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Belief Propagation Methods for Intercell Interference Coordination in Femtocell Networks Sundeep Rangan, Member, IEEE, Ritesh Madan, Member, IEEE Abstract—Interference coordination is a fundamental challenge in emerging femtocellular deployments. This paper considers a broad class of interference coordination and resource allocation problems for wireless links based on utility maximization with a general linear mixing interference model suitable for complex femtocellular systems. The resulting optimization problems are typically hard to solve optimally even using centralized algorithms but are an essential computational step in implementing rate-fair and queue stabilizing scheduling policies in wireless networks. We consider a belief propagation framework to solve such problems approximately. In particular, we construct approximations to the belief propagation iterations to obtain computationally simple and distributed algorithms with low communication overhead. Notably, our methods are very general and apply to, semi-static and dynamic interference coordination problems including the optimization of transmit powers, transmit beamforming vectors, fractional frequency reuse (FFR) and subband allocations to maximize the above objective. Numerical results for femtocell deployments demonstrate that such algorithms compute a very good operating point in typically just a couple of iterations. Index Terms—Interference coordination, cellular systems, wireless communications, belief propagation, femtocells.

I. I NTRODUCTION NTERFERENCE coordination has re-emerged as a fundamental challenge for next-generation cellular wireless systems. Traditional macrocellular deployments are likely to be supplemented with smaller femtocells and relays, with mixtures of restricted and open access, often deployed in an ad hoc manner [1], [2]. Such deployments may create much stronger and highly variable (in time and space) interference conditions than those experienced in current macrocellular networks, and traditional cellular power and rate control may not be adequate [3]. To address this challenge, a key focus of the current Third Generation Partnership Program Long Term Evolution, 3GPP LTE, and LTE-Advanced standardization efforts is on the design of an interference coordination framework for such unplanned cellular deployments of base-stations with widely different transmission powers [4], [5]. LTE offers a large range of inter-cell interference coordination (ICIC) mechanisms including backhaul-based signaling, sub-band scheduling and

I

Manuscript received 10 March 2011; revised 1 September 2011. This material is based upon work supported by the National Science Foundation under Grant No. 1116589. The material in this paper was presented in part at the IEEE International Conference on Computer Communications (INFOCOM), Shanghai, China, April 2011. As this paper was co-authored by a guest editor of this issue, the review of this manuscript was coordinated by Senior Editor David Lee. S. Rangan is with Polytechnic Institute of New York University, Brooklyn, NY (e-mail: [email protected]). R. Madan is with Qualcomm, Bridgewater, NJ (e-mail: rmadan@qual comm.com). Digital Object Identifier 10.1109/JSAC.2012.120412.

beamforming. More recently, in Release 10 of LTE, methods for ICIC for macrocellular networks (see e.g., [6]) were augmented with a mechanism to allow for the coordination of almost blank subframes between macro and femtocells [7]. During an almost blank subframe, the base-station does not transmit data; it transmits only a few control signals essential for legacy mobiles and thus limits the interference to mobiles in neighboring cells. A new reference signal (RS) called CSIRS was added for 8x8 MIMO and more accurate inter-cell measurements [8]. Along with the design of mechanisms for ICIC, algorithms to exploit such mechanisms are an active area of research as well, see for example, [9], [10]. Mathematically, interference coordination is a complex distributed optimization problem involving scheduling decisions at the transmitters of multiple interfering links. In this work, we consider a general linear mixing interference model where the scheduling decisions in each link are represented as vector (e.g., transmission powers on different sub-bands in frequency, beamforming weights) and the interference on each link is a linear combination of the scheduling vectors on the other links. Associated with each link at a given time is a utility function which describes the benefit to a link as a function of the scheduling vector from the serving transmitter and interference from the other transmitters. The linear mixing model is extremely general and can apply to a large class of interference models and objectives. Computing maximum weighted matching for queue stability [11] and maximization of sum utility of average rates for fairness [12] are special cases of this formulation. In this paper, we derive distributed heuristics based on approximate belief propagation (BP) to compute approximate solutions to the interference coordination optimization problem. A. Previous Work In the past few years, algorithms for special cases of the above interference coordination problem have been extensively studied. In many cases, algorithms with provable desired properties have been obtained – for example, there is a rich literature on distributed power control methods to achieve a desired SINR for each link [13] and to maximize a certain class of utility functions of SINRs [14], [15], approximation algorithms for maximum weight matching for combinatorial interference model were obtained in [16], [17], efficient methods to compute optimal beamforming vector for multiuser downlink [18] and maximizing sum utility of SINRs on uplink [19], stabilizing policies for collision sense multiple access (CSMA) type of models based on simulated annealing were derived in [20]. More generally, heuristic algorithms have been constructed to solve certain specific problems approxi-

