AM

=˚

M×Gnet. A rule for selecting a(t)∈ A is referred to as

apolicy πand, obviously, multiple policies can be constructed,

as will be described in Section III.

We denote with hk,l(t)the state, at slot t, of the wireless

channel between rooftop kand lamppost l. The channel is

assumed to be constant for the duration of the slot but may

change between slots. Although rooftop and lamppost antennas

have multiple radiating elements, hk,l(t)is still scalar instead

of a vector since there is a single RF chain per antenna (due

to the purely analog beamforming), which implies that any

pilot-based channel estimation procedure will yield a non-

vector result. By virtue of the antennas being stationary and

predominantly experiencing LOS channel conditions (due to

the elevated rooftops), hk,l(t)typically varies slowly in time.

Following the literature (e.g., [29]), we write

hk,l(t) = qβk ,l(t)ξk,l (t),(1)

where βk,l(t)captures large-scale fading effects (i.e., path loss,

LOS/NLOS conditions, shadowing from intervening objects,

gain variations etc.) and ξk,l(t)captures small-scale fading

(i.e., multipath caused by reﬂections from objects in the

vicinity of the lampposts) effects. No other assumption is made

regarding the model producing the values of βk,l(t), ξk,l (t),

so that any model for large- and small-scale fading can be

employed. The decomposition of (1) into large- and small-

scale fading is common in the literature (where it is typically

presented as an additive decomposition in dB scale) and is our

starting point, since it allows us to clearly incorporate the ef-

fect of antenna gain (which affects propagation and, therefore,

falls under large-scale effect) as the important control variable

in our problem. Thus, we can “isolate” its contribution from

other non-controllable factors towards computing the SINR at

each lamppost. It should also be noted that our conceptual

incorporation of a control variable (i.e., antenna gain) into the

channel is, essentially, a modeling decision, which also appears

in the literature [30], [31].

An important remark is that beam selection in mmWave

affects large-scale fading much more than small-scale fading,

due to the less pronounced mmWave multipath effects (par-

ticularly for elevated lampposts). Hence, ξk,l(t)is assumed to

be invariant w.r.t. beam selection.

The transmitted power Ptx,k(t)from rooftop kand re-

ceived power Prx,l(t)at lamppost l, when beam indices

bk(t),ˇ

bl(t), respectively, are selected, are related through

Prx,l(t)

Ptx,k(t)=βk,l (t)|ξk,l (t)|2. Friis equation [32] then implies

that Prx,l(t)

Ptx,k(t)∝Gbk(t)(˙

φl,k)ˇ

Gˇ

bl(t)(¨

φk,l), where ˙

φl,k (resp. ¨

φk,l)

is the relative angle of lw.r.t. k(resp. relative angle of

kw.r.t. l) and the proportionality factor is independent of

beam selection. Since ξk,l(t)is also invariant w.r.t. the beam

selection2, it holds

βk,l(t) = Gbk(t)(˙

φl,k)ˇ

Gˇ

bl(t)(¨

φk,l)˜

βk,l(t),(2)

where ˜

βk,l(t)captures only the path loss and shadowing

effects but not directional transmissions. Note that although

the selected beam bk(t)varies with time, its gain is always

computed at the ﬁxed azimuth value ˙

φl,k (similarly, for the

second term in (2)). Combining (2), (1) yields

hk,l(t) = qGbk(t)(˙

φl,k)ˇ

Gˇ

bl(t)(¨

φk,l)˜

hk,l(t),(3)

which is similar to the (scalar) channel model used in [33],

[34] (to describe both LOS and NLOS environments), where

˜

hk,l(t)M

=q˜

βk,l(t)ξk ,l(t)can be estimated using omnidi-

rectional transmission/reception. Following standard literature

on stochastic network optimization [35]–[37] we assume that

Cntrl can perfectly estimate ˜

hk,l(t).

Denoting with xk(t)the signal transmitted by rooftop k, the

received signal yl(t)at lamppost lis given by

yl(t) = X

k∈[Nro]

hk,l(t)xk(t) + wl(t),(4)

where wl(t)∼ CN (0, N0)is AWGN thermal noise. Note

that whether lamppost lreceives actual information intended

for it at slot tdepends on whether lis actually scheduled

under the assignment S(t), i.e., whether there exists some

(k→l)∈ S(t). Hence, using (3), (4), the SINR at lamppost

2This invariance assumption is only made for simplicity and to allow for

lighter notation. If this assumption was removed and the small-scale fading

also depended on the speciﬁc patterns used by k, l (written as ξk,l,bk,ˇ

bl(t)),

our analysis and Lyapunov optimization would still hold. Under Assumption

2 (to appear later), the theoretical result in Lemma 3 would also hold, albeit

for a larger ϑthat now accounts for beam variations in small-scale fading.

6

(SI N R)lS(t),Gb(t),ˇ

Gˇ

b(t)

| {z }

a(t)

=

Pt|˜

hk,l(t)|2Gbk(t)(˙

φl,k)ˇ

Gˇ

bl(t)(¨

φk,l)

N0+Pk06=kPt|˜

hk0,l(t)|2Gbk0(t)(˙

φl,k0)ˇ

Gˇ

bl(t)(¨

φk0,l)if ∃ks.t. (k→l)∈ S(t),

0otherwise.

(5)

lunder any valid assignment S(t)∈˚

Mand beam selection

Gb(t),ˇ

Gˇ

b(t)∈ Gnet is easily computed in (5) (top of page).

Knowing the SINR at each lamppost lfor control a(t)∈ A,

the achievable throughput Rl(a(t)) for lamppost lis upper

bounded by

Rl(a(t)) = Wlog2(1 + (SI N R)l(a(t))) ,(6)

where Wis the bandwidth used. Since the performance

of different policies for selecting the control a(t)will be

compared to each other, the exact value of Wis unimportant,

as it only provides a constant multiplicative factor for all

policies. Hence, W= 1 is assumed (note that, for general

W, the arrivals Al(t)should also be suitably scaled by W).

The Nla queues at Cntrl now evolve as

Ql(t+ 1) = max(Ql(t)−Rl(a(t)),0) + Al(t),(7)

which expresses the fact that the number of actually trans-

mitted bits cannot exceed the number of available bits in the

queue [35]. Notice that all Al(t)exogenously arriving bits are

admitted into the network, so it is possible for Ql(t)to grow

without bound for sufﬁciently high arrival rates λl. Also, due

to the stochastic nature of Al(t)(and Rl(a(t)), which depends

on the stochastic ˜

hk,l(t)), the queue levels Ql(t)are stochastic

processes, which are crucially affected by the applied policy.

