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Backhaul-Constrained Small Cell Networks: Refunding and QoS Provisioning Yufei Yang, Student Member, IEEE, Tony Q. S. Quek, Senior Member, IEEE, and Lingjie Duan, Member, IEEE
Abstract—Small cell access points (SAPs) can offload macrocell traffic, improve indoor coverage and cell-edge user performance, and boost network capacity. In this paper, we investigate the problem faced by the mobile network operator (MNO) on how to properly incentivize the existing private SAPs to serve extra roaming macrocell users. We propose a refunding framework for small cell networks with limited-capacity backhaul, where small cell holders (SHs) receive refunding from the MNO and then admit macrocell users. Specifically, we formulate a two-stage refunding-admission game with MNO being the leader and SHs being the followers. Our results can be summarized as follows: 1) we formulate a revenue maximization problem by allowing the MNO to set individualized refunding and interference temperature constraints to SAPs. We propose a lookup table approach to solve it; 2) for small cells with guaranteed QoS provisioning, we consider access-based refunding and propose a near-optimal joint user admission and power allocation algorithm to solve the utility maximization problem at each SAP; and 3) for small cells with best-effort QoS provisioning, we consider usage-based refunding and propose a majorization method-based power allocation algorithm. Extensive numerical results show that our proposed framework and algorithms yield significant improvements on the MNO’s net revenue and SHs’ utilities compared with a non-refunding case. Our research highlights the possibility of enhancing the MNO’s net revenue without changing the current network structure and the importance of incentivizing SHs by taking the limited-capacity backhaul into account. Index Terms—Small cell network, limited-capacity backhaul, refunding, two-stage game, admission and power control.
I. I NTRODUCTION
S
MALL cell access points (SAPs) are low-power access points operated in licensed spectrum, which encompass femtocells, picocells, microcells, and carrier Wi-Fi access points. Nowadays, SAPs are increasingly installed in public Manuscript received September 1, 2013; revised January 12, 2014; accepted May 5, 2014. Date of publication May 22, 2014; date of current version September 8, 2014. This work was supported in part by the SRG ISTD 2012037, SUTD-MIT International Design Centre under Grant IDSF1200106OH and in part by the A∗STAR SERC under Grant 1224104048. This paper was presented, in part, at the IEEE International Conference on Acoustics, Speech and Signal Processing, Vancouver, BC, Canada, May 2013, and the IEEE/CIC International Conference on Communications in China, Xi’an, China. The associate editor coordinating the review of this paper and approving it for publication was Z. Wang. Y. Yang and L. Duan are with the Engineering Systems and Design Pillar, Singapore University of Technology and Design, Singapore 138632 (e-mail:
[email protected];
[email protected]). T. Q. S. Quek is with the Information Systems Technology and Design Pillar, Singapore University of Technology and Design, Singapore 138632, and also with the Institute for Infocomm Research, Singapore 138632 (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TWC.2014.2326401
areas as well as residential sites to cope with the rapid growth of mobile data traffic [1]–[3]. The commercialized SAPs are either deployed by mobile network operators (MNOs) or purchased by customers. The MNOs deploy small cell networks on the existing macrocell networks to offload data and extend service coverage. It is easier for a centralized management and network optimization. While the majority of small cells are purchased by customers or enterprises to meet their data requirements. Therefore, the small cell standard supports three usage models according to different deployment scenarios, i.e., open access, closed access, and hybrid access [4]. Till now, the small cell technique has emerged to be a market success. It is predicted that the worldwide market for small cells will be worth $14.3 billion by 2017 according to an Allied Business Intelligence (ABI) Inc. report. The LTE small cells sold will surpass the number of LTE macrocells, forecasting at 127, 000, as early as 2014. Multitudes of scholarly works on the economic issues of femtocells are considered under various pricing and spectrum sharing schemes in [5]–[9]. In [6], the authors consider pricing issues for femtocells under different usage models. In [7], the authors investigate the economic incentive for a MNO to add femtocell service on the top of its existing macrocell service. In [8], the authors propose a utility-aware refunding framework for hybrid access femtocell networks. On the other hand, there are still many technical constraints, which may potentially barricade the further prosperity of small cell market. Among which, backhaul constraint is arguably the key challenge for small cells [10]. Currently, the majority of backhaul are finite wired links, e.g., 70%–80% in U.S. and 40% worldwide in 2010. It is predicted that mobile networks will require 10× fatter backhaul in 2016. In literature, the limited-capacity backhaul usually appears as constraints to the sum-rate or transmission delay [11]–[14]. With respect to different usage models, the joint admission and power control (J-APC) problem and sum-rate maximization (SRMax) problem are closely related to the Quality of Service (QoS) of subscribers. In literature, the main objective for the J-APC problem is to find the maximum admission set while minimizing the total transmission power. The state of the art on J-APC has focused on gradual admission [15], gradual removal [16], linear programming (LP) deflation [17], semidefinite relaxation [18], and distributed prime-dual implementation [19]. For SRMax problem, a complete survey can be found in [20]. It has been shown that the SRMax problem is generally a nonconvex problem in terms of transmission power. In high SINR regime, the sum-rate can be approximated as a convex problem in the form of geometric programming [21]. In [22], the authors propose a more general multiplicative linear fractional
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YANG et al.: BACKHAUL-CONSTRAINED SMALL CELL NETWORKS: REFUNDING AND QoS PROVISIONING
Fig. 1.