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mately in, for example, [21], [22]. While these algorithms perform well in practice in spite of no provable guarantees, the insights and approximations used to obtain these algorithms are very specific to the problem under consideration. Algorithms for ICIC in LTE macrocells have been considered in a large number of works in both the uplink and downlink [23]–[25]. These works are generally based on adaptive subband scheduling and fractional frequency reuse (FFR) methods [21], [26] and exploit statistics over large numbers of mobiles per macrocell. Interference mitigation in femtocells has focussed on similar techniques as well as frequency planning, power control [27], or semi-static resource allocation [9], [10], [28]. As we will demonstrate in the simulations, the methods presented here can also be used for adaptive subband scheduling as a special case of the linear mixing interference model. Also, much of the ICIC work has considered slowly varying allocations that don’t change over few 10s to 100s of milliseconds. Due to the low messaging overhead, it is possible that our approach based on approximate BP can also be used for more dynamic interference management in femtocell deployments, where there is high variability in load and interference from one timeslot to another. For scheduling based on more dynamic traffic statistics such as queue lengths and head-of-line delays, variants of maximum weight scheduling can be used [29], [30]. Unfortunately, computing a maximum weight schedule is generally NP-hard, and much work has thus focused on approximate algorithms. In addition to the works mentioned above, the works [29] and [31] proposed randomized linear complexity (but centralized) algorithms, and [32]–[34] present simple distributed algorithms for combinatorial interference models. Greedy maximal weight matching for such interference models has been considered in [35]–[37]. However, many of the above works apply a hard constraint interference models, where neighboring links cannot transmit simultaneously. Cellular systems in contrast permit multiple interfering links to transmit simultaneously and then use rate control to adapt to the resulting signal-to-interference and noise ratio (SINR). Thus, the degradation in rate with interference is gradual, and are difficult to capture in the combinatorial interference model. In contrast, “soft” interference effects can be easily modeled in the BP utility framework. However, we note that there is an important theoretical connection between the methods in [32], [33] and the BP method considered here. As we will discuss below, BP arrives at the scheduling decision by estimating the marginals of a certain joint probability distribution function given in (7). The CSMA-type methods in [32], [33] can be seen as a simulated annealing (SA) method for selecting a random scheduling vector from precisely the same distribution in the context of a constraint combinatorial interference model. SA can be seen as an asymptotically exact but slow method [38] for solving the optimization problem. In constrast, BP is approximate, but potentially faster. General overviews of BP can be found in a number of works including [39], [40]. In the context of wireless scheduling, theoretical guarantees have been obtained for on-off channels and the combinatorial contention graph model [41], [42]. The

methods here can be seen as a generalization of these methods to soft interference models with larger class of scheduling vectors. In [43], exact BP was used to compute optimal beamforming vectors for a network whose factor graph is a tree; our approximate BP methods have lower complexity, and we consider a more general class of problems and topologies. B. Contributions In this paper, we make the following contributions: BP Framework: We consider a belief propagation (BP) framework for a very general wireless scheduling and interference coordination optimization problem. The underlying optimization problem is posed as a problem of estimating marginals of a joint probability distribution. BP provides a systematic and general approach to obtain distributed algorithms; it can be used with arbitrary nonlinear utility functions and scheduling vectors sets, which enable the algorithm to be applied a range of complex scheduling problems including power control, subband scheduling and distributed beamforming. Also, while we do not obtain any theoretical guarantees, in practice a few iterations of BP generate a good operating point. Thus, it typically has faster convergence than gradient based algorithms (e.g.,10s of iterations in [19]) or simulated annealing [38]. Approximation Algorithms: It is well known that implementing BP for distributed optimization problems entails high computational complexity and communication overhead. Exploiting the linear mixing interference model and applying Gaussian and first-order approximations similar to [44]–[48], we develop an approximate BP method that has low complexity, distributed implementation and minimal messaging. Along many links, messages can be carried in small payloads and can be broadcast without separate unicast transmissions, which is particularly crucial for wireless systems. Moreover, the resulting algorithm has a natural interpretation has a “soft” RTS/CTS type handshaking. The approximate BP algorithm is also similar to the recent approximate message passing (AMP) algorithm in [46] and this connection may be useful for further analysis. Numerical Results: Through simulations for femtocell deployments, we demonstrate that approximate BP provides good performance for sub-band and power allocation to maximize utilities of rates, sub-band and power allocation to maximize a weighted sum of rates, and beamforming optimization to maximize utilities of average rates. Also, although the BP algorithm requires multiple exchange of messages before each scheduling decision, our simulations indicate good performance with only two rounds of messaging. Thus, the approximate BP approach is a promising paradigm for new emerging cellular deployments with large interference variations in time and space compared to current predominantly macro-only deployments. II. P ROBLEM F ORMULATION A. System Model We consider a network of n femtocells where each cell consists of one base-station and one mobile. We assume that the association of the mobiles to the base-stations is given

RANGAN and MADAN: BELIEF PROPAGATION METHODS FOR INTERCELL INTERFERENCE COORDINATION IN FEMTOCELL NETWORKS

– the associations can be based on, for example, closedsubscriber group constraints, relative path loss to femto and macro base-stations. We focus on interference coordination between the femtocells; we note that our algorithms can be extended to interference management between femtocells and macrocells. We assume certain time-slots or subframes are allocated to femtocells using, for example, almost blank subframes in LTE [7]. Thus, the interference from macro basestations to mobiles associated with the femto base-stations is mitigated. The goal is to design an distributed algorithms for interference management between femtocells. Specifically, we focus on algorithms with low complexity and low overhead to enable dynamic coordination at a fast-time scale. The coordination between the femtocells can be over a wired backhaul (using the X2 interface, for example) or over-the-air using wireless channels. Both approaches requires changes to the LTE standard – we briefly discuss implementation aspects for LTE femtocells in Sec. V. We assume that the channel gains (which may capture pathloss, shadowing, wall penetration) remain constant during the time required for coordination for each transmission; since femtocells are typically deployed in low mobility environments, this assumption is reasonable. We also assume that the femtocells are synchronized using for example, network listen to the macrocellular network [49] or network timing [50]. Note that synchronization is necessary even to implement even the almost-blank subframe interference negotiation scheme recently added to LTE [7]. B. Interference Coordination Optimization Problem We abstract the setting of n femtocells (with one basestation and mobile per cell) as n links, where each link has a transmitter and a receiver. The transmitter of each link j = 1, . . . , n, denoted TX j, must select some scheduling vector xj ∈ X ⊆ Rnx , which contains nx parameters related to link i. Examples of these parameters will be given below. The selection of the scheduling vectors results in an interference vector zi ∈ Rnz at the receiver of each link i, denoted RX i. The interference is assumed to be a linear function of the scheduling vectors of the other links, zi =

n 

Aij xj

(1)

j=1

for some matrices Aij ∈ Rnz ×nx . We assume that Aii = 0, so that link i does not interfere with itself. We will let x and z be the column vectors with entries xj and zi , x z



= [x1 · · · xn ] ∈ Rnx n 

= [z1 · · · zn ] ∈ Rnz n ,

and write z = Ax where A is the block matrix with entries Aij . We call A the interference matrix. Also, we let Ai denote the ith row of the A so that zi = Ai x. Associated with each link i, is some utility function fi (xi , zi ) of the scheduling vector xi and interference vector zi . The scheduling problem is to maximize the overall utility max F (x), x

F (x) =

n  i=1

fi (xi , zi ).

