Joint User Association and Data Rate Allocation in ...

1

Joint User Association and Data Rate Allocation in Heterogeneous Wireless Networks Ali Rıza Ekti, Member, IEEE, Xu Wang, Student Member, IEEE, Muhammad Ismail, Member, IEEE, Erchin Serpedin, Fellow, IEEE and Khalid A. Qaraqe, Senior Member, IEEE

Abstract—In this study, the problem of joint data rate allocation and mobile terminal (MT) assignment is investigated in a heterogeneous wireless network (HetNet) environment that consists of wireless local area network (WLAN) access points (APs) and cellular base stations (BSs) for a best effort service. MTs are equipped with multiple radio interfaces and have multihoming capabilities. As a result, MTs can connect simultaneously to more than one wireless network (e.g., cellular network BS and WLAN AP) and aggregate the offered bandwidths from these networks to support applications with high required data rates. Unlike the existing research, in order to account for the MT’s limited number of radio interfaces and the abundant wireless network options, the joint MT assignment and data rate allocation problem is formulated to select the optimal subset of networks for each MT and allocate the optimal data rate share from this subset to maximize the HetNet total utility. The problem is formulated as a mixed integer non–linear program (MINLP) and due to its intractability and computational complexity, we transform the problem into a convex optimization problem via a binary variable relaxation approach. Based on the mathematical analysis of the problem, we present a heuristic algorithm for joint MT assignment and data rate allocation. Numerical results demonstrate that the proposed solution achieves a near optimal MT assignment and data rate allocation at reduced computational complexity. Index Terms—Heterogeneous wireless network, MT assignment, data rate allocation, multi-homing.

I. I NTRODUCTION The agile evolution of wireless communications systems has led to the emergence of new concepts in terms of quality of service (QoS) and system efficiency in the next generation wireless networks (NGWNs). Earlier cell phones were used only for voice transmissions along with limited text messaging applications. However, contemporary cell phones are capable of transmitting multimedia along with an operating system running on. In order to support the inevitable dynamic changes in NGWNs, heterogeneous wireless networks (HetNets) have become an integral part of NGWNs, where several wireless technologies can co–exist such as wireless local area network Copyright (c) 2015 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected]. Ali Rıza Ekti is with the Department of Electrical and Computer Engineering, Gannon University, Erie, PA, U.S.A. e-mail: [email protected] Xu Wang and Erchin Serpedin are with the Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX, U.S.A. e-mails: [email protected], [email protected]. Muhammad Ismail and Khalid A. Qaraqe are with the Department of Electrical and Computer Engineering, Texas A&M University at Qatar, Doha, Qatar. e-mails: {m.ismail, khalid.qaraqe}@qatar.tamu.edu. This work was supported by NSF Award CCF-1318338.

(WLAN), 3rd Generation Partnership Project (3GPP) systems of the Universal Mobile Telecommunications System (UMTS) and Long Term Evolution (LTE). HetNets comprise several cellular base stations (BSs) and WLAN access points (APs) with overlapped coverage that can improve energy consumption, network capacity, data rate, and coverage [1], [2]. Thus, new radio resource allocation mechanisms should be investigated to provide an efficient usage of all available networks in HetNets. Cooperation among different wireless technologies enable them to complement each other and provide seamless data services and connections. The radio resource allocation problem in a HetNet can be categorized into two types: (a) Single network accessallocation where mobile terminals (MTs)1 can access only the required data rate from a single network, and this single network is the best available network at the MT location, and (b) Multi–homing network-allocation where MTs can simultaneously utilize all the available networks and aggregate the offered data rate from these networks to improve the achieved data rate [3]. Specifically, each MT is covered by a set of overlapped networks which consist of a combination of cellular BSs and WLAN APs [4]–[6]. MT manufacturers, like Apple, LG, Blackberry and Samsung, provide standard built-in WLAN and cellular technologies. For instance, Apple’s iPhone operating system (iOS) 7 supports the multiple–connection transmission control protocol (MCTCP) which allows users to utilize both LTE and WLAN connections simultaneously [7]. Another example of “multi–homing” is the concept of “Open Garden” app which enables all devices to find the best available network combination [8]. Currently, MTs are equipped with multiple radio interfaces such as cellular and WLAN in order to efficiently use all the available networks. Additionally, an MT can maintain simultaneous connections from different access networks using its cellular and WLAN interfaces to provide an increased aggregated data rate with multi-homing capability to support applications that require higher data rates. Furthermore, due to the fact that at least one radio interface is active, it will provide seamless mobility support and reduce the call blocking rate [9]. Therefore, “multi–homing” has gained significant attention recently.

1 Each user has one MT device and this device is equipped with two interfaces, WLAN and cellular.

2

A. Related Work There are many studies dedicated to the radio resource allocation problem in HetNets. Existing studies can be divided into two categories single-network resource allocation and multi–homing network-allocation, respectively. In what concerns the first category, the data rate resource allocation methods are studied in [10]–[13]. Data rate allocation and call admission control algorithms are proposed for different classes of services in [10]. The work in [11] develops a distributed resource allocation method based on a convex optimization mechanism in order to find the optimal data rate for a minimum required data rate. However, the authors of [11] consider only a single network connection. The authors in [12] introduce a utility function based resource allocation scheme which exploits a convex optimization mechanism for code division multiple access (CDMA) and WLAN networks. The authors in [13] utilize a stochastic programming method to handle the probabilistic nature of demand uncertainty in HetNets. The major drawback of considering a single network connection is that it causes call dropping if there are no other networks such as WLAN and/or cellular networks in the area due to the fact MT cannot be satisfied with the required data rate. The bandwidth resource allocation methods belonging to the second category are studied in [4]–[6], [14]–[19] where novel algorithms are proposed to allocate the radio bandwidth resource to different traffic types based on a specific utility of the service supported over all the available networks. Utility fairness is considered in [14] to accomodate the bandwidth for different traffic types such as variable bit rate (VBR) and constant bit rate (CBR). The authors in [15] and [17] use non-cooperative game theory to allocate the bandwidth in a HetNet where the requested bandwidth is collected from all the available networks. The works in [16] and [18] propose a cooperative game theoretic approach to create an alliance among different types of networks. In [4], the authors consider different traffic types and MT types to maximize the utility function while maintaining QoS. The utility maximization problem is solved optimally via a convex optimization method for radio resource allocation in a distributed manner. The work in [19] proposes an opportunistic user association for HetNets to address a resource allocation problem for machineto-machine (M2M) traffic under a cooperative Nash bargaining solution method. In [5], optimal centralized and suboptimal decentralized resource allocation algorithms are proposed to account for both single network and multi-homing service and their performance is compared. In [6], a decentralized resource allocation algorithm is proposed to reduce the resource allocation complexity in the HetNet while considering the arrivals of new calls and service requests. Therefore, MTs with multihoming capabilities can further optimize the utilization of the resources of the HetNets [20]. B. Contributions In HetNets, researchers mainly assume that an MT connects to all existing networks in a multi-homing fashion. However, this vision overlooks the fact that the MT is equipped with

only a limited number of radio interfaces for each network type, i.e., one cellular and one WLAN interfaces. Hence, the MT has to select one cellular BS and one WLAN AP from all the available ones to get its required data rate in a multihoming fashion. In order to account for the limited number of interfaces for MTs, we formulate a joint MT assignment and data rate allocation problem to support MTs with multihoming capabilities. The contributions of this study are summarized below: •

