A Distributed Method of Inter-Cell Interference ...

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A Distributed Method of Inter-Cell Interference Coordination (ICIC) Based on Dual Decomposition for Interference-Limited Cellular Networks Chrysovalantis Kosta, Bernard Hunt, Atta UI Quddus, and, Rahim Tafazolli, Member, IEEE Abstract—In this paper we present a novel distributed InterCell Interference Coordination (ICIC) scheme for interferencelimited heterogeneous cellular networks (HetNet). We reformulate our problem in such a way that it can be decomposed into a number of small sub-problems, which can be solved independently through an iterative subgradient method. The proposed dual decomposition method can also address problems with binary-valued variables. The proposed algorithm is compared with some reference schemes in terms of cell-edge and total cell throughput. Index Terms—Inter-cell RRM, interference avoidance/coordination, ICIC, linear programming optimization, dual decomposition.

I. I NTRODUCTION

E

ACH new mobile cellular system generation may provide a breakthrough in maximum user data throughput. However, no mobile system so far has guaranteed an achievable high data rate for the most disadvantaged members of the network, namely, cell-edge users. Cell-edge users experience not only a high propagation loss in their own cell, but also they receive considerable inter-cell interference from the neighboring cells. Inter-cell interference can be mitigated by an intelligent radio resource management (RRM) which spans across all cells in a network [1]. By virtue of optimization, the intercell RRM can be seen as a knapsack problem which is a non-convex optimization problem [2]. Generally, solutions that have exponential requirements (resource, time & complexity) with only a constant increase in the problem size, are unattractive. Thus, the search for an efficient sub-optimal algorithm that may solve a non-convex problem in a polynomial way is considered very important. A centralized formulation of inter-cell RRM through InterCell Interference Coordination (ICIC) which achieves a global optimum for OFDMA-based networks is proposed in [3] based on an ICIC approach in [4]. However, the global ICIC scheme is unsuitable for large-scale systems, such as heterogeneous network (HetNets), because of the complexity of coordination across macro and small cell layers (pico, femto) due to unplanned deployment of the latter. Several distributed approaches proposed in the literature may simplify the intercell RRM problem, e.g. by dividing the user scheduling into two phases (i.e., cell-edge and cell-centre user scheduling) [5]. Another way to simplify the problem is by allowing a small ratio of active cells (clusters) to transmit simultaneously in order to mitigate inter-cluster interference [5]. Recently, the

Manuscript received January 30, 2013. The assciate editor coordinating the review of this letter and approving it for publication was I. Guvenc. The authors are with the Department of Electronic Engineering, Centre for Communication Systems Research (CCSR), University of Surrey, Guildford, United Kingdom (e-mail: [email protected]). Digital Object Identifier 10.1109/LCOMM.2013.040913.130224

employment of discrete [7] rather than continuous [8] power control has become a promising and effective way to control and mitigate inter-cell interference. However, because a multilevel power control technique requires large information exchange [7], it may not be appropriate in a random deployment of small-cells. Another disadvantage of some distributed solutions [7], [8] in the literature is that they have a non-negligible performance gap compared with a centralized algorithm due to their convergence to a local optimum. Furthermore, they usually improve only one performance metric at a time, i.e., either the cell-edge throughput or the total cell throughput [6]. In this paper, we present a distributed ICIC algorithm applicable to HetNet cellular deployments, which is solved via a novel dual decomposition approach. Most dual decomposition methods [9] are effective only for linear problems with real variables, thus we propose a new method to address problems with binary-valued variables. Through extensive simulation results, we show that this novel approach minimizes the performance gap between the distributed and the semi-centralized solution as the number of iterations increases. In addition, we compare the performance of our distributed algorithm with a number of reference schemes in terms of cell-edge and total cell throughput. II. S YSTEM M ODEL We consider an OFDMA-based LTE downlink system. Let S denote the set of inter-connected eNodeBs (eNBs) by using an X2 communication interface [1], K denote the set of User Equipment (UEs) and N denote the set of Resource Blocks (RBs). The instantaneous Signal-to-Interference and i that can be achieved is calculated as Noise Ratio (SINR) γk,n follows: i Pni · Hk,n i, j ∈ S i =  j j , γk,n (1) n ∈ N, k ∈ K Pn · hk,n + Nw j=i

