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Distributed Sum-Rate Maximization Over Finite Rate Coordination Links Affected by Random Failures Paolo Di Lorenzo, Student Member, IEEE, Sergio Barbarossa, Fellow, IEEE, and Marco Omilipo

Abstract—The deployment of small cell networks is expected to yield a significant improvement of coverage and spectral efficiency, provided that radio resource management is properly handled. In particular, in view of a potential massive deployment of femto access points (FAP’s), it is advisable to consider decentralized resource allocation mechanisms able to keep interference below a suitable threshold. However, purely decentralized schemes might well be spectrally inefficient. In this paper, we consider a local coordination scheme, where nearby FAP’s exploit the availability of a wired backhaul (typically an x-DSL line) to exchange control data to enable a local coordination with the aim of improving spectral efficiency. Since the backhaul is prone to random (unpredictable) delay and packet drop and the exchanged data are necessarily quantized, we propose a distributed resource allocation mechanism that limits the effects of the impairments coming from a realistic backhaul model. In particular, using results from stochastic approximation theory, we propose a distributed Robbins-Monro scheme with provable convergence properties. Numerical results are provided to validate the theoretical findings. Finally, in the case where the packet drop probabilities are known (estimated), we show how to counteract the effect of failures through a proper weighting of the messages exchanged among neighbor FAP’s. Index Terms—Dynamic radio access, femtocell networks, quantized data, random graph, stochastic approximation.

I. INTRODUCTION

I

T is widely recognized that the most significant improvement of cellular networks efficiency comes from the dense deployment of small cell networks (SCN), to enable a pervasive spatial reuse of frequency channels [1]. To make this possibility economically feasible, it is necessary to deploy low-cost, low-complexity and low-power base stations, able to cover cells with a radius in the order of tens of meters. Femtocell networks are just an example of SCN, which provide enhanced indoor coverage through the use of femto-access points (FAP’s) or Manuscript received May 04, 2012; revised August 31, 2012; accepted October 23, 2012. Date of publication November 27, 2012; date of current version January 14, 2013. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Prakash Ishwar. This work was supported by FREEDOM Project, Nr. ICT-248891. Part of this work was presented at the Fourth International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), San Juan, Puerto Rico, December 13–16, 2011. P. Di Lorenzo and S. Barbarossa are with the Department of Information, Electronics, and Telecommunications (DIET), Sapienza University of Rome, 00184 Rome, Italy (e-mail: [email protected]; sergio@infocom. uniroma1.it). M. Omilipo is with the Page Europa s.r.l., 00144 Rome, Italy (e-mail: marco. [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/TSP.2012.2230170

Home eNode B (HeNB) [1]–[3]. These are low power base stations, typically user-installed, connected to the cellular operators network via a digital subscriber line (DSL) or cable modem. Femtocell networks are fully compliant with the cellular standard. This facilitates hand-over and it guarantees a Quality of Service (QoS) not achievable with common Wi-Fi. In a femtocell network, the femto mobile user communicates with a FAP via a wireless link, while the FAP forwards traffic to the macro base stations through a wired line. Unlike macro networks, femtocells are typically installed by subscribers and they are deployed and maintained without global planning. Hence, a potential massive deployment of FAP’s might induce an intolerable interference from femto to macro users, as well as from femto to femto users. Interference management is then arguably the major challenge to be faced from femtocell networks. A possible scenario for femtocell networks is depicted in Fig. 1. In this example, femto-user equipments (FUE’s) are directly connected through a wireless channel (solid link) to a FAP and perceive interference (dashed link) from other FAP’s and (dotted link) from the macro base station. Due to the limited transmit power and the attenuation resulting from wall penetration, femto mobile users are generally within the coverage radius of a reduced number of FAP’s so that only nearby FAP’s interfere with each other. In this context and in view of a potentially massive deployment of FAP’s, it is particularly important to devise decentralized mechanisms to adapt resource allocation dynamically in order to limit interference and get the advantages offered by the capillary deployment of FAP’s. A useful tool to devise innovative decentralized resource allocation strategies is game theory. Game theoretic approaches have been proposed for the multicarrier interference channel [4]–[9] and, lately, for cognitive radios [10]. These works focused on a competitive approach where every user is interested in maximizing his own transmission rate, under power budget constraints, or in minimizing the transmission power necessary to guarantee a desired rate [12]. Particular attention has been devoted to provide conditions for the existence and uniqueness of a Nash equilibrium (NE) and several algorithms have been suggested to reach a NE. Game theoretic approaches have already been proposed for femtocell networks as well [13]. In [14] it has been proposed an optimal joint time-frequency power allocation for femtocell networks, where the interference activity of primary users (Macro base stations) has been modeled using a statistical Markov model. However, because of the competitive nature of the underlying game, a NE might be highly inefficient. To improve the efficiency of a NE, it is useful to introduce some kind of coordination among FAP’s through a limited exchange of information among players. A possible way to move out of inefficient NE’s is to use repeated games [16], which enforce the Pareto optimality of the equi-

