Determining Frequency Reuse Feasibility in Device-to-Device Cellular Networks

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1 c 20xx IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. Determining Frequency Reuse Feasibility in Device-to-Device Cellular Networks Markus Klügel Technische Universität München, Arcisstr. 21, Munich, Germany Wolfgang Kellerer Technische Universität München, Arcisstr. 21, Munich, Germany Abstract Device-to-Device (D2D) communication in cellular networks allows dynamic frequency reuse even within a cell, which can increase the frequency reuse factor beyond one. To perform dynamic reuse, it must be feasible for all links to maintain a certain signal quality despite the presence of interference. This can be ensured using reuse feasibility tests. However, state of the art testing algorithms show erroneous behavior when infeasibility occurs due to limited transmission powers. In this paper we develop two testing algorithms, based on nonnegative matrix theory, that explicitly take limited transmission powers into account and show how these can be implemented in protocols. The protocols are compared with the state of the art in terms of control channel usage, speed and accuracy of decision making. We find that the proposed protocols estimate reuse feasibility more accurately than the state of the art and are comparable in terms of speed and control channel usage. I. INTRODUCTION Device-to-Device (D2D) communication has been proposed to offload local cellular traffic, i.e., traffic where source and sink are spatially nearby, from the base station [1]. One major benefit seen in D2D communication is that, due to the expected short range of links, transmission powers can be reduced, allowing the reuse of transmission resources even within a cell. However, this reuse leads to problems because it introduces a new form of interference, intracell-interference, into the cellular system. While reusing transmission resources, all communication links must maintain a certain signal to interference and noise ratio (SINR) to achieve a satisfactory link quality. This is commonly achieved by introducing spatial reuse constraints or interference power thresholds, c.f. [2]. However, these approaches often neglect the existing work on dynamic power control [3], which provides insights on inherent properties of interference and hence cannot take full advantage of reuse possibilities. To work efficiently, scheduling and reuse link selection mechanisms must take into account for which link combinations it is at all possible to achieve their signal quality targets while performing reuse. Due to the dynamics expected for D2D communication, this information needs to be acquired within tens of transmission time intervals (TTI). We investigate the problem how to check fast whether a set of links can reuse This work has been funded by the German Research Foundation (DFG) under grant number KE 1863/2-1 as part of the SPP COIN. resources and achieve signal quality targets under dynamic power control, which we call reuse feasibility. We develop two protocols to check for reuse feasibility and evaluate them in terms of speed, accuracy and control overhead. A. State of the Art The problem of determining reuse feasibility resembles call admission control problems, for which solution approaches exist. However, these have not been transferred to context of D2D-communication so far. In general, frequency reuse is possible if all links can achieve a certain SINR in the presence of mutual interference, and hence strongly depends on the used transmission powers. From a power control point of view, frequency reuse is possible if the set of transmission powers that can achieve certain SINR constraints is not empty. This condition is called feasibility of the power control problem and depends on the spectral radius of the Foschini-Matrix, a matrix derived from the channel gains and desired SINRs [3]. Determining reuse feasibility of a set of D2D links corresponds to determining the feasibility of their power control problem. Prior work on power control feasibility in wireless networks [4] [8] has been developed as a basis for calladmission control. The well-known Foschini-Miljanic power control algorithm (PCA) [9] is often assumed in this context because it is known to converge if and only if the reuse situation is feasible. Papers [4] [7] assume a set-up where several links are active already, whose PCA has converged, and new links seek admission to a channel. In [4], Bambos et al. propose that all links which want