To quantify this, we use the concepts of stability and stability

region [35], [38].

Deﬁnition 2: For a given (mean) arrival rate λl, the queue Ql

is (mean-rate) stable under the application of a policy π(i.e., a

rule for selecting a(t)) iff it holds limt→∞

E[Qπ

l(t)]

t= 0, where

the πsuperscript emphasizes the fact that the queue evolution

explicitly depends on policy π.

Deﬁnition 3: The network is stable for a given arrival vector

rate λM

= (λl:l∈[Nla]) under a policy πiff all queues Qπ

l(t),

for l∈[Nla], are stable under π. The stability region Rπof

a policy πis the set of arrival rates λfor which the network

is stable under π.

Deﬁnition 4: The network stability region ¯

Ris the set of

arrival rates λwhich can be stabilized under some policy π.

Equivalently, ¯

R=Sπ∈ΠRπ, where Πis the set of all possible

policies.

Obviously, Rπ,¯

Rare Nla-dimensional regions (see Fig. 3

for an illustration for Nla = 2).

B. Problem statement

This paper will answer the following questions:

1) Does there exist an optimal policy πopt that dominates

any other policy, in the sense that Rπopt ⊇ Rπfor any

policy π∈Π? Clearly, if πopt policy exists, it holds

¯

R=Rπopt .

l1

l2

0

A1A3

A2

B1

B3

B2

!"boundary

!#$boundary

!%$boundary

Fig. 3. Stability regions of policies π1,π2,π3. Each lampost arrival rate

ratio λ1:λ2(equivalently, λ1

λ2) corresponds to a ray emanating from the

origin. Policy π3dominates π1,π2, as its stability boundary is outside that

of π1,π2for all possible rays.

2) Is there a systematic way to compare the stability regions

of two different (non-optimal) policies or determine how

“far away” from optimal a policy performs?

The ﬁrst question is answered in the afﬁrmative via a standard

Lyapunov drift minimization argument while a simulation-

driven bisection method will be employed to determine the

boundary of any policy’s stability region and answer the

second question. Especially for one of the suboptimal policies

(i.e., LBS), a theoretical worst-case performance bound will

also be derived which, however, may be pessimistic in some

cases. This fact further supports the need for extensive sim-

ulations to determine the actual performance of a policy. We

next describe the optimal policy as well as some suboptimal

policies of lower computational complexity.

III. PROPOSED POLICIES

We ﬁrst provide the optimal Lyapunov-based CB policy,

hereafter termed LCB:

Policy LCB: at slot t,Cntrl observes Ql(t), for l∈

[Nla], and ˜

h(t)M

=˜

hk,l(t) : k∈[Nr o], l ∈[Nla]and se-

lects control aLCB(t)∈ A as aLCB(t) = arg maxa(t)∈A

PNla

l=1 Ql(t)Rl(a(t)).

The following Lemma, proved in [35] (Chapter 3, Section

3.1), shows that LCB is indeed the optimal policy πopt.

Lemma 1: Policy LCB is (throughput) optimal, i.e., it holds

RLCB =¯

R.

Due to the discrete nature of the control set Aand

the complicated way in which a(t)affects Rl(a(t)) via

(5), the obvious solution of LCB via exhaustive search

is computationally expensive, since the number of pos-

sible control choices (i.e., |A|) is given by |A| =

Nla!

(Nla−Nro )! N(pat)

ro Nro N(pat)

la Nro

.3To this end, we next

propose suboptimal policies that search over a smaller set

˘

A ⊆ A when trying to optimize the objective function of LCB,

3For the case Nro ≤Nla, it sufﬁces to consider only the patterns of

the Nro lampposts actually served by the rooftops, which explains the Nro

exponent in the last term.

7

as well as a Round-Robin based policy that uses Sum Rate as

the maximization objective. Furthermore, in the high SINR

regime (i.e., assuming negligible thermal noise compared to

interference), it is possible to cast the optimization problem

of LCB into an exact Geometric Programming formulation

combined with a Branch-and-Bound approach to handle the

essentially discrete nature of A. This approach requires an

exhaustive search only over the (much smaller) set ˚

Mand

produces an -optimal solution to the optimization in LCB.

Furthermore, it can also be applied to the other policies

proposed in this paper. Due to space constraints, this will not

be further discussed here but full details are provided in [28].

The ﬁrst suboptimal policy, termed Serve-Longest-Queues

(SLQ), is based on the intuitive idea of serving, at each slot t,

only the Nro lampposts with the longest queues (i.e., largest

Ql(t)). However, the exact assignment of which rooftop will

serve each of these Nro lampposts as well as which beam will

be used is determined via enumeration.

Policy SLQ: at slot t,Cntrl observes Ql(t), for l∈[Nla],

and ˜

h(t)and performs the following:

1) Out of Nla lampposts, ﬁnd these with the Nro largest

queues Ql(t)and collect them into the set L(t)

(i.e., L(t)thus contains the actually served lampposts

at slot t).

2) Construct the set ˚

MSLQ(t)containing the valid rooftop-

to-lamppost assignments SSLQ(t), where SSLQ (t)M

=

{(k→l) : k∈[Nro], l ∈ L(t)}.

3) Deﬁne ˘

ASLQ(t)M

=˚

MSLQ(t)× Gnet and select control

aSLQ(t)∈˘

ASLQ(t)as aSLQ (t) = arg maxa(t)∈˘

ASLQ(t)

PNla

l=1 Ql(t)Rl(a(t)).

It is easy to see that

˘

ASLQ(t)

=Nro!N(pat)

ro Nro

N(pat)

la Nro

, which offers signiﬁcant computational savings

compared to LCB.

Along with reducing the number of available controls

by considering a limited set of lamppost assignments, it

is also possible to limit the number of beams/patterns for

each rooftop-to-lamppost assignment examined by a policy.

This approach is employed by the next suboptimal policy

(termed Limited-Beam-Set, LBS) and is based on the intuition

that, for a given assignment S, only the beams “strongly”

pointing along the selected rooftop-lamppost direction need

to be examined as the other beams are likely to yield a much

lower SINR (see Fig. 4 for an illustration where, for graphical

simplicity, directional rooftops and omnidirectional lampposts

are assumed).