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librium. An alternative strategy to improve efficiency is to incorporate a pricing mechanism [17]–[19]. Pricing amounts to incorporate, in each players strategy, some kind of care about a socially meaningful performance parameter, rather than being purely selfish. In [17], [18], for instance, it has been shown that, if instead of maximizing his own rate, a user maximizes the sum of all the rates, this optimization can be still achieved in a decentralized manner, provided that the users exchange the so called interference prices, i.e., parameters measuring the impact that each user’s action, essentially his power allocation, has on the other users rates. Both approaches require some exchange of information among the players. Differently from cognitive radios, where this exchange of information is not possible, in femtocell networks a local coordination among FAP’s is possible through the backhaul link, as proposed in [13]. In most previous works, see e.g., [4]–[19], each user is supposed to interfere with all other users. In practice, however, given the limited transmit power and the attenuation resulting from wall penetration, only nearby FAP’s interfere with each other. To capture the interference mechanism among FAP’s, we introduce the so called interference graph, i.e., the graph whose vertices are the FAP’s and where there is an edge between two vertices only if the relative FAP’s interfere with each other. Given the limited transmit power, the interference graph is typically a sparse graph. Then, in this work, we consider a distributed pricing mechanism where each FAP needs to exchange control data (prices) only with few neighbors, according to an interference graph. A critical point in this exchange of data is that the backhaul link connecting the FAP’s is typically an Internet link, operating under a best-effort protocol. Hence, control packets sent through the backhaul might experience large delays, because of retransmissions of packets corrupted by errors. This random delay and the associated delay jitter could jeopardize the potential benefits of coordination. It is then of interest to examine a protocol that simply discards packets that are not received within a maximum delay. In such a case, some packets (prices) are randomly dropped. It is then of interest to investigate the effect of random packet drop on the final performance and to devise appropriate algorithms to counteract the effect of packet drops. We analyze this problem by modeling the graph describing the interaction among FAP’s as a random graph, where each link is on with a probability equal to the occurrence that the packet is correctly delivered within the maximum delay. Furthermore, we take into account the constraint that the price coefficients need to be quantized before transmission over the finite rate backhaul link. The quantization error introduces a further source of randomness in the data exchanged among FAP’s. These sources of randomness, e.g., delay and quantization, introduce stochastic noise in the pricing mechanism that needs to be properly handled to ensure convergence of the distributed algorithm. The effect of random links and quantization noise has been deeply analyzed, for example, in consensus protocols [25] and swarming models for decentralized radio access [27]. Our aim, in this paper, is to study the effect of random packet drops and quantization on distributed pricing mechanisms. We propose a projection-based Robbins-Monro (RM) stochastic approximation scheme [20] and we prove the almost sure convergence of such procedure to a local solution dependent on the mean graph of the network, even in the presence of such imperfect communication scenario. The use of sto-

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chastic approximation methods has a long tradition in stochastic optimization for both differentiable and nondifferentiable problems, starting with the work of Robbins and Monro [20] for differentiable problems and Ermoliev [21], [22] for nondifferentiable problems. Other works on stochastic approximation techniques include also [23], [24]. Recently, stochastic approaches for the distributed computation of equilibria, arising in monotone stochastic Nash games over continuous strategy sets, have also been proposed in [29], [30], where the authors considered the application of regularized projection-based stochastic approximation schemes. The main contributions of this paper are the following: 1) we introduce the network interference graph in the study of distributed pricing mechanism applied to the dynamic resource allocation problem in femtocell networks; 2) we study the effect of random link failures and quantization noise in the exchange of prices among FAP’s; 3) we propose a distributed resource allocation algorithm and we prove its convergence properties. The paper is organized as follows. In Section II, we introduce the system model, the random link failures model and the assumptions on the quantization mapping that will be used throughout the paper. Then, we formulate the optimization problem as the distributed maximization of the network sum-rate, considering the presence of such stochastic disturbances. In Section III, we describe the proposed distributed simultaneous RM stochastic approximation method and provide the supporting convergence result, exploiting known results from supermartingale theory. Section VI shows some simulation results aimed to confirm our theoretical results and to quantify the effect of random link failures on the network sum-rate. In particular, simulation results show how, reducing the probability to establish a communication link among FAP’s, the system performance decreases due to the lower coordination to mitigate interference. Then, we show how to counteract the effect of random link failures through a proper weighting of the price coefficients coming from the neighbors. Finally, Section VIII draws some conclusions. II. SYSTEM MODEL AND PROBLEM FORMULATION In this section we introduce the system model and formulate the optimization problem as the distributed maximization of the network sum-rate, where the users cooperate over a realistic channel affected by random packet drop and quantization. A. System Model We consider the downlink channel with FAP’s transmitting to as many FUE’s over the same set of physical resources, e.g., time and bandwidth. Each FAP is serving its own FUE and, in the lack of a centralized control, each FUE might receive interference signal from nearby FAP’s and Macro Base Stations (MBS). We will refer to each transmitter-receiver pair as a “user”. No multiplexing strategy is imposed a priori so that, in principle, each user interferes with each other. In practice, however, given the limited transmit power and the attenuation resulting from wall penetration, only nearby FAP’s interfere with each other. To study the interference mechanism, it is useful to introduce what we call the interference graph , defined as the directed graph whose vertices are FAP’s. Furthermore, there is a directed edge from node to node if the FUE connected to FAP receives an interference from FAP . This

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Fig. 1. Femtocell scenario.

condition holds true if the FUE connected to FAP is far from FAP less than an interference coverage radius, which is dictated by the FAP’s transmit power and the fading channel between FUE and FAP . We denote by the set of incoming neighbors of user , i.e., the set of FAP’s interfering at the -th FUE. Similarly, we denote by the set of outgoing neighbors of user , i.e., the set of FUE’s which the -th FAP interferes with. In the lack of any source of randomness, which we will incorporate later on, the interference graph is described by its adjacency matrix , whose entries are defined as follows otherwise.