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The organization of this paper.
programming-based power allocation algorithm to solve the long-standing SRMax problem globally, which considers minimum data rate and peak power constraints for users. Since the macrocell networks have become more and more congested, it often happens that the MNO cannot satisfy some of its macrocell subscribers’ data rates. These subscribers will stop paying to the MNO if they cannot receive a desired QoS. As a result, it reduces the MNO’s net revenue and reputation on service provisioning. Therefore, we are interested in how to incentivize the existing SAPs to serve these subscribers. Since SAPs are usually configured as closed access by SHs, the MNO needs to provide refunding to SHs as incentives to encourage them to admit extra macrocell users. In the sequel, we refer pre-registered small cell subscribers as home users (HUs) and macrocell subscribers as guest users (GUs). To increase net revenue, the MNO is willing to provide certain amount of refunding to SHs. While for SHs, with expectation of receiving refunding, they are willing to switch closed access to hybrid access and admit a limited number of GUs. To investigate the interactions between the MNO and SHs, we will formulate a Stackelberg game with MNO being the leader and SHs being followers. At SAPs, we consider guaranteed QoS provisioning and best-effort QoS provisioning, respectively. For guaranteed QoS provisioning, the optimization problem at SAPs is similar to J-APC problem. But the key difference is that in our problem we aim at balancing the number of admitted GUs with HU’s QoS triggerd by refunding. For best-effort QoS provisioning, the optimization problem at SAPs is similar to SRMax problem. But the key difference is that in our problem we aim at balancing the GUs’ achievable sum-rate with HU’s QoS triggerd by refunding. The organization of this paper is illustrated in Fig. 1 and the main contributions are summarized as follows: • Two-Stage Refunding-Admission Game Formulation: We formulate a novel refunding-admission game combining economic refunding and technical specifications. At the refunding game in Stage I, the MNO maximizes net revenue subject to the aggregate interference constraint. At the admission game in Stage II, each SAP tradeoffs between refunding and HU’s QoS subject to transmission power, interference temperature, signal to interference and noise
ratio (SINR), and backhaul capacity constraints. Backward induction is used to obtain subgame perfect equilibrium. • Access-Based Refunding with Guaranteed QoS Provisioning [23]: Each SAP is to tradeoff between refunding in terms of the number of admitted GUs and HU’s QoS metric, i.e., proportional fairness throughput. We propose a near-optimal joint GU admission and power control algorithm. Three simplified cases are considered, i.e., equal fading gain, equal target SINR, and LP approximation, to further understand the admission process. • Usage-Based Refunding with Best-Effort QoS Provisioning [24]: Each SAP is to tradeoff between refunding in terms of GUs’ achievable sum-rate and HU’s proportional fairness throughout. We obtain the optimal power allocations for GUs through majorization theory. • Individualized Refunding and Interference Temperaure: We propose a lookup table approach to solve the revenue maximization problem in Stage I. The MNO decides individualized refunding and interference temperature constraints to different SAPs. The layout of this paper is as follows. In Section II, we elaborate the system model and performance metrics. In Section III, we present the MNO refunding framework with guaranteed QoS provisioning. In Section IV, we present the MNO refunding framework with best-effort QoS provisioning. Numerical results are presented in Section V and conclusions are given in Section VI. Throughout the paper, we use the following notations and parameters listed in Table I unless otherwise stated. • Boldface uppercase letters denote matrices, boldface lowercase letters denote column vectors, italics denote scalars. For a matrix X (or a vector x), XA (or xA ) denote the submatrix (or subvector) with index set A. • For a vector x in RN , we denote its ordered coordinates by x(1) ≥ x(2) ≥ · · · ≥ x(N ) and x[1] ≤ x[2] ≤ · · · ≤ x[N ] . For x and y in RN, we say x majorizes y, i.e., x M y as ki=1 x(i) ≥ ki=1 y(i) , ∀k = 1, 2, · · · , N − 1 and N N N i=1 x(i) = i=1 y(i) . For a function f : R → R is said to be strictly Schur-convex if x M y implies f (x) ≥ f (y) for all x and y which are not a permutation of each
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TABLE I PARAMETERS U SED T HROUGHOUT THE PAPER
Note that Nm can vary during different time slots. The instantaneous SINR of the HU in the m-th SAP is given by m SINRm 0 (p )
= Nm
hmm pm i i +
i=1
pm hmm 0 0 M N n n=1,n =m
k=0
n 2 hmn k pk + σ m
(1)
where hmn denotes the fading gain, including slow and fast i fading gains from the i-th user belonging to the n-th SAP to m m m m T the m-th SAP, pm 0 and p = [p1 , p2 , . . . , pNm ] are the HU’s 2 is the and GUs’ transmission powers in the m-th SAP, and σm variance of the additive white Gaussian noise (AWGN) at the m-th SAP. In the denominator of (1), the first term is the aggregate interference generated by GUs in the m-th SAP and the second term is the aggregate interference generated by all the other users in M − 1 SAPs. Similarly, the instantaneous SINR of the i-th GU (∀i ∈ Nm ) in the m-th SAP can be expressed as m SINRm i (p )
= N m j=0, j =i
hmm pm j j +
pm hmm i i M
N n
n=1,n =m
i=0
n 2 hmn k pk + σ m (2)
In this work, we assume that the interference generated by a SAP to other SAPs is upper bounded by a threshold Nm m and we n nm m model it as max ( N i=0 hi pi ) ≤ gm ( i=0 pi ) ≤ Im , n∈M\{m}
∀m ∈ M, where gm =
Fig. 2. System Model: one macrocell with two SAPs network. The first SH admits 1 GU and rejects 1 GU. The second SH admits 1 GU and rejects 2 GUs.