•

•

•

•

The multi-homing radio resource allocation problem is formulated as a non-convex mixed integer non–linear program (MINLP) [21]–[27] to jointly perform MT network assignment and data rate allocation for a set of MTs with multi-homing capabilities within an overlapped coverage of WLAN APs and cellular BSs for best effort service. We show that the multi-homing radio resource allocation problem can be converted into a convex optimization problem after applying relaxation on the binary MT assignment variable and reparameterization of data rate variable. A Lagrangian decomposition approach is proposed to solve the relaxed convex optimization problem by dividing the problem into four sub-problems. We derive a lower bound and an upper bound for the optimal value of the non-convex MINLP problem. A closed-form upper bound is derived using a modified Lagrange duality method. However, the relaxed convex optimization problem does not necessarily provide a binary solution and therefore, the relaxed convex problem cannot perform the MT assignment. In order to ensure the binary assignment, we propose a heuristic method. We first assign the users and then allocate the data rate based on the selected MT assignment. Using such an approach, a lower bound is derived. Furthermore, it is also illustrated that under certain conditions, the lower-bound coincides with the upper bound and thus it achieves the optimal value of the MINLP. In this way, the computational complexity is dramatically reduced. Numerical results of the proposed heuristic algorithm are compared with the commercially available software: General Algebraic Modeling System (GAMS)/Branch– And–Reduce Optimization Navigator (BARON) [28]. The GAMS/BARON incorporates the branch and bound method and obtains global optimality while also utilizing the reduction tests. However, complexity and time consumption of GAMS/BARON increase dramatically when we consider a system with a large number of networks and MTs that compete on the available data rate at different networks. In literature, many studies utilize the relaxation method and Lagrangian multiplier approach in order to reduce the computational complexity of MINLP problems [21], [29]–[33]. However, these studies assume that the time sharing property is in place [25] due to the fact that the orthogonal frequency division multiplexing (OFDM) based system has many subcarriers. When the number of subcarriers is significantly large then the relaxed problem approaches the optimal solution [25]. Reference [34]

3

converts the non-convex MINLP problem into a convex problem by utilizing the successive convex approximation (SCA) method after binary variable relaxation in order to evaluate the energy efficiency of the problem. Then, the problem is solved in two steps by using the branch and bound method for user association, channel allocation and power allocation. Furthermore, [35] proposes a joint resource allocation and user association approach to provide an upper bound by utilizing numerical analysis. The optimal resource allocation which is converted into a convex problem can be solved using the methods in [36]. However, it is assumed that the user can occupy non-integer number of OFDM subcarriers. In our work, we show that regardless of the number of MTs or BSs/APs, our proposed approach exhibits tight bounds for the optimal solution. Additionally, we show that the relaxation approach does not provide the optimal solution in all cases; therefore, we provide the optimality rules and based on them, we derive a heuristic solution that achieves a near-optimal allocation and in some cases, the optimal solution. The remainder of this paper is structured as follows. The system model is presented in Section II. The multi-homing radio resource allocation problem is formulated and solved in Section III. Numerical results and discussions are given in Section IV. Finally, conclusions are drawn in Section V.

for two different calls. Therefore, we assume that the MT can have multiple interfaces of the same type, but the MT can only utilize one interface of the same type at a given moment of time. Hence, the MT can utilize one radio interface from I1 and one radio interface from I2 and aggregate the offered data rate from these two radio interfaces to support its ongoing call. The binary MT assignment variable is denoted by xnmi . Each AP in WLAN is assigned a single channel and MTs share this channel via an enhanced version of the distributed coordination function (DCF) which enables MTs to avoid transmission collisions [17], [41]. Macro Base Station

Micro Base Station

Femto Base Station

WLAN

II. S YSTEM M ODEL We consider a HetNet where a combination of WLAN APs and cellular BSs present an overlapped coverage as depicted in Fig. 1. The cellular BSs can be a mixture of different cellular BSs types such as macro base station (MBS), micro base station (MiBS) and femto base station (FBS). The network sets corresponding to WLAN APs and cellular BSs are denoted by N1 = {1, 2, · · · , N1 } and N2 = {N1 + 1, · · · , N1 + N2 }, respectively. The total network set is denoted by N = N1 ∪N2 , where N1 ∩ N2 = ∅. The candidate set of connected cellular BSs and WLAN APs that the MT selects from are the ones with the highest received signal strength, and hence the MTs are already within their coverage area. However, the question is how to assign MTs to these networks and allocate data rates to different MTs’ radio interfaces in a way that maximizes the total data rate of the heterogeneous network. The wireless networks are operated in separate frequency bands by different service providers, and hence no interference exists among these networks. Interference management techniques (e.g., frequency reuse [37]–[40]) are adopted for interference mitigation among BSs of the same network. The set of MTs located within this HetNet is denoted by M = {1, 2, · · · , M }. Each MT is equipped with WLAN and cellular interfaces. The set of interfaces is denoted by I = I1 ∪ I2 , where I1 and I2 represent the WLAN and cellular interfaces, respectively, and I1 ∩I2 = ∅. The allocated data rate from network n ∈ N to radio interface i ∈ I of m ∈ M MT is denoted by rnmi . Even though some of smartphones with dual sim card present more than one cellular radio interface, these interfaces cannot be used simultaneously

Mobile terminal with Overlapped Cellular and WLAN Networks Coverage

Fig. 1: Cellular network and WLAN overlapped coverage.

III. P ROBLEM F ORMULATION In this section, the multi-homing resource allocation problem is formulated. Mathematical modeling of resource allocation and scheduling algorithms has been widely investigated in literature [42]. One prominent feature of these algorithms is to employ the utility functions [14], [36], [42]–[47]. Many criteria can be considered in the utility functions based on the layers of the open systems interconnection (OSI) model, such as co-channel interference, channel gain, and fading, which are physical layer parameters. However, in this study, only the connection-level data rate allocation problem is considered in the upper layers (network layer) under the assumption of a successful call admission protocol [48]. Therefore, we do not consider the physical layer constraints [4], [45], [49]– [52] and instead we consider in this first step of research the maximization of a utility function which depends on the achieved data rate. We adopt a utility function perspective in order to account for the proportional fairness among users [53], [54]. Let unmi (rnmi ) denote the utility function of network n allocating data rate rnmi to i ∈ I of m ∈ M. Then, the utility function can be defined as: unmi (rnmi ) = xnmi ln (1 + ηrnmi ) ,

(1)

4

where η is used for scalability of rnmi and xnmi ∈ {0, 1} stands for the binary MT assignment variable for interface i ∈ I of MT m ∈ M to network n ∈ N . The overall resource allocation objective of all the networks resumes to finding the optimum allocation rnmi , ∀n ∈ N , ∀m ∈ M, ∀i ∈ I that maximizes the total utility in the region, expressed as U=

X X X

unmi (rnmi ) .

(2)

n∈N m∈M i∈I

For each network n, the allocated resources should be such that the total load in its coverage area is within the network capacity limitation Zn , i.e., X X

xnmi rnmi ≤ Zn , ∀n ∈ N .

(3)

m∈M i∈I

The data rate resource allocation problem is formulated under the assumption of proportional fairness in the overlapped WLAN APs and cellular BSs. Using (1), (3) and binary assignment variable, xnmi , the primary data rate resource allocation problem, (P1 ), can be expressed in the form of the following non-convex MINLP max r,x

s.t.

X X X

xnmi ln(1 + ηrnmi )

(P1 )

is smaller than the number of available BSs/APs, there is no need to do a resource allocation, an MT can connect to any available BS/WLAN AP. However, in practice this is not the case. The number of MTs is much greater than the number of BSs/WLAN APs. Therefore, pairing is not possible and instead MTs have to share the radio resources of these BSs/APs. The question is what is the optimal MT assignment and data rate share for each MT when the number of MTs is significantly larger than the number of available BSs/APs which is the case in our study. In this paper, a lower bound and an upper bound are derived for the value of objective function (P1 ). Specifically, the binary variable is relaxed such that the MINLP problem resumes to a convex optimization formulation and a closed-form upper bound is derived using a modified Lagrange duality method. However, the relaxed convex optimization problem does not necessarily have a binary solution, and therefore, it might not be able to perform the MT assignment. Motivated by the modified Lagrange duality method, a heuristic method is proposed to first assign the users and then allocate the data rate based on the selected MT assignment. In this way, a lower bound is derived in a closed-form. Furthermore, it is also illustrated that under certain conditions, the lowerbound coincides with the upper bound and thus it achieves the optimality of the MINLP.