(1) It is worth emphasizing that the interference graph depends on the position of the FUE’s. Hence, it is a dynamic graph and it can change through time because of the FUE mobility. As an example, the adjacency matrix of the interference graph of the femtocell scenario illustrated in Fig. 1 is given by:

(2)

where the links in Fig. 1 corresponding to a “1” entry in the adjacency matrix in (2) are the FAP-FUE interference links, which are depicted using dashed lines. As we can notice, matrix (2) is

very sparse, having only a few non-zero elements. Based on the adjacency matrix, the in-degree and out-degree of node are and , respectively. The vector of data received at the -th FUE has the following structure (3) where

is the vector transmitted by source , is the direct channel of link , is the cross-channel matrix between source and destination , is the interference coming from the MBS’s, is a zero-mean circularly symmetric complex white Gaussian noise vector with covariance matrix , and is the number of subcarriers. The second term on the right-hand side of (3) represents the Multi-User Interference (MUI) received by the -th FUE and caused by neighbor FAP’s. We focus on transmission schemes where no interference cancelation is performed and multiuser interference is treated as additive colored noise from each receiver. Consequently, encoding/decoding on each link is performed independently of the other links. Each channel is modeled as an FIR filter of maximum order , which is assumed to change sufficiently slowly to be considered constant during each transmitted block. A cyclic prefix (CP) of length is incorporated in each transmitted block , as in standard OFDM, where is the maximum channel order. , but it facilitates This entails a rate loss by a factor symbol recovery. For practical systems, is sufficiently larger than , so that the loss due to the CP insertion is negligible. Thanks to the CP insertion, the channel matrix , resulting

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after having discarded the guard interval at the receiver, is a Toeplitz circulant matrix. Thus, the matrix can be diagonalized as , with denoting the normalized inverse fast Fourier transform (IFFT) matrix, i.e., , and trix , where

is a

diagonal ma-

is the frequency response of

the channel between source and destination , including the path loss with exponent and normalized fading coeffi, with denoting the distance between source cient and receiver . Perfect channel state information is assumed; each receiver is also assumed to measure with no errors the covariance matrix of the noise plus macro interference and MUI generated by the neighbor nodes. We consider the following physical constraints required by the application: 1) For each transmitter , the total average transmit power is (4) denotes the trace operator, where is the covariance matrix of the transmitted vector , and is the maximum average transmitted power in units of energy per transmission; 2) Radio spectrum regulations introduce mask constraints, i.e, (5) , where denotes the m-th component of the absolute value of vector , and represents the maximum power that user is allowed to allocate on the -th frequency bin. Constraints in (5) attempt to limit the amounts of interference generated by each transmitter over some specified bands. We introduce the notation , where denotes the power vector of user , whose element is the power transmitted by node over the -th subcarrier. Under the previous assumptions, the rate of user is given by (6)

is the channel transfer function of the -th subwhere channel between the -th transmitter and the -th receiver, is the variance (power) received on the -th subchannel including receiver noise and power coming from the MBS, and is the set of FAP’s interfering at the -th receiving FUE. B. Problem Formulation In previous works, see, e.g., [6]–[13], the authors considered the maximization of the rate of each user assuming a purely competitive game strategy. In such a case, every player competes with all the others for the available resources, and the optimality criterion consists in reaching a Nash equilibrium (NE). However, a NE can be inefficient, depending on channels, interference levels, initialization, and so on. It is then of interest to look at mechanisms to improve upon the potential inefficiency of NE by allowing some exchange of information among the

players (FAP’s). In femtocell networks, this exchange is possible through the wired backhaul. Typically, this backhaul operates under a best-effort mechanism and then it has limited quality, but nonetheless it is of interest to look at what advantages we can achieve by incorporating some kind of interaction among FAP’s, even if only through a limited quality backhaul. Within this wider perspective, rather than maximizing each rate separately, our goal is to maximize a network utility function given by the sum rate of the users under power constraints, i.e.,

(7) where

and

, with and denoting, respectively, the power budget of user in (4) and the mask constraint that limits the maximum transmit power over each channel in (5). Before proceeding, it is worth emphasizing that the objective function in (7) is not concave in the power allocation and, as a consequence, the problem may have multiple local optima. Any local optimum of problem (7) is a regular point1 and, as a consequence, it satisfies the Karush-Kuhn-Tucker (KKT) conditions [35]. The solution of (7) requires in general a centralized approach. Nevertheless, we will show next that the solution can also be reached by allowing a very limited exchange of control data among the users. In [17], [18], the sum-rate maximization problem in (7) was recast in a distributed manner using a game theoretic approach, but modifying the utility function of each player by incorporating a social utility term. This requires the players to exchange information among each other, in the form of interference prices. The approach followed in [17] considered a fully interfering network and perfect communications among secondary users. Furthermore, the method assumed that the prices are constant. The major extension of our work with respect to [17]–[19] is that in our case the exchange of prices occurs through a random graph, modeling the realistic situation where the packets carrying the price coefficients sometimes are randomly dropped and the prices are quantized before transmission. To handle this kind of problem, we propose a distributed stochastic gradient projection approach that will be detailed in the next section. Before proceeding with the proposed optimization algorithm, we clarify the effect of the random interference graph topology on the solution of (7). We basically follow a gradient approach so that every user modifies its own power vector moving along the gradient ascent curve in order to maximize the sum-rate. Introducing the sum-rate , the partial derivacan be written as tive of the sum-rate with respect to (8) where (9) 1A feasible point is said to be regular if the equality constraints gradients and the active inequality gradients are linearly independent [35].