other. For a function g : I N → R, where interval I ⊂ R, is said to be a separable convex function if g is in the form of g(x) = N i=1 fi (xi ), where fi is a convex function on I. Any separable convex function is Schur convex function. II. S YSTEM M ODEL AND P ERFORMANCE M ETRICS We consider an uplink two-tier network, which consists of one macrocell operated by the MNO overlaid with M SAPs installed by different SHs indexed as M = {1, 2, . . . , M }, shown in Fig. 2. All the SAPs are connected to the macrocell through finite capacity-limited backhaul links. The MNO refunds SHs whenever they allow GUs to share their SAPs. The macrocell and SAPs operate in two separate frequency bands so there is no cross-tier interference [25], [26]. However, SAPs are densely deployed and all HUs and GUs are sharing a common bandwidth, thus resulting in intra-cell interference.1 In the following, we assume in the m-th SAP there is one HU2 and Nm GUs who are requesting to connect to it at each time slot. The HU is indexed as 0 and GUs are indexed as Nm = {1, 2, . . . , Nm }. 1 While it is possible to allocate orthogonal frequency bands to GUs to avoid interference, this will decrease the spectrum efficiency of SAP. Thus, this work aims to maximize the number of GUs to be admitted into each SAP while maintain the HU’s QoS. 2 It can be exteneded to multiple HUs case via TDMA or OFDMA scheme.
max
n∈M\m,i∈Nm ∪{0}
{hnm i } is the maxi-
mum fading gain from users in the m-th SAP to the other (M − 1) SAPs. The MNO has the authority to set {Im }, where M Im ≤ Q. With the above assumption, the second term m=1 Nn mn m M k=0 hk pk in (1) and (2) can be approximated n=1,n =m M as n=1,n =m In . The main motivation of this approximation is to enable the SAPs to perform distributed optimization as shown later in Section III. With Rayleigh fading assumption on }, the outage probability of the HU in the m-th SAP is {hmm i given by [27] m m m m Pm out (p ) = P (SINR 0 (p ) ≤ γ0 )
pm hmm 0 0
= 1 − P N m ≥ γ0m mm pm + n h m mi=1 i i −1 γ nm N m − 0pm γ m pm 0 =1 − e 1 + 0 mi p0 i=1
(3)
where γ0m ≥ 1 is the fixed target SINR of HU and nm = M 2 n=1,n =m In + σm . In the calculation of (3), the fading gains mm {hi } in each SAP are assumed to be independent and identically exponentially distributed random variables with unit mean. Furthermore, since SAPs have a relative smaller coverage radius, we assume that the slow fading gains for GUs within a common SAP are the same and normalized to 1. For the HU, the chosen QoS performance metric is throughput, which is a function of outage probability, and, in turn, depends on the transmission powers pm [28]. Specifically, the HU’s throughput is given by m m cm = [1 − Pm out (p )] log (1 + γ0 )
(4)
YANG et al.: BACKHAUL-CONSTRAINED SMALL CELL NETWORKS: REFUNDING AND QoS PROVISIONING
and the HU’s QoS utility is then defined as [29] Tm = log(cm ) = −
γ m pm log 1 + 0 mi + Am p0 i=1
Nm
III. ACCESS -BASED R EFUNDING W ITH G UARANTEED QoS P ROVISIONING (5)
m where Am = −(γ0m nm /pm 0 ) + log[log(1 + γ0 )] is a constant for a given {Im } and Tm can be viewed as proportional fairness throughput of the HU. It is apparent that each GU contributes to the HU’s QoS utility degradation separately, e.g., if the i-th GU is rejected by the HU, the corresponding term − log(1 + m (γ0m pm i /p0 )) vanishes without affecting the HU’s utility. For SAPs with guaranteed QoS provisioning, we consider in terms of the number of a linear refunding function ΦGQoS m admitted GUs as follows:
ΦGQoS = φGQoS |Ωm |, m m
m∈M
(6)
where φGQoS is the access-based refunding and |Ωm | is the m cardinality of the admission set Ωm such that Ωm ⊂ Nm . The total refunding is proportional to the number of admitted GUs, which implies that the MNO treats all the GUs within the same SAP equally. From the HU’s perspective, it prefers to admit the GUs causing the least amount of HU’s QoS utility degradation, i.e., requiring the least amount of transmission powers. Therefore, we will capture this tradeoff in the design of the utility function for SH later. On the other hand, for SAPs with best-effort QoS provisionis defined in terms of GUs’ ing, the refunding function ΦBQoS m sum-rate as follows: ΦBQoS = φBQoS Rm (pm ), m m
m∈M
(7)
where φBQoS is the usage-based refunding and Rm (pm ) is the m GUs’ approximated sum-rate as Nm mm m