n∈N m∈M i∈I

X X

xnmi rnmi ≤ Zn , ∀n ∈ N

(Cp1 )

m∈M i∈I

X X

xnmi ≤ 1, m ∈ M

(Cp2 )

xnmi = 0, m ∈ M

(Cp3 )

xnmi ≤ 1, m ∈ M

(Cp4 )

n∈N1 i∈I1

X X n∈N1 i∈I2

X X

A. Upper-bound: A Convex Relaxation Approach In order to convert the problem (P1 ) into a convex optimization problem, we adopt the binary relaxation approach [21]. The binary constraint xnmi ∈ {0, 1} is modified by allowing xnmi to take any fractional value in the interval [0, 1]. In addition, a new variable wnmi = xnmi rnmi is introduced such that the relaxed optimization problem resumes to

n∈N2 i∈I2

X X

xnmi = 0, m ∈ M

(Cp5 )

max w,x

n∈N2 i∈I1

rnmi ≥ 0, m ∈ M, n ∈ N , i ∈ I xnmi ∈ {0, 1}, m ∈ M, n ∈ N , i ∈ I

(Cp6 ) (Cp7 )

The objective function is given in (P1 ), the constraint (Cp1 ) ensures that the allocated resource cannot exceed the capacity limit, the constraints (Cp2 ) and (Cp3 ) guarantee that an MT can only connect to WLAN networks using WLAN interfaces. Moreover, the constraints (Cp4 ) and (Cp5 ) assure that an MT can only establish a connection to the cellular BS using only its cellular interfaces. Furthermore, the constraint (Cp6 ) secures that the allocated data rate is always a positive quantity, and (Cp7 ) describes the binary nature of the assignment variable. It can be seen that (P1 ) is a non-convex MINLP problem which involves both binary variables, xnmi and real–valued positive data rate variables, rnmi . Due to the computational complexity and mathematical intractability, MINLP is a Non– deterministic Polynomial–time (NP)-hard problem [55], [56]. For instance, without considering the number of interfaces, i.e., |N | = 2, |M| = 50 and |I| = 1, there will be a total of 250 network and MT assignments. If the number of MTs

s.t.

X X X n∈N m∈M i∈I

X X

xnmi ln(1 + η

wnmi ) xnmi

wnmi ≤ Zn , ∀n ∈ N

(P2 ) (Cr1 )

m∈M i∈I

Cp2 − Cp5 ,

(Cr2 )

wnmi ≥ 0, m ∈ M, n ∈ N , i ∈ I

(Cr3 )

0 ≤ xnmi ≤ 1, m ∈ M, n ∈ N , i ∈ I

(Cr4 )

It is illustrated in Appendix A that problem (P2 ) is a convex optimization problem. In addition, it is easy to verify that there exists an interior point in the feasible region. Thus, the Slater’s condition holds and the problem presents a zero duality gap [57]. In this way, the optimal solution of problem (P2 ) can be derived using the Lagrange duality method. However, such an optimal solution is not necessarily binary, and it may not satisfy the binary constraint in problem (P1 ). In the following context, the optimal objective value for the relaxed problem (P2 ), which is denoted as VU∗ , is derived in a closed-form using a modified Lagrange duality method. Consequently, if we denote the optimal objective value of (P1 ) ∗ ∗ as v ∗ , vU presents an upper-bound for v ∗ , i.e., v ∗ ≤ vU , since (P2 ) is optimized over a larger constraint set.

5

The Lagrangian of problem (P2 ) can be expressed as max   X X X wnmi L(w, x, λ) = xnmi ln 1 + η xnmi n∈N m∈M i∈I ! X X X − λn wnmi − Zn , n∈N

x

X X X

xnmi A(λn )

n∈N1 m∈M i∈I2

s.t.

X X

(P4 )

xnmi = 0, m ∈ M

n∈N1 i∈I2

(4)

0 ≤ xnmi ≤ 1, ∀n ∈ N1 , m ∈ M, i ∈ I2 .

m∈M i∈I

+

where λn ∈ R stands for the Lagrange multiplier associated with the network data rate constraint, (Cr1 ). The dual function is therefore

max x

X X X

xnmi A(λn )

n∈N2 m∈M i∈I1

s.t.

X X

(P5 )

xnmi = 0, m ∈ M

n∈N2 i∈I1

g(λ) = max L(w, x, λ) w∈D1

0 ≤ xnmi ≤ 1, ∀n ∈ N2 , m ∈ M, i ∈ I1 .

(5)

s.t. Cr2 − Cr4 ,

max x

where D1 denotes the feasible domain of constraint (Cr1 ). Since the problem is convex with zero duality gap, the Karush–Kuhn–Tucker (KKT) condition holds [57]. Taking the derivative of L(x, w, λ) over w yields  wnmi =

1 1 − λn η

+ xnmi ,

(6)

where it is assumed that λn 6= 0. If we assume λ1n − η1 < 0 (λn > η) for any network n ∈ N , then wnmi = 0, ∀m ∈ M, i ∈ I, n ∈ N , which does not meet the complementary slackness requirement ! X X λn wnmi − Zn = 0, ∀n ∈ N . (7)

Plugging (8) into (5) leads to

x

X X X

xnmi A(λn ) +

n∈N m∈M i∈I

X

λ n Zn

n∈N

(9)

s.t. Cr2 , Cr4 , where A(λn ) = λn /η − ln λn + ln η − 1. It can be verified that A(λn ) is a monotonically decreasing function over λn ∈ (0, η] with limλn →0 A(λn ) = ∞ and A(η) = 0. Thus, A(λn ) ≥ 0, ∀λn ∈ (0, η]. The problem above can be decomposed into four sub– problems2 max x

X X X

X X

xnmi A(λn )

0 ≤ xnmi ≤ 1, ∀n ∈ N2 , m ∈ M, i ∈ I2 . It is observed that the only feasible solution for problems (P4 ) and (P5 ) is all zeros. Therefore, only (P3 ) and (P6 ) need to be solved in this case. Towards this end, we assume that N1 WLAN networks and N2 cellular networks are available. Moreover, for the sake P of brevity, a new variable snm is defined as snm = xnmi , n ∈ N1 . Thus, problem (P3 ) resumes to i∈I1

! X

s.t.

X

X X

xnmi ≤ 1, m ∈ M

n∈N1 i∈I1

n∈N

λn Zn is neglected since it is optimized over x

(10)

snm ≤ 1, m ∈ M

(11)

n∈N1

0 ≤ snm ≤ 1, ∀n ∈ N1 , m ∈ M In order to obtain a better understanding of this optimization problem, matrix S = [snm ] is introduced as follows:   s11 s12 · · · s1M  s21 s22 · · · s2M    S= . (12) . .. .. ..  ..  . . . sN1 1

sN1 2

···

sN1 M

In this way, the constraint (11) can be interpreted as: the summation of all elements in each column is less PMthan 1. Also, the term associated with A(λn ) in (10), i.e., m=1 snm , can be interpreted as the summation of all elements in nth row. It is observed that the maximum of the objective function (10) depends on the value of λn . However, it is proved in Appendix B that all λn are equal, i.e., λ1 = · · · = λN1 .

(P3 )

(13)

Based on (13), the objective function (10) is reduced to: ! X

P

snm

m∈M

0 ≤ xnmi ≤ 1, ∀n ∈ N1 , m ∈ M, i ∈ I1 . 2

X

A(λn )

n∈N1

n∈N1 m∈M i∈I1

s.t.

(P6 )

xnmi ≤ 1, m ∈ M

n∈N2 i∈I2

max

Therefore, it can be seen that 0 < λn ≤ η, ∀n ∈ N and the relationship (6) between wnmi and xnmi can be simplified to   1 1 wnmi = xnmi . (8) − λn η

xnmi A(λn )

n∈N2 m∈M i∈I2

s.t.

m∈M i∈I

g(λ) = max

X X X

n∈N1

A(λn )

X m∈M

snm

! = A(λ1 )

X

X

m∈M

n∈N1

snm

.

6

It turns out that it is maximized when X

snm = 1, m ∈ M,

(14)

The optimal primal points x∗ are chosen such that conditions (14) and (15) are met. In terms of x, they are expressed as

n∈N1

and it admits the optimal P value M A(λ1 ). Defining hnm = xnmi , n ∈ N2 and following similar

X X n∈N1 i∈I1

i∈I2

X X

steps yield the optimality condition for problem (P6 ): X

x∗nmi = 1, m ∈ M, x∗nmi = 1, m ∈ M.