Policy LBS:Cntrl performs the following ofﬂine procedure

once to determine the limited beam/pattern sets to be used. For

integers Dro,Dla >0such that Dr o ≤N(pat)

ro and Dla ≤

N(pat)

la :

1) For each k∈[Nro]and each l∈[Nla ], collect the

indices bkof the rooftop patterns with the Dro highest

values of Gk(˙

φl,k)into set Bk→land, similarly, collect

the indices ˇ

blof the lamppost patterns with the Dla

highest values of ˇ

Gl(¨

φk,l)into set ˇ

Bk→l.

R1 R2

R3

L1

L2

L3

L4

Fig. 4. Assignment-dependent beam selection under LBS, assuming omnidi-

rectional lampposts: for the assignment S={(1 →2),(2 →1),(3 →3)},

out of the 5 beams per rooftop, only the D= 3 beams (shown in stripes)

“closest” to the direction of transmission (equivalently, the beams with the D

highest gain values in this direction) selected under Sare considered.

2) For each S ∈ ˚

M, construct sets

Gro(S)M

=×

k∈[Nro]

(k→l)∈S

{Gbk(·) : bk∈ Bk→l},

Gla(S)M

=×

l∈[Nla]

(k→l)∈S

ˇ

Gˇ

bl(·) : ˇ

bl∈ˇ

Bk→l,(8)

and deﬁne Gnet(S)M

=Gro(S)× Gla(S).

Note that Gro(S),Gla(S)essentially contain the striped beams

for Sshown in Fig. 4 (each beam corresponding to a distinct

pattern), which capture, for each S, the Dro, Dla “best LOS”

patterns (in terms of angular alignment between the respective

pattern’s main beam and the rooftop-to-lamppost direction,

leading to the Dro, Dla, respectively, highest gain values

in this direction). Although we could pick Dro =Dla =

1, this choice may be restrictive towards countering cross-

transmission interference; the most meaningful non-trivial

choice is Dro =Dla = 3 which, for each S, selects for each

rooftop/lamppost the “best LOS” pattern and the two other

patterns on either side of the angular domain in Fig. 4. Notice

that, when it holds Dro =N(pat)

ro and Dla =N(pat)

la ,LBS

becomes formally identical to LCB.

At each slot t,Cntrl observes Ql(t), for

l∈[Nla], and ˜

h(t)and selects control aLBS(t)as

aLBS(t) = arg maxa(t)∈˘

ALBS PNla

l=1 Ql(t)Rl(a(t)),

where ˘

ALBS M

=n(S,Gnet(S)) : S ∈ ˚

Mo. It follows that

˘

ALBS

=Nla!

Nro!(Nla −Nro !) DNro

ro DNro

la , which offers even higher

(compared with SLQ) computational savings w.r.t. LCB for

small Nro,Nla.

Note that the essential approach of LBS, namely limiting the

search only among the patterns in Gnet(S)for each assignment

S, is a general principle that can be applied to any policy.

Hence, we apply this to SLQ to construct a parameterized

version, named SLQK(where Kis a positive integer), as

follows:

Policy SLQK:

1) Same as step 1 of LBS by setting Dro =Dla =K

(i.e., ﬁnd the K“best” patterns for all rooftops and

lampposts).

2) Same as step 1 of SLQ.

3) Same as step 2 of SLQ, i.e., construct ˚

MSLQ(t).

8

4) Deﬁne the search space ˘

ASLQK(t)M

=

n(S,Gnet(S)) : S ∈ ˚

MSLQ(t)o, where Gnet (S)is

constructed by (8), and maximize the same objective as

in step 3 of SLQ over ˘

ASLQK(t).

Essentially, SLQKsearches at slot tover the limited patterns

as well, albeit only for the valid assignments dictated by SLQ,

and can be considered a “hybrid” between SLQ and LBS. It

follows immediately that

˘

ASLQK(t)

=Nro!K2Nr o .

In all of the above policies, once the action a(t)is deter-

mined, the corresponding lampposts are served and queues are

updated via (7).

Finally, to examine the effect of ignoring queue levels

when selecting control a(t), we propose a baseline policy

(termed RRSM) as a combination of Round-Robin and Sum

Rate maximization.

Policy RRSM: at slot t,Cntrl observes ˜

h(t)and performs

the following actions:

1) Select the rooftop-lamppost assignment SRRSM(t)∈˚

M

in a (temporal) Round-Robin fashion (i.e., enumerate all

elements of ˚

Mand sequentially select one of them).

2) Select the radiation patterns Gb(t),ˇ

Gˇ

b(t)RRSM

as

Gb(t),ˇ

Gˇ

b(t)RRSM

= arg max

(Gb,ˇ

Gˇ

b)∈Gnet

Nla

X

l=1

RlSRRSM(t),Gb,ˇ

Gˇ

b.

(9)

3) Update the queues via (7).

Since RRSM only performs an optimization over Gnet,

its search space ˘

ARRSM is equal to Gnet and much

smaller compared to LCB, i.e., it holds

˘

ARRSM

=

N(pat)

ro Nro N(pat)

la Nro

.

Clearly, RRSM allows for inefﬁcient mappings in ˚

Mto be

used (e.g., a rooftop serving a far-away lamppost instead of

another one close by). It is possible to create more “intelligent”

versions of RRSM, say by selecting, in Round-Robin, only

the subsets of the lampposts to be served in the current

slot and then computing the rooftop assignments and pattern

selections by enumeration. Such a policy (named RRSMi)

has comparable complexity to SLQ and the optimization

performed at each slot tis similar to that in SLQ, albeit

without the queue terms. However, simulation results (omitted

due to lack of space) indicate that RRSMi does not always

outperform RRSM and is, in fact, dominated by SLQ, which

offers far superior performance at the same complexity as

RRSMi. This fact removes the need for more involved Round-

Robin-based policies and we retain RRSM as a typical queue-

agnostic policy for comparison purposes. Finally, RRSM can

be parameterized in a similar fashion to SLQ to yield a new

policy, named RRSMK, that solves (9) over the smaller search

space ˘

ARRSMKM

=Gnet (SRRSM(t)) (see [28]).

IV. THEORETICAL PERFORMANCE BOUNDS AND POLICY

CO MPAR IS ON

Although our model makes minimal assumptions for the

large- and small-scale fading distributions and characteristics,

we impose some additional conditions in the theoretical anal-

ysis:

Assumption 1: The large-scale fading coefﬁcients ˜

βk,l(t)

(see description after (2)) are time-invariant and proportional

to d−α

k,l , where dk,l is the Euclidean distance between rooftop

kand lamppost land α≥2is the path-loss exponent (free

space conditions correspond to α= 2).