p h i i log 1 + Nm (8) Rm (pm ) ≈ mm pm + n h m j j=1,j =i j i=1 M 2 pm where n m = hmm 0 0 + n=1,n =m In + σm . At the m-th SAP, we define the total utility function as the summation of refunding from the MNO and HU’s QoS utility. For guaranteed QoS provisioning, it is written as GQoS = ΦGQoS + ν m Tm , Um m
m∈M
(9)
and, for best-effort QoS provisioning, it is written as BQoS Um = ΦBQoS + ν m Tm , m
m∈M
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(10)
where νm is the unit converter between monetary utility (ΦBQoS ) and QoS utility Tm . Given a νm , serving GUs ΦGQoS m m can receive refunding from the MNO but, as a consequence, cause HU’s QoS utility degradation. However, rejecting GUs can secure a better HU’s QoS at the expense of refunding. If νm is extremely large, the SH is unlikely to serve any GUs since it weighs the HU’s QoS more. On the contrary, if νm is a small, the SH tends to serve as many GUs as possible, i.e., receiving as much refunding as possible regardless of the HU’s QoS.
A. Two-Stage Refunding-Admission Game Formulation In Stage I, the refunding game at the MNO is formulated as ⎧ M GQoS ⎪ ⎪ max π−φGQoS |Ωm | UMNO = ⎪ m ⎪ GQoS ⎪ φ ,{I } m=1 ⎪ ⎨{ m } m GQoS s.t. Im ≥ gmpm 0 , m∈M PMNO (11) := M ⎪ ⎪ ⎪ Im ≤ Q ⎪ ⎪ ⎪ m=1 ⎩ 0 ≤ φGQoS ≤ π, m ∈ M m where π is the payment by the admitted GUs and the GUs only need to pay to the MNO when they are connected to the SAPs; Q is the upper bound of the aggregated interference generated by all SAPs. The first constraint ensures that Im is at least equal to gm pm 0 , which allows for the transmission of HUs since they have subscribed to the small cell service. If the MNO sets a larger Im , it means the corresponding m-th SAP can potentially admit more GUs. If the MNO sets a smaller Im , it means the MNO prohibits the m-th SAP to admit GUs. Since the MNO has been authorized to manage its own licensed spectrum, it can assign {Im } to different SAPs which are operated on the MNO’s licensed spectrum. The second constraint ensures that the total interference generated by all SAPs should be }, the MNO less than or equal to Q. Together with {φGQoS m GQoS is the controls the admission preocess at each SAP and UMNO MNO’s net revenue, which consists of two terms: the first term, M π|Ω |, is the total gain due to GU admission and the m m=1 M GQoS second term, m=1 φm |Ωm |, is the total refunding to SHs. , Im )} to maximize its In (11), the MNO determines {(φGQoS m net revenue. Since |Ωm | is an implicit integer function of φGQoS m GQoS and Im , PMNO is a mixed integer optimization problem and it is NP hard. In Stage II, the admission game at the m-th SH (∀m ∈ M) is formulated as ⎧ GQoS max Um = φGQoS |Ωm |+νm Am m ⎪ ⎪ Ωm ,pm ⎪ ⎪ ⎪ N m ⎪ ⎪ ⎪ −ν log (1+am pm m ⎪ i ) ⎪ ⎪ i=1 ⎪ ⎪ MAX ⎨ s.t. , ∀i ∈ Ωm 0 ≤ pm i ≤p GQoS m m m := (12) Pm SINR (p ) ≥ γ i i , ∀i ∈ Ωm ⎪ ⎪ ⎪ ⎪ m m ⎪ ⎪ gm pi +p0 ≤ Im ⎪ ⎪ ⎪ i∈Ωm ⎪ ⎪ ⎪ ⎪ ⎩ log (1+γ m ) ≤ C¯m i∈Ωm
i
m where am = γ0m /pm 0 , p0 is the HU’s fixed transmission power, MAX is the maximum transmission power, γim is the i-th GU’s p target SINR, and C¯m = Cm − log(1 + γ0 ) is the remaining backhaul capacity after serving the HU. The first constraint is the transmission power constraint. The second constraint is the SINR constraint, where the SH will guarantee each admitted GU’s target SINR. The third constraint is the interference temperature constraint, where the aggregate interference generated by the m-th SAP should be less than or equal to Im . The last constraint is the backhaul constraint, where the GUs’ total data
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rate should be less than or equal to the remaining backhaul GQoS consists of two terms: capacity. The total utility function Um GQoS the first term, φm |Ωm |, is the total refunding from the MNO, m m and the second term, −νm N i=1 log(1 + am pi ) + νm Am , is the HU’s QoS utility. The goal is to tradeoff between refunding and performance of HU by choosing Ωm and pm . Since {Im } decouple the intra-cell interference, it allows each SAP to make decision non-cooperatively and distributedly. If we directly work on the instantaneous SINRs in (1) and (2), it may incur ping-pong effect among SAPs. Consider a simple example of two SAPs A and B: both of them have two feasible admission set choices {A1 , A2 } and {B1 , B2 } such that |A1 | = |A2 | and |B1 | = |B2 |. If we use the instantaneous SINR in (1) and (2), B the ping-pong effect may happen