i Hk,n

denotes the desired channel gain that UE k Here, experiences from eNB i at RB n including all long-term and short-term channel-fading characteristics. In a similar way, hjk,n denotes the interfering link. Pni and Pnj are the transmit powers allocated to RB n by eNB i and j, respectively. Nw stands for the average thermal noise power in each RB. To i convert the achievable SINR γk,n into an effective data rate i rk,n , we assume a log-linear function [10] as follows:   i i , (2) = B · log2 1 + γk,n rk,n where B is the bandwidth allocated in a RB. III. P ROPOSED ICIC M ECHANISM In order to reduce the complexity with minimum loss to the optimality, we divide the multi-cell RRM problem into two independent problems (ref. Appendix A) i.e., the ‘assignment

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problem’ and the ‘transmit power problem’. The assignment problem decides which user is to be scheduled to which RB. The transmit power problem decides the level of transmit power P i and interfering transmit power P j that best satisfies the given constraint. The complication regarding the P i and P j is as follows; once an optimal P i is reached, it may no longer be optimal when the P j has adjusted to the changes resulting from inter-cell interference.

In a multi-user interference-limited system, the binary power allocation policy maximizes the total data rate [11]. As the transmit power of each RB can be allocated to either maximum transmit power or zero transmit power, we may substitute the two complex variables P i and P j with one joint equivalent variable. Therefore, we consider only these two cases (transmit/not transmit) to calculate the achievable user data rate. We assume that the served users are able to estimate the separate levels for strong sources of interference by employing cell-specific orthogonal reference sequences in the LTE standard [1]. In addition, let m ∈ M denote the index up to which (0th , 1st , 2nd , ) dominant interfering eNB is mitigated. Then, let U denote a utility metric which is formulated as follows: rˆ i,m i,m Uk,n = rk,n · dk ; dk = ; m ∈ M, (3a) rˆk where rˆ is the average throughput across all UEs1 , rˆk is the average throughput of UE k and dk is the user demand to provide a network-wide fairness. In this way, the utility metric i,m can describe the additional gain by mitigating up to Uk,n the m dominant interfering eNBs. Afterwards, the assignment problem for each RB n is solved by selecting the best user K  with the highest utility price as  follows:  

K = arg max k∈K

Uki,m

.

(3b)

Next, the transmit power problem is formulated for each RB n as follows:   maximize U i,m · ρi,m ; (4a) i∈S m∈M i,m ,m

= 1;   ρ = min ρj,m + ρk,m , 1 ; j,k∈  i,m j,m k,m ρ ,ρ ,ρ , ρm ∈ {0, 1}. ρ

B. Linear problem formulation and relaxation In order to reduce the complexity involved with the nonlinear expression in (4c) we may reformulate it to this: ρ,m = ρj,m + ρk,m . (5) j,k∈

A. Problem formulation

subject to

eNB i does not transmit (ρi,m=1 = 0). The optimization problem in (4) will maximize the network utility based on the user selection in (3b).

+ρ

,m

(4b) (4c) (4d)

m i,m

Now the reformulated problem is a special case (i.e., a problem with totally unimodular matrix specifications) whose relaxed solution is also the optimal solution to the initial problem. This means that there is a small integrality gap2 between the reformulated problem and its relaxation. Linear programming relaxation is a problem that arises when binary variables are replaced with real variables belonging to interval [0, 1]. Therefore, the variables in (4d) may be relaxed as:  ρi,m , ρj,m ,ρk,m , ρm ∈ [0, 1]. (6) m