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(10)

work model, where the link failures are independent both over space and time [26]. Using the interference graph, the gradient computed by node at time is then

where and is a logical function is true, or zero, otherwise, and denotes equal to one, if the set of FUE’s which FAP interferes with. As in [17], [18], we introduce the price coefficients as

(11) is always nonnegative and is proportional to The price the marginal decrease of user ’s rate because of an increase of the -th node’s transmit power, as:

(14) where the prices vary dynamically with the iteration index, according to (11), as we let the interference level and the power vector vary as a function of in (11). Hence, to evaluate the gradient (14), user needs to know the price vectors , and the cross channels , to be communicated by its neighbors. This is the only exchange of data that must occur over the backhaul to enable every user to maximize the sum-rate rather than the individual rate. C. Finite Rate Coordination

(12) For our ensuing derivations, it is useful to express the sumrate gradient in terms of the adjacency matrix of the interference graph introduced in (1). In practice, at each iteration of the gradient approach, the prices exchanged among FAP’s travel along a best effort link, whose delay is sometimes unpredictable because of the unknown traffic over the Internet link. Some packets may be received with errors and in such a case a retransmission is required. This adds a further source of delay. But resource allocation cannot take too long or otherwise it would make the overall rate optimization useless. It is then of interest to consider a simple coordination protocol such that if the packets with prices are not received within a maximum delay, they are simply dropped. This random dropping can be incorporated in our scheme by making the unit coefficients of the adjacency matrix introduced in (1) to become binary random variables that, at each iteration, are equal to one if the packet from FAP to FAP is received on time or equal to zero otherwise. With such a model, the adjacency matrix becomes a dynamic random matrix, whose entries change from one iteration to the next, depending on the probability that packets are delivered on time. For our purposes, it is useful to express the adjacency matrix at iteration as (13) denotes the adjacency matrix of the mean where graph and is a zero-mean sequence of statistically inde. More specifically, pendent matrices, with entries the entry of the expected adjacency matrix represents the probability with which FAP successfully communicates with FAP . This model is fairly general and it can incorporate different sources of randomness. Although the link failures, and so the adjacency matrices, are independent at different times, during the same iteration, the link failures can be spatially dependent, i.e., correlated. This model subsumes the erasure net-

A further impairment affecting the coordination among FAP’s is that the control data (prices) must travel over a finite rate (backhaul) link. This implies that these control data must be quantized with a finite number of bits. We assume that each inter-node communication channel uses a uniform quantizer, which is defined by the following vector mapping, , (15) where the entries of the vector and the error satisfies

are the quantized values

(16) with denoting an -dimensional column vector composed of all ones. The quantization alphabet is (17) Conditioned on the input, the quantization error is deterministic. This strong correlation of the error with the input influences the statistical properties of the error and can influence the convergence of the algorithm. To avoid these effects, we consider dithered quantization [31], [32], which endows the quantization error with some useful statistical properties. The dither added to randomize the quantization effects satisfies a special condition, namely the Schuchman conditions, as in subtractively dithered systems, see [33]. Then, at every time instant , adding to each component a dither sequence of i.i.d. uniformly distributed random variables on independent of the input sequence, the resultant error sequence becomes (18) is now an i.i.d. sequence of uniformly The sequence distributed random variables on , which is independent of the input sequence. Thus, by properly randomizing the

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input to the uniform quantizer, the quantization error is independent of the input and uniformly distributed on . Taking into account all the sources of randomness, expression (14), evaluated at , takes the form

where

and

is the vector collecting the interference prices of the entire network. The matrix , where denotes the Hadamard product, is a multidimensional weighted adjacency matrix, where .. .

(19) , , where and are contributions of dithered quantization noise that the -th FAP receives from the -th FAP on the -th subchannel, at time . To rewrite (19) in compact form, we introduce the and random vectors , having the -th element of the -th vector component given by

(21) is the maximum gain of the where cross-channels across the network. From (21), we get

(22) It is now useful to express the previous quantities in vector form. To this end, we introduce the -dimensional graph adjacency defined by (23) where denotes the Kronecker product and is the -dimensional identity matrix. Using this notation and exploiting expression (14), the gradient of the sum-rate can be written in compact vector form as .. .

(24)

(25)

reprewith senting diagonal matrices that contain the cross channels between users and . Expression (24) shows the dependence of the sum-rate gradient with respect to the network graph and the interference prices, in the case of an ideal communication scenario. Then, using (19), the sum-rate gradient with respect to the power allocation in the presence of random link failures and dithered quantization can be written in compact form as (26) Now, expanding the multidimensional weighted adjacency matrix as in (13), expression (26) is given by the sum of a plus a random function deterministic function :

(20) The vectors and are aggregated contributions of quantization noise and dithering. Furthermore, from the conditions on the dither, it follows that

.. .

(27) where (28) (29) To solve the problem in (7), in the following section we introduce a projection-based stochastic approximation algorithm, where every user modifies its own power vector moving along the gradient ascent curve in order to maximize the sum-rate. Due to the presence of additive randomness on the gradient expression (26), the problem in (7) can be solved using a stochastic approximation approach. In particular, the problem can be converted into the search, inside the feasible set , for the (the local maxima of zeros of a deterministic function the mean sum-rate gradient) in (28) whose value, measurable at each time instant , is corrupted by the additive random disin (29). turbance III. DISTRIBUTED STOCHASTIC PRICING ALGORITHM In the previous section, we showed how the gradient of the sum-rate is affected by the randomness introduced by the link failures and by the quantization error present on the data exchange between FAP’s. To find a solution to problem (7) in the presence of random disturbances, we propose an algorithm based on stochastic approximation. In the remainder of this section we introduce a stochastic approximation scheme for solving the problem in (7) in a distributed manner, providing the supporting convergence results. The problem is amenable for distributed solutions because the optimization set is given by Cartesian product of sets , allowing the parallel

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1) Persistence: To obtain convergence, we assume that the step size satisfies the following conditions:

TABLE I DISTRIBUTED STOCHASTIC PRICING ALGORITHM (DSPA)