(22)

n∈N2 i∈I2

hnm = 1, m ∈ M.

(15)

n∈N2

Similarly, the optimal value is expressed as M A(λN1 +1 ), where λN1 +1 = · · · = λN1 +N2 .

In the meantime, all the variables associated with (P4 ) and (P5 ) are zeros. Moreover, since w is expressed in terms of x in (6), the constraint associated with variable w, (Cr1 ), also needs to be checked

(16)

X X

Combining these results, the dual function g(λ) is expressed as: g(λ) = M A(λ1 ) + M A(λN1 +1 ) + λ1 Zw + λN1 +1 Zc ,

wnmi = Zn , ∀n ∈ N ,

(23)

m∈M i∈I

where the equality is achieved due to the complementary slackness in (7). Specifically, the optimal points w∗ can be expressed as

where

Zw Zc

=

=

N1 X

∗ wnmi =

Zl

l=1 NX 1 +N2

(17)

1 1 − λ∗n η



x∗nmi =



Zw x∗nmi /M, n ∈ N1 Zc x∗nmi /M, n ∈ N2

(24)

and (23) resumes to Zj .

(18)

j=N1 +1

X X

Equations (17) and (18) express the total data rate assigned by the WLAN and cellular networks, respectively. It is also assumed that the same type of networks have the same data rate capacity Z1 = Z2 = · · · = ZN1 and ZN1 +1 = ZN1 +2 = · · · = ZN1 +N2 , e.g., the situation where each WLAN network assumes 54 Mbps and each cellular network admits 50 Mbps. The dual problem resumes to minimize g(λ) with respect to λ min M A(λ1 ) + M A(λN1 +1 ) + λ1 Zw + λN1 +1 Zc λ

s.t. 0 < λ1 ≤ η, 0 < λN1 +1 ≤ η.

(19)

Mη , M + ηZw Mη = · · · = λ∗N1 +N2 = . M + ηZc

X X

x∗nmi

m∈M i∈I2

In this case, since problem (P2 ) presents a zero duality gap, it follows that (21)

∗ where λ∗1 and λ∗N1 +1 are given in (20). Moreover, vU represents an upper-bound for the original MINLP problem, i.e., ∗ v ∗ ≤ vU .

M M Zn = , n ∈ N2 . = Zc N2

(25)

n∈N1

h∗nm

=

1, m ∈ M

(27)

s∗nm

=

M Zn M = , n ∈ N1 , Zw N1

(28)

h∗nm

=

M Zn M = , n ∈ N2 , Zc N2

(29)

n∈N2

X X

(20)

M Zn M = , n ∈ N1 Zw N1

In a more compact form, (22) and (25) can be considered in terms of s∗nm and h∗nm such that the optimality conditions for problem (P2 ) can be depicted in a two-dimensional space as follows X s∗nm = 1, m ∈ M (26)

m∈M

λ∗1 = · · · = λ∗N1 =

∗ vU = M A(λ∗1 ) + M A(λ∗N1 +1 ) + λ∗1 Zw + λ∗N1 +1 Zc ,

x∗nmi =

m∈M i∈I1

X

The dual problem (19) is a convex optimization problem and achieves the closed-form solution:

λ∗N1 +1



m∈M

where 0 ≤ snm ≤ 1 and 0 ≤ hnm ≤ 1. The solution to the convex problem (P2 ) does not lead to a binary assignment solution, since x is relaxed and no longer binary. It just gives an upper bound for the objective function value. Therefore, we need to find an assignment and data rate solution. Therefore, we study the structure of (P2 ) using the Lagrangian approach and from its optimality conditions, we derive the heuristic approach which leads to a near-optimal solution.

7

we select S∗ and H∗ as follows:

B. Lower-bound: A Heuristic Method

If a binary snm and hnm satisfying the P above constraints can be found, then any xnmi with snm = xnmi , n ∈ N1 i∈I1 P and hnm = xnmi , n ∈ N2 (the rest x are all zeros) is an i∈I2

optimal solution for the original MINLP problem (P1 ). This argument can be proved in two steps: first, such an x satisfies the optimality conditions (26) - (29) for problem (P2 ) and thus achieves a larger objective value than v ∗ ; second, such an x is a feasible solution for problem (P1 ) and consequently achieves a lower objective value than v ∗ . In this way, the argument is proved and the corresponding optimal w is given by (24). Combining (26) with (28) and (27) with (29), the problem whether a binary optimal solution exists can be interpreted as two matrix assignment problems: (1) If an N1 × M matrix S∗ with binary elements can be found such that the summation of all elements in each column is 1 and the summation of all M . (2) If an N2 × M matrix H∗ elements in each row is N 1 with binary elements can be found such that the summation of all elements in each column is 1 and the summation of M all elements in each row is N . However, it can be seen that 2 generally such two matrices do not exist and only fractional optimal solutions can be obtained from problem (P2 ). Therefore, in this section, a heuristic method is proposed based on the optimality conditions (26) - (29). Specifically, the proposed heuristic method first assigns users and then optimizes the data rate based on the selected MT assignments. We first round the RHS of (28) and (29) to the largest integers not greater than them and relax (26) and (27) as follows

X

s∗nm

≤ 1, m ∈ M

(30)

h∗nm

≤ 1, m ∈ M

(31)

n∈N1

X n∈N2

X

s∗nm

m∈M

X m∈M

h∗nm



 M = = , n ∈ N1 , N1     M Zn M = = , n ∈ N2 , . Zc N2 M Zn Zw



 1| ·{z · · 1}  L1  · · 0} 0| ·{z  ∗  L S = 1  ..  . 0 · · · 0 | {z } L1

0| ·{z · · 0}

···

1| ·{z · · 1}

···

.. . 0| ·{z · · 0}

..

L1 L1

. ···

L1

|

{z M

 0| ·{z · · 0}  K2  · · 0} 0| ·{z  K2 H∗ =   .  .  . 0 · · · 0 | {z } K2

0| ·{z · · 0}

···

0| ·{z · · 0}

0| ·{z · · 0}

···

1| ·{z · · 1}

.. . 1| ·{z · · 1}

. ..

L2

where L1 =

L2

L2

L2

···

L2

|

.. . 0| ·{z · · 0} L2

{z M

j

     0| ·{z · · 0} 0| ·{z · · 0}       L1 K1    0| ·{z · · 0} 0| ·{z · · 0}  L1 K1  N , (34)  1 .. ..    . .    1| ·{z · · 1} 0| ·{z · · 0}      L1 K1  }      1| ·{z · · 1}     L2    0| ·{z · · 0}   L2  N2 (35) ..     .    0| ·{z · · 0}      L2  }

k j k M M N1 , L2 = N2 , and K1 , K2 are the

M M and N , respectively. In this selection, remainders of N 1 2 even though the last K1 users are not assigned any WLAN data rate, they are compensated with the cellular data rate. Similarly, the first K2 users are not assigned any cellular data rate, they are compensated with the WLAN data rate. Thus, the MT assignment is made with respect to snm and hnm . Then, the interface can be randomly Pspecific P ∗ picked such that s∗nm = x∗nmi , n ∈ N1 and h∗nm = xnmi , n ∈ N2 . i∈I1

i∈I2

To this end, what remains is to allocate data rate based on the selected MT assignments. It can be obtained by solving the following convex optimization problem: X X X wnmi max x∗nmi ln(1 + η ∗ ) w xnmi n∈N m∈M i∈I X X s.t. wnmi ≤ Zn , n ∈ N m∈M i∈I



(32) (33)

Then, any binary snm and hnm satisfying (30), (31), (32) and (33) is a feasible solution for problem (P1 ) since (30) and (31) guarantee the feasibility of (Cp2 ) and (Cp4 ), and the binary property meets (Cp7 ). Equivalently, the binary snm and hnm are obtained by alternatively finding two matrices S∗ and H∗ such that for each matrix, the summation of all elements in each column is less j than k 1 and j the k summation of all elements M M in each row is N and N2 , respectively. It is easy to 1 verify that such two binary matrices always exist. Specifically,

wnmi ≥ 0 where x∗nmi stands for the MT assignments based on (34) and (35). It is easy to verify that the above optimization problem is maximized when the data rate of each network is equally allocated for the assigned users. In a matrix form, it is represented in (36) and (37) where W1∗ and W2∗ stand for the data rate allocation for WLAN and cellular networks, respectively. Therefore, using this heuristic method, the MT is assigned based on the matrices (34) and (35). Based on the MT assignment in (34) and (35), the data rate is allocated according to the matrices (36) and (37). Since the proposed heuristic only provides a feasible solution of the MINLP, a closedform utility function value can be calculated based on (34), (35), (36) and (37) as a lower bound of the MINLP, which can be expressed as

8

Z

Z1 1 ···  L1 L | {z 1}  L 1    0···0  | {z }  ∗ L1 W1 =    ..  .    0···0  | {z } L1

0| ·{z · · 0}

···

L1

Z1 Z1 ··· L1 L | {z 1}

0| ·{z · · 0} L1

···

0| ·{z · · 0} L1

L1

.. .

