Distributed Stochastic Pricing for Sum-Rate Maximization in Femtocell Networks with Random Graph and Quantized Communications Paolo Di Lorenzo, Marco Omilipo and Sergio Barbarossa Sapienza University of Rome, DIET, Via Eudossiana 18, 00184 Rome, Italy e-mail: {dilorenzo, sergio}@infocom.uniroma1.it, [email protected]

Abstract—In this paper, we consider a pricing mechanism aimed at maximizing the sum-rate of a femtocell network in a distributed manner, thanks to a limited exchange of information among neighbor femto access points (FAPs). In a femtocell network, coordination among FAPs is possible exploiting the IPbased backhaul link. In particular, we consider the case where the exchange of information among FAPs is quantized and happens through a network graph (typically a sparse graph), whose links fail randomly across iterations. Using results from stochastic approximation theory, we propose a distributed projection based Robbins-Monro (RM) scheme that converges almost surely (a. s.) on a final allocation equilibrium dependent on the mean graph of the network, even in the presence of such imperfect communication scenario. Numerical results show how the system performance reduces due to the effect of link failures, which cause a lower coordination among FAPs to mitigate interference. Nevertheless, supposing to know the probability with which each link fails, we show how to counteract the effect of failures through a proper weighting of the price coefficients received by the neighbor FAPs. The distributed allocation algorithm is then robust to channel imperfections, whose effect is only to slow down the convergence process.

I. I NTRODUCTION Femtocell networks [1]- [2] are composed of cells having a coverage radius in the order of tens of meters, which provide enhanced indoor coverage through the use of femtoaccess points. These stations are miniature, typically userinstalled, low power base stations connected to the Internet and the cellular operators network via a digital subscriber line (DSL) or cable modem. Unlike macro networks, femtocells are typically installed by subscribers and they are deployed and maintained without global planning. Hence, a potential massive deployment of FAPs might induce an intolerable interference from femto to macro users, as well as from femto to femto users. In this context, devising decentralized mechanisms, able to self-organize and adapt resource allocation in order to limit interference adequately and get the advantages offered by the capillary deployment of femtoaccess points, has a clear importance. An important tool to devise decentralized resource allocation strategies is game theory. Game theoretic approaches have been proposed for the multicarrier interference channel [3] and, lately, for cognitive radios [4]. More recently, game theory has been shown to be This work has been supported by FREEDOM Project, Nr. ICT-248891.

useful for femtocell networks as well [5]. 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. However, purely decentralized approaches might lead to highly inefficient systems. Hence, a more viable approach consists in endowing the radio nodes with the capability to exchange information only with their immediate neighbors, to find out the most appropriate radio resources. In a femtocell network, this is possible exploiting the backhaul link. In this paper we consider a pricing mechanism, which requires a limited exchange of information among FAPs and then it is a perfectly viable solution for femtocells, even under busy backhaul conditions. As shown in [6]- [7], pricing is a way to incorporate, in each players strategy, some kind of care about a socially meaningful performance parameter, rather than being purely selfish. In previous works [3]- [7], each user interferes with all other users. In practice, however, given the limited transmit power and the attenuation resulting from indoor propagation, only nearby FAPs interfere with each other. Then, in this work, we consider a distributed pricing mechanism where each FAP needs to exchange interference prices only with a few neighbors, thus remarkably reducing the signalling in the network. The backhaul link among the FAPs is an Internet connection, which delivers packets in the network using 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 received with a delay exceeding a given threshold. We analyze the effect of this protocol by modeling the graph describing the interaction among FAPs as a random graph, where each link is on with a probability equal to the probability that the packet is correctly delivered within the given maximum delay. Furthermore, we take into account the quantization of the information exchanged among the FAPs. These sources of randomness introduce stochastic noise in the pricing mechanism that needs to be handled to insure convergence of the distributed algorithm. We propose a projection-based RobbinsMonro (RM) stochastic approximation scheme [8] and we

prove the almost sure convergence of such procedure to a final allocation even in the presence of such imperfect communication scenario. The final result is shown to depend only on the expected value of the graph. Then, building on this result, we show how to counteract the effect of random link failure through a proper weighting of the price coefficients. II. S YSTEM