Assumption 2 ([39]): There exists some constant ϑ≥1

such that the small-scale fading coefﬁcients ξk,l(t)satisfy 1

ϑ≤

|ξk,l(t)|2≤ϑ, for all slots tand all rooftops k∈[Nr o]and

lampposts l∈[Nla], with ϑ= 1 corresponding to a non-

multipath fading environment.

To ease the notation and given that our system model mainly

applies to urban scenarios, with small distances between

rooftops and lampposts, we concentrate on the interference-

limited regime (i.e., assuming thermal noise to be negligible

compared to interference), so that we hereafter consider SIR

instead of SINR. However, it is worth noting that our analysis

can be easily extended to an SINR setting with simple modi-

ﬁcations. In the remainder of the manuscript, we denote with

sl(S)the serving rooftop for lamppost lunder assignment S,

so that sl(S) = 0 if lamppost lis not served under assignment

S. We deﬁne the SIR achieved at lamppost lunder controls

S,Gb,ˇ

Gˇ

bwhen small-scale fading is ignored as

˜σlS,Gb,ˇ

Gˇ

bM

=

d−α

sl(S),lGbsl(S)˙

φl,sl(S)ˇ

Gˇ

bl¨

φsl(S),l

Pk06=sl(S)d−α

k0,lGbk0˙

φl,k0ˇ

Gˇ

bl¨

φk0,l

×I[sl(S)6= 0] ,

(10)

where I[·]is an indicator function for the event (·).

For notational convenience, we also deﬁne σS,Gb,ˇ

Gˇ

b

M

=σlS,Gb,ˇ

Gˇ

b:l∈[Nla], and ¯

σS,Gb,ˇ

Gˇ

bM

=

1

ϑ2˜

σS,Gb,ˇ

Gˇ

b,¯

σS,Gb,ˇ

Gˇ

bM

=ϑ2˜

σS,Gb,ˇ

Gˇ

b.

Corollary 1: For any small-scale fading values ξsatisfy-

ing Assumption 2 and a large-scale fading path-loss expo-

nent αaccording to Assumption 1, the actual SIR vector

σξ,S,Gb,ˇ

Gˇ

bsatisﬁes the condition: ¯

σS,Gb,ˇ

Gˇ

b≤

σξ,S,Gb,ˇ

Gˇ

b≤¯

σS,Gb,ˇ

Gˇ

b[28].

To provide a deterministic guarantee for the stability region

of any policy, the following Lemma from [35] is used:

Lemma 2: Consider a policy πwhich, at each slot t, selects

a control aπ(t)that satisﬁes

Nla

X

l=1

Ql(t)Rl(aπ(t)) ≥υ·W SR∗(t)−C, (11)

for some constants 0< υ ≤1and C > 0, where W SR∗(t)is

the weighted sum rate at slot tof the optimal LCB solution.

Then, πstabilizes all arrival rates within υ¯

R, i.e., Rπ⊇υ¯

R.

Since Deﬁnition 4 implies Rπ⊆¯

R, we conclude that any

policy πthat satisﬁes Lemma 2 also satisﬁes υ¯

R⊆Rπ⊆¯

R

so that, the closer υis to 1, the better the performance of

π. Hence, for any of the proposed suboptimal policies SLQ,

LBS, we should examine whether it satisﬁes Lemma 2 and

determine the corresponding υfor it.

Determining the υthat satisﬁes Lemma 2 for SLQ is

extremely challenging since any bounding procedure for υ

9

W SRLBS (t)≥

min

S∈ ˚

M

min

Fro⊆[Nr o],

Fla⊆[Nla ]

min l∈[Nla]:

sl(S)∈Fro

log21+ $l(S)γl(Fro ,S)

ϑ2·¯

˚σl(S)

log2(1+¯

˚σl(S))

z }| {

Υ (Fro,S) Ψ (Fla,S)

| {z }

minl∈Fla log2(1+∆l(S)·¯

˚σl(S))

log2(1+¯

˚σl(S))

·W SR∗(t),(13)

γl(Fro,S)M

= max

Gb0∈Mb(S,Fro)min

(Gb,ˇ

Gˇ

b)∈ES(Gnet)

Pκ6=sl(S)d−α

κ,l Gbκ˙

φl,κˇ

Gˇ

bl¨

φκ,l

Pκ6=sl(S)d−α

κ,l Gb0

κ˙

φl,κˇ

Gˇ

bl¨

φκ,l,(14)

will require proving deterministic bounds for the queue sizes

or their ratios (as SLQ selects the Nro largest queue sizes at

each slot), which is typically a very difﬁcult task. Notice that

Deﬁnition 2 on stability contains an expected value only and

theoretically even allows for a queue to slowly grow to inﬁnity

and still remain stable. However, such queue size bounds are

not required for LBS, which makes the analysis more tractable.

Speciﬁcally, the following parameters, which will appear in

subsequent results and are collected here for conciseness, can

be computed a priori based only on network topology and

radiation pattern information.

Deﬁnition 5: For any rooftop-to-lamppost assignment S, any

rooftop κand any lamppost lthat is served under S, we deﬁne

the following scalar parameters:

ζκ,l

M

=

(Dla)-th largest in ˇ

Gˇ

b¨

φκ,l

(Dla + 1)-th largest in ˇ

Gˇ

b¨

φκ,l,

$l(S)M

=

(Dro)-th largest in Gb˙

φl,sl(S)

(Dro + 1)-th largest in Gb˙

φl,sl(S),

Zl(S)M

=ζsl(S),l,˚

Zl(S)M

= min

κ6=sl(S)ζκ,l,

∆l(S)M

=Zl(S)˚

Zl(S)

ϑ2.

(12)

Finally, the theoretical performance bound for LBS is cap-

tured in the following Lemma.

Lemma 3: In the interference-limited regime under As-

sumptions 1, 2, policy LBS computes at each slot ta

weighted sum rate W SRLBS (t)that satisﬁes (13) (top of

page), where W SR∗(t)is the weighted sum rate of the

LCB policy, γl(Fro,S)is deﬁned in (14), with ¯

˚

σ(S)M

=

min(Gb,ˇ

Gˇ

b)∈ES(Gnet)¯

σS,Gb,ˇ

Gˇ

band, for any rooftop

beam conﬁguration band any sets Fro ⊆[Nro]and Fla ⊆

[Nla], we deﬁne Mb(S,Fro)M

={Gb0:Gb0∈ Gro(S),

(b0

k=bk,∀k∈ Fro)}, with Gr o (S)being the set of rooftop

patterns that are valid for LBS under assignment S. Moreover,

ES(Gnet)is the set of possible gain conﬁgurations under LCB.