(2)

633

We will sometimes call the optimization problem (2), an optimization with linear mixing to stress the linear dependence of the interference on the transmit vectors. III. L INEAR M IXING U TILITY E XAMPLES The linear mixing formulation above is extremely general and can incorporate a large class of utility functions and interference models. A general treatment of utility functions for wireless scheduling can be found in [51], [52]. In our simulations, we will consider utility maximization for both static and time-varying problems. For static optimization, the scheduling vectors xj are selected once for a long time period and the utility function is typically of the form fi (xi , zi ) = Ui (Ri (xi , zi )),

(3)

where Ri (xi , zi ) is the long-term rate as a function of the TX vector xi and interference zi and Ui (R) is the utility as a function of the rate. The problem formulation above can incorporate any of the common utility functions including: Ui (R) = R which results in a sum rate optimization; Ui (R) = log(R) which is the proportional fair metric and Ui (R) = −βR−β for some β > 0 called an β-fair utility. Penalties can also be added if there is a cost associated with the selection of the TX vector xj such as power. To accommodate time-varying channels and traffic loads, many cellular systems enable fast dynamic scheduling in time slots in the order of 1 to 2 ms. For these systems, the utility maximization can be re-run in each time slot. One common approach is that in each time slot t = 0, 1, 2 . . ., the scheduler uses a utility of the form fi (t, xi , zi ) = wi (t)Ri (t, xi , zi ),

(4)

where wi (t) is a time-varying weight given by the marginal utility ∂Ui (Ri (t)) wi (t) = , (5) ∂R and Ri (t) is exponentially weighted average rate updated as i (t),  Ri (t + 1) = (1 − α)Ri (t) + αRi (t, x zi (t)),

(6)

i (t) is the TX vector and  where x zi (t) is the interference for link i at time t. Any maxima of the optimization (2) with the weighted utility (4) is called a maximal weight matching. A well-known result of stochastic approximation [12] is that if α → 0, and the scheduler performs the maximum weight matching with the marginal utilities (5), then for a large class of processes, the resulting average rates will maximize the total utility i Ui (Ri (t)). The above utilities are designed for infinite backlog queues. For delay sensitive traffic, one can take the weights wi (t) to be the queue length or head-of-line delay. Maximal weight matching performed with these weights generally results in socalled throughput optimal performance [11]. These results also apply to multihop networks with the so-called backpressure weights. In addition to incorporating general utilities, an appealing feature of the linear mixing framework is that a large class

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of interference models can also be considered, including, for example: • Flat fading with power control: In this case, xj is a scalar representing the transmit power, and Aij is the gain from TX j to RX i, so that zi is the total interference at RX i. The rate, Ri (xi , zi ) can then be described as a function of the SINR gi xi /zi , where gi is the channel gain along link i. Arbitrary SINR to rate mappings may be used. Note that a special case of on-off channels where xj is zero or a maximum transmit power can be used. • Multiple subbands: The above example is easily extended to the case of multiple subbands. As described in the Introduction, subband scheduling is one of the key motivating features of LTE, but the optimization is difficult. To handle multiple subbands, we simply let xj and zi be the vectors of transmit and interference powers in each subband and Aij be a diagonal matrix with channel gains in each subband. • Beamforming and linear precoding: The linear mixing formulation can also incorporate problems with transmit beamforming or linear precoding. For example, suppose a link has N transmit antennas and one receive antenna. If each transmitter TX j uses a beamforming vector bj ∈ CN , and gij ∈ CN is the channel from TX j to RX i, the interference at RX i is given by   gij bj bj gij , zi = j=i

which is linear in the rank one matrices bj bj . Hence, if 2 we let xj ∈ CN be the column vector with entries of  the matrix bj bj , the interference zi can be represented as a linear combination of the vectors xj . The idea can also be generalized to the case where each transmitter can use one of a (pre-specified) set of precoding matrices with multiple transmit streams. IV. B ELIEF P ROPAGATION A. Standard BP We begin by briefly reviewing how we would apply standard BP to the optimization (2). Let u > 0 and define the probability distribution p(x) =

n 1  1 exp(uF (x)) = exp(ufi (xi , zi )), Z Z i=1

(7)

where Z is a normalization constant called the partition function (it is a function of u). BP can be seen as a method to estimate the marginal distributions of the distribution p(x) with respect to the variables xj . From these marginals, one can j = E (xj ). A standard compute the marginal expectations x result of large deviations [53] is that as u → ∞, under suitable conditions, p(x) concentrates around the maxima of F (x) and  = arg max F (x). lim x

u→∞

x

So, if we can estimate the marginal expectations of the probability distribution (7) for large u, we can recover a good estimate for the maximization of (2).

