1

Orchestrating Resource Management in LTE-Unlicensed

Systems with Backhaul Link Constraints

Tuan LeAnh, Student Member, IEEE, Nguyen H. Tran, Member, IEEE,

Duy Trong Ngo, Member, IEEE, Zhu Han, Fellow, IEEE,

and Choong Seon Hong, Senior Member, IEEE.

Abstract

LTE-Unlicensed, an extension of Long Term Evolution Advanced (LTE-A) to unlicensed spectrum,

can provide high performance and seamless user experience. To reap the full benefits of the LTE-

Unlicensed deployment, efficient resource allocation and interference management are critical to ensuring

a harmonious coexistence between LTE-Unlicensed and WiFi systems. In this paper, we study a resource

orchestration scheme for an LTE-Unlicensed network where small cells share the same unlicensed

spectrum with a WiFi system. An optimization problem for channel and power allocations is formulated

to maximize the overall network utility, which is an NP-hard problem. The problem is constrained on

meeting the desired data rate demands of the served small cell users, the capacity-limited backhaul links,

and the maximum tolerable interference at the WiFi access point. To solve this challenging problem, a

distributed solution based on Lagrangian relaxation is proposed to assist the LTE-Unlicensed network in

making decisions on channel allocation and transmit power. Furthermore, low-complexity solutions are

devised upon applying the one-to-one matching game theory. Simulation results with practical parameter

settings show that the proposed algorithms converge to the suboptimal solution after a small number of

iterations in the considered examples.

Tuan LeAnh and C. S. Hong are with the Department of Computer Science and Engineering, Kyung Hee University, Korea

(email: latuan, , cshong@khu.ac.kr).

N. H. Tran is with the School of Information Technologies, The University of Sydney, N.S.W. 2006, Australia (e-mail:

nguyen.tran@sydney.edu.au).

D. T. Ngo is with the School of Electrical Engineering and Computing, The University of Newcastle, Callaghan, N.S.W. 2308,

Australia (e-mail: duy.ngo@newcastle.edu.au).

Z. Han is with the Electrical and Computer Engineering Department, University of Houston, Houston, Texas, USA (email:

zhan2@uh.edu).

Index Terms

Dual decomposition, matching theory, LTE-Unlicensed, resource allocation.

I. INTRODUCTION

According to the latest mobility report from the industry, the number of mobile-connected

devices is growing exponentially and it is projected to reach 11.5 billion in 2020 [1]. Additionally,

many new applications and services that run on smart mobile devices are being integrated into

wireless networks. As a result, data traffic congestion will soon be expected in the current LTE-

A systems that utilize the licensed spectrum from 700 MHz to 2.7 GHz. To address the issues

of data traffic congestion and spectrum scarcity, LTE-Unlicensed technology is proposed as an

extension of the LTE-A solution to improve the network capacity while continually providing

excellent user experience to the customers [2]. Using carrier aggregation (CA), it integrates

the unlicensed spectrum (e.g., the WiFi’s frequency bands) into the cellular network’s licensed

spectrum [3].

Coexistence with WiFi in the unlicensed spectrum is a critical element in establishing a busi-

ness case for the LTE-Unlicensed technology. Current projects on the LTE-Unlicensed include

LTE-U, licensed assisted access (LAA), and MuLTEfire [4], [5]. These specific implementations

depend on the frequency allocation policies for the 5GHz band, the regulatory requirements,

and the WiFi band allocations of individual countries. A relatively simple mechanism for early

deployment, the LTE-U does not require modifications to the existing LTE air interface pro-

tocol [6]. LTE-U specifically targets markets without listen-before-talk (LBT) regulation in the

unlicensed spectrum such as in the U.S.A., China, and South Korea. In another way, LAA is

the 3rd Generation Partnership Project (3GPP)’s effort to standardize the LTE operation in the

WiFi bands. It uses a contention protocol known as LBT, which is mandated in some European

countries and Japan, to coexist with WiFi devices on the same unlicensed band [4], [5], [7]. In

both LTE-U and LAA, the reliable licensed radio links are used for the signaling and control

messages, whereas the unlicensed links are for data transmission only. This is different from the

MuLTEfire where unlicensed spectrum is used for signaling and control messages. In our work,

a coexistence mechanism is designed according to the LBT regulation [6], [8], in which WiFi

access point’s interference limit is guaranteed from the LTE-Unlicensed deployment [7]–[11].

A. Related Works and Motivations

It is important to meet the transmission quality for both LTE-Unlicensed users and WiFi

users in the face of co-channel interference [6]. The works of [12]–[14] confirm the perfor-

mance degradation when the LTE cellular networks operate in the WiFi band. Meanwhile,

the Carrier-Sensed Multiple Access (CSMA) protocol in the WiFi system is only designed

for low interference scenarios. In traditional WiFi transmission, the interference level must

be less than the energy detection threshold to avoid backoff processes for the WiFi users.

Specifically, with limited unlicensed bands, some cellular network operators deploying LTE-

Unlicensed may use unlicensed bands of the WiFi access points of other network operators.

In this case, LTE-Unlicensed transmissions should not cause considerable interference on the

existing WiFi transmissions. Also, the interference from the co-channel WiFi users and other

LTE-Unlicensed users may deteriorate the LTE-Unlicensed users’ performance.

To address the coexistence issues of multiple LTE-Unlicensed/WiFi systems, interference

management and efficient unlicensed spectrum utilization are key [6]–[8]. In the blank sub-frame

technique, when LTE-Unlicensed/WiFi users operate at the same time on the same unlicensed

channel, the WiFi users are designed to access the channel during the LTE blank sub-frame [15],

[16]. On the other hand, the channel selection technique enables the LTE-Unlicensed users to seek

the empty unlicensed channels as well as allowing the WiFi APs to search for the least congested

channels [17]. When there is no clean channel being available, the LTE-Unlicensed network

can perform power control at the LTE-Unlicensed base stations and users to mitigate undue

interference at the WiFi system [18]. Here, the LTE-Unlicensed users control their transmission

power based on information from the presence and proximity estimations of the WiFi users,

to avoid strong interference to the WiFi system. Thus, it is critical to efficiently allocate the

unlicensed band and to adjust the transmit power of multiple LTE-Unlicensed users, to achieve

the highest utilization, while satisfying service demands for both cellular and WiFi users.

References [7], [9]–[11] ensure the possible coexistence between the LTE-Unlicensed and

WiFi systems in the unlicensed spectrum by avoiding interference at the WiFi access points.

The requirements of the LAA in LTE-Unlicensed systems can be met by the underlay cognitive

radio solution [19], [20]. Here, LTE-Unlicensed users operate in the unlicensed band of the

WiFi system while the overall interference generated by the LTE-Unlicensed users on the same

channel is kept below a given threshold. Similarly, the works of [18] and [10] investigate the

interference-controlled power allocation problem in LTE-Unlicensed systems. In [18], a power

control plan with an interference-aware power operating point is devised to balance the LTE-

Unlicensed and WiFi performances in the uplink data transmission. In [10], a successive cap-

limited water-filling method is proposed to handle the interference to WiFi system. However, the

problem of subchannel allocation in the unlicensed channels is not considered in these studies.

In [9], a coordinated hierarchical game is formulated to model the multi-operator spectrum

sharing in LTE-Unlicensed systems. A multi-leader multi-follower Stackelberg game framework

is developed to analyze the interaction among multiple operators and subscribers in the unlicensed

spectrum. In this case, the operators profit from operating on unlicensed resources while the users

choose which unlicensed bands to transmit based on the interference penalty price. Additionally,

the adopted game strategies make sure the interference to the WiFi access point be kept under

a tolerable level.

Matching game theory [21], [22] has been used for resource allocation in LTE-Unlicensed [7],

[11]. Its aim is to find proper and stable unlicensed user partners for the cellular users. With the

matching games, the competition and negotiation among distinct user sets of the LTE-Unlicensed

and WiFi systems can be well modeled, where distributed solutions can be devised. In [11], the

interaction between LTE-Unlicensed and WiFi users is modeled by utilizing the the matching

game-based student-project allocation. Preference lists for these two types of users are generated

to avoid the interference between them, while the time division multiple access (TDMA) method

is utilized to avoid the interference among the co-channel LTE-Unlicensed users. However, [11]

does not consider power control to mitigate interference at unlicensed users. In [23] a dynamic

spectrum sharing solution among multiple operators in the unlicensed spectrum with time-varying

traffic was proposed. Similarly, in [24] the matching theory framework was employed to solve

the dynamic resource allocation problem for the LTE-Unlicensed. Apart from the distinguished

formulated optimization problem, our matching-based solution is also different from the studies

in [23], [24], in which a sequence of the one-to-one matching game is utilized to combine

with the Lagrangian relaxation-based solution [25], [26] for finding decisions on unlicensed

subchannels and transmit power.