between A and B that A1 −→ A B A B B1 −→ A2 −→ B2 −→ A1 , where A1 −→ B1 means if A chooses A1 , B will choose B1 . Hence, one SAP’s decision can trigger oscillations between A and B. However, the MNO may not be aware of the ping-pong effect since it only cares about the cardinality of admission sets. By introducing {Im }, we can alleviate ping-pong effect at the expense of net revenue. Finally, from our refunding framework, we can observe that the network performance improvement rates to the GUs’ aggregate is equal m log(1 + γ ). from all the SAPs, i.e., M i m=1 i∈Ωm GQoS GQoS The problem PMNO and Pm form a two-stage refundingadmission game and the subgame perfect equilibrium (SPE) is defined as follows. Definition 1: Denote (φGQoS , I ) be a feasible solution in Stage I and P be a feasible solution in Stage II. Then, the point (φGQoS , I , P ) is a SPE for the formulated Stackelberg game if for any other feasible solution (φGQoS , I , P ), the following conditions are satisfied: GQoS GQoS GQoS φGQoS , I , P φ , I , P ≥ UMNO UMNO GQoS GQoS GQoS GQoS φ φ Um , I , P ≥ Um , I , P , ∀m ∈ M (13) where GQoS
I = [I1 , I2 , . . . , IM ]T ,
P = [p1 , p2 , . . . , pM ],
and
T [φGQoS , φGQoS , · · · , φGQoS ] . 1 2 M
= To obtain SPE, the two-stage refunding-admission game is analyzed by backward induction and the goal is to obtain the subgame perfect equilibrium. The Stage II problem is solved for a given (φGQoS , I) first. Then we can achieve the SPE by solving the Stage I problem based on the best response functions from Stage II. φ
B. Joint GU Admission and Power Control Algorithm We propose a joint GU admission and power allocation algorithm. Firstly, we have the following theorem given a feasible Ωm . Theorem 1: For any feasible admission set Ωm , the minimum transmission powers are m −1
1|Ωm | (14) pm Ωm = nm HΩm where [Hm ]ij =
m hm i /γi m −hj
if i = j . if i = j
Proof: For any feasible admission set Ωm , it receives a |Ωm | from the MNO. Then, we simply fixed refunding φGQoS m need to obtain the minimum sum-log power of the following optimization problem min log (1 + am pm i ) m pΩ GQoS m i∈Ω m m pΩm := Pm (15) m
s.t. Hm Ωm pΩm nm 1|Ωm | . Minimizing a concave function is NP hard and the optimal solution lies in the extreme points of the feasible do m −1 main. It is easy to verify that pm Ωm = nm (HΩm ) 1|Ωm | is the extreme point which minimizes the objective function of GQoS (pm Pm Ωm ) [30]. Theorem 1 states the minimum transmission powers for any feasible admission set in (11). Then in the following theorem, we derive a power update rule by adding a new GU to the current admission set. Theorem 2: For simplicity, denote K, instead of Ωm , as the current admission set (|K| = K), if the SH adds a new GU K + 1 to the set K (we assume all the constraints in (11) are still valid), then the transmission power for each GU is updated as follows: m T m ¯ K + n m (16) pm K+1 (K + 1) = Dm (hK ) p hm K+1 m ¯m pm p (K +1) , k ∈ K (17) K+1 (k) = p K (k) 1+ n m K+1 m m T ¯m where hK = [hm 1 , h2 , . . . , h K ] , p K are the transmission powers before adding the new (K + 1)th GU, pm K+1 are the updated transmission powers after adding the new (K + 1)th GU, and m m
m T m −1 ¯ m −1 Dm = ((hm K+1 /γK+1 ) − (hK+1 /nm )(hK ) (HK ) p K) . Proof: See Appendix A. Theorem 2 states how to update transmission powers if the SH adds Gus gradually. From Theorem 2, by adding one new GU, the transmission powers of previous GUs are amplified by
m the same scaling factor, i.e., 1 + (hm K+1 /nm )pK+1 (k + 1), and the transmission power of the newly admitted GU is jointly m m ¯m determined by hm K+1 /γK+1 , p K and HK . In spite of other constraints, the transmission power becomes infeasible when m m
m T m −1 ¯ m hm K+1 /γK+1 ≤ (hK+1 /nm )(hK ) (HK ) p K . It is difficult to obtain a simple admission criterion. To cope with this difficulty, we propose a gradual GU admission and power allocation algorithm. The idea is to add GUs in terms of the least increment of sum-log-power until we reach the suboptimal solution. Algorithm 1 consists of two-phases: it first sorts feasible GUs in terms of least increment of sum-log-powers and then GQoS . determines the optimal admission set which maximizes Um Algorithm 1 significantly reduces computational complexity 2 ). It also avoids from O(2Nm ) (enumeration search) to O(Nm recalculation of the matrix inverse when adding a new GU.