..

0| ·{z · · 0}

···

L1

.

.. . Z1 Z1 ··· L L | 1 {z 1} L1

|

 0···0 | {z }  K2    0 · · · 0 | {z }  W2∗ =  K2   .  ..   0 · · · 0 | {z } K2

{z M

0 · · 0} | ·{z

···

0 · · 0} | ·{z

···

L2

L2

.. . ZN1 +1 ZN1 +1 ··· L2 L2 {z } |

.

..

···

L2

|

    ZN1 +1 ZN1 +1     0| ·{z · · 0} ···  L2 L2    | {z }  L2     L2    ZN1 +1 ZN1 +1    0| ·{z · · 0} ···   L2 L2  | {z } L2 N2  L2    .. ..   . .       0| ·{z · · 0} 0| ·{z · · 0}      L2 L2     {z }

(36)

(37)

M

    ηZ1 ηZN1 +1 ∗ vL = N1 L1 ln 1 + + N2 L2 ln 1 + . L1 L2 (38) The difference between the original MINLP and the heuristic method is bounded by ∗ ∗ ∗ v ∗ − vL ≤ vU − vL .

       0| ·{z · · 0}     K1         0| ·{z · · 0}    K1  N1 ,    ..    .        0| ·{z · · 0}      K1    } 

(39)

In this way, even though we are not able to get the difference between our heuristic method and the MINLP analytically since v ∗ is generally not traceable, an upper-bound is obtained to measure how our heuristic method works. As discussed earlier, if two binary matrices S∗ and H∗ can be found such that conditions (26) - (29) are met, then the optimality of problem (P1 ) is achieved. This occurs in our heuristic method when K1 = 0 and K2 = 0, or equivalently, the remainders of M/N1 and M/N2 are zeros. In this case, our heuristic method is optimal based on the MT assignments ∗ (34), (35) and data rate allocations (36), (37). Furthermore, vL ∗ in (38) coincides with vU in (21) and thus achieves v ∗ in a closed-form.

For (P2 ), the gradient method is employed to solve the dual problem of the convex optimization [21], [58]. With regard to the problem (P2 ), the total number of computations needed for resource allocation is N M I, and the number of dual variables is N . Therefore, the overall complexity to solve (P2 ) using the gradient method is O(N 2 · N M I) = O(N 3 M I) [58], which is polynomial (cubic) in number of networks, N and linear dependence with respect to MTs, M , and radio interfaces, I. In this paper, we have further investigated the properties of the dual problem and derived a closed-form Lagrange dual function in terms of λ as shown in (19). In this way, the convex optimization problem (P2 ) is solved in a closed-form and the upper bound is given in (21). Finally, we have proposed a heuristic method, which achieves a closed-form near-optimal solution for the MINLP (P1 ). Specifically, the users are assigned based on (34) and (35), and the corresponding data rate is allocated based on (36) and (37). Thus, it significantly reduces the computational complexity compared with the resource allocation problem (P1 ). IV. N UMERICAL R ESULTS

C. Complexity Analysis The original problem (P1 ) is a MINLP, which is NP-hard [55], [56]. After relaxing the binary variable in (P1 ), we obtain a convex optimization problem (P2 ). Solving problem (P2 ) leads to an upper bound of the original problem (P1 ).

In this section, our proposed method is validated via numerical simulations. In particular, the heuristic solution is compared with the solution yielded by the GAMS/BARON [28]. The proposed heuristic solution is implemented in MATLAB. We consider a scenario in which different wireless network

9

Fig. 2 and Fig. 3 show the comparison of optimal objective function values obtained by GAMS/BARON for the original MINLP, the relaxed convex optimization, (21) and our proposed heuristic method, (38). It can be seen that the numerical results coincide very well with GAMS/BARON and provide near optimal results. As discussed earlier, the upper bound is obtained by optimizing over a larger feasible domain, which results in a slightly greater objective function values over GAMS/BARON as shown in Fig. 2 and Fig. 3. On the other hand, the heuristic method takes less values than GAMS/BARON since it is calculated by a feasible solution of (P1 ). One interesting observation in Fig. 2(a) is the two-network case where N1 = 1 and N2 = 1. In this case, since either S∗ and H∗ has only one row, all the elements of S∗ and H∗ are ones to ensure the condition that the summation of all elements in each column is 1. Following this particular structure, for both matrices, the requirement (28) and (29) for the summation of all elements in each row is always met. Therefore, the validation of this special case is also confirmed in the matrix interpretation. Basically, remainders are always zero when N1 = N2 = 1. Hence, by applying the modified Lagrange duality method, the relaxed problem (P2 ) is ensured to have a binary optimal solution with the proposed heuristic method, which is also globally optimal for the non-convex MINLP problem (P1 ).

72 69

Objective Function

66

GAMS/BARON Lower Bound Upper Bound

63 60 57 54 51 48 45 42 39 36 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 Number of Mobile Terminals

Objective Function

(a) |N1 | = 1 and |N2 | = 1.

GAMS/BARON 112 Lower Bound 108 Upper Bound 104 100 96 92 88 68 84 80 76 72 68 64 60 56 64 52 48 44 40 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 Number of Mobile Terminals (b) |N1 | = 2 and |N2 | = 2.

Objective Function

technologies are deployed: IEEE 802.11 WLAN based on an enhanced version of the DCF [41] and 4G cellular BSs such as MBS, MiBS and FBS. We consider six service area scenarios which have overlapped coverage. In area 1, service from one WLAN and one cellular network is available in Fig. 2(a). In area 2, services from two WLANs and two cellular networks, which consist of a MBS and a FBS, respectively, are present in Fig. 2(b). In area 3, two WLANs and three cellular networks, which consist of a MBS, a MiBS and a FBS, are available in Fig. 2(c). In area 4, services from three WLANs and two cellular networks which consist of a MBS and a FBS are available in Fig. 3(a). In area 5, services from four WLANs and three cellular networks which consist of a MBS, a MiBS and a FBS are available in Fig. 3(b). In area 6, services from five WLAN and three cellular networks which consist of a MBS, a MiBS and a FBS are available in Fig. 3(c). We assume that the set of candidate WLAN APs and cellular BSs are the ones that provide the highest received signal strength to the MTs for the given coverage area. The cellular BS is based on the 4G orthogonal frequency division multiple access (OFDMA) access technology and assumes a maximum data rate capacity of 50 Mbps [59], [60]. In addition, a WLAN AP is based on the IEEE 802.11 with enhanced DCF media access control (MAC) and assumes a maximum data rate capacity of 54 Mbps [41]. The number of MTs with multi-homing capability is between [10; 50] and η is chosen to be 1. Variable η is used so that the outcome of the utility function scales with the bandwidth data rate unit. In our simulation setting, both WLAN and cellular have same units [Mbps] so all assume the same η = 1.

GAMS/BARON 124 120 Lower Bound 116 Upper Bound 112 108 104 100 96 92 88 64 84 80 76 72 68 64 60 56 52 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 Number of Mobile Terminals (c) |N1 | = 2 and |N2 | = 3.

Fig. 2: Bounds on objective function compared with optimal solution of GAMS/BARON.