The above mentioned works and other existing works on LTE-Unlicensed do not consider the

important issue of backhaul protection, which should be rigorously dissected in dense small cell

deployment situations with different backhaul solutions for both wireless and wired connections,

e.g., non-line-of-sight (NLOS) microwave, wireless mesh networks, point-to-multipoint (PMP)

topology, and virtualized wireless networks [27]–[29]. Note that many current applications

require a large amount of upload capacity such as the Internet of Things (IoT), virtual office,

Mobile Edge Computing (MEC), web meeting, and connected vehicle safety applications [?],

[30]. It is argued that the backhaul network will be a major performance bottleneck in small

cell networks [31]. The backhaul constraint is also an important issue in the coexistence of

LTE-Unlicensed/WiFi systems [6], [32]. To the best of our knowledge, our work is the first that

considers the issues of backhaul capacity when performing joint channel and power allocation

for the coexistence of LTE-Unlicensed/WiFi systems.

Furthermore, to adapt the inevitable traffic explosion, it is necessary to enhance the uplink

data rate in the future mobile networks [33]. This is a direct result of communication scenarios

with significant uplink traffic loads such as the integration of cloud-enabled technologies, the

proliferation of IoT systems, machine-to-machine, and machine-to-cloud platforms in wireless

networks. To guarantee the service demands of the LTE-Unlicensed users, it is necessary to

consider uplink communications [20], [34].

B. Research Contributions

This paper develops an orchestration scheme of channel and transmit power for the uplink of

an LTE-Unlicensed system to enable coexistence with a WiFi system. In particular, interference

threshold protection for the WiFi system is studied. The chunk-based channel allocation approach

(i.e., subcarrier aggregation) is used to allocate the unlicensed spectrum for the uplink transmis-

sions in an orthogonal frequency division multiple access (OFDMA)-based system [?]. The

considered joint channel and power allocation complicates any optimization-based design due to

several coupled constraints: i) power allocation, ii) backhaul limitation, and iii) guaranteeing

coexistence of LTE-Unlicensed/WiFi systems. To obtain a suboptimal solution, we develop

distributed algorithms based on the Lagrangian relaxation approach. In addition, we take the

competitive behavior of selfish and rational network entities into consideration by modeling

the problem as a matching game [21], [22]. Our key research contributions are summarized as

follows.

• We formulate an optimization problem of channel and power allocations for the uplink LTE-

Unlicensed. The goal is to maximize the overall network utility while guaranteeing WiFi

access point’s interference limit, protecting limited backhaul capacity links, and meeting

the data rate requirements of the served small cell users.

• To solve the formulated NP-hard problem, we employ Lagrangian relaxation to decompose it

into tractable subproblems that separate the power allocation and the channel assignment. An

analytical framework is developed to find a locally optimal solution for channel and power

allocations. Two low-complexity solutions are further devised by combining Lagrangian

relaxation with the matching game. In the proposed designs, small cell users compete to get

matched with the chunk-based channels in the formulated one-to-one matching game. After

that, we develop distributed algorithms that decide the assignment of channel and the power

allocation for small cell users in a distributed manner. We demonstrate that the proposed

algorithms converge to the suboptimal solutions with low computational complexity.

• Simulation results with practical parameters confirm that the proposed approach gives

suboptimal solutions and it only takes a small number of iterations to converge.

The remainder of this paper is organized as follows. In Section II, we describe the system

model of the coexistence of LTE-Unlicensed and WiFi systems, and the problem formulation. The

formulated problem solving based on the Lagrangian relaxation method is presented in Section

III. Two suboptimal solutions by combining Lagrangian relaxation with one-to-one matching

game are devised in Section IV. The computational complexity of the designed algorithms are

analyzed in Section V. Section VI gives simulation results to verify the effectiveness of the

suggested algorithms. Concluding remarks for the whole paper are given in Section VII.

II. SYSTEM MODEL AND PROBLEM FORMULATION

A. System Model

Let us consider a small cell network (SCN) as shown in Fig. 1, in which a set M =

1, 2, ...,M of small cell base stations (SBSs) are deployed to serve a set of small cell users

(SUEs) for the uplink transmission. As SBS m serves a set of Nm = 1, 2, ..., Nm SUEs,

the set of the total number of SUEs in the SCN is denoted as N = ∪m∈MNm. We assume

all the SUEs have access to sufficient licensed resources to maintain a predefined data rate of

Rlicensednm , ∀n ∈ N , m ∈ M. Besides, the SBSs and SUEs can also operate in the unlicensed

radio spectrum to further enhance the uplink data rate, supporting a minimum data transmission

rate Rminnm > 0, ∀n ∈ N , m ∈M.

In the unlicensed spectrum, an LTE-Unlicensed manager (LTE-UM) is assumed to operate as

a third party in a virtualized wireless network model to provide radio resources on the unlicensed

channels to different small cell network operators [9], [29]. The LTE-UM can collect unlicensed

Internet

: Small-cell user (SUE)

: Small-cell base station (SBS)

Optical Fiber

: WiFi user

Wireless

backhaul

Wireline backhaul

LTE-UM

... ...

1 2 k 1k

l

...

...1k

kl

1k2l 3l l| |c

12 ...

Radio Access

Core Network

l...

...l1 l

Wireline backhaul

WiFi AP

K

| |c

| |c

Fig. 1: System architecture of an LTE-Unlicensed network.

channel state by using the LBT procedure. Besides, the LTE-UM can bundle the unlicensed

channel c into chunks before selling them to the small cell network provider. Here, the unlicensed

channel c is sensed to be busy if it is being used by the WiFi access point (WiFi-AP). The

unlicensed channel c is divided into a set of Lc orthogonal narrowband flat-fading sub-bands,

with each sub-band spanning a bandwidth of ∆l = Bc

|Lc| Hz. We assume that the sub-bands are

deterministic during the optimization period. Here, Bc is the bandwidth of the unlicensed channel

c. These sub-bands are grouped into a set of Kc chunks, and each chunk is aggregated by a set

of Lc,k sub-bands. Depending on the frequency distance between sub-bands, the occupied band

of licensed channel c and the number of sub-bands in each chunk k, the LTE-UM requests a

payment φc,k from SUEs in return for accessing to the chunk k. Each SUE and each SBS are

only permitted to access at most one chunk in the unlicensed spectrum. The transmit powers can

be dynamically adapted on each sub-band of the chunk. Without loss of generality, we assume

that the SBSs are operated on a single unlicensed channel c with non-overlapping coverage areas.

The co-channel interference among small cells is negligible due to the wall penetration loss and

low power of SBSs [35], [36]. In our model, it is included in the therm noise term.

B. Problem Formulation

At first, we describe all system and design constraints. Then, a resource orchestration problem

is formulated for LTE-Unlicensed in coexistence with the WiFi system.

1) Data Transmission Model: When an SUE n ∈ Nm transmits data on the unlicensed

spectrum, its data rate is given by

Runlicensednm (ψ,P ) =

∑k∈Kc

ψknm∑l∈Lc,k

rl,knm(P l,knm), (1)

where the binary variable ψknm = 0, 1 represents the chunk-based channel allocation decision

for transmission from SUE n to SBS m on chunk k; ψ ∆= [ψknm]N×M×K , ψknm = 1 means that

SUE n ∈ Nm is assigned to chunk k, and ψknm = 0 otherwise; rl,knm is the data rate of SUE

n ∈ Nm on sub-band l of chunk k which is determined as:

rl,knm(P l,knm) = ∆l log2(1 + γl,knmP

l,knm), (2)

where γl,knm = gl,knm

I(l,k)WiFi,m+σ2

; I(l,k)WiFi,m = g

(l,k)WiFi,m

∫ dcl,k+∆l/2

dcl,k−∆l/2JI|Lc|(w))dw is the interference from the

WiFi system to SBS m on sub-band l ∈ Lc,k [10], [37]; JI|Lc|(w) is the power spectral

density of WiFi c’s signal after |Lc|-Fast-Fourier-Transform (FFT) processing; g(l,k)WiFi,m is channel

gain from WiFi AP to SBS m on sub-band l; gl,knm represents the instantaneous channel power

gain on sub-band l of chunk k from SUE n to SBS m; P l,knm represents the transmit power on

sub-band l of chunk k; σ2 is the background noise. In the considered fading channel model,

gl,knm = Gl,knmF

l,knm where Gl,k

nm and F l,knm are the mean channel power gain from SUE n to SBS

m and the fast fading gain in sub-band l of the kth chunk, respectively. In this paper, all

channel fading gains are assumed independent and identically distributed (i.i.d.). We further

define P ∆= P l,k

nm, ∀n,m, l, k as the power allocation matrix of all SUEs.