C. Special Cases 1) Equal Fading Gain: Within the coverage of the m-th SAP, ˜ m. we assume all the users experience the equal fading gain h
YANG et al.: BACKHAUL-CONSTRAINED SMALL CELL NETWORKS: REFUNDING AND QoS PROVISIONING
Algorithm 1 Gradual GU Admission and Power Control
GQoS Initialize Ω m = ∅, Φm = Nm , Um [0] = Am , and pm i = m m (γi nm )/hi , ∀i ∈ Nm for i = 1 to Nm do MAX m ¯ if pm , (Im /gm )−pm 0 } or log(1+γk ) > C k > min{p then pm k =∞ end if end for Find k = arg mini∈Φm pm i if pm k = ∞ then Ω m = Ω m ∪ {k}, Φm = Φm \ {k} else return No GU can be served. end if for i = 1 to Nm − 1 do Find the GU l in Φm with the minimum increment of logsum-power using (16) and (17). if all the constraints in (12) are valid then Ω m = Ω m ∪ {l}, Φm = Φm \ {l} end if end for for i = 1 to |Ω m | do GQoS [i] in (12) Calculate Um end for GQoS [i]. Find t = arg max Um i=0:1:|Ωm |
GQoS GQoS return Um = Um [t] and Ωm = {the first t ordered GUs in Ω m }
Then the outage probability of HU in (3) is rewritten as ˜ m pm h 0 Pm pm ≤ γ0m out (ˆ K ) =P ˜ m pˆm + nm h K γ0m nm = 1 − exp − m (18) p0 − γ0m pˆm K where pˆm ˆm K is the sum-power of GUs in K and p K is given by nm 1 pˆm K =
i∈K
1−
GQoS Pm
Then the problem ⎧ max ⎪ ⎪ ⎪ Ωm ,pm ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ s.t. EFG Pm := ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
1+1/γim
i∈K
˜m h
1 1+1/γim
.
(19)
becomes νm γ0m nm m ˆm pm 0 −γ0 p K +νm log log (1+γ0m )
GQoS Um = φGQoS |K| − m
K ⊂ Nm MAX 0 ≤ pm , ∀i ∈ K i ≤p m m ≥ γ , ∀i ∈ K SINR i i log (1+γim ) ≤ C¯m i∈K m p0 Im m . pˆm ≤ min , −p m 0 K γ gm
(20)
0
In this case, we propose a simple algorithm to obtain the EFG optimal solution of Pm , which is denoted as Algorithm 2.
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It terminates if any constraints in (20) become invalid or adding GQoS . It chooses a subset a new GU begins to decrease Um m m m to maximize of ordered GUs with γ[1] ≤ γ[2] ≤ · · · ≤ γ[N m] GQoS . Intuitively, by adding a GU, the refunding increment Um while the difference pˆm ˆm is a constant φGQoS m K+1 − p K is an increasing function, hence, there exists a unique admission set GQoS . which maximizes Um Theorem 3: Algorithm 2 attains the optimal set K , where the admitted GUs have smaller target SINRs compared with rejected GUs. The cardinality of the optimal set is equal to [K] ≤ φGQoS < K if the access-based refunding satisfies φGQoS m m GQoS GQoS m 2 m pm − p ˆ ))/ φm [K +1], where φm [K] = (νm (γ0 ) nm (ˆ K+1 K m m m m ((pm ˆK+1 )(pm ˆK )). 0 − γ0 p 0 − γ0 p Proof: See Appendix B. 2) Equal Target SINR: We assume that all GUs have the same target SINR γ m . The transmission power of the (i)th GU in K is given by pm (i) =