(34) Condition (34) ensures that the step-size decays to zero, but not too fast. It is standard in stochastic adaptive signal processing and control; it is also used in consensus with noisy communications in [25]. An example of step-size sequence that satisfies (34) is (35) 2) Independence: The stochastic sequences , and are mutually independent. Before providing our main theorem on the convergence of the DSPA, we also need to introduce two further lemmas. Lemma 2: (Descent Lemma [36]): If is continuously differentiable and its gradient is Lipschitz continuous with constant , i.e., (36) then, implementation of the algorithm. Since the exchange of interference prices needs some kind of synchronization among FAP’s, we consider a possible solution implementing a simultaneous update of the FAP’s power profiles. Let us now focus on a simultaneous implementation of the distributed stochastic pricing algorithm (DSPA), where every FAP updates its power profile at the same time, as described in Table I. The structure of the problem allows also an asynchronous implementation, which takes in consideration the possible lack of synchronization among FAP’s, for achieving distributed minimization. The only difference between the distributed simultaneous solution and the asynchronous one is that in the second step of the DSPA, at each time , a subset of FAP’s is randomly selected to update its power profile, given the neighbors’ power profiles and price vectors, according to (31). In the following, we will study the stochastic convergence of both simultaneous and asynchronous implementation of the DSPA. Throughout the paper, we denote by the filtration and the stoof the -algebra generated by the initial point , and for , i.e., chastic errors (32) To prove the convergence of DSPA, we use a known result from supermartingale theory, which we recall here below for convenience. Lemma 1: Let , , be three random sequences is non-negative for all . If almost surely such that , and (33) or else then, almost surely, either verges to a finite value and . Proof: The proof can be found in [23], Lemma 1. Furthermore, we consider two assumptions:

con-

(37) be a sequence of random variables Lemma 3: Let being measurable, and suppose that with each and , where is some deterministic constant. Then, the sequences (38) converge to finite limits (with probability 1), if the step size satisfies conditions (34). is a martingale whose variProof: Since ance is bounded by , it must have a finite limit by the martingale convergence theorem. Furthermore, (39) converges to a finite limit which shows that with probability 1. We are now able to state and prove our main result. Theorem 1: Let be the sequence generated by the DSPA in (31), or by its asynchronous implementation, with stepsize satisfying the conditions in (34). Then, the sum rate sequence converges almost surely to a finite value , i.e., (40) where denotes the probability of the event . Furthermore, let be an accumulation point of the sequence , almost surely the optimal local solution is a fixed point of the mapping , such that .

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Proof: To prove the convergence of the sum-rate , we exploit Lemma 2. We will first consider the case of a simultaneous update of the algorithm, extending then the proof to the asynchronous case. 1) Simultaneous Implementation: In the -th iteration of the algorithm, each user simultaneously updates its power profile. One sufficient condition for Lipschitz continuity is that the -norm of the Hessian matrix of is bounded, in which case this bound can be used for the Lipschitz constant . It can be shown this is true for and, specifically, there exists a positive constant which upper bounds the -norm of the . Hessian matrix of Applying the descent Lemma to , we get

(41) Now, we recall some basic properties of projection mappings from [36]. Let be a nonempty, closed and convex set, it holds that: 1) For every , there exists a unique that minimizes over all , and will be denoted as ; , a vector is equal to if 2) Given some and only if for all ; 3) The mapping defined by is continuous and nonexpansive, that is, for all . Applying property 2 of the projection mapping and using the expression in (31), we have

Now, using the non-expansivity of the projection operator and substituting the expression in (27), we get

(46) Since is the mean gradient of with respect to , evaluated for the expected graph , and each user’s total power can be upper bounded by a positive conis bounded, stant . Now, considering the expression of in (29) and exploiting assumption 2, we have

(47) take values from a bounded set, Since the matrices resorting to the Gershgorin circle theorem [38], the maximum eigenvalue of can be upper bounded by a positive constant . Then, according to the previous assertions and exploiting (22), we can upper bound (47) as (48) Now, recalling the expression of the interference prices, we have

(42) (49)

which yields

(43) Substituting the last expression in (41), we get

(44) In the notation of Lemma 1, expression (44) can be recast as in (33), where

By choosing a positive step-size , the sequence is nonnegative for all . Moreover, exploiting the expression in (31) , we have and the fact that

(45)

, , where is the signal to noise plus interference ratio on the -th subcarrier of the -th FAP. It is then easy to check that, for every power allocation vector , it holds the upper bound , , , with denoting the minwhere imum noise variance across the network. As a consequence of the previous bound, we have . Substituting this last bound in (48), we get . Now, resorting to the bound in (45) on the sequence and on , we have

(50) Exploiting the conditions in (34), the first term of the right hand side of (50) has a finite limit. Moreover, due to the fact that and , lemma 3 applies and also the second term of the right hand side of (50) has a finite limit. Then, we conclude that the series , having a finite limit with probability 1.

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All the conditions of Lemma 1 are then satisfied and the result applies. Hence, almost surely, either or else converges to a finite value and . Since we are optimizing over a compact set, the sequence cannot diverge to and it has to converge almost surely to a finite value. Furthermore, almost surely it must be that

(51)

and it has to converge almost surely to a finite value. Furthermore, almost surely it must be that

(55) satisfies (34), the series However, if the step size is divergent. Then, since , it follows that (56)

where . However, if the step size satisfies (34), the sequence is divergent. Then, since , it must follow that (52) is an accumulation point of the sequence, we can Thus, if almost surely guarantee subsequence convergence such that . This concludes the proof of convergence of the simultaneous algorithm. 2) Asynchronous Implementation: The convergence of the asynchronous algorithm can be proved following the same guidelines of the previous proof. In the -th iteration of the algorithm, a subset of users is chosen at random to update its power profile. Thus, in this case, the filtration of the -algebra generated by the initial point and the stochastic errors , and , keeps into account the fact that at each time only a subset of users updates its power allocation. The total utility can be considered as a because only the power profile function of of users belonging to the subset is updated. The sum-rate function is Lipschitz continuous and, specifically, there exists a positive constant that upper bounds the -norm of the Hessian matrix of , , independently of other’s power profile. Applying the descent Lemma to , we get