The small integrality gap of the reformulated problem can be reduced by introducing a number of tighter cuts3 as follows: ρ ≤ b, ρ ∈ [0, 1],  ∈ S, b = || /2 , (7) where  is the set of eNBs where the cutting plane method is applied and b is the lower integral part of the cardinality (| |) of  divided by 2. Finally, a slack variable y may be added to the inequality constraint in (7) to transform it to an equality as follows: (8) ρ + y = b, y ∈ [0, 1]. IV. P ROPOSED D UAL D ECOMPOSITION M ETHOD One main disadvantage of the semi-centralized problem is that is still highly dependent upon a centralized processing entity which requires a large amount of the network information exchange. Therefore, we may distribute further the complexity of the relaxed semi-centralized problem in (4) into a number of independent sub-problems via dual decomposition [9] as follows:    i,m i,m i,m  i,m ,m  U ·ρ −λ ρ +ρ −1  m∈M ; L(ρ, λ)=    z i i∈S −λ ρ + y − b =



    i Li ρi,m , y i , λi,m , λz + λ + λz · b ,

i∈S

i∈S

(9) where λi , λz > 0 is the Lagrange multipliers or dual variables associated with the constraints in (4b) and (8), respectively. Let z denote the index of the cutting plane equations. Therefore, λz is the dual variable associated with the equation z. λi,m is calculated using the dual variable λi and λj with respect to i,m as follows: λi m=0 λi,m = λj (10) m = 0 .

Let ρ denote the binary variable whether the eNB i may transmit (ρi,m = 1) or not (ρi,m = 0) in the case where up to the m dominant interfering eNBs are mitigated. Also, let i,m denote the set of different eNBs where the eNB i is in conflict when they are mitigating up to the m dominant interfering eNBs. For example, i,m = {j, k} refer that both eNB j, k ∈ S find the eNB i as their 1st dominant interfering eNB. Note that for simplicity the i,m is referred as . Consequently, eNB j and eNB k may transmit in the case m = 1 (ρ,m=1 = 1) only when their dominant interfering

The main network problem is now reformulated as:

1 The user demand may also be defined by using the average throughput across served UEs, however this provides only local fairness. In order to avoid inter-cell signaling, the average throughput across all UEs may be a constant network parameter, which can be optimized over time.

2 The Integrality gap is defined as the maximum ratio between the solution quality of the binary/integer problem and of its relaxation 3 Cuts are a number of additional constraints that may be added to the relaxed problem in order to restrict fractional solutions.

j∈(i,m)

KOSTA et al.: A DISTRIBUTED METHOD OF INTER-CELL INTERFERENCE COORDINATION (ICIC) BASED ON DUAL DECOMPOSITION . . .

TABLE I M AIN S YSTEM S IMULATION PARAMETERS PARAMETER

TABLE II P ERFORMANCE IN M ACROCELLS (G AIN /L OSS )

ASSUMPTION OR VALUE

Femto deployment Total bandwidth

10 MHz (50 RBs)

Total eNB/HeNB Power

43 dBm / 10 dBm

Outdoor path loss model

L = 128.1 + 37.6 log 10 (R), R[KM]

Indoor path loss model

5TH –ILE USER THROUGHPUT IN KBPS 334.4 (38.4%)

SECTOR THROUGHPUT

ZHANG [7]

NUMBER OF ITERATIONS (t) ∞

ABAII [8]

∞

13.52 (16.3%)

335.3 (38.8%)

9.69 (-16.7%)

333.5 (38.1%)

SCHEME

2-tier tri-sectorized sites (19-sites) with a total of 57 eNBs Block type: 5x5 grid House dimensions: 10x10 m2

Macro deployment

3

DYN FR3

N/A

13.55 (16.5%)

13.89 (19.4)

338.8 (40.3%)

∞ 20

13.74 (16.6%)

333.8 (38.2%)

13.72 (16.5%)

333.5 (38.1%)

L = 127 + 30 log 10 (R), R[KM]

10

13.65 (15.9%)

331.7 (37.3%)

External wall penetration loss

30dB

5

13.37 (13.5%)