All quantities deﬁned above are strictly positive and it holds

ζκ,l ≥1for all κ, l and Zl(S),˚

Zl(S), $l(S)≥1for all S. On

the other hand, it is possible for ∆l(S),Υ (Fro,S),Ψ (Fla,S)

to take values lower than 1.

Proof: See the Appendix for an outline of the proof and

[28] for full details.

We can now directly plug (13) into Lemma 2 to prove the

following result:

Corollary 2: It holds RLBS ⊇υ¯

R, where υis the factor in

parentheses in (13).

A careful study of the proof of Lemma 3 in [28] reveals that

this bound represents the worst-case scenario of the actual

performance of LBS. However, since average case statistics are

not easily attainable in theory, we have to resort to simulations

to determine the actual performance of LBS and the other

policies for which no theoretical bounds are available. Thus,

we next describe a simulation-based approach for comparing

the performance of different policies.

A. Policy performance comparison

Since Lemma 1 shows policy LCB to be optimal, albeit at

high computational cost, and the bound of Corollary 2 may

be loose, the main question becomes how to compare two

distinct policies performance-wise in terms of their stability

regions. Once this question is answered, we can then answer

the related question of “how much better than RRSM,SLQ,

LBS,CLU is LCB and is the associated performance beneﬁt

worth the higher complexity?”

Clearly, policy π1is “better” than π2if Rπ1⊇ Rπ2;

however, as Fig. 3 shows, it is possible that, for two speciﬁc

policies, neither Rπ1⊇ Rπ2nor Rπ2⊇ Rπ1holds. In this

case, we can only compare π1,π2for the same arrival rate

vector ratio λ1:λ2:. . . :λNla . Speciﬁcally, for any policy

π, let the arrival rate vector λ%

M

=%(λ1, λ2, . . . , λNla )M

=%λ0,

parameterized over %, and increase %while keeping λ0ﬁxed

until the network becomes unstable, i.e. deﬁne %∗

π>0as

%∗

π= max{% > 0 : λ%∈ Rπ}.(15)

There is a unique solution to (15) due to the convexity of

Rπ[35]. Graphically, the point λ%∗

πis the intersection of the

boundary of Rπand a ray emanating from the origin along

a line, whose direction is determined by the arrival rate ratio.

Fig. 3 shows 2 such rays (one for each rate ratio), where the

intersecting boundary points for each policy, shown in circles

and marked as A1–A3and B1–B3, are the arrival rates λ%∗

π.

We can now compare the two policies by simply comparing

scalars %∗

π1,%∗

π2; the higher value indicates the corresponding

policy being superior for the given arrival rate ratio. Hence,

π2is better than π1for the upper ray while π1is better than

π2for the lower ray in Fig. 3, and π3is better than both π1,

π2for all possible rays.

10

Although (15) can be cast into a linear program [35]

(where the unknown variables are the long-term probabilities

of a stationary policy selecting the same controls as policy

π), this still requires an exponential number of unknowns

and a priori knowledge of channel statistics, which may

not be available. Instead, we opt for an approach combining

simulation and theory as follows. Theorem 2.4 in [35] states

that, under mild conditions which hold in our case, queue

Qlis mean-rate stable under πiff it holds λl≤¯rl

M

=

limt→∞ 1

tPt−1

τ=0 Rl(aπ(τ)), where aπ(τ)is the control se-

lected under π. An equivalent condition for network stability

is a vectorized form of this equation for all queues. Since

we can only simulate policy πfor a ﬁnite time Ts, we must

estimate ¯rlas ˆ

¯rl

M

= (1/Ts)PTs−1

τ=0 Rl(aπ(τ)). Obviously, it

holds ˆ

¯rl→¯rlas Ts→ ∞.

Since ˆ

¯rlcan be computed via simulation and ¯rlcannot, we

determine stability of Qlaccording to the slightly modiﬁed

condition λl(1 −)≤ˆ

¯rl. To better distinguish the cases of

ﬁnite and inﬁnite horizon, we use the term “numerical” stabil-

ity (resp. instability) to refer to the condition λl(1 −)≤ˆ

¯rl

(resp. λl(1 −)>ˆ

¯rl) and the term “true” stability (resp. in-

stability) to refer to the condition λl≤¯rl(resp. λl>¯rl). We

apply a vectorized version of the above for vector rates λ.

We can now ﬁnd a sufﬁciently large %0such that λ%0

is numerically unstable under π. Starting from the interval

[0, %0], we can then use bisection search to create a sequence

of intervals [¯

%, ¯%]such that λ

¯

%,λ¯%are numerically stable and

unstable, respectively, under π.4Bisection search stops when

(¯%−¯

%)/¯

%<δ, for some δ > 0, at which point we estimate %∗

π

as the midpoint %mid of the last interval [¯

%, ¯%]. The following

result, proved in [28], shows that using the modiﬁed ﬁnite-

horizon based concept of numerical stability instead of true

stability incurs a controllable deviation of %mid from the true

(unknown) %∗

π.

Lemma 4: For any policy πand any > 0, there exists

some N>0such that the combination of bisection search

and simulation of πfor a ﬁnite time Ts> Nconverges to an

interval [¯

%e,¯%e]whose midpoint %mid satisﬁes

%mid−%∗

π

%∗

π

<

δ(1 + )+2.

V. PERFORMANCE EVALUATION

A. Simulation scenarios

We consider a region of 120 m ×120 m, where rooftops and

lampposts are deployed in a manner emulating a dense urban

environment. We have examined two different deployment

modes, termed “Peripheral” and “Interior” (abbreviated as Per

and Int) but, due to space constraints, we present the results

for the Per mode only (with Int results appearing in [28]

and offering similar insights and conclusions to Per). The

difference between the two modes is that, under Per, a rooftop

is placed at an exterior region so that all lampposts are on

the same half-plane w.r.t. the rooftop (hence, the rooftop need

only provide 180◦azimuth coverage) whereas, under Int, the

4At a slight abuse of terminology, we consider ¯

%, ¯%to be (numerically) sta-

ble/unstable according to whether the corresponding λ

¯

%,λ¯%are (numerically)

stable/unstable. If %mid = (¯%+¯

%)/2is (numerically) stable, next bisection

interval is [%mid,¯%]. Otherwise, next interval is [¯

%, %mid].