To compute the marginal distributions, BP associates with the interference matrix A a bipartite graph G = (V, E) called the factor or Tanner graph. The vertices V consists of n transmitter nodes associated with the transmitters TX j, and n receiver nodes associated with the receivers RX i. There is an edge (i, j) ∈ E if and only if i = j or Aij is non-zero – that is TX j has some influence on the interference or signal at RX i. We let Nrx (i) and Ntx (j) be the neighbors sets of the nodes RX i and TX j in graph G, respectively. With this graph, BP iteratively passes beliefs along the edges of the graph that represent estimates of the marginal distributions of p(x) with respect to the variables xj . In the context of the wireless scheduling problem, we can interpret the iterations as rounds, where computations are first performed at the receivers and then at the transmitters. We index the round by t, and let pi→j (t, ·) : X → R denote the belief message from RX i to TX j in the receiver half of the round. The reverse belief message from TX j to RX i is denoted pi←j (t, ·) : X → R. pi←j (t, xj ) and pi←j (t, xj ) denote the values of the beliefs at xj . After some fixed number of rounds, the algorithm is stopped and a final scheduling decision, meaning a selection of the TX vectors xj , is made by the transmitters. The steps for BP are as follows: 1) Initialization: Set t = 0 and for all (i, j) ∈ E, let pi←j (t, xj ) be some initial distribution on xj . This distribution could be, for example, the uniform distribution on the set X . 2) RX node update: In the RX half of the round, each RX i sends a belief message to the transmitters TX j with j ∈ Nrx (i) given by pi→j (t, xj ) = E [exp(ufi (xi , zi )) | xj ] ,

(8)

where zi = Ai x as in (1) and the expectation is over independent xk ∼ pi←k (t, xk ), ∀k ∈ Nrx (i). 3) TX node update: In the TX half of the round, each TX j sends a belief message back to the receivers RX i with i ∈ Ntx (j) given by  1 pi←j (t + 1, xj ) = p→j (t, xj ), (9) Z ∈Ntx (j)=i

where Z is a normalization constant and the product is over all  ∈ Ntx (j) with  = i. The iteration number is incremented, t = t + 1, and we return to step 2 until a sufficient number of rounds have been performed. 4) Final solution: The final estimate for the marginal distribution of xj is given by pj (t + 1, xj ) =

1 Z



pi→j (t, xj ).

(10)

i∈Ntx (j)

The scheduling vector can be selected as the maximum of this marginal distribution. In the case when the graph G has no cycles, it can be shown that the pj (t, xj ) converges to the true marginal distribution of p(x) in (7) with respect to the variable xj . However, for general G, BP is approximate. A complete treatment of BP is beyond the scope of this work – the reader is referred to the references above.

RANGAN and MADAN: BELIEF PROPAGATION METHODS FOR INTERCELL INTERFERENCE COORDINATION IN FEMTOCELL NETWORKS

Implementation of the above BP algorithm in wireless networks is challenging due to two reasons: • High computational complexity: The expectation in (8) requires integration over all the variables xr with r ∈ Nrx (i) and r = j. This computation grows exponentially in |Nrx (i)|, which is the number of transmitters interfering with RX i. If this set is large, the computation is prohibitive. • High messaging overhead: Passing the beliefs requires unicast messages between each RX and each TX (which are neighbors as per graph G) as opposed to single broadcast message. Also, in each round t, the messages comprise of the beliefs pi→j (t, xj ) and pi←j (t, xj ) for all values xj ∈ X . If X is large, the messaging overhead may be significant, and it grows with the number of transmitters interfering at a receiver.

635

where Zi0 (·) is the partition function  Zi0 (xi ,  sii , Qsii ) = exp [uLi0 (xi , zi ,  sii , Qsii )] dzi , (16) and 1 Li0 (xi , zi ,  sii , Qsii ) = fi (xi , zi ) − (zi −  sii ) Q−s si ) ii (zi −  2 (17) s and Q−s ii is the matrix inverse of Qii . Next consider the case i = j. A similar argument as above shows that, conditional on xj , the distribution of zi can be approximated by the Gaussian sij (t), Qsij (t)/u), zi = N ( where  sij (t)

=



(18)

i←r (t) + Aij xj Air x

r∈Nrx (i)=j

B. Gaussian Approximation Qsij (t)

We first consider the simplification of the RX node update (8). We describe the simplification in log domain. Let Δi→j (t, ·) and Δi←j (t, ·) be log likelihood functions, meaning any functions such that 1 log [pi→j (t, xj )] + const, (11a) u 1 log [pi←j (t, xj )] + const. (11b) Δi←j (t, xj ) := u where the constants do not depend on xj (although they may depend on t and the indices i and j). Observe that we can recover the probabilities from the log likelihoods by the relation Δi→j (t, xj )

:=

pi→j (t, xj ) ∝

exp [uΔi→j (t, xj )]

(12a)

pi←j (t, xj ) ∝

exp [uΔi←j (t, xj )] .

(12b)

We now consider the simplification of the RX node update (8) under two cases: when i = j and when i = j. We begin with the case when i = j. The expectation in (8) is to be evaluated with zi given by (1) with the variables xj i←j (t) and being independent and xj ∼ pi←j (t, xj ). Let x Qxi←j (t) be the mean and 1/u times the variance of the distribution pi←j (t, ·). Then, under the simplifying assumption that the summation in (1) consists of a large number of independent terms, we can apply the Central Limit Theorem and approximate the distribution of zi with sii (t), Qsii (t)/u), zi = N ( where  sii (t)

=



(13)

i←j (t) Aij x

(14a)

Aij Qxi←j (t)Aij ,

(14b)

j∈Nrx (i)

Qsii (t)

=



i←j (t)) = sii (t) + Aij (xj − x  x = Air Qi←r (t)Air , =

r∈Nrx (i)=j Qsii (t) − Aij Qxi←j (t)Aij .

(19a)

(19b)

Then, applying the Gaussian approximation (18) along with (11b) to the expectation (8) shows that we can write Δi→j (·) in (11a) as   1 Δi→j (t, xj ) ≈ log Zi (Δi←i (t, ·),  sij (t), Qsij (t)) , (20) u where    Zi (Δ(·),  sij , Qsij ) = exp uLi (xi , zi ,  sij , Qsij ) dxi dzi , and Li (xi , zi ,  sij , Qsij ) = Li0 (xi , zi ,  sij , Qsij ) + Δ(xi ).