MODEL

We consider an open access system using OFDMA, with Q links sharing the same set of physical resources, e.g., time and bandwidth. No multiplexing strategy is imposed a priori so that, in principle, each FAP might interfere with each other. In practice, however, given the limited transmit power and the attenuation resulting from wall penetration, only nearby FAPs interfere with each other. To study the interference mechanism, we introduce what we call the interference graph GI = (VI , EI ), defined as the graph whose vertices are FAPs and where there is an edge between two vertices only if the relative FAPs interfere with each other. Typically, the interference graph is a sparse graph (i.e., with a number of links much smaller than the maximum number of possible links). We denote by Nqi the set of interfering neighbors of FAP q. We also introduce the notation p = col{pq }Q q=1 , where pq = (pq1 , . . . , pqN ) denotes the power vector of FAP q, whose element pqm is the power transmitted by node q over the mth subcarrier. Under the previous assumptions, resorting to Shannon capacity expression, the rate of FAP q is given by ! N qq 2 q X |Hm | pm P Rq (p) = log2 1 + 2 (1) rq 2 r σqm + r∈N i |Hm | pm m=1 q

rq Hm

where is the channel transfer function of the m-th subchannel between the r-th transmitter and the q-th receiver, and 2 σqm is the variance (power) received on the m-th subchannel including receiver noise and power coming from the macro users. We are interested in a cooperative approach where FAPs pursue a common social objective. In particular, the optimization problem we would like to solve is the maximization of the sum rate of the Q FAPs under power constraints, i.e., PQ max q=1 Rq (p) p (2) s.t. p∈P QQ PN where P = q=1 Pq and Pq = {pq ∈ RN : m=1 pqm ≤ Pq , 0 ≤ pqm ≤ pmax (m), m = 1, . . . , N }, with Pq q and pmax (m) denoting, respectively, the power budget of q user q and the mask constraint that limits the maximum transmit power over each channel. In general, the objective function in (2) is not concave in the power allocation p and, as a consequence, the problem may have multiple local optima. A local solution can be found in a centralized manner using standard optimization algorithms. However, we focus on distributed solutions where a low bit rate local coordination among FAPs is allowed. In this work, we take into account the presence of an imperfect communication scenario, considering the following assumptions on the stochastic processes affecting the algorithm.

1) Random link failure model: The exchange of control data among FAPs is described by a communication graph Gc = (Vc , Ec ), i.e., the graph whose vertices are FAPs and where there is an edge between two vertices only if the relative FAPs exchange data with each other. We denote by A = {aqr } the adjacency matrix of graph Gc , having nonnegative entries aqr ≥ 0. Random packet drops are incorporated by modeling the network as a time-varying random graph, where the presence of an edge (link) depends on the packet error rate over the corresponding communication link. The set of adjacency matrices, at different times k, is modeled as a sequence of i.i.d. matrices {A[k]}. At each time k, we write the graph adjacency matrix as ¯ + A[k] ˜ A[k] = A

(3)

¯ = E{A[k]} denotes the adjacency matrix of the exwhere A ˜ pected graph and A[k] is a zero-mean sequence of independent identically distributed (i.i.d.) matrices. 2) Dithered quantization: We assume that the control data sent through the backhaul are quantized, with quantization step ∆ > 0. We indicate the quantization function as q(·) : R → Q . Conditioned to the input, the quantization errors are deterministic. To simplify the analysis and ensuring convergence of the proposed algorithms, we add a dither sequence {ν[k]}k≥0 of i.i.d. uniformly distributed random variables on [−∆/2, ∆/2) to the input y[k], with ν[k] independent of y[k]. The resulting error sequence {ǫ[k]}k≥0 becomes ǫ[k] = q(y[k] + ν[k]) − (y[k] + ν[k]).

(4)

Thanks to dithering, the sequence {ǫ[k]}k≥0 is a sequence of i.i.d. random variables, uniformly distributed on [−∆/2, ∆/2), independent of the input sequence. In the following, we will evaluate the effect of the random communication graph and of the price quantization on the gradient of the sum-rate. Introducing the notation R(p) = PQ R (p), the partial derivative of the sum-rate with respect q q=1 to pqm can be written as X ∂Rr (p) ∂R(p) ∂Rq (p) = + . (5) q q ∂pm ∂pm ∂pqm i r∈Nq

Following the same steps as in [6]- [7], it is useful to introduce the price coefficients as r πm (p) = −

∂Rr (p) , r ∂Im

(6)

r where Im is the interference perceived by FAP r, over subr channel m. The price πm (p) is always nonnegative and is proportional to the marginal decrease of user r’s rate due to an increase of the q-th node’s transmit power, as: r r ∂Rr (p) ∂Rr (p) ∂Im ∂Im r r qr 2 = = −π (p) = −πm (p)|Hm | . m r ∂pqm ∂Im ∂pqm ∂pqm (7) Then, substituting expression (7) in (5), we get Q

∂R(p) ∂Rq (p) X qr 2 r = − aqr |Hm | πm (p) q ∂pm ∂pqm r=1

(8)

where the interference graph is explicitly taken into account through the coefficients aqr of the adjacency matrix A. To evaluate expression (8), FAP q needs to know the price vectors r N π r (p) = col{πm }m=1 ∈ RN , ∀r ∈ Nqi , which are transmitted by the neighbors through the backhaul link. Collecting all the contributions in (8), the gradient of the sum-rate with respect to p can be written in compact vector form as ∇p R(p) = r(p) − AH π(p) (9) h iQ where r(p) = col ∇pq Rq (p) ∈ RQN , π(p) = q=1