2) SUE QoS Guarantee: Each SUE requires a minimum data rate for its individual services.

However, the data rate achievable on the licensed channel may not be enough to meet the user

demand. In this case, when Rlicensednm ≤ Rmin

nm , each SBS m allows its SUE n to access the

unlicensed spectrum resources managed by LTE-UM to further improve SUE n’s data rate [7].

Hence, to satisfy the QoS requirement Rminnm for SUE n ∈ Nm, the following constraint must be

met:

Rnm∆= Rlicensed

nm +Runlicensednm (ψ,P ) ≥ Rmin

nm . (3)

3) Backhaul Link Constraint: In our model, the backhaul links of SBSs are capacity-limited.

To avoid overcrowding data traffic at the backhaul links, the data rate aggregated from all SUEs

has to meet the following constraint:∑n∈Nm

Rnm(ψ,P ) ≤ Zm,bh, ∀m, (4)

where Zm,bh is a predefined parameter representing the maximum backhaul link capacity of SBS

m. The value Zm,bh may vary for different SBSs. The minimum backhaul link capacity of SBS

m is assumed to satisfy Zm,bh ≥∑

n∈NmRminnm , ∀m.

4) Interference Introduced by SUEs’ Signal Transmission and WiFi Protection: The interfer-

ence at one spectrum pool is caused by the side lobes of the Orthogonal Frequency-Division

Multiplexing signal [37]. Besides, the signal on each channel of the considered WiFi standard

is a rectangular non-return-to-zero signal [10]. In the considered network model, the unlicensed

spectrum is divided into multiple orthogonal sub-bands that are utilized in the LTE-Unlicensed

network. A coexistence on the same unlicensed channel can lead to the mutual interference

between the LTE-Unlicensed and WiFi AP due to the non-orthogonality of their respective

transmitted signals. To this end, the interference power from a set of orthogonal sub-bands L to

the WiFi system on the channel c is modeled by considering the power spectral density of the

signal, as follows.

When SUE n ∈ Nm emits transmit power P l,knm on sub-band l of chunk k, the power spectral

density of sub-band l is given as [10]:

Φl,knm(f) = P l,k

nmTs

(sin πfTsπfTs

)2

, (5)

where Ts is the symbol duration of an ideal Nyquist pulse.

The interference at the WiFi AP introduced by the transmission on sub-band l in chunk k of

SUE n ∈ Nm is the integral of the power spectral density of sub-band l across the WiFi AP’s

band as [10], [37]:

I(Bc)l,k (dcl,k, P

l,knm) =

∫ dcl,k+Bc/2

dcl,k−Bc/2

gl,knm,cΦl,knm(f)df, (6)

where dcl,k represents the spectral distance between sub-band l of chunk k and occupied band

Bc of the WiFi AP; gl,knm,c = Gl,knm,cF

l,knm,c where Gl,k

nm,c and F l,knm,c are the mean channel power

gain from SUE n ∈ Nm to the WiFi AP and the fast fading gain in sub-band l of the kth chunk,

respectively.

To meet the performance in the WiFi system, it is assumed that the LTE-Unlicensed network

can use the unlicensed band c whenever the total interference introduced from SUEs to the WiFi

AP’s band does not exceed IBc,thWiFi , i.e.,∑

m∈M

∑n∈Nk

∑k∈Kc

∑l∈Lc,k

ψknmI(Bc)l,k (dcl,k, P

l,knm) ≤ IBc,th

WiFi , (7)

where I(Bc)l,k (dcl,k, P

l,knm) = P l,k

nmHl,knm,c is determined from (6). For simplicity, let

H l,knm,c = gl,knm,cTs

∫ dcl,k+Bc/2

dcl,k−Bc/2

(sin πfTsπfTs

)2

df (8)

be the interference factor at the WiFi AP for sub-band l in chunk k on unlicensed band c from

SUE n ∈ Nm.

5) Resource Orchestration for Uplink LTE-Unlicensed in Coexisting LTE-WiFi systems (ROC):

The ROC problem is formulated as follows:

(ROC)

max.(ψ,P )

∑m∈M

∑n∈Nm

φrRnm(ψ,P )−∑m∈M

∑n∈Nm

∑k∈Kc

φc,kψknm, (9)

s.t. (3), (4), (7),∑n∈Nm

ψknm ≤ 1, ∀m ∈M, k ∈ Kc, (10)

∑k∈Kc

ψknm ≤ 1, ∀n ∈ Nm, m ∈M, (11)

∑k∈Kc

ψknm∑l∈Lc,k

P l,knm ≤ Pmax

nm , ∀n ∈ Nm, m ∈M, (12)

P l,knm ∈ [P l

min, Plmax],∀n ∈ Nm,m ∈M, k ∈ Kc, (13)

ψknm = 0, 1 ,∀m ∈M, n ∈ Nm, k ∈ Kc. (14)

Here, the parameter φr is a reward for each data rate unit. In this paper, in order to guarantee

a positive network profit from each user in the unlicensed channel, we assume the channel

utilization cost φc,k is no greater than φrRnm, ∀m ∈M, n ∈ Nm, k ∈ Kc. The network utility

in (9) is the net profit, defined as the total network throughput discounted by the total channel

utilization costs that the small cell network provider has to pay to the LTE-UM. Constraint (10)

ensures that each chunk is allocated to at most one SUE in its SBS. Constraint (11) guarantees

that each SUE can be allocated to at most one chunk. Constraints (12) and (13) mean that the

transmit power of the SUEs on sub-bands are adjusted within the permitted range of the total

transmitted power and on each sub-band, respectively.

Since the ROC contains binary variables ψ and continuous variables P , it is a mixed integer

non-linear optimization problem, which is generally NP-hard. In the following sections, we

propose solving the ROC problem in a distributed manner based on the Lagrangian relaxation

and matching game approaches.

III. LAGRANGIAN RELAXATION SOLUTION FOR RESOURCE ORCHESTRATION

Herein, the solution to the ROC is detailed based on the Lagrangian relaxation method. Then,

we propose the distributed algorithms to find suboptimal solution of the ROC problem.

A. Lagrangian Relaxation-Based Solution

Let us denote the non-negative multipliers associated with constraints (3), (4), (7) and (12) by

λnm, βm, νc and κnm, respectively. By jointly considering the objective function and constraints,

the partial Lagrangian of the ROC can be derived as follows:

(15)

L(ψ,P ,λ,β, νc,κ) = φr∑m∈M

∑n∈Nm

∑k∈Kc

ψknm∑l∈Lc,k

rl,knm(P l,knm)−

∑m∈M

∑n∈Nm

∑k∈Kc

φc,kψknm

+∑m∈M

∑n∈Nm

λnm(∑k∈Kc

ψknm∑l∈Lc,k

rl,knm(P l,knm)−Rmin

nm )

−∑m∈M

βm(∑n∈Nm

(∑k∈Kc

ψknm∑l∈Lc,k

rl,knm(P l,knm))− Zm,bh)

− νc(∑m∈M

∑n∈Nk

∑k∈Kc

∑l∈Lc,k

ψknmI(Bc)l,k (dcl,k, P

l,knm)− IBc,th

WiFi )

−∑m∈M

∑n∈Nm

κnm(∑k∈Kc

ψknm∑l∈Lc,k

P l,knm − Pmax

nm ),

where λ = [λnm]1×(NmM), β = [βm]1×M , and κ = [κnm]1×(NmM).

Then, the dual problem of the ROC is given by:

min.(λ0,β0,νc≥0,κ0)

D(λ,β, νc,κ), (16)

where the Lagrange dual function D(λ,β, νc,κ) is

D(λ,β, νc,κ) = max(ψ,P )

L(ψ,P ,λ,β, νc,κ), (17)

s.t. (10), (11), (13), and (14).

Proposition 1. Using the Lagrangian in (15), the maximization problem (17) can equivalent to

ROC-D :

max(ψ,P )

∑m∈M

∑n∈Nm

∑k∈Kc

ψknm

[Ωknm(P k

nm)− φc,k], (18)

s.t. (10), (11), (13),

in which

(19)

Ωknm(P k

nm) = φr∑l∈Lc,k

rl,knm(P l,knm) + λnm

∑l∈Lc,k

rl,knm(P l,knm)

− βm∑l∈Lc,k

rl,knm(P l,knm)− νc

∑l∈Lc,k

P l,knmH

l,knm,c − κnm

∑l∈Lc,k

P l,knm,

where P knm = [P l,k

nm]1×|Lc,k| is the power vector of chunk k at SUE n ∈ Nm.

Proof: See Appendix A.