n m
hm (i) 1 +
1 γm
−K
,
i∈K
(21)
which implies that the admitted GUs have better channel gains. GQoS can be rewritten as Then, Pm ⎧ GQoS max Um = φGQoS K + νm Am ⎪ m ⎪ ⎪ K ⎪ ⎪ K ⎪ ⎪ am nm ⎪ −νm log 1 + hm 1+ 1 −K ⎪ ⎪ ) ⎪ γm (i) ( i=1 ⎪ ⎨ ¯m C ESINR s.t. K ≤ min Nm , log(1+γ m ) Pm := ⎪ ⎪ ⎪ ⎪ 0 ≤ hm 1+nm1 −K ≤ pMAX , ∀i ∈ K ⎪ ⎪ ( ) γm (i) ⎪ ⎪ ⎪ K ⎪ nm ⎪ ⎪ ≤ gIm − pm ⎩ 0 m hm 1+ γ1m −K ) i=1 (i) ( (22) We can find the optimal set K using a similar method as the equal fading gain case. The SH prefers to serve the GUs with better channel gains since all GUs’ target SINR are the same. It is equivalent to choosing a subset of ordered GUs with hm (1) ≥ m m GQoS . h(2) ≥ · · · ≥ h(Nm ) which maximizes Um 3) LP Approximation: We assume that am pMAX 1, then m GQoS log(1 + am pm i ) can be approximated as am pi . Then, Pm can be approximated as
⎧ N m ⎪ GQoS ⎪ max = φGQoS um Um ⎪ m i ⎪ ⎪ um , pm i=1 { i } { i } ⎪ ⎪ ⎪ N ⎪ m ⎪ ⎪ ⎪ −νm a m pm i +νm Am ⎪ ⎪ i=1 ⎪ ⎪ ⎪ ⎪ s.t. 0 pm pMAX 1 ⎪ ⎨ 0 um 1 LP := Pm m m −1 pm 1−um ⎪ i hi +(ωi ) ⎪ Nm m m( i ) ≥ γim , ∀i ∈ K ⎪ ⎪ p hj +nm ⎪ ⎪ j=i j ⎪ ⎪ N m ⎪ ⎪ Im m ⎪ pm ⎪ i ≤ gm −p0 ⎪ ⎪ i=1 ⎪ ⎪ ⎪ N ⎪ ⎪ m ¯ ⎩ um i log (1+γi ) ≤ Cm i=1
(23)
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 9, SEPTEMBER 2014
Algorithm 2 Equal Power Gain
GQoS Initialize K = ∅, Πm = Nm and Um = νm Am for i = 1 to Nm do K+{k} Find k = arg min γi and calculate psum in (19) i∈Πm
if all constraints in (20) are valid then GQoS calculate Um = φGQoS (|K| + 1) − (νm γ0m nm / m K+{k} (p0 − γ0m psum )) + νm log log(1 + γ0m ) else return K end if GQoS GQoS ≥ Um then if Um GQoS GQoS = Um , K = K + {k}, Πm = Πm − k Um else return K end if end for
Algorithm 3 Lookup Table Approach • Each SH feedbacks {hmm } and the MNO estimates {gm }; i • On behalf of each SH, the MNO calculates the best response function table Ωm for different pair of (tΔπ, I0 + lΔI) using Algorithm 1, t = 1, 2, . . . , T and l = 1, 2, . . . , L; • Then, it searches for the near-optimal strategies by any fast search method; • Finally, the MNO feedbacks the refunding and GU admission results to the corresponding SHs.
m m T where u = [um 1 , u2 , . . . , uNm ] are auxiliary variables to indim cate the GUs admission status i is an auxiliary constant and ω m m MAX m hj + n m ), ∀i ∈ Nm . such that 0 ≤ ωi ≤ 1/γi ( j =i p LP is a LP, we can solve it globally and efficiently. Since Pm The GUs are gradually removed if um i = 1. For iterative GU removal, the performance is heavily dependent on the criterion to remove GUs. In [18] and [19], they remove the user with the largest gap to its target SINR. In this case, it is equivalent to removing the GU with the smallest um i .