(53) In the notation of Lemma 1, expression (53) can be recast as in (33), where

(54) By choosing a positive step size , the sequence is nonnegative for all . Furthermore, by using similar arguments as before, we have that has a finite limit with probability 1. Then, the result of Lemma 1 applies, hence, almost surely, either or else converges to a finite value and . Since we are optimizing over cannot diverge to a compact set, the sequence

where . Then, if each FAP is activated infinitely often (i.e., there is no finite time after which , letting be some user is not updated anymore) as an accumulation point of the sequence, we can almost surely guarantee subsequence convergence such that , thus concluding the proof of convergence of the asynchronous algorithm. Remark: The effect of the diminishing step-size in (34) is to drive to zero the variance of the additive noise term in (29). Then, the final power allocation at convergence is determined only by the mean part of the gradient in (28), which depends on the expected graph . In the next section, numerical results will show the effect of the graph randomness on the resource allocation performance. IV. SIMULATION RESULTS Numerical Example 1—Effect of the interference graph on system performance: The performance of femtocell networks is strongly dependent on how FAP’s interfere with each other. Throughout the paper, this interaction has been described using the concept of interference graph, i.e., the graph whose vertices are FAP’s and where there is an edge between two FAP’s if one FAP generates an undue interference toward the FUE served by the other FAP. Usually, in a femtocell scenario, due to the low transmit power of FAP’s and the strong attenuation due to wall penetration, the interference graph is typically a sparse graph. In this first example, we show the effect of the interference graph’s connectivity on the network performance, considering ideal communications of interference prices among FAP’s. We assume the presence of 30 FAP’s, each one interfering with a subset of neighbors . The physical layer is OFDMA, with subchannels and a channel order . We assume for simplicity that each FAP covers a disk with a common radius and serves a single FUE. If a FUE falls within the coverage areas of nearby FAP’s, it receives interference from them. The FAP’s are uniformly deployed over an area of . In Fig. 2 we report the average behavior of the network sum-rate versus the coverage radius of each FAP, averaged over 500 independent FAP’s deployments, considering the values achieved at convergence of our iterative algorithm. The situation analyzed in Fig. 2 assumes ideal communications among FAP’s, so that there are neither dropped packets nor quantization errors. In Fig. 2, we compare the sum-rates achieved with or without coordination, i.e., with or without the exchange of prices. In the first case, each FAP aims to maximize the sum-rate, whereas in the second case each FAP can only maximize its own rate. From Fig. 2 we notice

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Fig. 2. Sum-rate versus FAP’s covering radius, for the cooperative and noncooperative case.

Fig. 3. Network interference graph. Undirected edges (2-way) are plotted using solid lines while directed edges (1-way) are depicted using clock-wise dashed lines.

that, as the coverage radius of each FAP increases the potential interference increases, as more FAP’s can interfere at each FUE. As a consequence, the sum rate decreases as increases. At high values of , the sum-rate curve tends to become flat because the interference graph becomes fully connected. It is evident from the simulation how a cooperative approach among FAP’s provides a rate gain with respect to the non cooperative case, in both cases of low interference scenarios (at low values of ) and of high interference scenarios (at high values of ). In particular, we can notice from Fig. 2 how the pricing-based method yields a larger gain in the case of high interference scenario, thanks to the cooperation among FAP’s. Numerical Example 2—Effect of random links on system performance: In this second example, we assume the presence of 20 FAP’s, interfering according to a sparse graph. The directed interference graph is depicted in Fig. 3, where undirected edges (2-way) are plotted using solid lines while directed edges (1-way) are depicted using clock-wise dashed lines. Because of the randomness introduced by the backhaul link, a packet has a certain probability to be delivered correctly on time. The values to be exchanged are also affected by dithered quantization noise, supposing the presence of a 6

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Fig. 4. Stochastic simultaneous algorithm: Sum rate versus iteration index, for different probabilities of link failures.

bit encoder. We consider a number of subchannels and a channel order . In Fig. 4 we show a numerical example of sum-rate behavior as a function of the iteration index, achieved by the simultaneous DSPA in (31), considering different values of probability . The ideal case corresponds to and it is shown as a benchmark. We also report the , corresponding to the behavior of the classical case (non-cooperative) iterative water filling algorithm (IWFA) without pricing. Furthermore, assuming two intermediate values and , we compare the ideal behaviors obtained using the mean graphs associated to these probability values, but without any random disturbance, and the average behaviors, averaged over 500 independent realizations, given by the simultaneous DSPA in (31) in the presence of random link failures and quantization noise. The total average transmit power is for all FAP’s. The step size of the deterministic algorithms is fixed and has been chosen to guarantee the convergence to a local minimum. Instead, the step size of the stochastic algorithm has been chosen as , with and , in order to satisfy (34), and its effect is to reduce the convergence speed of the algorithm with respect to the correspondent ideal case. It is interesting to notice that the stochastic algorithm converges to the same value of the correspondent ideal case evaluated for the expected graph. As expected, the link failures determine a performance loss due to the degraded coordination among FAP’s. However, even if we consider a low probability to establish a communication link among FAP’s, we still get a rate gain with respect to the IWFA case without pricing. A second example is given in Fig. 5, where we show the sum-rate behavior as a function of the iteration index, achieved by the asynchronous implementation of the DSPA, considering different values of probability . As we can notice, the results achieved for the simultaneous case hold also for the asynchronous stochastic algorithm. However, as expected, the asynchronous algorithm needs more time to converge with respect to the simultaneous update. Numerical Example 3—Compensation of the random link failures: As shown in the previous example, the effect of link failures is to reduce the system performance due to the degraded coordination among FAP’s to mitigate the interference.