Disregarding the Lagrangian multipliers λ, β, νc, and κ, the ROC-D problem is combinatorial

optimization problem concerning ψ for a fixed P . Additionally, it can be seen that (18) is a

concave function with respect to P l,knm. The channel and power allocations can be found by

solving two subproblems as follows.

1) Power Control Phase: When channel allocation and Lagrangian multipliers values are

fixed, problem (18) is concave with respect to P . Moreover, due to the independent channel

fading in different sub-bands, the ROC-D problem can be further decomposed into |Lc| sub-

problems as follows:

maxP

∑m∈M

∑n∈Nm

∑k∈Kc

∑l∈Lc,k

ψknm

[ωl,knm(P l,k

nm)− φc,k], (20)

s.t. (13),

in which

(21)ωl,knm(P l,knm) , (φr + λnm − βm)rl,knm(P l,k

nm)

− (νcHl,knm,c + κnm)P l,k

nm.

Proposition 2. The optimal power allocation of (20) can be determined based on the Karush-

Kuhn-Tucker (KKT) conditions [26] as

P l,k∗nm =

[∆l(φr + λnm − βm)

(νcHl,knm,c + κnm) ln 2

− 1

γl,knm

]P lmax

P lmin

, (22)

where [x]ba := max(min(x, b), a).

Proof: See Appendix B.

The result in (22) confirms our intuition that transmit power on the sub-bands of chunk k

is reduced whenever the backhaul of SBS m is congested (i.e., backhaul price βm increases).

This is also the case when the total interference at the WiFi AP is violated, i.e., the interference

price νc increases. Additionally, when the total transmit power at the SUE is greater than a given

threshold, the power price increases which also leads to reducing the transmit power on sub-

bands at that SUE. Moreover, whenever the data rate of SUE n ∈ Nm is less than a threshold at

iteration t, λnm goes up at the iteration (t+ 1). Hence, the transmit power on each subchannel

l ∈ Lc,k also goes up in next iteration (t + 1). Consequently, this leads to increasing the data

rate of the SUE n ∈ Nm on the chunk k at the next iteration (t + 1). These steps are looped

until satisfying the SUE’s QoS constraint.

2) Channel Allocation Phase: By fixing P and the Lagrangian multipliers, problem (18) is

combinatorial in the variable ψ. Moreover, because the mutual interference among small cells

is not taken into account in our work, the ROC-D problem can be decomposed into M sub-

problems, each of which is in the following form.

ROC-Dm :maxψ

∑n∈Nm

∑k∈Kc

ψknm

[Ωknm(P k

nm)− φc,k], (23)

s.t. (10), (11), and (14).

The optimal chunk-based channel allocation in (23) becomes a maximum weighted matching

problem. The chunk k∗ is thus allocated to SUE n ∈ Nm according to

ψk∗

nm = 1, k∗ = arg max∀n∈Nm

[Ωknm(P k

nm)− φc,k]. (24)

By observing (23) and (24), we see that the SBS prefers to allocate the chunk to its SUEs

who maximize the utility in terms of the benefit Ωknm(P k

nm) minus the channel cost φc,k. To find

the optimal chunk-base channel allocation in (24), the Hungarian method [38] can be centrally

performed at each SBS to solve this bipartite matching problem with a computational complexity

of O ((KNm)3).

3) Lagrangian Multiplier Update: To determine the Lagrangian multipliers λ, β, νc, κ, we

solve the primal problem (16) given the suboptimal solutions P ∗ and ψ∗ from (22) and (24).

By substituting (22) and (24) back into (17), the dual objective is as

(25)

D(λ,β, νc,κ) =∑m∈M

∑n∈Nm

∑k∈Kc

ψk∗nm

[ ∑l∈Lc,k

((φr + λnm − βm)rl,knm(P l,k∗nm )

− (νcHl,knm,c + κnm)P l,k∗

nm )− φc,k]−∑m∈M

∑n∈Nm

λnmRminnm

+∑m∈M

βmZm,bh +∑k∈Kc

∑l∈Lc,k

νIBc,thWiFi +

∑m∈M

∑n∈Nm

κnmPmaxnm .

λnm(t+ 1) =

λnm(t)− s1(t)

(∑k∈Kc

ψknm(t)∑l∈Lc,k

rl,knm(P l,knm(t))−Rmin

nm

)+

(27)

βm(t+ 1) =

βm(t) + s2(t)

( ∑n∈Nm

∑k∈Kc

ψknm(t)∑l∈Lc,k

rl,knm(P l,knm(t))− Zm,bh

)+

(28)

νc(t+ 1) =

[νc(t) + s3(t)

( ∑m∈M

∑n∈Nk

ψknmHl,knm,cP

l,knm(t)− IBc,th

WiFi

)]+

(29)

κnm(t+ 1) =

κnm(t) + s4(t)

(∑k∈Kc

ψknm(t)∑l∈Lc,k

P l,knm(t)− Pmax

nm

)+

. (30)

Algorithm 1 ROCH: Resource orchestration for LTE-Unlicensed system.

Initialization: N , M, Kc, L, P (0), ψ(0), λ(0), β(0), ν(0)c , κ(0).

Repeat:

* Algorithm at the WiFi AP:

1: Update and broadcast the interference price νc(t+ 1) as (29).

* Algorithm at the SBS m ∈M:

2: Allocate chunk-based channels to SUEs to obtain (24) using the Hungarian searching method.

3: Update and broadcast the congestion price βm(t+ 1) as (28).

* Algorithm at SUE n ∈ Nm:

4: Update the QoS price λnm(t+ 1) and power price κnm(t+ 1) as (27) and (30), respectively.

5: Update the transmit power P knm(t+ 1) as (22).

Until: |βm(t+ 1)− βm(t)|≤ ξ1, |νc(t+ 1)− νc(t)|≤ ξ2, |λnm(t+ 1)− λnm(t)|≤ ξ3, |κnm(t+ 1)− κnm(t)|≤ ξ4simultaneously.

Since (25) is an affine function of λ, β, νc, and κ, the dual problem is convex. By the projected

gradient-descent method [26], the optimal Lagrangian multipliers can be found according to (27),

(28), (29), and (30), where [a]+ = maxa, 0. Here, the step sizes si(t) (i = 1, 2, 3, 4) are chosen

such that∞∑t=0

si(t)2 <∞ and

∞∑t=0

si(t) =∞, i = 1, 2, 3, 4, (26)

to guarantee convergence of the algorithm [26].

B. Proposed Algorithm for the ROC problem.

The above derivation allows us to now propose the distributed algorithm ROCH that solves

the originally formulated problem ROC using the Hungarian method for the channel allocation.

Specifically, the information exchange among the SUEs and SBSs is realized using a feedback

mechanism. The total interference on unlicensed band c is estimated at the WiFi AP. Then,

the interference price will be updated and broadcast to the SBSs. After that, the interference

prices are forwarded by the SBSs to their serviced SUEs. It is assumed that there exists virtual

unlicensed users to manage the unlicensed band on the WiFi AP [11].

Proposition 3. The sequence of primal-dual variables updated by the ROCH algorithm converges

to a locally optimal solution P ∗ and ψ∗ of the ROC problem.

Proof: Since the dual problem ROC-D is convex in the variable P for a fixed ψ, (22)

gives an optimal solution P ∗ for a fixed ψ. As well, (24) gives an optimal solution ψ∗ using

the Hungarian method for a fixed P . The channel and power allocation policies are the unique

strategy guaranteeing a optimal solution of the ROC-D in both P and ψ. For given (P ,ψ), the

optimal Lagrangian multipliers of the primal problem can be found by the projected gradient-

descent method [26] according to (27), (28), (29) and (30) with non-negative step sizes s1(t),

s2(t), s3(t) and s4(t), respectively. By updating the sequence of primal-dual variables according

to the ROCH algorithm, convergence to a unique solution is guaranteed. The duality gap is non-

zero due to the ROC problem is non-convex. Hence, the ROCH algorithm can only guarantee

a locally optimal solution P ∗ and ψ∗ of the ROC problem [26].

An optimal channel allocation is obtained by using the Hungarian method. The issue with

this method is that the SBS and/or LTE-UM would need to have information on all chunks from

the SUEs. Here, the channel information of each chunk k is the value of (Ωknm(P k

nm) − φc,k),

∀k ∈ Kc. In addition, the LTE-UM would need to broadcast the channel price φc,k to all SUEs,

∀k ∈ Kc. A centralized unit is needed to run the Hungarian method. Also, the solution based on

the Hungarian method may require significant overhead for message exchanges in the network.

The Hungarian method execution faces selfish and rational SUEs and SBSs that care about

their own utility in the channel allocation phase. In next section, the competitive behaviors of

the SUEs and SBSs for channel allocation are carefully examined by taking advantage of the

one-to-one matching game [22], [39].