323.2 (33.9%)

User Noise figure/ thermal noise

9 dB/ −174 dBm

3

12.97 (10.1%)

310.6 (28.6%)

Simulation time/drops

500 TTIs / 200 drops

2

12.52 (6.29%)

298.1 (23.5%)

Num of eNB/HeNB UEs per host

10 / 1

1

11.78 (1.22%)

280.1 (16%)

Femto deployment /activation ratio

0.1/1.0

0

11.63

241.5

Traffic Scenario

Full buffer

maximize subject to

g(λ) =

 i∈S z

g i (λ) +

λ, λi , λ ≥ 0,



 λi + λz · b ;

CENTRAL [3]

IN MBPS

REF2 - SEMI

PROP

REF1- FR1

TABLE III P ERFORMANCE IN F EMTOCELLS (G AIN /L OSS )

(11a)

i∈S

(11b)  i,m = Li ρi,m ), y∗i (λz ), λi,m , λz . where g i (λ) ∗ (λ ρ∗ i,m (λi,m ) and y∗ i (λz ) are the optimal primal variables of the independent subproblem Li and are unique due to strict concavity of the log-transformed U which can be calculated   as: U i,m·ρi,m−λi,m ρi,m+ρ,m−1 i,m i,m  ρ∗ λ   . arg max −λz ρ + y i − b (C) = (12) &    m y∗i λi ρ = 1, (C) : ρi,m ∈ [0, 1]; y i ∈ [0, 1];

m∈M

To calculate λ, the following subgradient may be used as,     λi (t + 1) = λi (t)−β·g i + ; λz (t + 1) = λz (t) − β · g i +, (13) where [ ]+ is the projection on the non-negative orthant and is a positive iterative step size. The advantage of using the dual decomposition method to distribute the semi-centralized algorithm is to limit the intercell signalling required to converge the algorithm. Theorem 1: The iterative algorithm will converge to the semi-centralized solution with a large number of iterations. Proof : The log-linear operating mapping function in (2) is i i ∗ exists for γk,n . In a similar way, convex [10], so a unique rk,n the solution of the log-transformed U in (12) is unique. Then, if the iteration step size β is sufficiently small, as t → ∞, the duality gap between the relaxed problem and its dual will converge to zero. Algorithm 1: To be executed by eNB i for each RB n. Initialization: Calculate the U i,m , ρ , b, λi , λi,m and λz . For each iteration 1, ..., Maximum iteration step : • Solve the equation in (12). • Calculate subgradient step in (13). • Update the subgradients ( λi and λz ). • Exchange subgradients to all eNBs involved. Termination: Stop if the maximum iteration step is reached. V. S IMULATION S TUDY & R ESULTS The simulation study is performed in the downlink using a modified version of the LTE-based system-level simulator [12]. Apart from the outdoor eNB network, a closed-access

ZHANG [7]

NUMBER OF ITERATIONS (t) ∞

ABAII [8]

∞

SCHEME

DYN FR3 CENTRAL [3] REF2 - SEMI

PROP

REF1- FR1

N/A

CELL THROUGHPUT IN MBPS 20.08 (11.3%)

5TH –ILE USER THROUGHPUT IN KBPS 1245 (1794%)

18.52 (2.66%)

1166 (1674%)

12.76 (-29.3%)

800 (1117%)

21.12 (17.07)

1515 (2205%)

∞ 20

21.10 (17.0%)

1496 (2177%)

21.03 (16.6%)

1470 (2138%)

10

20.91 (15.9%)

1447 (2102%)

5

20.7 (14.8%)

1422 (2064%)

3

20.45 (13.4%)

1417 (2057%)

2

20.11 (11.5%)

1411 (2047%)

1

19.48 (7.99%)

1224 (1763%)