Fig. 5. Available radiation patterns at rooftops for Scenario B with Half

Power Beamwidth (HPBW) and Side Lobe Level (SLL) indicated.

rooftop is placed arbitrarily and can be surrounded by multiple

lampposts (so that it should offer 360◦azimuth coverage).

These two distinct deployment modes can coexist in a network

topology (examples of this coexistence and corresponding

results are found in [28]) and can be used to abstractly emulate

greenﬁeld and brownﬁeld deployment, respectively.

We select Nro and Nla values in the sets {3,4}and

{4,5,6,7,8}, respectively, so that Nro ≤Nla holds and for

each of the 10 valid (Nro, Nla )pairs, we simulate multiple

network topology instances. We consider two different sce-

narios for the employed radiation patterns:

• Scenario A corresponds to a strong interference environ-

ment, where lamppost antennas are assumed to be omnidi-

rectional and each rooftop antenna is a linear microstrip array

which generates the 5 patterns shown in Fig. 2 with beamwidth

of 15-20◦. For each (Nro, Nla )pair, 40 topology instances are

generated.

• Scenario B assumes that all rooftops and lampposts use

directional patterns conceptually similar to the Terragraph

system [40]. Speciﬁcally, each lamppost (resp. rooftop) is

equipped with an 8x8 (resp. 12x12) rectangular microstrip

array so that, in Per mode, 9 patterns are needed by each

rooftop to provide the necessary azimuth coverage (as shown

in Fig. 5), whereas each lamppost employs 18 patterns (9

of them are shown in Fig. 6, with the other 9 being their

mirror images with respect to azimuth; this symmetry is used

for illustration purposes only and to provide a challenging

scenario in terms of search space size). Due to the much higher

search space sizes, we simulated 40 topology instances for

each pair (Nro, Nla)with Nro = 3 and 20 topology instances

for each pair with Nro = 4.

All rooftops and lampposts are placed at a height of 10 m

and 4 m, respectively. Standard thermal noise corresponding

to 400 MHz bandwidth is assumed (note that, due to the dense

deployment and wider beams in Scenario A, the network can

be approximately treated as interference-limited). Each rooftop

antenna transmits at power Pt= 20 dBm and a path-loss

exponent of 2 is assumed, to capture the predominant LOS

11

Fig. 6. Available radiation patterns at lampposts for Scenario B with Half

Power Beamwidth (HPBW) and Side Lobe Level (SLL) indicated. The mirror

images of these patterns (w.r.t. azimuth) are also considered, bringing the total

number of patterns per lamppost to 18.

conditions in this setting.

The policies described in Section III are simulated for 5000

slots as described in Section IV-A with δ= 0.04,= 0.02

and, based on the output, an estimate of %∗

π(denoted as ˆ%∗

π)

is produced as the midpoint of the ﬁnal bisection interval for

given lamppost arrival rate ratios.

For Scenario A, policies LCB,SLQ,LBS,RRSM are simu-

lated where LBS uses Dro = 3,Dla = 1 (the other policies

consider all available patterns). For Scenario B, the search

space of LCB quickly becomes intractable so that only SLQK,

LBS,RRSMKare simulated as follows:

•For Nro = 3, we compare the performance of SLQ9,

RRSM9 and LBS, where the latter considers either Dro =

Dla = 3 or Dro =Dla = 5. We refer to these two

versions of LBS as LBS3 and LBS5, respectively.

•For Nro = 4, we compare the performance of SLQ7,

RRSM7,LBS3,LBS5.

We use lower values of Kfor SLQK,RRSMKbetween

Nro = 3 and Nr o = 4 as these policies are intended for

online implementation, which implies that, for a ﬁxed amount

of computational resources, Kmust be reduced for higher

Nro in order to ensure that the policies reach their decisions

at each slot within the allocated time. We also simulate LBS3

and LBS5 for both Nro = 3 and Nro = 4 in order to determine

the effect of Nro on their relative performance.

Both equal and unequal lamppost arrival rate ratios are con-

sidered, with Table I listing the exact arrival rates examined.

The above values are selected for illustration purposes only,

as the system model presented in Section II is very general

and can accommodate a wide range of channel models [41]

and arbitrary rate ratios.

To determine how “far away” a policy πperforms from

the optimal LCB policy, we deﬁne the normalization factor

θπ

M

=%∗

LCB/%∗

π, where θπ≥1(and lower values of θπ

indicate that πbetter approaches LCB performance-wise),

as well as the estimate ˆ

θπ

M

=ˆ%∗

LCB

ˆ%∗

π. Notice, that since, by

TABLE I

LAM PPO ST A RRI VAL RATE RAT IOS (F OR U NEQ UAL R ATES)

Nla Arrival rate ratio

4 1:2:2:1

5 1:2:2:2:1

6 1:2:1:2:1:2

7 1:2:2:1:2:2:1

8 1:2:2:1:1:2:1:2

Fig. 7. Theoretical performance bound of LBS for Scenario A and Nro = 3.

Lemma 4, we cannot determine %∗

πwith perfect accuracy via

simulation, it is possible that ˆ

θπ<1. However, this occurs

very rarely in the performed simulations and, even in these

rare cases, it typically holds ˆ

θπ≥0.9. Intuitively, ˆ

θπindicates

a multiplicative throughput gain per user achieved by the

optimal LCB compared to π(equivalently, πachieves 1/ˆ

θπ

times the performance of LCB per user). In a similar fashion,

we can directly compare (for a given rate ratio) policies π1,

π2by forming ˆηπ2

π1

M

=ˆ%∗

π1

ˆ%∗

π2

. Policy π1then outperforms π2

for this rate ratio if ˆηπ2

π1≥1. Finally, to numerically examine

the tightness of the theoretical LBS bound as determined by

the value of vin (13), we plot in Fig. 7 the quantity 1/v

for Scenario A and the case Nro = 3. We choose to plot

1/v instead of vbecause 1/v is semantically identical to ˆ

θLBS

appearing in Fig. 8 and Fig. 9. It is apparent from Fig. 7–

Fig. 9 that the bound in (13) is pessimistic (by a factor of

5-10), as expected, since (13) provides worst-case guarantees.

The bound’s looseness adds further value to the simulations

that follow, since they are a more accurate indicator of the

achieved performance.