Objective Function

Objective Function

10

128 124 120 116 112 108 104 100 96 92 88 84 80 76 72 68 64 60 56 52 10 12 14

V. C ONCLUSION GAMS/BARON Lower Bound Upper Bound

In this study, the joint MT assignment and optimal data rate allocation problem for HetNets is investigated under the set-up where the MTs present multi–homing capability. In particular, the problem of maximizing the overall network data rate capacity is considered. First, we formulate the primal problem as a non-convex MINLP (P1 ) and transform it into a convex optimization problem (P2 ) using a change of 80 variables and the binary variable relaxation approach. Then, the modified Lagrange duality method is adopted and the problem is decomposed into four-subcomponents. It is shown that the optimal solutions of the relaxed convex optimization problem yields to upper bound for the original non-convex MINLP and the proposed heuristic method further provides a lower bound which also performs near optimal solution of 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 the original non-convex MINLP. In addition, the difference Number of Mobile Terminals between the original non-convex MINLP and the heuristic (a) |N1 | = 3 and |N2 | = 2. method is determined analytically. The validity of the proposed method is confirmed by numerical results. GAMS/BARON In future research, we will investigate the physical layer Lower Bound constraints to develop a cross layer design approach that Upper Bound maximizes the allocated data rate and accounts for channel gain and fading parameters. Taking into account the fading effects in the optimization approach is a major problem that is beyond the scope of this paper.

150 144 138 132 126 120 114 108 102 96 90 78 84 78 72 66 60 54 48 42 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 Number of Mobile Terminals

Objective Function

(b) |N1 | = 4 and |N2 | = 3.

162 156 150 144 138 132 126 120 114 108 102 96 90 84 78 72 66 60 54 48 42 10 12 14

GAMS/BARON Lower Bound Upper Bound

90

16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 Number of Mobile Terminals (c) |N1 | = 5 and |N2 | = 3.

Fig. 3: Bounds on objective function compared with optimal solution of GAMS/BARON.

R EFERENCES [1] J. Andrews, “Seven ways that hetnets are a cellular paradigm shift,” IEEE Commun. Mag., vol. 51, no. 3, pp. 136–144, Mar. 2013. [2] M. Bennis, M. Simsek, A. Czylwik, W. Saad, S. Valentin, and M. Debbah, “When cellular meets wifi in wireless small cell networks,” IEEE Commun. Mag., vol. 51, no. 6, pp. 44–50, Jun. 2013. [3] K.-K. Yap, T.-Y. Huang, M. Kobayashi, Y. Yiakoumis, N. McKeown, S. Katti, and G. Parulkar, “Making use of all the networks around us: A case study in android,” in Proc. 2012 ACM SIGCOMM Workshop Cellular networks: operations, challenges, and future design, Helsinki, Finland, Aug. 2012, pp. 19–24. [4] M. Ismail and W. Zhuang, “A distributed multi-service resource allocation algorithm in heterogeneous wireless access medium,” IEEE J. Sel. Areas Commun., vol. SAC-30, no. 2, pp. 425–432, Feb. 2012. [5] ——, “Decentralized radio resource allocation for single-network and multi-homing services in cooperative heterogeneous wireless access medium,” IEEE Trans. Wireless Commun., vol. WC-11, no. 11, pp. 4085–4095, Nov. 2012. [6] M. Ismail, A. Abdrabou, and W. Zhuang, “Cooperative decentralized resource allocation in heterogeneous wireless access medium,” IEEE Trans. Wireless Commun., vol. WC-12, no. 2, pp. 714–724, Feb. 2013. [7] M. Beasley, “iOS 7 becomes first commercial software to support multipath TCP, allowing simultaneous Wi-Fi and cell network connections,” Available: http://9to5mac.com/2013/09/19/ios-7becomes-first-commercial-software-to-support-multipath-tcp-allowingsimultaneous-wi-fi-and-cell-network-connections/, Accessed: Jul. 5, 2015. [8] M. Benoliel, S. Shalunov, and G. Hazel, “Open Garden-You are the Internet,” Available: https://opengarden.com/home, Accessed: Jul. 5, 2015. [9] K. Chebrolu and R. Rao, “Bandwidth aggregation for real-time applications in heterogeneous wireless networks,” IEEE Trans. Mobile Comput., vol. MC-5, no. 4, pp. 388–403, Apr. 2006. [10] W. Shen and Q.-A. Zeng, “Resource management schemes for multiple traffic in integrated heterogeneous wireless and mobile networks,” in Proc. IEEE 2008 ICCCN 17th Int. Conf. Comput. Commun. Networks, St. Thomas, US Virgin Islands, Aug. 2008, pp. 1–6. [11] I. Blau, G. Wunder, I. Karla, and R. Sigle, “Decentralized utility maximization in heterogeneous multicell scenarios with interference limited and orthogonal air interfaces,” EURASIP J. Wireless Commun. and Netw., vol. 2009, pp. 1–12, Jan. 2009.

11

[12] X. Pei, T. Jiang, D. Qu, G. Zhu, and J. Liu, “Radio-resource management and access-control mechanism based on a novel economic model in heterogeneous wireless networks,” IEEE Trans. Veh. Technol., vol. VT59, no. 6, pp. 3047–3056, Jul. 2010. [13] A. E. Taha, H. Hassanein, and H. Mouftah, “On robust allocation policies in wireless heterogeneous networks,” in Proc. IEEE 2004 QSHINE 1st Int. Conf. Quality of Service Heterogeneous Wired/Wireless Networks, Dallas, TX, USA, Oct. 2004, pp. 198–205. [14] C. Luo, H. Ji, and Y. Li, “Utility-based multi-service bandwidth allocation in the 4g heterogeneous wireless access networks,” in Proc. IEEE 2009 WCNC Wireless Commun. and Netw. Conf., Budapest, Hungary, Apr. 2009, pp. 1–5. [15] D. Niyato and E. Hossain, “Bandwidth allocation in 4g heterogeneous wireless access networks: A noncooperative game theoretical approach,” in Proc. IEEE 2006 GLOBECOM Global Telecommun. Conf., San Francisco, CA, USA. [16] ——, “A cooperative game framework for bandwidth allocation in 4g heterogeneous wireless networks,” in Proc. IEEE 2006 ICC Int. Conf. Commun., Istanbul, Turkey. [17] ——, “A noncooperative game-theoretic framework for radio resource management in 4g heterogeneous wireless access networks,” IEEE Trans. Mobile Comput., vol. MC-7, no. 3, pp. 332–345, Mar. [18] M. A. Khan, C. Truong, T. Geithner, F. Sivrikaya, and S. Albayrak, “Network level cooperation for resource allocation in future wireless networks,” in Proc. 2008 WD 1st IFIP Wireless Days, Dubai, UAE, Nov. 2008, pp. 1–5. [19] D. Liu, Y. Chen, K. K. Chai, T. Zhang, and M. Elkashlan, “Opportunistic user association for multi-service hetnets using Nash bargaining solution,” IEEE Commun. Lett., vol. COMML-18, no. 3, pp. 463–466, Mar. 2014. [20] M. Ismail, W. Zhuang, E. Serpedin, and K. Qaraqe, “A survey on green mobile networking: From the perspectives of network operators and mobile users,” IEEE Commun. Surveys Tuts., vol. COMST-PP, no. 99, pp. 1–1, Nov. 2014. [21] M. Hajiaghayi, M. Dong, and B. Liang, “Jointly optimal channel and power assignment for dual-hop multi-channel multi-user relaying,” IEEE J. Sel. Areas Commun., vol. SAC-30, no. 9, pp. 1806–1814, Sep. 2012. [22] H. Soury, F. Bader, M. Shaat, and M. S. Alouini, “Joint sub-carrier pairing and resource allocation for cognitive networks with adaptive relaying,” EURASIP J. Wireless Commun. and Netw., vol. 2013, no. 1, pp. 1–15, Nov. 2013. [23] D. Bertsekas, G. Lauer, N. Sandell, and T. Posbergh, “Optimal shortterm scheduling of large-scale power systems,” IEEE Trans. Autom. Control, vol. AC-28, no. 1, pp. 1–11, Jan 1983. [24] M. F. Uddin, C. Assi, and A. Ghrayeb, “Joint optimal af relay assignment and power allocation in wireless cooperative networks,” Elsevier Computer Networks, vol. 58, pp. 58–69, Sept. 2013. [25] W. Yu and R. Lui, “Dual methods for nonconvex spectrum optimization of multicarrier systems,” IEEE Trans. Commun., vol. COM-54, no. 7, pp. 1310–1322, Jul. 2006. [26] S.-J. Kim and G. Giannakis, “Scalable and robust demand response with mixed-integer constraints,” IEEE Trans. Smart Grid, vol. SG-4, no. 4, pp. 2089–2099, Dec. 2013. [27] G. Giannakis, V. Kekatos, N. Gatsis, S.-J. Kim, H. Zhu, and B. Wollenberg, “Monitoring and optimization for power grids: A signal processing perspective,” IEEE Signal Process. Mag., vol. 30, no. 5, pp. 107–128, Sep. 2013. [28] N. Sahinidis, “GAMS: Baron MINLP Solver,” Available: http://www. gams.com/dd/docs/solvers/baron/index.html, Accessed: Jul. 5, 2015. [29] Y. F. Liu and Y. H. Dai, “On the complexity of joint subcarrier and power allocation for multi-user ofdma systems,” IEEE Trans. Signal Process., vol. SP-62, no. 3, pp. 583–596, Feb. 2014. [30] F. Jin, R. Zhang, and L. Hanzo, “Resource allocation under delayguarantee constraints for heterogeneous visible-light and rf femtocell,” IEEE Trans. Wireless Commun., vol. WC-14, no. 2, pp. 1020–1034, Feb. 2015. [31] H. Zhang, C. Jiang, N. C. Beaulieu, X. Chu, X. Wen, and M. Tao, “Resource allocation in spectrum-sharing ofdma femtocells with heterogeneous services,” IEEE Trans. Commun., vol. COM-62, no. 7, pp. 2366–2377, Jul. 2014. [32] D. T. Ngo, S. Khakurel, and T. L. Ngoc, “Joint subchannel assignment and power allocation for ofdma femtocell networks,” IEEE Trans. Wireless Commun., vol. WC-13, no. 1, pp. 342–355, Jan. 2014. [33] Y. Fu and Q. Zhu, “Joint optimization methods for non-convex resource allocation problems of decode-and-forward relay based ofdm networks,” IEEE Trans. Veh. Technol., vol. VT-PP, no. 99, pp. 1–1, Aug. 2015.