0

18.04

65.7

low-power HeNB (Home eNB) network is implemented to simulate a HetNet scenario. Table I gives the main simulation parameters used. We observe two performance metrics, i.e. the cell throughput and the 5th percentile (5th -ile) point of Cumulative Distribution Function (CDF) of UE throughput. Table II and Table III present the performance of the proposed algorithm (‘P ROP ’) for macrocells and femtocells, respectively, where some comparison (reference) schemes are also included. The ‘R EF 1’ employs full frequency reuse and its performance is equivalent when the number of iterations executed by our proposed algorithm is set to zero. Because its performance is increased if the number of iterations is also increased, this relative gain is displayed in the table with respect to the ‘R EF 1’. On the other hand, ‘S EMI ’ is the semi-centralized solution without the dual decomposition. According to the theorem 1, the gap between ‘S EMI ’ and the proposed iterative scheme should converge to zero for a large number of iterations (infinity). Other comparison schemes include: dynamic FR3 (‘DYN FR3’), the centralized solution (‘C ENTRAL’) as proposed in [3], the algorithms ‘Z HANG ’ [8] and ‘A BAII ’ [9] which can also be implemented in a distributed way. The latter two algorithms employ inter-cell power control (continuous [7] or discrete [8]) to achieve a maximum radio resource utilization and quality of service (QoS). As mentioned earlier, performance degradation can be seen in dense interference-limited systems using a nonbinary power allocation plan. This can be observed particularly

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TABLE IV S IGNALING & C OMPLEXITY C OMPARISON (P ER I TERATION )

ZHANG [7]

INFORMATION EXCHANGE OVER X2 INTERFACE 4 Variables (d k , Pi, L 1 , L 2 )

REQUIRED COMPUTATIONS |K| + |S|˜2

PROP

2 Variables (Ȝ, ȡȚ,m)

|M|+1

ALGORITHM

in the case of femtocells. Additionally, it can be observed that their performances are proposed for macro deployments. However, we can also observe that the compared distributed algorithms may converge to a local optimum rather than the global optimum. Our algorithm shows its performance both in macro and femto deployments. The relative gain indicates the adaptability of the algorithm to the current deployment. In the case of femtocells, 1 or 2 iterations may be sufficient to mitigate the detrimental inter-cell interference. However, in the case of macrocell performance improvement can be seen with a higher number of iterations which can vary from 3 to 5. The ‘C ENTRAL’ scheme shows the peak performance that can be achieved by using a centralized algorithm. By assuming a semi-centralized approach, only a negligible performance gap can be seen (between ‘C ENTRAL’ and ‘S EMI ’). This is due to the small correlation of the ‘assignment problem’ with the ‘power transmit problem’. VI. C OMPLEXITY A NALYSIS We provide a brief overview of the complexity and signaling overhead of the evaluated algorithms. For simplicity, we do not account for the complexity in channel status information since all schemes require this information. By default, all centralized algorithms (i.e. ‘C ENTRAL’ & ‘DYN FR3’) must have a common processing entity in order to reach the final solution and distribute this to all parts of the network. The distributed algorithms in [7], [8] are implemented without a centralized entity at the cost of excessive inter-cell signaling. As a result, it is required to compute multiple sub-versions of the centralized problem (own cell & neighboring cells). For example, the distributed algorithm in [7] exchanges four variables to all eNBs over X2 interface (i.e. user demand dk , eNB transmit power P i and two SINR components (L1 & L2 )) in order to reconstruct the weighted sum rate (utility metric). Furthermore, it is required to solve two sub-problems at each iteration (i.e. user assignment and transmit power (on/off) which requires |K| and |S|·2 computations, respectively). Table IV gives a signalling and complexity comparison of the proposed dual algorithm and Zhang’s algorithm in terms of information exchange and required computations (per iteration). The complexity and the signaling overhead of our distributed algorithm is reduced to a minimum by collecting only the necessary information in order to converge to the semi-centralized solution. Note, that the network-wide user demand dk is incorporated into the dual λ variable. Each eNB solves its own problem with respect to this exchanged variable which requires |M | + 1 computations (including the case ‘do not transmit’). VII. C ONCLUSION In this paper, a novel distributed ICIC has been presented for heterogeneous multi-cell cellular networks. The proposed