(21)

Then, the RX and TX node update steps of the BP algorithm with Gaussian approximation of the interference at the RX are: • RX node update: The above equations are used to simplify the standard BP algorithm as follows. Each RX i i←j (t) and Qxi←j (t) of the receives mean and variances x vectors from the transmitters TX j with j ∈ Nrx (i). RX i also receives the function Δi←i (t, ·) from its serving transmitter, TX i. RX i then computes the interference means and variances in (14) and (19). Then, for each TX j it sends back the log likelihoods Δi→j (t, ·) by evaluating the log partition functions (15) and (20). • TX node update: It can be verified that converting the update (9) to log domain yields  Δi←j (t + 1, xj ) = Δ→j (t, xj ). (22) ∈Ntx (j)=i

j∈Nrx (i)

which have the interpretation as a mean and variance of the interference at RX i. Applying the Gaussian approximation (13) to the expectation (8) shows that we can write Δi→i (·) in (11a) as 1 sii (t), Qsii (t))] , (15) Δi→i (t, xi ) ≈ log [Zi0 (xi ,  u

Each TX j can first computes the log likelihoods Δi←j (t + 1, xj ) from the log likelihoods Δ→j (t, xj ) from the receivers RX  with  ∈ Ntx (j),  = i. Then, using the log likelihoods Δi←j (t+1, xj ) , TX j computes i←j (t+1) and Qxi←j (t+1) from the mean and variance x the probability distribution pi←j (t+1, xj ) in (12b). Then, TX j sends messages to the receivers as described in the RX node update above.

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After a sufficient number of rounds, TX j can compute the final log likelihood  Δj (t + 1, xj ) = Δ→j (t, xj ), (23)

where the expectation is with respect to the conditional distribution sii (t), Qsii (t)) px,z|bs,Qs (xi , zi |  1 exp [uLi (xi , zi ,  sii (t), Qsii (t)] , := Zi

∈Ntx (j)

and then compute the final scheduling vector as the one which maximizes the above function. We have therefore simplified the standard BP algorithm by eliminating the exponential complexity of the RX update (8), and replaced this computation with a Gaussian approximation. C. First Order Approximations Unfortunately, the Gaussian approximation above does not significantly reduce the messaging overhead. Each RX and TX must still send separate unicast messages to every TX or RX in its neighbor set. Also, although the TX must only send a mean i←j (t) and Qxi←j (t), the receivers and a covariance matrix x must send the entire log likelihood functions Δi→j (t, xj ). The messaging overhead can be reduced via selective use of first order approximations as follows: Divide the edges (i, j) ∈ E with i = j into two sets – weak and strong – depending on whether Aij is small or large. Along a strong edge (i, j), RX i and TX j exchange the full unicast messages described above. However, for the weak edges, the messages can be replaced with a first order approximation described below. The precise classification rule between weak and strong edges is an algorithm parameter that can be used to trade off complexity and accuracy. To describe the first order approximation along the weak ˆ j (t) edges, suppose Aij is small for some edge (i, j). Let x be the mean value of the distribution corresponding to the log likelihood Δj (t, ·) in (23). Now consider the log likelihood Δi→j (t, xj ) in (20). Applying (19) Δ , i→j (t, xj ) ≈

  1 log Zi (Δi←i (t, ·),  sij (t), Qsij (t)) u

(a)

1 log [Zi (Δi←i (t, ·), u sii (t) + Aij (xj − x j (t)), Qsii (t))] ,

(b)

uij (t)xj + const

≈ ≈

(24)

where (a) follows from (19) and the approximation that i←j (t) ≈ x j (t) when Aij is small; Qsij (t) ≈ Qsii (t) and x and (b) follows from taking a Taylor’s approximation with  j (t), (t) − Aij Di2 (t)Aij x uij (t) = Aij Di1

(25)

where for r = 1, 2, Dir (t), are the derivatives Dir (t) :=

1 ∂r log [Zi (Δi←i (·),  sii (t), Qsii (t))] . u ∂sr

(26)

The constant in (24) is independent of xj . Using standard properties of the cumulant function [40], it can be shown that the derivative is given by

r 

∂

sii (t), Qs (t), Δi←i (t, ·) , f (x , z ) Dir (t) = E i i i ii

∂zri (27)

(28)

and Li (·) is defined in (21) and (17). The dependence on Δi←i (·) in (27) is implicit in (21). Note that the derivatives Dir (t) in (27) can be interpreted as a sensitivity of the expected utility fi (xi , zi ) to changes in the interference zi . The computation of xi←j (t) can also be simplified as j (t) are the expectation i←j (t) and x follows: Recall that x with respect to the likelihood functions Δi←j (t, ·) in (22) and Δj (t, ·) in (23), respectively. Therefore, we can write them as i←j (t) x j (t) x

=

E [xj |Δi←j (t, ·)]

(29a)

=

E [xj |Δj (t, ·)] ,

(29b)

where we have used the notation E(g(x)|Δ(x)) to denote the expectation of g(x) with respect to the probability distribution pΔ (x) ∝ exp [uΔ(x)] . It is easy to check that for small perturbations of (x) of Δ(x), we have the first order approximation E(g(x)|Δ + ) ≈ E(g(x)|Δ) + u [E(g(x)(x)|Δ) − E(g(x)|Δ)E((x)|Δ)] . (30) Therefore, (a)

i←j (t + 1) = E [xj |Δj (t + 1, ·) − Δi→j (t, ·)] x (b)

≈ −

E(xj |Δj (t + 1, ·)) − uE(xj Δi→j (t, xj )|Δj (t + 1, ·)) uE(xj |Δj (t, ·))E(Δi→j (t, xj )|Δj (t + 1, ·))