B. Simulation results

The above simulation scenarios produce a large dataset,

since either 40 or 20 different topology instances are generated

for each (Nro, Nla)pair, so that the results are statistically

summarized via boxplots.5Also, the control sets (equivalently,

the search spaces) of all policies are known a priori to

Cntrl, which implies that the optimization problems in LCB,

SLQ,LBS,RRSM and their parameterized versions can be

5In a boxplot, the closed rectangle signiﬁes (from bottom to top) the ﬁrst

quartile (Q1), median and third quartile (Q3), respectively, while outliers are

shown in + markers. In case of tightly concentrated values, some of the

quartiles may coincide with the median, collapsing the boxplot (in the extreme

case where Q1, Q2 and the median are all equal to each other, the boxplot

degenerates to the single median line and all other values become outliers).

12

TABLE II

SIZ E OF SE AR CH SPAC E ˘

AπIN OPTIMIZATION PROBLEM SOLVED BY POLICY π∈ {LCB,LCQ,LBS3, RRSM}AT EACH S LOT F OR SCENARIO A.

Nro = 3 Nro = 4

Nla

˘

ALCB

˘

ASLQ

˘

ALBS3

˘

ARRSM

˘

ALCB

˘

ASLQ

˘

ALBS3

˘

ARRSM

4375,000 93,750 17,496 15,625 9,375,000 9,375,000 157,464 390,625

5937,500 93,750 43,740 15,625 46,875,000 9,375,000 787,320 390,625

61,875,000 93,750 87,480 15,625 140,625,000 9,375,000 2,361,960 390,625

73,281,250 93,750 153,090 15,625 328,125,000 9,375,000 5,511,240 390,625

85,250,000 93,750 244,944 15,625 656,250,000 9,375,000 11,022,480 390,625

TABLE III

SIZ E OF SE AR CH SPAC E ˘

AπIN OPTIMIZATION PROBLEM SOLVED BY POLICY π∈ {LCB,LCQ,LBS3, RRSM}AT EACH S LOT FO R SCENARIO B.

Nro = 3 Nro = 4

Nla

˘

ALCB

˘

ASLQ9

˘

ALBS3

˘

ALBS5

˘

ARRSM9

˘

ALCB

˘

ASLQ7

˘

ALBS3

˘

ALBS5

˘

ARRSM7

41.02E8 3.19E6 1.75E4 3.75E5 5.31E5 1.65E10 1.38E8 1.57E5 9.38E6 5.76E6

52.55E8 3.19E6 4.37E4 9.38E5 5.31E5 8.26E10 1.38E8 7.87E5 4.69E7 5.76E6

65.10E8 3.19E6 8.75E4 1.88E6 5.31E5 2.48E11 1.38E8 2.36E6 1.41E8 5.76E6

78.93E8 3.19E6 1.53E5 3.28E6 5.31E5 5.79E11 1.38E8 5.51E6 3.28E8 5.76E6

81.43E9 3.19E6 2.45E5 5.25E6 5.31E5 1.16E12 1.38E8 1.10E7 6.56E8 5.76E6

solved in parallel by partitioning the search spaces among

different CPU cores. Although such hardware acceleration

and parallelization techniques can further decrease the running

time of the policies, they are not pursued in this paper since

our goal is a comparative analysis of the different policies and

any acceleration techniques will provide a uniform speedup to

all considered policies.

The size of the search spaces of the various policies are

shown in Table II and Table III for Scenario A and B,

respectively, where it becomes apparent that LCB quickly

becomes intractable (see the bold numbers in Table III) and

its use is solely as a yardstick against which the other policies

will be evaluated.

Fig. 8 summarizes the results for the case Nro = 3 in

Scenario A and shows the values of ˆ

θπfor the different policies

π, for both equal and unequal rate ratios. We observe that LCB

consistently offers a throughput gain higher than 2X per user

compared to RRSM, which implies that the actual sum rate

increases by a factor of more than (2Nla)X. Additionally, the

throughput gain of LCB over RRSM has a generally increasing

trend vs. Nla, which agrees with intuition, as the presence

of more lampposts in close proximity to each other generally

exacerbates interference and, therefore, increases the potential

beneﬁts of CoMP schemes. No other meaningful quantitative

comparisons can be performed between different Nla values

(for a ﬁxed Nro), since they correspond to different dimensions

of stability regions.

The heuristic policy SLQ performs only 20-30% worse than

LCB, but also offers a 2X-16X computational speedup6, and

LBS performs even better than SLQ (albeit at a 2X-4X speedup

w.r.t. LCB), since the majority of the cases achieve ˆ

θLBS = 1

(i.e., optimal performance). This is evidenced in Fig. 8 by the

fact that some LBS boxplots have their main box collapsed

around its median value, which implies an extremely tight

6Although the computational speedup can be inferred by the relation

between the search space sizes of LCB,SLQ, the speedup values mentioned

here are based on actual wall time measurements.

(a) equal rate ratios.

(b) unequal rate ratios.

Fig. 8. Policy performance for Per deployment and Nro = 3: equal and

unequal rate ratios for Scenario A.

concentration of values around the median (in fact, the median,

Q1, Q3 values are identical whenever the box collapses to a

single line), where all other values that do not coincide with

the median appear as outliers (e.g., out of the 40 values in

each LBS boxplot of Fig. 8, there are only 4 outliers for

Nla = 6 and 3 outliers for Nla = 7 and the remaining 36, 37,

respectively, values have coalesced onto the horizontal line of

the median). The same interpretation is valid for all Figures

which show a collapsed instead of a “regular” boxplot box.

13

(a) equal rate ratios.

(b) unequal rate ratios.

Fig. 9. Policy performance for Per deployment and Nro = 4: equal and

unequal rate ratios for Scenario A.

These observations are consistent for both equal and unequal

rate ratios and the same overall trends are also observed in

Fig. 9 for Scenario A and Nro = 4 (where a 60X, 7X speedup

of SLQ and LBS, respectively, is attained over LCB), which

indicates the robustness of the proposed policies. Note that

for the case Nro =Nla = 4 in Fig. 9, SLQ perfectly matches

the performance of LCB (i.e., all 40 values are equal to 1,

with no outliers). This is not a coincidence, since A,˘

ASLQ are

identical sets in this special case so that LCB,SLQ solve the

same problem.

Finally, we also include the performance of SLQg, which is

a “greedy” version of SLQ proposed in [28] and designed to

greedily optimize individual transmissions from each rooftop

to its served lamppost while ignoring cross-transmission in-

terference (i.e., no interference management is considered).