algorithm solves the problem iteratively by employing a revised dual decomposition method which can address binary variables. Simulation activities show the effectiveness of the algorithm with a small number of iterations especially in the case of femtocell, which would otherwise suffer heavily from inter-cell interference due to their asymmetric deployment. A PPENDIX A D ECOMPOSITION OF THE C ENTRALIZED RRM P ROBLEM The complexity of a centralized problem may be shifted to other network entities (i.e. eNBs) at the cost of intercell signaling. Given that the intra-cell (adjacent-channel) interference is avoided in OFDMA systems, our problem is limited to the inter-cell (co-channel) interference. For simplicity of illustration, the notation of the RB n is omitted. Therefore, the centralized problem, which includes all the network information, is as follows:    i,m i,m U k · ρk Umax = max i∈S m∈M k∈K

Umax =

max



Assignment i,m =Uki,m Problem→ U k∈K



max 

 

 U

i,m

·ρ

i,m

i∈S m∈M



Power transmit problem



The centralized problem is equal to the maximization of the maximized subproblems as shown above. We may reduce the signaling overheads with minimum loss of optimality by selecting the best user (i.e. K = K  ) in each power transmit problem with the argmax in (3b). R EFERENCES [1] Sesia, Toufik, Baker, LTE - The UMTS Long Term Evolution: From Theory to Practice. Wiley, 2009. [2] S. Martello and P. Toth, Knapsack Problems: Algorithms and Computer Implementation. John Wiley and Sons, 1990. [3] M. Rahman and H. Yanikomeroglu, “Inter-cell interference coordination in OFDMA networks: a novel approach based on integer programming,” in Proc. 2010 IEEE VTC – Spring, pp. 1–5. [4] M. Rahman and H. Yanikomeroglu, “Enhancing cell-edge performance: a downlink dynamic interference avoidance scheme with inter-cell coordination,” IEEE Trans. Wireless Commun., vol. 9, no. 4, pp. 1414– 1425, Apr. 2010. [5] M. Pischella and J.-C. Belfiore, “Distributed weighted sum throughput maximization in multi-cell wireless networks,” in Proc. 2008 IEEE PIMRC, pp. 1–5. [6] S. G. Kiani, G. E. ien, and D. Gesbert, “Maximizing multicell capacity using distributed power allocation and scheduling,” in Proc. 2007 IEEE WCNC, pp. 1690–1694. [7] H. Zhang, L. Venturino, N. Prasad, L. Peilong, S. Rangarajan, and X. Wang, “Weighted sum-rate maximization in multi-cell networks via coordinated scheduling and discrete power control,” IEEE J. Sel. Areas Commun., vol. 29, no. 6, pp. 1214–1224, Jun. 2011. [8] M. Abaii, Y. Liu, and R. Tafazolli, “An efficient resource allocation strategy for future wireless cellular systems,” IEEE Trans. Wireless Commun., vol. 7, no. 8, pp. 2940–2949, Aug. 2008. [9] W. Yu and R. Lui, “Dual methods for nonconvex spectrum optimization of multicarrier systems,” IEEE Trans. Commun., vol. 54, no. 7, pp. 1310–1322, Jul. 2006. [10] D. Leith, V. Subramanian, and K. Duffy, “Log-convexity of rate region in 802.11e WLANs,” IEEE Commun. Lett., vol. 14, no. 1, pp. 57–59, Jan. 2010. [11] A. Gjendemsjoe, D. Gesbert, G. Oien, and S. Kiani, “Binary power control for sum rate maximization over multiple interfering links,” IEEE Trans. Wireless Commun., vol. 7, no. 8, pp. 3164–3173, Aug. 2008. [12] J. C. Ikuno, M. Wrulich, and M. Rupp, “System level simulation of LTE networks,” in Proc. 2010 IEEE VTC – Spring, pp. 1–5.