to (re)use a channel send a low-power probing tone and wait for the active PCAs to re-converge. From the interference produced by the reconverged PCAs, reuse feasibility is determined. The authors of [5] and [6] follow this approach, using a discriminant [5] and the spectral radius [6] as feasibility criterion. Ku cera et al. [7] investigate the dynamics of the Foschini-Miljanic PCA and use observations of its convergence speed to calculate the spectral radius before convergence of the PCA. The algorithm developed in [7] is shown to outperform those in [4] [6]. Lin et al. [8] show how the feasibility problem can be transformed into a linear program. The work on call admission control can also be used to test for reuse feasibility in the D2D scenario. However, current work evaluates reuse feasibility incorrectly

2 when constrained transmission powers make reuse impossible. This recognized in [5], [6], but not solved, and [4], [7] and [8] completely neglect power constraints. The protocols developed in this paper are shown to evaluate reuse feasibility correctly in all cases. P P 2 SINR 2 γ 2 F SINR 1 γ 1 II. SYSTEM MODEL & BASIC PROPERTIES We consider a set L ofn wireless links, which can be either cellular up-/downlink or D2D users. The sender of link j L transmits with a power P j and is received at the receiver of link i with power h ij P j, where h ij is a channel gain that takes account for path-loss and fading effects. The SINR of link i is: h ii P i SINR i = j i h, j = 1,2,..,N (1) ijp j +η i P P p P h P where η i is the noise power at receiver i. We assume that the quality of link i is acceptable if SINR i γ i for a given minimum SINR γ i. To take account for maximum transmission powers and that powers are nonnegative, we further require 0 P i P max,i i, where P max,i is the maximum transmission power of link i s sender. TheN SINRconstraints can be rewritten in matrix form, which leads to a compact formulation [3]: (I F) P u, (2) where P = [P 1,...,P N ] T is the vector of transmission powers, u = [γ 1 η 1 /h 11,...,γ N η N /h NN ] T is the vector of normalized noise powers, I is the N N identity matrix and F is the Foschini-Matrix with (i,j) entry F ij = γ i h ij /h ii if i j and 0 otherwise. In (2) and further on, the inequalities are taken element-wise. We now define P = { P : 0 P P max }, the set of transmission powers achievable by all transmitters with P max the vector of maximum transmission powers. Further, we define F = { P 0 : (I F) P u}, i.e., the set of power vectors fulfilling all SINR-requirements. For frequency reuse to be possible, it must be ensured that: F P. (3) Equation (3) implicitly contains that F. We now re-state some known properties of F and introduce some formulations that will be useful later on. As all channel gains and SINRconstraints are nonnegative, so are the entries of F, hence F is a nonnegative matrix. According to the Perron-Frobenius Theorem for nonnegative matrices [3], F is non-empty if and only if the spectral radius of F, i.e., its maximum modulus eigenvalue, ρ(f) 1. Each row of (2) defines a half-space of power vectors which meet the SINR constraints of the corresponding link. Hence, F is known to be a cone defined by the intersection of N such half-spaces. Because of this structure, we can decompose any P F into P = P p + P h, where P p 0 is a unique vector which solves (2) with equality and P h 0 is any vector that solves (2) for the noiseless case u =0. Pp is at the tip of the cone F and can be seen as the particular solution to (2). P h is the homogeneous solution and P P 1 Fig. 1. Structure of F and P with SINR-constraints for two links that both have P as maximum power. points from the tip of F into F. The structure of F and its relation to P p, P h and P max is depicted in Figure 1. From the decomposition of P into P p and P h we see that two cases can make frequency reuse impossible. First, F can be empty. This is the case when ρ(f) > 1, i.e., the halfspaces have no intersection for P 0. Second, P p can violate the maximum power constraint. In this case, the situation is feasible in principle but at least one sender cannot use enough transmission power to assume a state in F. We call F = the first type of infeasibility and F, F P = the second type of infeasibility, respectively, and note that the second type holds if and only if P p P max. For both types, it does not make sense to let the corresponding links reuse resources because at least one link cannot have a satisfactory signal quality. The state of the art solutions discussed in section I-A aim