IV. MATCHING GAME-BASED LOW-COMPLEXITY SOLUTIONS FOR ROC PROBLEM

Hereafter, we design channel allocation by exploiting the advantages of the matching game

theory [22], [39]. Moreover, the both cases of with and without channel utilization cost sharing

are considered. Based on these assumptions, we propose two distributed algorithms to solve

the ROC-Dm problem using the one-to-one matching game approach. We first investigate the

competitive channel allocation without sharing the channel price to the SUEs. Then, we consider

the channel allocation with sharing channel price from LTE-Unlicensed to the SUEs. In the

formulated games, each SUE competes with others to get matched with the most preferred chunk-

based channel in its preference list. Meanwhile, the SBS prefers to match the most preferred

SUE on each chunk.

A. Low Computational Complexity Solution based on Matching Game without Channel Utiliza-

tion Cost Sharing

1) Matching game for channel allocation without channel utilization cost sharing: First,

we study a scenario where the SUEs do not know about channel utilization cost information.

We consider a one-to-one matching game at each SBS m ∈ M, which is defined by a tuple

(Nm,Kc,Nm ,Kc). Here, the preference relations of the SUEs and chunks in SBS m are

denoted by Nm= nmn∈Nm and Kc= kmk∈Kc , respectively. The definition of the

modeled matching game is stated as follows:

Definition 1. A matching game with two setsNm and Kc for the channel allocation is represented

by a function ϕm: Nm 7→ Kc with

(i) n = ϕm(k)↔ k = ϕm(n), ∀n ∈ Nm, k ∈ Kc;

(ii) |ϕm(k)|≤ 1 and |ϕm(n)|≤ 1, n ∈ Nm, k ∈ Kc.

In the matching ϕm, SUE n preferring chunk k to k′ is denoted by k nm k′ (k, k′ ∈ Kc).

Meanwhile, a chunk k preferring SUE n to n′ is denoted by n km n′ (n, n′ ∈ Nm). For

the two-sided matching game, the aim is to seek a stable matching ϕm such that there exists

no blocking pair in the matching ϕm. A pair (n, k) is a blocking pair for ϕm if there exists

k nm k′, (k, k′ ∈ Kc) and n km n′, (n, n′ ∈ Nm).

The utility functions Unm(k) and Ukm(n) that, respectively, form the preference relations nmand km of SUEs and the SBS in small cell m are stated as follows:

Utility function for the SUE. At each SBS m, the SUE estimates a corresponding utility on

chunk k based on (19) as

Unm(k) = Ωknm(P k

nm). (31)

With (31), each SUE forms its preference list of all chunks. Then, it only gives a request to its

SBS for occupying the most preferred chunk in its preference list.

Utility function for each chunk at the SBS. In response to the requests from a set of N reqm

SUEs to occupy a certain chunk, each SBS desires to maximize a utility function on each chunk

stated as

Ukm = maxΩknm(P k

nm)− φc,kn∈N reqm, (32)

which is the net profit received by SUE n on chunk k. Then, based on (32), each SBS only

accepts the SUE on the chunk with the most preferred SUE in its preference list.

Proposition 4. The ROC-Dm problem can be modeled as an one-to-one matching game in

Definition 1.

Proof: In Definition 1, the conditions |ϕm(k)|≤ 1 and |ϕm(n)|≤ 1 correspond to the

constraints (10) and (11), respectively. Moreover, the utility definitions in (31) and (32) also

capture the objective function (23) of the ROC-Dm. Therefore, the ROC-Dm problem can

be modeled as a one-to-one matching game. The tuple (Nm,Kc,Nm ,Kc) corresponds to the

channel allocation by SBS m to its SUEs.

Using Proposition 4, we now present a distributed algorithm MCAW in Algorithm 2 for

channel allocation in a single SBS without channel utilization cost sharing. Our goal is to

find a suboptimal channel allocation under a stable matching ϕ∗m. At first, each SUE forms its

preference list based on (31) [cf. Line 1]. In the matching process, each SUE gives a request

bn→k = 1 to occupy chunk k that takes the highest utility [cf. Lines 2, 3 and 4]. On the SBS

side, the SBS collects and creates a preference list on chunks [cf. Line 6]. The SBS decides to

assign chunk-based channels to SUEs that give the highest utility value on each chunk [cf. Line

7]. The SUEs that are replaced or rejected by the SBS will be updated in the reject list of the

SBS. Then, the SUEs’ preference relations are also updated by removing the chunks that are

rejected by the SBS [cf. Line 8]. It is noted that the acceptance or rejection process of applicants

(i.e., the SUEs) is executed similarly to the conventional deferred acceptance algorithms [21],

Algorithm 2 MCAW: Matching-based Channel Allocation Without channel utilization cost

sharing.Initialization: Nm,Kc, ϕm = ∅.

* Discovery and utility evaluation:

1: Each SUE n ∈ Nm builds nm based on (31).

* Find stable matching ϕ∗m:

2: while∑∀k,n

bn→k 6= 0 do

3: Algorithm at SUE n ∈ Nm:

4: Send a bid for its SBS to chunk k∗ = arg maxk∈nm

Unm(k).

5: Algorithm at the SBS m ∈M:

6: Construct km based on (32) .

7: Update n∗ = ϕm(k)|n∗ = arg maxn∈km

Ukm(n).

8: Update the reject list and nm.

9: end while

Results: Convergence to stable matching ϕ∗m.

[22]. Hence, the MCAW algorithm converges to the stable matching ϕ∗m where there exists no

blocking pair between the SUEs and SBSs.

Theorem 1. The stable matching ϕ∗m gives a locally optimal solution of the ROC-Dm problem.

Proof: See Appendix C.

2) Algorithm design for the ROC problem based on the matching game without channel

utilization cost sharing: A distributed algorithm utilizing the matching game to find a suboptimal

solution to the ROC problem is presented in the ROCM-W algorithm (Algorithm 3). Since there

is no interference in data transmission among small cells in the considered model, one assignment

of a chunk does not generate interference to other small cell users. Consequently, the preference

of other users does not change in the channel allocation phase given a fixed transmit power level.

Hence, the ROCM-W guarantees a convergence to the stable matching at each outer-iteration of

Algorithm 3.

To implement a distributed channel and the power allocation, we assume that the channel

and power allocations occur on different time-scales. In the ROCM-W algorithm, a distributed

channel allocation based on the MCAW algorithm is implemented instead of the centralized

Hungarian method in the ROCH counterpart. In the former case, SBSs and SUEs share partial

Algorithm 3 ROCM-W: Low-complexity solution for ROC problem based on Matching game

Without channel utilization cost sharing.

Initialization: N , M, Kc, L, P (0), ψ(0), λ(0), β(0), ν(0)c , κ(0).

Repeat:

* Power allocation phase:

- Algorithm at the WiFi AP:

1: Update and broadcast the interference price νc(t+ 1) as (29).

- Algorithm at the SBS m ∈M:

2: Update and broadcast the congestion price βm(t+ 1) as (28).

- Algorithm at SUE n ∈ Nm:

3: Update the QoS price λnm(t+ 1) and the power price κnm(t+ 1) as (27) and (30), respectively.

4: Update the transmit power P knm(t+ 1) as (22).

* Channel allocation phase:

5: Allocate chunk-based channels to SUEs (i.e., ψknm(t+ 1)) using the MCAW algorithm.

Until |βm(t+ 1)− βm(t)|≤ ξ1, |νc(t+ 1)− νc(t)|≤ ξ2, |λnm(t+ 1)− λnm(t)|≤ ξ3, |κnm(t+ 1)− κnm(t)|≤ ξ4are simultaneously satisfied.

information to obtain a suboptimal solution.

Theorem 2. The sequence of primal-dual variables updated by the ROCM-W algorithm con-

verges to a suboptimal solution of the ROC problem.

Proof: The problem ROC-D is a convex in the variable P for a given ψ. Hence, at each

iteration t of the ROCM-W algorithm, the solution in (22) gives an optimal solution P (t) for a

given ψ(t). Then, given P (t), the preference lists of Nm (t) and Kc (t) are determined based

on (31) and (32) at each iteration t, respectively. Hence, the channel allocation phase based on

MCAW algorithm guarantees a convergence to the stable matching ϕ∗m(t) at each iteration t.

Since ϕ∗m(t) only gives a locally optimal solution in the channel allocation phase at iteration t

as proved in Theorem 1, the Slater condition is not satisfied, and hence there is a nonzero duality

gap. For given allocations ψ(t) and P (t), the optimal value of the Lagrangian multipliers of

the primal problem can be obtained by the projected gradient-descent method [26] according to

(27), (28), (29) and (30) with non-negative step sizes of s1(t), s2(t), s3(t) and s4(t), respectively.