[34] Q. Chen, G. Yu, R. Yin, and G. Y. Li, “Energy-efficient user association and resource allocation for multi-stream carrier aggregation,” IEEE Trans. Veh. Technol., vol. VT-PP, no. 99, pp. 1–1, Aug. 2015. [35] D. Fooladivanda and C. Rosenberg, “Joint resource allocation and user association for heterogeneous wireless cellular networks,” IEEE Trans. Wireless Commun., vol. WC-12, no. 1, pp. 248–257, Jan. 2013. [36] R. Madan, S. Boyd, and S. Lall, “Fast algorithms for resource allocation in wireless cellular networks,” IEEE/ACM Trans. Netw., vol. NET-18, no. 3, pp. 973–984, Jun. [37] D. L. Perez, A. Valcarce, G. de la Roche, and J. Zhang, “Ofdma femtocells: A roadmap on interference avoidance,” IEEE Commun. Mag., vol. 47, no. 9, pp. 41–48, Sep. 2009. [38] A. Mahmud and K. A. Hamdi, “A unified framework for the analysis of fractional frequency reuse techniques,” IEEE Trans. Commun., vol. COM-62, no. 10, pp. 3692–3705, Oct. 2014. [39] V. Chandrasekhar and J. Andrews, “Spectrum allocation in tiered cellular networks,” IEEE Trans. Commun., vol. COM-57, no. 10, pp. 3059–3068, Oct. 2009. [40] H. C. Lee, D. C. Oh, and Y. H. Lee, “Mitigation of interfemtocell interference with adaptive fractional frequency reuse,” in Proc. IEEE 2006 ICC Int. Conf. Commun., Cape Town, South Africa. [41] J. Choi, J. Yoo, S. Choi, and C. Kim, “Eba: an enhancement of the ieee 802.11 dcf via distributed reservation,” IEEE Trans. Mobile Comput., vol. MC-4, no. 4, pp. 378–390, Jul. 2005. [42] L. Wang and G. S. Kuo, “Mathematical modeling for network selection in heterogeneous wireless networks a tutorial,” IEEE Commun. Surveys Tuts., vol. COMST-15, no. 1, pp. 271–292, First Quarter 2013. [43] G. Song and Y. Li, “Utility-based resource allocation and scheduling in OFDM-based wireless broadband networks,” IEEE Commun. Mag., vol. 43, no. 12, pp. 127–134, Dec. 2005. [44] C. Rong, W. XiuJuan, C. QianBin, and S. Tommy, “Utility-based bandwidth allocation algorithm for heterogeneous wireless networks,” Springer Science China Information Sciences, vol. 56, no. 2, pp. 1–13, Feb. 2013. [45] O. Ormond, J. Murphy, and G. M. Muntean, “Utility-based intelligent network selection in beyond 3g systems,” in Proc. IEEE 2006 ICC Int. Conf. Commun., Istanbul, Turkey. [46] V. Gazis, N. Alonistioti, and L. Merakos, “Toward a generic “always best connected” capability in integrated WLAN/UMTS cellular mobile networks (and beyond),” IEEE Wireless Commun. Mag., vol. 12, no. 3, pp. 20–29, Jun. 2005. [47] Q. Song and A. Jamalipour, “Network selection in an integrated Wireless LAN and UMTS environment using mathematical modeling and computing techniques,” IEEE Wireless Commun. Mag., vol. 12, no. 3, pp. 42–48, Jun. 2005. [48] W. Song, Y. Cheng, and W. Zhuang, “Improving voice and data services in cellular/wlan integrated networks by admission control,” IEEE Trans. Wireless Commun., vol. WC-6, no. 11, pp. 4025–4037, Nov. 2007. [49] H. Chan, P. Fan, and Z. Cao, “A utility-based network selection scheme for multiple services in heterogeneous networks,” in Proc. IEEE 2005 Int. Conf. Wireless Netw., Commun. Mobile Comput., Maui, HI, USA. [50] Q. T. N. Vuong, Y. G. Doudane, and N. Agoulmine, “On utility models for access network selection in wireless heterogeneous networks,” in Proc. IEEE 2008 NOMS Network Operations Manage. Symp., Salvador, Bahia, Apr. 2008, pp. 144–151. [51] W. H. Kuo and W. Liao, “Utility-based resource allocation in wireless networks,” IEEE Trans. Wireless Commun., vol. WC-6, no. 10, pp. 3600– 3606, Oct. 2007. [52] ——, “Utility-based radio resource allocation for qos traffic in wireless networks,” IEEE Trans. Wireless Commun., vol. WC-7, no. 7, pp. 2714– 2722, Jul. 2008. [53] F. Kelly, “Charging and rate control for elastic traffic,” Wiley European Trans. Telecommun., vol. 8, no. 1, pp. 33–37, Jan. 1997. [54] H. Shen and T. Basar, “Differentiated internet pricing using a hierarchical network game model,” in Proc. IEEE 2004 American Control Conf., Boston, MA, USA, Jun. 2004, pp. 2322–2327. [55] P. Bonami, M. Kilinc, and J. Linderoth, “Algorithms and software for convex mixed integer nonlinear programs,” in Mixed integer nonlinear programming. Springer, Nov. 2012, pp. 1–39. [56] G. Michael and J. David, “Computers and intractability: A guide to the theory of np-completeness,” WH Freeman & Co., San Francisco, Jan. 1979. [57] S. Boyd and L. Vandenberghe, Convex optimization. Cambridge University Press, Mar. [58] M. S. Alam, J. Mark, and X. S. Shen, “Relay selection and resource allocation for multi-user cooperative ofdma networks,” IEEE Trans. Wireless Commun., vol. WC-12, no. 5, pp. 2193–2205, May. 2013.