SLQg generally performs better than RRSM but systematically

much worse than LSQ,LBS. This result indicates the per-

formance loss inherent to policies lacking global interference

coordination control (such as SLQg) and warns against the

danger of “too greedy” solutions. For this reason, SLQg is

not further evaluated in the paper and was included only to

illustrate the limitations of such approaches.

Since the results for Scenario A indicate that LBS generally

achieves performance very close to LCB, we expect that

this trend will also hold for Scenario B which, by nature

of its narrower employed beams, exhibits a “less diffuse”

interference environment. Therefore, and since simulation of

LCB becomes computationally prohibitive, LBS5 can act as

the benchmark policy and performance of the other policies π

w.r.t. LBS5 can be evaluated via the quantity ˆηLBS5

π(deﬁned

(a) equal rate ratios.

(b) unequal rate ratios.

Fig. 10. Policy performance for Per deployment and Nro = 3: equal and

unequal rate ratios for Scenario B.

at the end of Section V-A; the higher its value, the worse π

performs w.r.t. LBS5).

The results for Scenario B are shown in Fig. 10, Fig. 11

for Nro = 3 and Nro = 4, respectively, where we have used

two different vertical axis scales for better visual distinction.

Speciﬁcally, RRSM7, RRSM9 use the left-side vertical axis,

whereas SLQ9, SLQ7, LBS3 and LBS5 use the right-side

vertical axis. It is apparent that the parameterized RRSM

policies still perform much worse than LBS5. However, it is

worth noting that SLQ9, SLQ7 and LBS3 perform extremely

close to LBS5, with all outliers being within 15% (in the

worst case) of the median. Examining Table III, this implies

that LBS3 achieves essentially the same performance as LBS5,

SLQ7, SLQ9 by using a search space that is 15X-900X smaller.

This indicates that once the 3 dominant patterns for each

rooftop-pattern assignment are selected under LBS3, adding

additional pattern in the search space (as happens in LBS5) or

using alternative policies (like SLQ7, SLQ9) offers diminishing

returns.

In summary, the simulation results show LBS consistently

performing within 20-30% (in the worst case) and much

closer on the average w.r.t. the optimal (but computationally

expensive) LCB policy at computational savings of 60X-900X,

depending on network size. SLQ offers similar performance to

LBS, although its computational complexity increases much

faster w.r.t. the number of patterns per antenna site than LBS.

Hence, SLQ is preferable to LBS when there is a high number

14

(a) equal rate ratios.

(b) unequal rate ratios.

Fig. 11. Policy performance for Per deployment and Nro = 4: equal and

unequal rate ratios for Scenario B.

of rooftops/lampposts but with few patterns each, whereas

LBS is preferable to SLQ when there is a large number of

available patterns per rooftop/lamppost. Considering that the

number of patterns is not impeded by any physical space

limitations so that it can grow much faster than the number

of rooftops/lampposts, we conclude that LBS offers the most

favorable performance/complexity trade-off, especially when

narrower beams are used. Finally, both SLQ,LBS perform

much better than the non-queue-aware RRSM policy, indicating

the beneﬁt of exploiting queue state information. The varying

extent of the performance/complexity trade-off of the above

policies can be exploited by the network designers to meet

their requirements.

VI. CONCLUSIONS

This paper proposed queue-aware CoMP-CB policies for

mmWave ﬁxed-wireless dense urban networks with purely

analog beamforming used at rooftop and lamppost anten-

nas. Based on the Lyapunov optimization framework, the

throughput-optimal (equivalently, having the “largest” stability

region) policy LCB was proposed, which jointly performs user

scheduling and antenna radiation pattern selection to mitigate

interference, while accounting for stochastic packet arrivals

and channel ﬂuctuations. The system model is very general

and the proposed policy can be extended to include additional

controls (e.g., rooftop transmission power) or some of the

policy controls (such as user scheduling) can be delegated to

MAC layer algorithms, if needed.

Due to the high computational complexity arising from the

exponential cardinality of the search space, two additional

lower-complexity policies SLQ,LBS were proposed and eval-

uated through simulations, using a non-queue-aware Round-

Robin-based policy RRSM for benchmarking. A theoretical

worst-case performance bound for LBS was also derived.

The performed simulations indicate LCB offering a consistent

average throughput gain of 2X per user w.r.t. RRSM while

LBS,SLQ perform only 20-30% worse than the optimal LCB

at huge computational savings of 60X-900X. Future work

entails the study of hybrid beamforming under the proposed

Lyapunov-based framework as well as incorporating controls

to allow for CoMP-JT operation.

APPENDIX

OUTLINE OF PROOF OF LEMM A 3

Consider a thought experiment where, at slot t, an oracle

provides us with the optimal action aLCB(t)∈ A under

LCB, i.e., the optimal rooftop-lamppost assignment SLCB(t)

and the optimal gain values GLCB

b(t),ˇ

GLCB

ˇ

b(t)for the rooftops,

lampposts. Our goal is to construct a control a0(t)∈˘

ALBS

whose corresponding weighted sum rate W SR0(t)satisﬁes a

condition of the form: W SR0(t)≥˚ν·WS R∗(t), where ˚νis

the multiplicative factor in (13). Denoting with W SRLBS(t)

the optimal weighted sum rate computed by LBS, we conclude

that it also holds W SRLBS (t)≥W SR 0(t), which implies the

desired inequality W SRLBS (t)≥˚ν·W SR∗(t).

Since it holds ˘

ALBS ⊆ A, it is possible that the oracle-

provided pattern conﬁgurations GLCB

b(t),ˇ

GLCB

ˇ

b(t)are not valid

under LBS (recall that LBS only selects a subset of available

patterns for each rooftop and lamppost). We focus on this

case and denote with Fro(t),Fla(t)the sets of rooftops and

lampposts, respectively, which have selected, under aLCB(t),

patterns that are not valid under LBS. For each rooftop in

Fro(t)and each lamppost in Fla (t), we now select arbitrary

valid patterns under LBS and quantify the resulting difference

in SIR achieved when using the modiﬁed patterns and the

original oracle-provided ones. Further details are provided in

[28].

ACK NOW LE DG EM EN T

The authors would like to thank Prof. Leonidas Georgiadis,

Department of Electrical and Computer Engineering, Aristotle

University of Thessaloniki and Center for Interdisciplinary Re-

search and Innovation, Thessaloniki, Greece, for the insightful

discussions and helpful suggestions.

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