(c)

j (t + 1) − Qxj (t + 1)uij (t) x

≈

(31)

where (a) follows from (29a) along with (22) and (23); (b) follows from (30); and (c) follows from (29b) and (24). We can use the above relations to define the following algorithm: 1) Initialization: Set t = 0. Each TX j broadcasts an initial j (t) and variance Qxj (t). These can scheduling vector x be based on the mean and variance of xj over the i←j (t) = xj (t) and set X . The receivers RX i sets x Qxi←j (t) = Qxj (t) for all j, and Δi←i (t, xi ) = 0 for all xi . 2) RX node update: In the RX half of the round, each RX i first computes the interference means and variances  sii (t) and Qsii (t) in (14) and sij (t) and Qsij (t) in (19). Then, RX i computes the log likelihood function Δi→i (t, xi ) in (15) and sends it as a unicast message to its serving transmitter TX i. Also, for each strong edge (i, j), RX i computes the log likelihood functions Δi→j (t, xj ) in (20) and sends it as a unicast message to the interfering TX j. For the weak edges, RX i simply computes the sensitivity Di1 (t) in (27) and broadcasts it to all other transmitters TX j. The receiver also computes uij (t) in (25) and stores it for the next round. 3) TX node update: In the TX half of the round, each transmitter TX j would have received the log likelihoods

RANGAN and MADAN: BELIEF PROPAGATION METHODS FOR INTERCELL INTERFERENCE COORDINATION IN FEMTOCELL NETWORKS

Victim RX3

Serving TX1

Interference

Desired RX1

Signal

Soft RTS xˆ1(t ), Q1x (t ) Soft CTS D31(t ), D32 (t )

Soft CQI s (t ) sˆ11(t ), Q11

Interfering TX2

Interference

Soft RTS xˆ 2 (t ), Q 2x (t )

Soft CTS D11(t ), D12 (t )

Fig. 1. One round of the BP messages along the weak edges interpreted as a “soft” RTS / CTS mechanism.

Δi→j (t, xj ) from the receivers RX i, along the edges (i, j) that were strong. For any weak edge (i, j), TX j can approximately compute the log likelihood from the sensitivity Di1 (t) using (24). TX j can then compute the log likelihoods Δi←j (t + 1, xj ) from (22) for all i ∈ Ntx (j) and the log likelihood Δj (t + 1, xj ) in (23). TX j sends the receiver RX j that it is serving the entire log likelihood Δj←j (t + 1, xj ). For the receivers RX i such that (i, j) is a strong edge, TX j computes the i←j (t + 1) and variance Qxi←j (t + 1) from the mean x log likelihood Δi←j (t + 1, xj ) and sends it as a unicast message to RX i. Each TX j also computes the mean j (t + 1) and Qxj (t + 1) from the log and variance x likelihood Δj (t + 1, xj ) and broadcasts the quantities to the other receivers. Any receiver RX i that is along a weak edge (i, j) can then approximately compute i←j (t) from (31). x The round number is incremented, t = t + 1, and we return to Step 2 until a fixed number of rounds have been performed. 4) Final solution: After the final round, each transmitter TX j takes the scheduling vectors to be the vector xj that maximizes the log likelihood Δj (t, xj ). The message flow along the weak edges has an appealing interpretation. Consider Fig. 1 where a transmitter TX1 attempts to send data to a receiver RX1. The receiver RX1 experiences interference from a second transmitter TX2, while the transmitter TX1 causes interference onto a victim receiver RX3. Fig. 1 shows the messages along the weak edges in one round of the BP algorithm to coordinate the interference. The transmitters TX1 and TX2 will broadcast the mean and j (t) and Qxj (t) of their intended transmit vectors. variance x These transmissions can be interpreted as “soft” request to sends (RTS). They are soft since the intended transmit vectors are signaled by a distribution. Based on the transmit vector distribution from the interfering TX2, the receiver RX1 replies s11 (t) to the serving TX1 with an estimate of the interference  and Qs11 (t) which can be interpreted as a “soft” channel quality indication (CQI). As victim receivers, RX1 and RX3 also compute the sensitivities to the interference level by the derivatives Dir (t). These values can be interpreted as soft clear to send (CTS) indications, since instead of a binary go/no go type CTS, they signal a soft cost on changes in the interference from other transmitters.

637

TABLE I M ESSAGE SIZE AND COMPUTATIONAL COMPLEXITY PER ROUND OF PER TX OR RX. Method RX i Message size Exact O (|X ||Nrx (i)|) Gaussian Approx. O (|X ||Nrx (i)|) First Order O(1) Computational complexity ” “ Exact Gaussian Approx. First Order

O |X ||Nrx (i)| O (|Nrx (i)|) O (|Nrx (i)|)

TX i O (|X ||Ntx (i)|) O (|Ntx (i)|) O(1) O (|X ||Ntx (i)|) O (|X ||Ntx (i)|) O (|Ntx (i)|)