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Fig. 5. Stochastic asynchronous algorithm: Sum rate versus iteration index, for different probabilities of link failures.

Fig. 6. Compensated stochastic simultaneous algorithm: Sum rate versus iteration index, for different probabilities of link failures.

However, from the theoretical results, validated in the previous numerical example, we know that the final convergence value of the stochastic algorithm depends on the expected graph of the network. Hence, if a FAP, let us say , knows, through preliminary estimation, the probability with which it , it is possible to communicates with its neighbor , counteract the effect of the graph randomness by weighting the price coefficient coming from the link with the inverse of . In formulas, this means that the sum-rate the probability gradient, affected by random links and dithered quantization, is compensated by each FAP as:

(57) , . Since , the effect of the compensation is to normalize the mean network graph in order to be coincident with the ideal graph in the case of absence of failures. The compensation increases the variance of the stochastic noise term. Nevertheless, since the noise remains zero-mean with bounded variance, the theoretical results apply also to the compensated algorithm that almost surely converges to a local optimum of the original problem (7). To check the validity of these arguments, in this last example, we consider the same network and parameters used in the previous simulation. In Fig. 6 we show the behavior of the sum rate as a function of the iteration index. In particular, we report the ideal case correspondent to , as a benchmark, compared to the average behaviors, averaged over 500 independent realizations, given by the simultaneous DSPA updated using (57), in the presence of quantization noise and for different probabilities to establish a communication link. The stochastic algorithm compensates for the effect of the link failures by weighting the interference prices for the inverse of the probability over each link. As we can notice, thanks to the compensation, the final equilibrium value of the stochastic algorithm coincides with the ideal case for every value of probability . However, reducing the probability to establish a communication link, the network

Fig. 7. Compensated stochastic asynchronous algorithm: Sum rate versus iteration index, for different probabilities of link failures.

requires more time to reach the final equilibrium state. A second example is given in Fig. 7, where we show the sum-rate behavior as a function of the iteration index, achieved by the DSPA compensated by (57) with asynchronous update of the users’ power profiles, considering different values of probability . Also in this case, the final convergence value of the stochastic algorithm coincides with the ideal case one. V. CONCLUSIONS In this paper we have proposed and analyzed a distributed stochastic projection-based algorithm aimed at maximizing the sum-rate of OFDMA based femtocell networks where local coordination among FAP’s occurs through a backhaul link affected by random packet drops and quantization errors. To handle the randomness introduced by the signaling channel, we have proposed a projection-based RM stochastic approximation scheme that converges almost surely to a final value dependent on the expected value of the interference graph. Numerical results show that even if the probability of delivering the control data (prices) across FAP’s is low, the method still gets some gain with respect to the uncoordinated case (e.g., the conventional iterative water-filling). More specifically, we have shown

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how, reducing the probability to establish a coordination link among FAP’s, the sum-rate of the network decreases due to the degraded coordination among FAP’s. An interesting by-product of our theoretical derivations is that the final convergence value of the stochastic algorithm coincides with the value achieved by the correspondent deterministic algorithm evaluated for the expected value of the interference graph. Hence, assuming to know the success probability on each link, we have shown how to counteract the effect of random link failures through a proper weighting of the price coefficients. The proposed resource allocation algorithm is then robust against the impairments of the backhaul link used to establish a coordination among FAPs, whose effect is only to slow down the convergence process. Even if the paper has been written using the terminology of femtocell networks, the proposed procedure can be used in all small cell or ad hoc networks as a way to improve the performance of decentralized radio resource allocation techniques whenever there is a signaling, albeit nonideal, channel that allows the exchange of messages among radio nodes competing over the same resources. REFERENCES [1] J. Hoydis, M. Kobayashi, and M. Debbah, “A cost- and energy-efficient way of meeting the future traffic demands,” IEEE Veh. Technol. Mag., vol. 6, pp. 37–43, Mar. 2011. [2] V. Chandrasekhar, J. G. Andrews, and A. Gatherer, “Femtocell networks: A survey,” IEEE Commun. Mag., vol. 46, pp. 59–67, Sep. 2008. [3] O. Simeone, E. Erkip, and S. S. Shitz, “Robust transmission and interference management for femtocells with unreliable network access,” IEEE J. Sel. Areas Commun., vol. 28, pp. 1469–1478, Dec. 2010. [4] W. Yu, G. Ginis, and J. M. Cioffi, “Distributed multiuser power control for digital subscriber lines,” IEEE J. Sel. Areas Commun., vol. 20, pp. 1105–1115, Jun. 2002. [5] Z. Luo and J. Pang, “Analysis of iterative waterfilling algorithm for multiuser power control in digital subscriber lines,” EURASIP J. Appl. Signal Process., vol. 2006, no. 1, Jan. 2006. [6] G. Scutari, D. P. Palomar, and S. Barbarossa, “Optimal linear precoding strategies for wideband non-cooperative systems based on game theory—Part I and II,” IEEE Trans. Signal Process., vol. 56, no. 3, pp. 1230–1267, Mar. 2008. [7] G. Scutari, D. P. Palomar, and S. Barbarossa, “Competitive design of multiuser MIMO systems based on game theory: A unified view,” IEEE J. Sel. Areas Commun., vol. 26, no. 7, pp. 1089–1103, Sep. 2008. [8] E. G. Larsson, E. A. Jorswieck, J. Lindblom, and R. Mochaourab, “Game theory and flat-fading Gaussian interference channel,” IEEE Signal Process. Mag., vol. 26, pp. 18–27, Sep. 2009. [9] A. Leshem and E. Zehavi, “Game theory and the frequency selective interference channel,” IEEE Signal Process. Mag., vol. 26, pp. 28–40, Sep. 2009. [10] G. Scutari, D. P. Palomar, and S. Barbarossa, “Cognitive MIMO radio: A competitive optimality design based on subspace projections,” IEEE Signal Process. Mag., vol. 25, no. 6, pp. 46–59, Nov. 2008. [11] Q. Zhao, L. Tong, A. Swami, and Y. Chen, “Decentralized cognitive MAC for opportunistic spectrum access in ad hoc networks: A POMDP framework,” IEEE J. Sel. Areas Commun., vol. 25, pp. 589–600, Apr. 2007. [12] J. S. Pang, G. Scutari, F. Facchinei, and C. Wang, “Distributed power allocation with rate constraints in Gaussian parallel interference channels,” IEEE Trans. Inf. Theory, vol. 54, no. 8, pp. 3471–3489, Aug. 2008. [13] S. Barbarossa, S. Sardellitti, A. Carfagna, and P. Vecchiarelli, “Decentralized interference management in femtocells: A game-theoretic approach,” in Proc. CrownCom, Cannes, France, 2010. [14] S. Barbarossa, A. Carfagna, S. Sardellitti, M. Omilipo, and L. Pescosolido, “Optimal radio access in femtocell networks based on Markov modeling of interferers’ activity,” in Proc. 36th Int. Conf. Acoust., Speech Signal Process. (ICASSP), Prague, May 2011.