Hence, the solution of the ROCM-W algorithm keeps improving their schedules according to the

weights resulting from P and dual variables λ, β, νc, and κ. Eventually, the channel allocation

in the ROCM-W algorithm converges to a unique solution. Therefore, by updating the sequence

of primal-dual variables according to the ROCM-W algorithm with a sufficiently small step sizes,

the convergence to a suboptimal solution (ψ∗,P ∗) is guaranteed.

B. Low Computational Complexity Solution based on a Matching Game with Channel Utilization

Cost Sharing

In this subsection, we study a scenario in which the SBS shares utilization cost on chunk-

based channels to the SUEs. A matching game is applied to find channel allocations. Unlike the

one-to-one matching game in Section IV-A, the demand function of the SUE is expressed as

Unm(k) = Ωknm(P k

nm)− φc,k. (33)

Responding to the demands to occupy chunks the SUEs for occupying chunks, each SBS

seeks to match each chunk with the SUE that maximizes the SBS’s utility function as

Ukm = maxΩknm(P k

nm)− φc,kn∈N reqm. (34)

After that, a locally optimal solution is obtained for the ROC-Dm problem based on the

matching game with channel utilization cost sharing as in the MCAW algorithm. A locally

optimal solution with channel utilization cost sharing can be found using the MCAS algorithm

which is obtained by substituting (31) and (32) in the MCAW algorithm by (33) and (34),

respectively.

By replacing the MCAW with the MCAS algorithm in the channel allocation phase of the

ROCM-W algorithm, we have the ROCM-S algorithm.

Theorem 3. The sequence of primal-dual variables updated by the ROCM-S algorithm converges

to a suboptimal solution of the ROC problem.Proof: The proof for Theorem 3 is similar to that for Theorem 2 and so it is omitted for

brevity.

V. COMPUTATIONAL COMPLEXITY ANALYSIS

To find the optimal (P ,ψ) in the ROCH algorithm, the Lagrangian multipliers need to

be updated via the sub-gradient method. Specifically, N QoS demand prices, M congestion

prices, one interference price, and N power prices are updated at each iteration with step sizes

(27), (28), (29) and (30), respectively. Thus, the computational complexity of the sub-gradient

Algorithm 4 ROCM-S: Low-complexity solution for ROC problem based on Matching game

with channel utilization cost Sharing.

Initialization: N , M, Kc, L, P (0), ψ(0), λ(0), β(0), ν(0)c , κ(0).

Repeat:

* Power allocation phase:

- Algorithm at the WiFi AP:

1: Update and broadcast the interference price νc(t+ 1) as (29).

- Algorithm at the SBS:

2: Update and broadcast the congestion price βm(t+ 1) as (28).

- Algorithm at the SUE:

3: Update the QoS price λnm(t+ 1) and power price κnm(t+ 1) as (27) and (30), respectively.

4: Update transmit power P knm(t+ 1) as (22).

* Channel allocation phase:

5: Allocate chunk-based channels to SUEs (ψknm(t+ 1)) to obtain (24) using the MCAS algorithm.

Until convergence:

6: |βm(t+1)− βm(t)|≤ ξ1, |νc(t+1)− νc(t)|≤ ξ2, |λnm(t+1)− λnm(t)|≤ ξ3, |κnm(t+1)− κnm(t)|≤ ξ4 are

satisfied, simultaneously.

method is O(

(N2M)2)

[26], here N = NmM . Additionally, the channel allocations need

to be determined based on the Hungarian method in each SBS. Since the computation in the

channel allocation phase is distributed to each SBS for each outer loop in the ROCH algorithm,

the computational complexity in the channel allocation phase is O(MN3mK

3). Thus, the total

computational complexity of the ROCH algorithm is O (MN3mK

3 +N4M2).

In the suboptimal solution using ROCM-W, the channel allocation computation is updated

using the MCAW algorithm at each SBS. The MCAW algorithm is hard to determine the

exact computational complexity since it depends on private information. Nevertheless, we can

determine an upper bound on the maximum number of communication message passings between

the SUEs-SBSs in the MCAW algorithm by analyzing the worst-case scenario. This scenario

occurs whenever all of the SUEs own the same preference relations in their preference list.

Moreover, the list order of of chunks at the SBS side is on exact contrary to the list order

at the SUEs side. Thus, when a SBS m has K chunks, in each loop of the MCAW algorithm,

Nm−1 SUEs will be rejected by the SBS m and this rejection is repeated for K−1 rounds. The

computational complexity for the channel allocation at SBS m in each outer loop of the ROCM-W

algorithm is bounded above by O(N2m(K−1)). Therefore, the computational complexity for the

ROCM-W algorithm is O(MN2m(K − 1) +N4M2). By similar arguments, the total complexity

of the ROCM-S algorithm is also O(MN2m(K − 1) +N4M2).

VI. PERFORMANCE EVALUATION

In this section, computer simulations are presented to verify the effectiveness of the proposed

algorithms.

A. Simulation Setup

We consider one WiFi AP and five SBSs (M = 5) in which each SBS has a coverage radius

of 30 m. The SBSs are located in a small indoor area of 150m × 200m to serve |N |= 10

SUEs, in which each SBS m has Nm = 2 SUEs. We setup an LTE-Unlicensed system using

12 sub-bands operating on the single frequency of the WiFi AP where each sub-band has a

bandwidth of 180 kHz. These sub-bands are further divided into 6 chunks to be assigned to

the SUEs. The channel power gains are supposed to be i.i.d. Rayleigh random variables with

unit mean. We adopt the log-distance pathloss model of [40]. In the pass-loss model of the

SUE-to-WiFi AP path-loss for distance a, we have La = 15.3 + 37.6 log10(a) + ρ. The wall

penetration loss ρ is taken as 10 dB. In the path-loss model between the SBS and its SUEs with

the distance a, we have La = 38.46 + 20 log10(a). The maximum tolerance interference power

on the channel c at the WiFi AP is set as −75 dBm. The noise power at any receiver is set to

−110 dBm. Each SUE has a minimum required data rate of 2.048 Mbps and a maximal power

constraint of 100 mW. We set φr = 10 per each unit data rate Mbps. Besides, we set the data rate

Rlicensednm = 0, ∀m ∈M,∀n ∈ Nm. Moreover, we use error tolerances ξi = 10−3, i = 1, 2, 3, 4

to terminate the proposed algorithms.

B. Simulation Results

Here, we first show the results given by the ROCH algorithm in a single snapshot based on

the above settings. After that, the results of the proposed algorithms over multiple snapshots will

be presented.

1) Evaluation of the ROCH algorithm within a single snapshot of the chunk-based channel

assignment and power allocations: We evaluate the proposed ROCH algorithm with 5 SBSs and

10 SUEs as shown in Fig. 2. We set φc,k equals to 5 for all chunks. The capacities of the backhaul

links are set as Z th1 = 6 Mbps, Z th

5 = 16 Mbps, and 8 Mbps for the remaining backhaul links.

X-Axis (m)0 100 200 300 400

Y-A

xis

(m)

0

50

100

150

200

250

300

350

400

SBS-1

SBS-2

SBS-3

SBS-4

SBS-5

1

2

3 4

56

78

910

WiFi AP

Fig. 2: Simulation model.

0 20 40 60 80 100 1200

100

200

300

400

500

Iteration

Net

wor

k U

tility

Network utility based on ROCH

Fig. 3: Convergence of the total utility of all

SUEs with K = 6 chunks.

The maximum interference power at the WiFi AP is kept at −75 dBm. The results of solving

the ROC problem by the ROCH algorithm are presented in Figs. 4 and 3. Fig. 3 shows that the

proposed algorithm converges in fewer than 40 iterations. The channel allocation and transmit

power are adapted based on the ROCH algorithm as shown in Figs. 4a and 4b. All backhaul

links are guaranteed to be less than pre-defined thresholds as shown in Fig. 4c. The WiFi AP

interference threshold is also guaranteed as shown in Fig. 4d. In the proposed algorithm, small

cell base stations 1, 2, and 4 force their SUEs to reduce transmission rate to avoid overloading at

their backhaul links. However, to maximize total network utility, the SUEs in small cells 3 and 5

increase their respective data rate. To protect the WiFi AP, the data traffic in small cells 3 and 5

only increase whenever the total interference power at the WiFi AP is kept below the predefined

threshold. Besides, small cell 4 has a higher rate than does small cell 3. This is because the

SUEs in small cell 3 produce a higher interference power level than that of the SUEs in small

cell 4 when they are allocated the same transmit power, i.e., the distance between the WiFi

system and small cell 3 is shorter than that of small cell 4 in Fig. 2. Intuitively, to maximize the

total network utility, the proposed algorithm prefers increasing transmission rate at small cell 4

to small cell 3.