[π 1 (p), . . . , π Q (p)] ∈ RQN is the vector collecting the interference prices of the entire network and AH ∈ RQN × RQN is a multidimensional adjacency matrix weighted by the cross channels between FAPs. Expression (9) shows the dependence of the sum-rate gradient with respect to the network graph in the case of an ideal communication scenario. Considering the presence of failures and dithered quantization noise, expression (8), evaluated at time k, takes the form ∂Rq (p[k]) ∂R(p[k]) = + (10) q ∂pm ∂pqm Q X qr 2 r qr + aqr [k]|Hm | (πm (p[k]) + νm [k] + ǫqr m [k]) r=1

qr m = 1, . . . , N , q = 1, . . . , Q, where νm [k] and ǫqr m [k] are contributions of dithered quantization noise that the q-th FAP receives from the r-th FAP on the m-th subchannel, at time k. To rewrite (10) in compact form, we introduce the QN random vectors Υ[k] = col{Υq [k]}Q and Ψ[k] = q=1 ∈ R Q QN col{Ψq [k]}q=1 ∈ R , which are aggregated contribution of dithered quantization noise. Then, the sum-rate gradient with respect to the power allocation p[k], in the presence of random link failures and dithered quantization, can be written in compact form as

Pq , allowing the parallel computation of the algorithm. Let us now focus on a distributed stochastic simultaneous solution, which is summarized as follows: 1) Each FAP q chooses an initial power profile pq [0] satisfying the power constraint; 2) Using (6), each FAP q computes the interference price vector π q , given the current power profiles, and transmits this to the neighbor FAPs; 3) At each time k, every FAP, given the neighbors’ power profiles and price vectors, simultaneously updates its power profile according to h i pq [k + 1] = pq [k] + α[k]∇pq R(p[k]) (14) Pq

=

T q (p[k]),

q = 1, . . . , Q,

k ≥ 0,

where [·]Pq denotes the projection over the feasible set Pq , α[k] is an iteration-dependent step size and ∇pq R(p[k]) ∈ RN is the q-th vector element of (11); 4) Go to step 2 and repeat until convergence. We consider two assumptions on the stochastic procedure (14): 1) Persistence : The step size sequence α[k] satisfies: α[k] > 0,

∞ X

α[k] = ∞,

k=0

∞ X

α2 [k] < ∞.

(15)

k=0

˜ 2) Independence : The stochastic sequences {A[k]} k≥0 , {Υ[k]}k≥0 and {Ψ[k]}k≥0 are mutually independent. Next, we provide the convergence result for the distributed stochastic algorithm in (14). Theorem 1: Let {p[k]} be the sequence generated by the distributed stochastic pricing algorithm in (14), with step-size satisfying the conditions in (15). Then, the sum rate sequence {R(p[k])}k≥0 converges a.s. to a finite value R∗ , i.e.,   ∗ Prob lim R(p[k]) = R = 1, (16) k→∞

∇p R(p[k]) = r(p[k]) − AH [k]π(p[k]) + Υ[k] + Ψ[k]. (11) Expanding now the multidimensional weighted adjacency matrix AH [k] as in (3), we can write expression (11) as the sum of a deterministic function F (p[k]), dependent on the expected ¯ plus a zero-mean random function Γ(p[k]), where graph A, F (p[k]) = Γ(p[k]) =

¯ H π(p[k]), r(p[k]) − A ˜ −AH [k]π(p[k]) + Υ[k] + Ψ[k].

III. D ISTRIBUTED

(12) (13)

STOCHASTIC PRICING ALGORITHM

To find a solution to problem (2) affected by random disturbances, it is useful to introduce stochastic approximation algorithms. In this section we introduce a stochastic approximation scheme for solving problem in (2) in a distributed manner. In particular, we consider a projection-based RobbinsMonro (RM) stochastic approximation procedure, based on a simultaneous update of the users’ power profiles. Then, we prove the convergence of the proposed algorithm. The problem is amenable QQfor distributed solutions because the optimization set P = q=1 Pq is given by the Cartesian product of sets

where Prob[E] denotes the probability of the event E. Furthermore, let p∗ be an accumulation point of the sequence {p[k]}, as k → ∞, the optimal solution p∗ is a fixed point of ∗ ∗ the mapping T (p) = col{T q (p)}Q q=1 , such that p = T (p ). Proof: The proof can be found in [9]. IV. S IMULATION