V. I MPLEMENTATION C ONSIDERATIONS Although the above BP-based ICIC algorithm is conceptually simple, its implementation in current cellular systems would require significant modifications to the standard and also place additional requirements on both the base stations and mobiles. To understand the scope of these changes, we briefly consider the necessary modifications for 3GPP LTE [54]: • Messaging: The proposed BP method requires one or two round of messaging between the base stations and mobiles before the scheduling decision in each subframe. The sizes of the messages per round is summarized in Table IV-C. BP with the linear approximation requires particularly small messaging: simply the broadcast of j (t) and Qxj (t) from each TX j and Di1 (t) and Di2 (t) x from each RX i. However, the current LTE standard does not have any channels to carry these messages. Following [34], one possible modification would be to reserve a period at the beginning of each subframe with multiple very short rounds of messaging between the mobiles and base stations before the data transmission. Since the propagation delays in femtocellular networks are likely to be small, it may be possible to make each messaging round short, thus enabling multiple rounds in one subframe. This messaging scheme will be assumed in the simulations below. However, to reduce the messaging overhead, one could also pipeline the messages over multiple subframes with the price of additional delay in coordination – this can also allow for the coordination between femto basestations to take place over a wireless backhaul. • Channel state information requirements: Along any weak edges between a TX j and RX i, both elements must know the interference matrix Aij . For the power control and subband interference coordination problems, the matrix represents a channel gain. In the beamforming problem, the matrix must also contain the complex phases. In the downlink, the mobiles can estimate such channel gains from the pilots of the base stations. However, for the base stations to obtain the channel gains, the mobiles would need to transmit the values in the uplink, possibly via some measurement report. Moreover, since both serving and interfering base stations need to know the channel gains, these gains would then need to be forwarded over the backhaul from the serving to interfering base stations. Signaling for such channel state information may be assisted by new reference signals (RS) developed for

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TABLE II S IMULATION PARAMETERS .

Carrier freq Bandwidth Wall loss Lognormal shadowing Path loss Femto BS TX power Femto UE noise figure

•

Value 3 × 3 apartment model, with active links in 5 of the 9 apartments. 2 GHz 5 MHz 0 or 10 dB 10 dB std. dev. 38.46+20 log10 (R)+0.7R dB, R distance in meters. 0 dBm 4 dB

0.9 0.8 Cumulative probability

Parameter Network topology

1

0.7 0.6 0.5 0.4 0.3 0.2

beamforming and cooperative multipoint [8]. Computational Complexity: The computational complexity per round of the BP algorithm is summarized in Table Table IV-C which shows the complexity as a function of the size of X , Nrx (i) and Ntx (i). While standard BP requires grows exponentially in the number of neighbor sets Nrx (i) and Ntx (i), both the Gaussian and linear approximations reduce the complexity significantly.

0.1 0 −1 10

1 0.9

The BP algorithm was simulated on a a simplified version of an industry standard model for LTE femtocell evaluation in [3]. The simulation parameters are shown in Table II. The network consists of a 3 x 3 grid of 10m x 10m apartments with active links in 5 of the 9 apartments. Each link consists of one femto BS transmitting to one femto mobile (called a UE, or user equipment, in 3GPP terminology). Due to restricted association, UEs connect to the femto BS in their apartment even if it is not the BS with the minimum path loss. As mentioned in the introduction, this scenario exposes many links to strong interference, and thus presents a good test scenario for advanced interference coordination algorithms. In the first simulation, we considered a time-varying simulation with a simple on-off model where, in each time slot, each link either transmits at the max power or is completely off. As described in Section III, for the time-varying problem, the utility maximization was rerun in each time slot with the weighted sum rate utility (4) with weights (5). We used the proportional fair utility Ui (R) = log(R). We generated independent flat fades on the links in each slot, and took a filter time constant in (6) was α = 0.1. For each random realization, or “drop”, of the femto network, we ran the simulation over 100 time slots and measured the average rate. The wall loss in the femto model was 10 dB. The top panel of Fig. 2 plots the cumulative distribution function (CDF) of the time-averaged rates for 5 links and 100 drops comparing various optimization methods for computing the maximum weighted matching optimization (2). The curve “reuse 1” is the case when all links transmit at max power. Two cases of BP are simulated: (i) 4 rounds of BP in each time slot using the Gaussian approximation in Section IV-B but no first order approximations; and (ii) using only two rounds of BP with both the Gaussian approximations and first order approximations on all interfering links less than 0 dB below the serving link. We see that even the linear approximated BP with 2 rounds does significantly better than reuse 1, and is not that far from the 4 round BP. The gap between BP and

Cumulative probability

0.8

VI. N UMERICAL S IMULATION

Reuse 1 BP, lin app (2 rounds) BP, no lin app (4 rounds) Optimal

0

10 Rate (Mbps)

1

10

Reuse 1 BP, lin app (2 rounds) BP, no lin app (4 rounds) Optimal

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −1 10

0

10 Geometric Mean Rate (Mbps)

1

10

Fig. 2. Downlink femtocell simulation with an on-off channel model and time-varying flat fading. The top panel shows the rate CDFs across all links and drops using various optimization algorithms for weight matching. The bottom panel is the CDF of system utility (represented as the geometric mean rate) across different drops.

reuse 1 particularly large at low rates. For example, for the 20% worst links, BP offers almost a factor of 5 improvement in rate over reuse 1. Also plotted is the rate CDF with optimal matching to maximize sum log utilities in each time slot based on an exhaustive centralized search. BP performs reasonably close to this curve, although there is still an obvious gap, again at low rates. The bottom panel of Fig. 2 shows the CDF of the total system utility over the 100 drops. Since the optimization used a log utility, the optimization is equivalent to maximizing the geometric mean rate over the 5 links. We see in this plot that the approximate BP algorithms are close to optimal and significantly better than reuse 1. As a second simulation, we considered the same problem but with a static single optimization and K = 4 subbands. Random fading was generated in each subband, so the simulation would be applicable in the case with frequency selective fading with coherence bandwidth roughly equal to the subband bandwidth. The optimization was over scheduling vectors

RANGAN and MADAN: BELIEF PROPAGATION METHODS FOR INTERCELL INTERFERENCE COORDINATION IN FEMTOCELL NETWORKS

1

1

Cumulative probability

0.8 0.7

Reuse 1 BP, lin app, 4 rounds, u=∞ BP, lin app, 4 rounds, u=1 Optimal

0.9 0.8 Cumulative probability

0.9

0.6 0.5 0.4 0.3

0.6 0.5 0.4 0.3 0.2

0.1

0.1 0

10 Rate (Mbps)

1

10

Fig. 3. Subband static optimization. Performance comparison for downlink femtocell rates with various optimizations for 4 independently faded subbands.