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Paolo Di Lorenzo (S’10) received the M.Sc. degree in 2008 and the Ph.D. degree in electrical engineering in 2012, both from University of Rome “La Sapienza,” Italy. He is currently a Postdoctoral Researcher with the Department of Information, Electronics and Telecommunications, University of Rome, “La Sapienza.” During 2010, he held a visiting research appointment with the Department of Electrical Engineering, University of California at Los Angeles (UCLA). He has participated in the European research project FREEDOM on femtocell networks. He is currently involved in the European projects SIMTISYS, on moving target detection through satellite constellations, and TROPIC, on distributed computing, storage and radio resource allocation over cooperative femtocells. His primary research interests are in statistical signal processing, distributed optimization algorithms for communication and sensor networks, graph theory, and adaptive filtering. Dr. Di Lorenzo received three Best Student Paper awards, respectively, at IEEE SPAWC’10, EURASIP EUSIPCO’11, and IEEE CAMSAP’11, for works in the area of signal processing for communications and synthetic aperture radar systems. He is recipient of the 2012 GTTI (Italian national Group on Telecommunications and Information Theory) award for the Best Ph.D. Thesis in information technologies and communications.

Sergio Barbarossa (S’84–M’88–F’12) received the M.Sc. degree in 1984 and the Ph.D. degree in electrical engineering in 1988, both from the University of Rome “La Sapienza,” Rome, Italy. He has held positions as a Research Engineer with Selenia SpA (1984–1986) and with the Environmental Institute of Michigan (1988), as a Visiting Professor with the University of Virginia (1995 and 1997) and with the University of Minnesota (1999). He has taught short graduate courses at the Polytechnic University of Catalunya (2001 and 2009). Currently, he is a Full Professor with the University of Rome “La Sapienza.” His current research interests lie in the area of signal processing for self-organizing networks, vehicular networks, bio-inspired signal processing, femtocell networks, graph theory, game theory, and distributed optimization algorithms. He is the author of a research monograph titled “Multiantenna Wireless Communication Systems.” He has been the scientific lead of the European project WINSOC on wireless sensor networks, the European Project FREEDOM on femtocell networks, and he is currently the scientific lead of the TROPIC project on cloud computing over small cell networks. He is also a principal investigator in the European Project SIMTISYS, on the radar monitoring of maritime traffic from satellites. Dr. Barbarossa has been nominated as an IEEE Fellow for his contributions to signal processing, sensor networks, and wireless communications. He received the 2010 EURASIP Technical Achievements Award for his contributions to synthetic aperture radar, sensor networks, and communication networks. He received the 2000 IEEE Best Paper Award from the IEEE Signal Processing Society. He is the coauthor of papers that received the Best Student Paper Award at ICASSP 2006, SPAWC 2010, EUSIPCO 2011, and CAMSAP 2011. From 1997 until 2003, he was a member of the IEEE Technical Committee for Signal Processing in Communications. He served as an Associate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING for two terms (1998–2000 and 2004–2006). He is now a member of the IEEE SIGNAL PROCESSING MAGAZINE Editorial Board. He has been the General Chairman of the IEEE Workshop on Signal Processing Advances in Wireless Communications (SPAWC) 2003. He has been the Guest Editor for Special Issues on the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, EURASIP Journal of Applied Signal Processing, EURASIP Journal on Wireless Communications and Networking, and the IEEE SIGNAL PROCESSING MAGAZINE. In 2012, he was nominated a IEEE Distinguished Lecturer from the Signal Processing Society.

Marco Omilipo received the M.Sc. degree in telecommunication engineering from the University of Rome “La Sapienza,” Rome, Italy, in 2009. During 2010 and 2011, he was involved on European project FREEDOM on femtocell networks. His primary research interests includes algorithms for distributed power allocation in femtocells networks, under Markovian interference models. At the end of 2011, he joined Page Europa S.r.l (General Dynamics U.K.) as a system engineer. He is currently working on design and integration of several communication systems such as TETRA, Airband VHF, data network, SDH, and satellite systems.