RESULTS

In this section we provide some numerical results to validate the theoretical findings and to assess the performance of the proposed algorithm. We assume the presence of 20 FAPs, interfering according to a connected and sparse graph. Because of the randomness introduced by the control channel, a link between two neighbors has a certain probability pc to be established correctly. The values to be exchanged are also affected by dithered quantization noise, supposing the presence of a 6 bit mapping. We consider a number of subchannels N = 16 and a multipath channel of order Lc = 4. In Fig. 1 we show a numerical example of sum-rate behavior as a function of the iteration index, considering different values of probability pc . The ideal case corresponds to pc = 1 and it is shown as

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Sum−rate (bps)

Sum−rate (bps)

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1360 1350 1340 p =0 c

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pc = 0.3 ideal case

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p = 0.3 compensated stochastic algorithm

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c

p = 0.3 stochastic algorithm c

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pc = 0.6 stochastic algorithm

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p = 1 ideal case 1300

c

c

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Fig. 1. Sum-rate vs. iteration index, for different probabilities of link failures.

a benchmark. We also report the case pc = 0, corresponding to the behavior of the classical iterative water filling algorithm (IWFA) without pricing. In particular, considering two intermediate values pc = 0.6 and pc = 0.3, we compare the ideal behaviors obtained having a mean graph dependent on these probability values and the average behaviors, averaged over 500 independent realizations, given by the algorithm in (14) in the presence of random links and quantization noise. As we can notice, the stochastic algorithm converges to the same equilibrium of the correspondent ideal case evaluated for the expected graph. The effect of the diminishing step size in (15) is to reduce the convergence speed of the algorithm with respect to the correspondent ideal case. As expected, link failures determine a performance degradation due to the lower coordination among FAPs to mitigate interference. However, from 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, assuming that the probability pc of establishing a communication over each link is known, through preliminary estimation, it is possible to counteract the graph randomness by weighting the price coefficients coming from each link with the inverse of the probability pc . In this way, we are ”normalizing” the mean network graph in order to make it coincident with the ideal graph in the case of absence of failures. Considering the same settings of the previous simulation, in Fig. 2 we show the behavior of the sum-rate as a function of the iteration index. In particular, we report the ideal case correspondent to pc = 1, as a benchmark, compared to the average behaviors, averaged over 500 independent realizations, given by the compensated stochastic algorithm in (14), in the presence of quantization noise and for different probabilities to establish a communication 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 the probability pc . However, reducing the probability to establish a communication link, the network requires more time to reach

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Fig. 2. Sum-rate vs. iteration index, for different probabilities of link failures.

the final equilibrium state. V. C ONCLUSIONS In this paper we have studied a decentralized stochastic pricing algorithm aimed at allocating power in OFDMAbased femtocell networks in the case where the communications among FAPs are affected by random link failures and quantization noise. We have proposed a projection-based RM stochastic approximation scheme that converges almost surely to a final allocation dependent on the mean graph of the network. Numerical results show how the network performance is affected by link failures. Finally, assuming to know the failure probability over each link, we have shown how to counteract the effect of random links, thus making the allocation algorithm robust to channel imperfections, whose effect is only to slow down the convergence process. R EFERENCES [1] V. Chandrasekhar, J. G. Andrews, A. Gatherer, “Femtocell networks: a survey,” IEEE Comm. Mag., vol. 46, pp. 59–67, Sept. 2008. [2] O. Simeone, E. Erkip, S. S. Shitz, “Robust transmission and interference management for femtocells with unreliable network access,” IEEE J. Select. Areas Commun., vol. 28, pp. 1469–1478, Dec. 2010. [3] 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. on Signal Processing, vol. 56, no. 3, pp. 1230–1267, March 2008. [4] G. Scutari, D. P. Palomar, S. Barbarossa, “Competitive design of multiuser mimo systems based on game theory: a unified view,” IEEE Jour. of Sel. Areas in Comm., vol. 26, no.7, pp. 1089–1103, Sept. 2008. [5] S. Barbarossa, S. Sardellitti, A. Carfagna, P. Vecchiarelli, “Decentralized interference management in femtocells: a game-theoretic approach,” Proc. of CrownCom, 2010, Cannes. [6] J. Huang, R. A. Berry, and M. L. Honig, “Distributed interference compensation for wireless networks,” IEEE J. on Sel. Areas in Comm., vol. 24, no. 5, pp. 1074–1084, May 2006. [7] C. Shi, R.A. Berry and M.L. Honig, “Distributed interference pricing for OFDM wireless networks with non-separable utilities,” Proc. 42nd CISS, Princeton, NJ, pp. 755–760, March 2008. [8] H. Robbins and S. Monro, “A stochastic approximation method,” The Annals of Mathematical Statistics 22, 1951, no. 3, pp. 400–407. [9] P. Di Lorenzo, S. Barbarossa and M. Omilipo, “Distributed sum-rate maximization in femtocell networks with random graph and quantized communications,” submitted to IEEE Trans. on Signal Processing.