12

[59] ETSI, “LTE; Requirements for further advancements for Evolved Universal Terrestrial Radio Access (E-UTRA) (LTE-Advanced) (3GPP TR 36.913 version 10.0.0 Release 10),” Available: http://www.etsi.org/deliver/etsi tr/136900 136999/136913/10.00. 00 60/tr 136913v100000p.pdf, Accessed: Jul. 25, 2015. [60] T. Rappaport, S. Sun, R. Mayzus, H. Zhao, Y. Azar, K. Wang, G. Wong, J. Schulz, M. Samimi, and F. Gutierrez, “Millimeter wave mobile communications for 5g cellular: It will work!” IEEE Access, vol. 1, pp. 335–349, May 2013.

A PPENDIX A In this appendix, it is proved that problem (P2 ) is a convex optimization problem. The Hessian matrix of the function nmi xnmi ln(1 + η w xnmi ) is first calculated as follows   H=

2 wnmi 2 x3nmi xnmi +1 w nmi 2  wnmi +1 x2nmi x

−  wnmi nmi



wnmi 

wnmi xnmi

 2

+1 x2nmi −  wnmi 1 2 +1 xnmi x

 

(40)

nmi

It is observed that the Hessian matrix (40) is positive semidefinite, which leads to the fact that the function xnmi ln(1 + nmi ηw xnmi ) is a concave function. Furthermore, the objective function in (P2 ) is concave since the summation of concave functions is also concave. Therefore, problem (P2 ) is a convex optimization problem due to the fact that all the constraints are affine [57]. A PPENDIX B In this appendix, it is proved that λ1 = · · · = λn . Assume not all λn , n = 1, · · · , N1 are equal to each other, then there exists a λk in this sequence such that λk > λmin , where λmin = minn=1,··· ,N1 λn . Since A(λn ) is a monotonically decreasing function, it follows that A(λk ) < A(λmin ). Therefore, in order to maximize (10), the kth row of S is assigned to be all zeros. In other words, if any positive value is assigned to the row corresponding to A(λk ), then that value can be added to the row corresponding to A(λmin ) which will yield a larger objective value. Consequently, it follows that skm = 0, m ∈ M, and moreover xkmi = 0, m ∈ M, i ∈ I1 . Due to (6), wkmi = 0, m ∈ M, i ∈ I1 .

(41)

It remains that X X

wkmi − Zk =

m∈M i∈I

+

X X

wkmi

m∈M i∈I1

X X

Ali Rıza Ekti (S’08-M’15) is from Tarsus, Turkey. He received his B.Sc. degree in Electrical and Electronics Engineering from Mersin University, Mersin, Turkey, (September 2002-June 2006), also studied at Universidad Politechnica de Valencia, Valencia, Spain in 2004-2005, received M.Sc. degree in Electrical Engineering from the University of South Florida, Tampa, Florida, USA (August 2008December 2009) and received Ph.D. in Electrical Engineering from Department of Electrical Engineering and Computer Science at Texas A&M University, College Station, TX, USA (August 2010-August 2015). He is currently a Visiting Professor at Gannon University Electrical and Computer Engineering Department. He is serving as a TPC member for the “LTE/LTE-A, 5G, and Wireless Heterogeneous Networks” track in the IEEE VTC 2016-Spring and “Wireless Communications Symposium” track in the IEEE ICC 2016. He has been a Technical Reviewer for IEEE conferences and journals. His current research interests include statistical signal processing, wireless communications in 4G and 5G systems and bioinformatics. He is an active member of IEEE.

wkmi − Zk = −Zk ,

(42)

m∈M i∈I2

where the last equality follows from (41) and the fact that k ∈ N1 (the cellular interface cannot connect to the WLAN network). Based on (7), λk = 0. Then, the assumption that λk > λmin leads to λmin < 0, which is contradictory to the nonnegativity property of the Lagrange multiplier λ. In addition, the argument that λk = 0 also voids the assumption λn 6= 0 when deriving (6). This completes the proof.

Xu Wang (S’15) was born in Dalian, China. He received his B.Sc. degree in Electrical and Electronics Engineering from Huazhong University of Science and Technology, China, in 2010. He received his M.Sc. degree in Electrical Engineering from Southern Methodist University, Dallas, TX, USA, in 2012. He is currently a Ph.D. candidate in Department of Electrical and Computer Engineering at Texas A&M University, College Station, TX, USA. His main research interests are the studies of convex, non-convex, and variational optimization with applications in wireless communications, statistical signal processing and bioinformatics.

Muhammad Ismail (S’10-M’13) received the B.Sc. and M.Sc. degrees in electrical engineering (electronics and communications) from Ain Shams University, Cairo, Egypt, in 2007 and 2009, respectively, and the Ph.D. degree in electrical and computer engineering from the University of Waterloo, Waterloo, ON, Canada, in 2013. He is an Assistant Research Scientist with the Department of Electrical and Computer Engineering, Texas A&M University at Qatar, Doha, Qatar. His research interests include distributed resource allocation, green wireless networks, cooperative networking, smart grid, and biomedical signal processing. He is a co-recipient of the Best Paper Award at IEEE ICC 2014, IEEE Globecom 2014, and SGRE 2015. He is a co-author of two research monographs by Wiley/IEEE Press and Springer. Dr. Ismail is an Associate Editor in IET Communications since Dec. 2013. He joined the International Journal on Advances in Networks and Services Editorial Board in January 2012. He served as a TPC Member for IEEE ICWMC 2010-2014 and IEEE ICC 2014, 2015, and 2016. He is serving as a TPC co-chair for the “Ad-Hoc, M2M, and Sensor Networks” track in the IEEE VTC 2016. He served as a Web Chair for the IEEE INFOCOM 2014 Organizing Committee. He was an Editorial Assistant of the IEEE Transactions on Vehicular Technology in the period January 2011 to July 2013. He has been a Technical Reviewer for several IEEE conferences and journals.

13

Erchin Serpedin (F’13) is a professor with the Department of Electrical and Computer Engineering, Texas A&M University, College Station. He received the specialization degree in transmission and processing of information from Ecole Superieure DElectricite (SUPELEC), Paris, in 1992, the M.Sc. degree from the Georgia Institute of Technology in 1992, and the Ph.D. degree from the University of Virginia in January 1999. Dr. Serpedin is the author of two research monographs, one textbook, ten book chapters, 120 journal papers, and 200 conference papers. His research interests include signal processing, computational biomedical engineering, bioinformatics, and systems biology. Dr. Serpedin is currently serving as an associate editor of the IEEE Signal Processing Magazine and as the editor-in-chief of EURASIP Journal on Bioinformatics and Systems Biology, an online journal edited by Springer. He served as an associate editor for more than 12 journals, including journals such as the IEEE Transactions on Information Theory, IEEE Transactions on Signal Processing, IEEE Transactions on Communications, IEEE Signal Processing Letters, IEEE Communications Letters, IEEE Transactions on Wireless Communications, Signal Processing (Elsevier), Physical Communications (Elsevier), EURASIP Journal on Advances in Signal Processing, and as a Technical Chair for six major conferences. His research work was recognized through numerous awards and research grants.

Khalid Qaraqe (SM’00) was born in Bethlehem. Dr Qaraqe received the B.S. degree in EE from the University of Technology, Bagdad, Iraq in 1986, with honors. He received the M.S. degree in EE from the University of Jordan, Jordan, Amman, Jordan, in 1989, and he earned his Ph.D. degree in EE from Texas A&M University, College Station, TX, in 1997. From 1989 to 2004 Dr Qaraqe has held a variety positions in many companies and he has over 12 years of experience in the telecommunication industry. Dr Qaraqe has worked on numerous GSM, CDMA, and WCDMA projects and has experience in product development, design, deployments, testing and integration. Dr Qaraqe joined the department of Electrical and Computer Engineering of Texas A&M University at Qatar, in July 2004, where he is now a professor. Dr Qaraqe research interests include communication theory and its application to design and performance, analysis of cellular systems and indoor communication systems. Particular interests are in mobile networks, broadband wireless access, cooperative networks, cognitive radio, diversity techniques and beyond 4G systems.