2) Evaluation of the proposed algorithms in multiple snapshots: Here we compare four

schemes: ROCH, ROCM-W, ROCM-S, and ROC-Greedy. For the ROC-Greedy scheme, we utilize

the ROCM-S algorithm in which the channel allocation phase uses a greedy algorithm that the

SUE always selects a chunk in its preference list with the highest utility. All the presented results

0 20 40 60 80 100 1200

20

40

60

80

100

Iteration

Pow

er (

mW

)

SUE−1

SUE−2

SUE−3

SUE−4

SUE−5

SUE−6

SUE−7

SUE−8

SUE−9

SUE−10

(a) Power control

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

SUE index

Sub

chan

nel i

ndex

(b) Channel allocation

0 20 40 60 80 100 1200

2

4

6

8

10

12

14

16

Iteration

Dat

a ra

te (

Mpb

s)

Z1

Z2

Z3

Z4

Z5

Z1,th

Z2,th

, Z3,th

, Z4,th

Z5,th

(c) Backhaul protection.

0 20 40 60 80 100 120−115

−110

−105

−100

−95

−90

−85

−80

−75

−70

−65

Iteration

Inte

rfer

ence

pow

er (

dBm

)

I0 with ROCH

I0th

MBS protection

(d) WiFi AP protection.

Fig. 4: A snapshot of the chunk-based channel assignment and power allocations resulting from

the the ROCH algorithm with 5 SBS, 10 SUEs, and 6 chunks.

are averaged over five hundred of independent simulation runs, each of which realizes random

locations of the SUEs inside the SBSs’ coverage area and random channel power gains.

Fig. 5 compares the total network utility achieved by different schemes, in which Nm = 4, ∀m,

Z thm = 20, ∀m, φc,k = 5 (k = 1, 2, 3, 4, 5), φc,6 = 25, and IB,thWiFi = −75 dBm. Cumulative distri-

bution functions with different schemes are captured from five hundred independent simulation

runs. It is observed that the ROCM-W and ROCM-S schemes perform more similarly to the

ROCH scheme than to the ROC-Greedy scheme. This is because of using the matching game

only obtains a locally optimal solution of channel allocation in each outer loop of the updated

primal-dual iteration meanwhile the Hungarian method obtains an optimal solution. Besides,

500 550 600 650 700 750 8000

0.2

0.4

0.6

0.8

1

Network Utility

Cum

ulat

ive

dist

ribut

ion

Empirical CDF

ROCM−WROCM−SROCHROC−Greedy

Fig. 5: CDF of the network utility with different

methods.

14 16 18 20 22 24500

550

600

650

700

750

Limited backhaul rate (Mbps)

Net

wor

k U

tility

ROCM−WROCM−SROCHROC−Greedy

Fig. 6: Average network utility versus the lim-

ited backhaul rate with different methods.

since the updated channel allocation in the matching game-based schemes is better than in

the greedy scheme, the dual gaps by the ROCM-W and ROCM-S algorithms are smaller than

those by the ROC-Greedy algorithm. Moreover, the ROCM-S scheme outperforms the ROCM-W

scheme. This confirms that the proposal side (i.e., SUEs) in matching operations with the channel

utilization cost sharing in the ROCM-S scheme is better than the without channel utilization cost

sharing in the ROCM-W scheme. Clearly, the proposed algorithms can efficiently operate with

the partially sharing network information, giving a near optimal solution.

Fig. 6 presents the average network utility versus the backhaul links capacity for Nm =

4, ∀m, φc,k = 5, k = 1, 2, 3, 4, 5, φc,6 = 25, and IB,thWiFi = −75 dBm. As the backhaul links

capacity increases, the average network utilities of all proposed schemes go up due to the growing

data traffic of the SUEs at SBSs. These average network utilities saturate for the high-capacity

backhaul due to the constraint on the WiFi AP’s interference threshold. It is clear from Fig. 6

that the utility value by the ROCM-S algorithm is very close to that by the ROCH algorithm for

low-capacity backhaul. Due to the total reward getting from SUEs is limited by the low-capacity

backhaul links, the total network utility does not change in the first term of (9). In this case,

the SUEs will be assigned to low-price chunks to improve the network utility from the channel

allocation phase of the ROCM-S algorithm. Here, chunk 6 with the highest channel utilization

cost in the matching processes of the ROCM-S algorithm is less likely to be chosen by SUEs

for low-capacity backhaul. Besides, due to the remaining chunks are the same price, the result

of the ROCM-S is similar to that of the ROCH algorithm when there is no SUE allocating on

chunk 6.

Moreover, Fig. 6 depicts that the network utilities by the ROCM-S and ROCM-W schemes

achieve the respective gains of 17.6% and 20.1% over the ROC-Greedy scheme for a backhaul

link capacity of 18 Mbps. Moreover, the average network utilities of the ROCM-W and ROCM-S

schemes achieve 4% and 2.1% lower than that of the ROCH algorithm for backhaul link capacity

of 18 Mbps, respectively. These results confirm that the proposed algorithms using matching

theory with different strategies of cost sharing can obtain good performances in terms of utility

values with low computational complexities in the considered simulation setup.

In Fig. 7, we show the average network utility depending on the number of SUEs for different

schemes with Z thm = 18 Mbps, ∀m, φc,k = 5, k = 1, 2, 3, 4, 5, φc,6 = 25 and IB,thWiFi = −75 dBm.

The SUEs are deployed uniformly inside the SBSs’ coverage. As seen the network utilities of

all proposed schemes increase with more SUEs due to the growing data traffic of SUEs at SBSs.

Also show that the net utility values of the ROCM-W and ROCM-S schemes are close to the

ROCH scheme for a small number of SUEs. Particularly, using the proposed matching processes

in the channel allocation phase in the ROCM-S scheme can obtain the results as the ROCH

scheme for five SUEs. This is because there is only one SUE in each SBS in this scenario.

Therefore, there are no rejections of SUEs at the SBS from the MCAS algorithm in the channel

allocation phase of the ROCM-S algorithm for occupying chunks. This confirms that the matching

process in the MCAS algorithm can obtain the same result as the Hungarian algorithm in the

channel allocation phase through numerical results.

Additionally, Fig. 7 shows that the average network utility of the ROCM-S and ROCM-W

schemes can reach up to 17.5% and 22.7% gains over the ROC-Greedy algorithm for 25 SUEs,

respectively. With more SUEs, there is more competition for occupying the chunks among the

SUEs. As such, the net utility in the channel allocation phase as well as the net utility in both

the ROCM-S and ROCM-W schemes are reduced. In Fig. 7, the average network utility of the

ROCM-W and ROCM-S schemes is 5.3% and 1.8% lower than that of the ROCH algorithm for

25 SUEs, respectively.

To estimate optimality gap of the proposed approach, we run a globally optimal exhaustive

search to find optimal solution of the ROC problem. An optimal solution can be found by

examining all possibilities of chunk assignments, followed by solving the associated convex

power-allocation problems. Due to the combinatorial nature of the problem, we limit to a small-

to-medium sized problem with maximum number of 3 SUEs for each SBS. Fig. 6 shows that

Number of SUEs5 10 15 20 25

Net

wor

k U

tility

100

200

300

400

500

600

700

800

ROC-Optimal

ROCM-W

ROCM-S

ROCH

ROC-Greedy

Fig. 7: Average network utility versus the num-

ber of SUEs.

5 10 15 20 25560

580

600

620

640

660

680

700

720

740

Channel usage cost

Net

wor

k U

tility

ROCM−WROCM−SROCHROC−Greedy

Fig. 8: Average network utility following the

channel utilization costs of chunk 6.

the ROCH scheme yields the solution close to that of the ROC-Optimal scheme, with a gap of

4.08% for a network with 3 SUEs a per each SBS.

The proposed algorithms are further investigated with respect to the channel utilization cost

for Z thm = 18 Mbps, ∀m, and Nm = 4 SUEs, ∀m, IB,thWiFi = −75 dBm. To evaluate the effects of

the channel utilization cost, we fix φc,k = 5 (k = 1, 2, 3, 4, 5). Then, the channel utilization

cost of chunk 6 is increased from 5 to 25 units. As can be seen from Fig. 8, the average network

utilities of the proposed schemes decrease as the channel utilization cost of chunk 6 increases.

Intuitively, the channel utilization cost of chunk 6 affects the ROC problem in both the ROCM-S

and ROCM-W algorithms as follows. We first consider the network utility when φc,6 = 5 for all

chunks. At this operation point, the preference relations for the proposal side in both cases of

with and without channel utilization cost sharing are not affected by channel utilization cost. As

such, the SUEs’ proposals for occupying chunks are the same preference list in the matching

operation in both the ROCM-S and ROCM-W algorithms. Similarly, the preference relations at

the acceptance side are also the same in both scenarios. The average network utilities by the

ROCM-S and ROCM-W algorithms are the same as shown in Fig. 8. However, the gap between

the ROCM-S and ROCM-W schemes widens as the channel utilization cost of chunk 6 increases.