xj ∈ CK with each component being on or off, so there are 2K − 1 = 15 possible non-zero scheduling vectors in each link. Optimal subband allocation is a well-known challenging but important problem for OFDMA systems like LTE. Under these assumptions, Fig. 3 shows the CDF of the rates under various optimization methods. Similar to the dynamic single subband case, we see that BP provides significant gains over simple reuse 1, even when we use linear approximations and only four rounds. Also, for most links, BP achieves a rate reasonably close to the optimal subband allocation found by exhaustive search over all subband allocations over all rates. However, for the lowest rate links, there is a significant gap between BP and optimal. Thus, even though BP outperforms reuse 1 significantly in this regime, there is significant room for improvement. The BP method can also be applied to beamforming (BF) problems. As a simple simulation, we consider again the femtocell deployment with transmit beamforming with two antennas with half wavelength spacing spacing and one receive antenna. We neglect scattering so the channel appears as a linear phase across the two antennas on all links. For the transmit vectors xj , we optimize beamforming angles over 10 angles uniformly spaced between 0 and π. A performance comparison of the rate CDFs under various optimization algorithms is shown in Fig. 4. In this case, we took a wall loss of 0 dB. The curve labeled “opt serving link only” is the case when the BF vector is chosen to maximize the signal strength from the serving link only without regard to interference. This simple method provides the baseline. We see that BP provides some gains over serving link optimization. For example, the median rate with BP is approximately 50% higher than using optimal BF on the serving link only. Also, BP appears to be reasonably close to the optimal BF selection based on exhaustive search. VII. C ONCLUSIONS We formulated a general wireless scheduling and interference coordination problem as an optimization problem with linear mixing utilities. This was cast in a BP framework

Opt serving link only BP, lin app (2 rounds) Optimal

0.7

0.2

0 −1 10

639

0 −1 10

0

10 Rate (Mbps)

1

10

Fig. 4. Beamforming optimization. Downlink femtocell rates for various optimizations with 2 TX antenna beamforming. Channel models assume no scattering and beamforming optimization is performed over linear phase beamforming vectors.

where the goal is to compute the marginal distributions of a joint probability function. Using Gaussian and linear approximations, we obtained a distributed interference coordination algorithms with low overhead. The algorithm has a natural interpretation as a soft RTS/CTS scheme. Numerical simulations demonstrated that the resulting algorithm is close to an optimal scheme for dynamic and static sub-band optimization as well as for beamforming coordination across cells. Thus, while good heuristics can be handcrafted for these specific optimization problems, our approach is general and provides close-to-optimal solutions to a general class of scheduling and interference management problems in wireless networks. Moreover, the results show that the algorithm computes a good operating point in just two to four iterations making it very attractive to be used in practical wireless cellular systems. In the future, we plan to explore connections between our work and the AMP framework in [46], [48] to obtain performance bounds for large random networks. R EFERENCES [1] V. Chandrasekhar, J. G. Andrews, and A. Gatherer, “Femtocell networks: A survey,” IEEE Commun. Mag., vol. 46, no. 9, pp. 59–67, Sep. 2009. [2] D. L´opez-P´erez, 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. [3] FemtoForum, “Interference Management in OFDMA Femtocells,” Whitepaper available at www.femtoforum.org, Mar. 2010. [4] 3GPP, “New Work Item Proposal: Enhanced ICIC for non-CA based deployments of heterogeneous networks for LTE,” RP-100372, 2010. [5] I. F. Akyildiz, D. M. Gutierrez-Estevez, and E. C. Reyes, “The evolution to 4g cellular systems: Lte-advanced,” Physical Communication, vol. 3, no. 4, pp. 217 – 244, 2010. [6] G. Fodor, C. Koutsimanis, A. Rcz, N. Reider, A. Simonsson, and W. Mller, “Intercell interference coordination in OFDMA networks and in the 3GPP Long Term Evolution system,” Journal of Commun., 2009. [7] Ericsson and ST-Ericsson, “Details of almost blank subframes,” R1105335 (3GPP), 2010. [8] Huawei and HiSilicon, “Remaining details on CSI RS,” R1-105840 (3GPP), 2010. [9] S. B. Kang, Y. M. Seo, Y. K. Lee, M. Z. Chowdhury, W. S. Ko, S. W. C. M. N. Irlam, and Y. M. Jang, “Soft QoS-based CAC scheme for WCDMA femtocell networks,” Adv. Commun. Tech., 2008. [10] K. Sundaresan and S. Rangarajan, “Efficient resource management in OFDMA femto cells,” ACM MobiHoc, 2009.

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Sundeep Rangan (M’02) received the B.A.Sc. at the University of Waterloo, Canada and the M.Sc. and Ph.D. at the University of California, Berkeley, all in Electrical Engineering. He has held postdoctoral appointments at the University of Michigan, Ann Arbor and Bell Labs. In 2000, he co-founded (with four others) Flarion Technologies, a spin off of Bell Labs, that developed Flash OFDM, one of the first cellular OFDM data systems. Flarion grew to over 150 employees with trials worldwide. In 2006, Flarion was acquired by Qualcomm Technologies where Dr. Rangan was a Director of Engineering involved in OFDM infrastructure products. He joined the ECE department at Poly in 2010. His research interests are in wireless communications, signal processing, information theory and control theory. Ritesh Madan (S’00-M’06) received a Ph.D. in 2006 and a M.S. in 2003 from Stanford University, and a B.Tech from the Indian Institute of Technology (IIT) Bombay in 2001, all in Electrical Engineering. At Stanford, he was a recipient of the Sequoia Capital Stanford Graduate Fellowship. He is currently at Qualcomm New Jersey Research Center (NJRC). His research interests include methods for resource allocation in wireless networks, stochastic control, and optimization.