D. Lookup Table Approach at the MNO GQoS The problem PMNO is generally difficult to solve since |Ωm | and Im . Therefore, we is an implicit integer function of φGQoS m propose a lookup table approach to determine individualized refunding and interference temperature constraints to different SHs. The MNO divides the feasible refunding interval [0, π] into T equal intervals with step size Δπ = π/T and the interference temperature interval [I0 , Q] into L equal intervals with step size ΔI = (Q − I0 )/L, where I0 = max(gm pm 0 ). Then, on behalf of each SH, the MNO calculates a table of Ωm in terms of different pairs of (tΔπ, I0 + lΔI), t =
1, 2, . . . , T and l = 1, 2, . . . , L. Finally, the MNO decides its strategy through a lookup table approach. The performance of Algorithm 3 is dependent on step size and search method, which is a tradeoff between optimality and computational complexity. As Δπ → 0 and ΔI → 0, the solution converges to the SPE. Implicitly, we assume that the MNO has enough computational resources to employ this lookup table approach. However, to ensure the scalability of the approach, the MNO can constrain on the number of SAPs that are available to admit GUs. IV. R EFUNDING W ITH B EST-E FFORT QoS P ROVISIONING In this section, we discuss the refunding with best-effort QoS provisioning, which is different from Section III. In Stage I, the refunding game at the MNO is formulated as ⎧ M ⎪ BQoS ⎪ max = π min C¯m , Rm (pm ) U ⎪ MNO ⎪ {I }, φBQoS ⎪ m=1 } m { m ⎪ ⎪ ⎪ ⎪ M ⎨ BQoS φBQoS Rm (pm ) − m PMNO := m=1 ⎪ ⎪ M ⎪ ⎪ ⎪ ⎪ s.t. Im ≤ Q ⎪ ⎪ ⎪ m=1 ⎩ BQoS I m ≥ g m pm ≤π 0 , 0 ≤ φm (24) BQoS The objective function of PMNO is the MNO’s net revenue. It consists of two terms: the first term is the total service gain, where the MNO can, at most, charge a total of π min{C¯m , Rm (pm )} from the GUs in the m-th small cell and the second term is the total refunding to the SHs. From our refunding framework, we can observe that the network performance improvement is equal tomthe GUs’ aggregate rates from all the SAPs, i.e., M m=1 Rm (p ). In Stage II, the admission game at the m-th SH is formulated as ⎧ BQoS max Um = φBQoS Rm (pm ) + νm Am ⎪ m ⎪ pm ⎪ ⎪ ⎪ N m ⎪ ⎪ ⎨ −νm log (1 + am pm i ) BQoS i=1 := Pm MAX ⎪ , ∀i ∈ K s.t. 0 ≤ pm ⎪ i ≤p ⎪ ⎪ ⎪ N m ⎪ Im ⎪ m ⎩ pm i ≤ g − p0 i=1
m
(25) is always less than or Lemma 1: In SPE, the sum-rate Rm equal to C¯m for all small cells. > C¯m , then we Proof: By contradiction, in SPE, if Rm GQoS
¯ > can always find a achievable Rm = Cm such that UMNO GQoS BQoS . Hence, the term min{C¯m , Rm (pm )} in PMNO is UMNO equivalent to inserting a backhaul constraint Rm (pm ) ≤ C¯m BQoS to Pm . A. Utility Maximization via Majorization BQoS We focus on solving Pm . m = hm Lemma 2: Denote xm i i pi , the GUs’ sum-rate m Rm (x ) is a strictly schur-convex function on the feasible m m m m MAX domain Dm = {xm |0 ≤ xm , N i ≤ hi p i=1 (xi /hi ) ≤ }. (Im /gm ) − pm 0
YANG et al.: BACKHAUL-CONSTRAINED SMALL CELL NETWORKS: REFUNDING AND QoS PROVISIONING
Algorithm 4 Utility Maximization via Majorization Sort the GUs with hm i in the descending order BQoS and Rm using (24) and (25) Initialize Um for i = 2 to Nm do if (i − 1)pMAX ≤ (Im /gm ) − pm 0 then for j = 1 to Nm do if j ≤ i − 1then MAX pm (j) = p else if j = ithen MAX MAX , (Im/gm)−pm } pm 0 −(i−1)p (j) = min{p else pm (j) = 0 end if end for end if BQoS BQoS
= Um and Rm = Rm (pm ) Calculate Um BQoS BQoS ≥ Um then if Um BQoS BQoS
= Um and Rm = Rm Um end if end for BQoS and Rm return Um m m m m Proof: Fix N i=1 xi = X , we can write Rm (x ) as N m m
m
+ nm )/(X − xi + nm )). It is easy to prove i=1 log((X m m m that Rm (x ) = N + n m )/(X m − xi + n m )) is i=1 log((X a separable convex function on [0, X m ]Nm , which implies that Rm (xm ) is a strictly schur-convex function on Dm . Based on the Schur-convexity of sum-rate function Rm (xm ) N m and concavity of logarithmic function, i.e., i=1 log(1 + ), we have the following theorem to obtain the optimal a m pm i transmission powers of GUs for a given Bm and Im . BQoS , Theorem 4: At the optimal solution of Pm 1) if more than two GUs can transmit, then Nm = Am 1 ∪ and the GUs’ transmission powers are {k} ∪ Am 2 ⎧ MAX m ⎪ ⎨p i ∈ A1 MAX pm (26) i=k 0 or min pMAX , gIm −pm i = 0 −lp m ⎪ ⎩ 0 i ∈ Am 2 m ≥ hmm ≥ hmm , ∀i ∈ Am where hmm 1 and ∀j ∈ A2 . i j k mm mm 2) If only one GU transmits, then hk ≥ hi , ∀i ∈ Nm and its transmission power pm k lies in one of the following three discrete power points
⎧ ⎡ ⎤min{pMAX , gIm −pm 0 } ⎪ m m
⎨ h − ν a n φBQoS m m m m (1) ⎦ , 0, ⎣ ⎪ ⎩ νm − φBQoS am hm m (1) 0
Im min pMAX , − pm 0 gm
(27)
BQoS . [a]cb is the projection of a onto which maximizes Um the interval [b, c].
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Proof: See Appendix C. Based on Theorem 4, we propose Algorithm 4 to maximize BQoS MAX . For the case (Im /gm ) − pm , no GU can Um 0