It is confirmed that the SUEs can have a better observation to choose low cost channels. The

average network utility with the ROCM-S algorithm is more efficient with the ROCM-W scheme

in terms of the channel utilization cost.

Particularly, in Fig. 8, we also show that the average network utility of the ROCM-S and

ROCM-W schemes can reach up to 18.5% and 17.0% gains, respectively, over results of the

ROC-Greedy scheme for the channel utilization cost of 20 for chunk 6. Additionally, the average

network utility of the ROCM-W and ROCM-S schemes also respectively achieve approximations

of 2.8% and 1.6% compared to the ROCH scheme for φc,6 = 20.

VII. CONCLUSIONS

In this paper, we have studied joint chunk-based channel and power allocation for an LTE-

Unlicensed network with limited backhaul link capacity. We have formulated an optimization

problem that maximizes the overall uplink network utility while guaranteeing the data rate

requirements of the served SUEs, avoiding congestion at the backhaul links, and protecting

the WiFi AP. A distributed framework based on Lagrangian relaxation has then been proposed

to analyze the interactions among the SUEs, SBSs, and WiFi AP. Under this framework, the

distributed algorithms have been introduced to enable the LTE-Unlicensed network to obtain

a suboptimal decision about chunk-based channel assignment and transmit power allocations.

Moreover, two distributed solutions with the limited information sharing and low computational

complexity have also been proposed based on the one-to-one matching game. Simulation results

have illustrated that the proposed algorithms converge after a few iterations. Additionally, the

results of the proposed algorithms using the matching game are shown to approach the suboptimal

solution. In addition, the results have also pointed out that the ROCM-S scheme outperforms

the ROCM-W scheme at the cost of some extra communication.

In the current proposed work, the intra-tier interference among small cells and admission

control mechanisms to guarantee a feasible solution of the proposed work have not been in-

vestigated. Nevertheless, our solution is still applicable to practical network settings without

intra-tier interference among small cells, i.e., the small cells are deployed to operate on a single

unlicensed channel with non-overlapping coverage areas or with non-overlapping sub-channels

among small cells.

APPENDIX

A. Proof of Proposition 1

The Lagrangian in (15) can be rewritten as (35). After grouping ψknm and substituting I(Bc)l,k (dcl,k, P

l,knm)

(35)

L(ψ,P ,λ,β, νc,κ) =∑m∈M

∑n∈Nm

∑k∈Kc

ψknmφr∑l∈Lc,k

rl,knm(P l,knm)−

∑m∈M

∑n∈Nm

∑k∈Kc

ψknmφc,k

+∑m∈M

∑n∈Nm

∑k∈Kc

ψknmλnm∑l∈Lc,k

rl,knm(P l,knm)−

∑m∈M

∑n∈Nm

λnmRminnm

−∑m∈M

∑n∈Nm

∑k∈Kc

ψknmβm∑l∈Lc,k

rl,knm(P l,knm) +

∑m∈M

βmZm,bh

−∑m∈M

∑n∈Nm

∑k∈Kc

ψknm∑l∈Lc,k

νcHl,knm,cP

l,knm + νcI

B,thWiFi

−∑m∈M

∑n∈Nm

∑k∈Kc

ψknmκnm∑l∈Lc,k

P l,knm +

∑m∈M

∑n∈Nm

κnmPmaxnm .

by P l,knmH

l,knm,c, we have:

L(ψ,P ,λ,β, νc,κ) =∑m∈M

∑n∈Nm

∑k∈Kc

ψknm

[φr∑l∈Lc,k

rl,knm(P l,knm) + λnm

∑l∈Lc,k

rl,knm(P l,knm)

− βm∑l∈Lc,k

rl,knm(P l,knm)− νc

∑l∈Lc,k

P l,knmH

l,knm,c − κnm

∑l∈Lc,k

P l,knm − φc,k

]−[ ∑m∈M

∑n∈Nm

λnmRminnm −

∑m∈M

βmZm,bh−νcIB,thWiFi−∑m∈M

∑n∈Nm

κnmPmaxnm

],

(36)

Let us define

Λknm(λ,β, νc,κ)∆=

[ ∑m ∈M

∑n ∈Nm

λnmRminnm −

∑m ∈M

βmZm,bh − νcIB,thWiFi −∑m ∈M

∑n ∈Nm

κnmPmaxnm

](37)

and

(38)

Ωknm(P k

nm)∆= φr

∑l ∈Lc,k

rl,knm(P l,knm) + λnm

∑l ∈Lc,k

rl,knm(P l,knm)

− βm∑l ∈Lc,k

rl,knm(P l,knm)− νc

∑l ∈Lc,k

P l,knmH

l,knm,c − κnm

∑l ∈Lc,k

P l,knm.

where P knm = [P l,k

nm]1×|Lc,k| is the power vector of SUE n ∈ Nm on sub-bands of chunk k.

Due to the fixed Lagrangian multipliers λ,β, νc and κ, Λknm(λ,β, νc,κ) is a constant value

in ψ and P . Hence, the results of ψ and P from the optimization problem (17) are equivalent

to results in (18). Therefore, the objective function (18) follows after eliminating the terms that

do not contain ψ and P .

B. Proof of Proposition 2

Taking the first-order derivative of (21) with respect to P l,knm, we have:

(39)∂ωl,knm(P l,knm)/∂P l,k

nm = (φr + λnm − βm)∆lγl,knm/((1 + γl,knmP

l,knm) ln 2)− (νcH

l,knm,c + κnm).

By setting ∂ωl,knm(P l,knm)/∂P l,k

nm = 0, we have:

(40)(φr + λnm − βm)∆lγl,knm/((1 + γl,knmP

l,knm) ln 2)− (νcH

l,knm,c + κnm) = 0.

Then, (40) is equivalent to

(41)γl,knmPl,knm =

(φr + λnm − βm)∆lγl,knm

ln 2(νcHl,knm,c + κnm)

− 1.

By dividing both sides of (41) by γl,knm, we obtain (22).

C. Proof of Theorem 1

Denote by τ the τ -th epoch time of the while loop in the MCAW algorithm.∑

k∈KcU

(τ)km =∑

k∈KcmaxΩk

nm(Pk,(τ)nm )− φ(τ)

c,kn∈N reqm

denote the total utility value of the ROC-Dm problem

that is captured at the end of each the τ -th epoch time. Let ϕ(τ)m be the formed matching at

iteration τ .

On the one hand, the decisions made by the SBS on chunks in the MCAW algorithm can be

seen as a sequential acceptance or rejection operation in the matching ϕ(τ)m as:

ϕ(0)m → ϕ(1)

m ...→ ϕ(τ)m → ϕ(τ+1)

m ... (42)

Since the MCAW algorithm is executed similarly to the conventional deferred acceptance algo-

rithm, it converges to a stable matching ϕ∗m, implying no blocking pair in the matching ϕ∗m [21],

[22]. Hence, the MCAW algorithm terminates at the stable matching ϕ∗m.

On the other hand, in the MCAW algorithm, the applicant SUE n ∈ Nm always seeks a

chunk k in its preference list (τ+1)nm to guarantee k∗ = arg max

k∈(t+1)nm

Unm(k),∀τ , where k ∈(t+1)nm .

At the acceptance side, the SBS m executes acceptance or rejection process on each chunk

to guarantee nm(τ+1) k n′m(τ) (n, n′ ∈ Nm), which means that U (τ+1)km ≥ U

(τ)km,∀k ∈

Kc,∀m ∈ M,∀τ . Hence,∑

k∈KcU

(τ+1)km ≥

∑k∈Kc

U(τ)km,∀m ∈ M, k ∈ Kc,∀τ . This leads to an

increment∑

k∈KcU

(τ+1)km (τ + 1)-th matching. Therefore, given the proposals from the SUEs,

every acceptance/rejection process from ϕ(τ)m → ϕ

(τ+1)m produces a dominance for allocating

chunks to the SUEs as follows:

ϕ(τ)m → ϕ(τ+1)

m ⇔∑

k∈KSARANG∈′∞∈cU

(τ+1)km >

∑k∈Kc

U(τ)km,∀m ∈M, k ∈ Kc,∀τ (43)

Therefore, from (42) and (43), we can see that the objective function (∑

k∈KcΩknm(P

k,(τ)nm )−

φ(τ)c,k) is dominated by the matching ϕ(τ)

m . In addition, the strategies of SUEs in stable matching

ϕ∗m give a locally optimal solution of the channel allocation in the ROC-Dm problem.

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