at calculating ρ(f) and do not solve the second type of infeasibility. This leads to wrong feasibility statements, as we will show in section V. For the rest of this paper, we assume that all links use a power-constrained Foschini-Miljanic PCA. According to this PCA, transmission powers are updated with the rule: P(k +1) = min{f P(k)+u, P max }. (4) The min{ } in (4) is taken element-wise. In the references, the PCA is controlled by the enb and hence centralized. However, it can easily be decentralized. In fact, approaches to adapt it to the D2D use-case exist already [10]. The updates can be calculated in a decentralized manner because (F P + u) i corresponds to P i γ i /SINR i [5]. Analog, (F P) i corresponds to P i γ i /, where is the signal to interference ratio

3 (SIR) of link i. The PCA in (4) is known to converge towards P p if F P and to some other value at the border of the maximum power bounds otherwise [5]. Without proof, we remark that the presented protocols could in principle use any PCA that converges towards P p, instead of (4). III. MATHEMATICAL SOLUTION We now discuss several mathematical possibilities to evaluate reuse feasibility with maximum power constraints. In Section IV, the approaches will be developed into protocols. A natural solution to determine reuse feasibility is to calculate ρ(f) and P p and then check for ρ(f) < 1 and P p P max. Efficient algorithms exist for the calculation of both, however, a lot of information needs to be obtained, namely all channel gains, noise figures, maximum transmission powers and SINR requirements, causing much overhead. We will consider this approach as brute-force reference although we deem it to be impractical for a real world setup. The first observation that we make is based on the following result from nonnegative matrix theory [11]: Define r F ( P) (FP) = min i, P i 0 P i s F ( P) (FP) = max i. (5) P i 0 P i r F ( P) is the minimal and s F ( P) the maximal factor by which F distorts the components of P, i.e., r F ( P) P F P s F ( P) P holds for P > 0. Then, for all P > 0 [11]: r F ( P) ρ(f) and s F ( P) ρ(f). (6) Equality is achieved in (6) for both, r F ( P) and s F ( P) when P is the dominant eigenvector of F, i.e., the eigenvector to the largest modulus eigenvalue. Applying this result, we use that (F P) i = P i γ i / in (5), which leads to: ρ(f) min γ i, We thus can state the following proposition: ρ(f) max γ i. (7) Proposition 1: If, for some P 0, all SIRs meet the SINR constraint then F. If no SIR meets the SINR constraint then F =. Proof: If all SIRs meet the SINR constraint, 1 max γi ρ(f), hence F. If no SIR meets the SINR constraint, 1 < min γi ρ(f), hence F =. Proposition 1 can be used to check if the half-spaces of F have an intersection or not and detect the first type of infeasibility. However, it cannot be used to determine the second type of infeasibility. For the second type, we need to check whether P p is in P or not. This can be done using the following proposition: Proposition 2: If, for some 0 P P max, all SINRconstraints are met, F P. If SINR i γ i i and P i = P max,i, SINR i < γ i for some i, F P =. Proof: We note that SINR i γ i i corresponds to (I F) P u. So if the first part holds, P is in F and P, hence F P. For the second part we observe that any P for which(i F) P u holds can be written as P = P p P h, which can also be seen from Figure 1. Then P P p because P h 0. If P i = P max,i and SINR i < γ i holds for some i, then P max,i < P p,i, hence P p P max and F P =. Proposition 1 and 2 will be combined into a protocol in Section IV. For the next observation we refer to the Foschini- Miljanic PCA in (4) and notice that it can be seen as a power iteration on F with a constant added on each iteration. Following this interpretation, we see that the spectral radius of a matrix can be calculated from a power iteration in combination with the Rayleigh-quotient [12]: Let P(k) be a power iteration of the form P(k + 1) = F P(k), then for k, P converges towards the dominant eigenvector of F: lim P(k) = lim k k Fk P(0) = lim k λk 1v 1, (8) for the maximum modulus eigenvalue λ 1 (= ρ(f)) and its corresponding eigenvectorv 1. In particular, it is known that after sufficient iterations the Rayleigh-quotient R F ( P(k)) provides a good estimate of the eigenvalue [12], because: lim R F( P T (k)fp(k) P(k)) = lim k k P(k) 2 = λ2k+1 1 v 1 2 λ 2k 1 v = λ (9) Here, ( ) T is the transpose operation and the euclidean norm. Applying this to the current scenario, R F ( P(k)) can be used to estimate ρ(f). This estimation has in principle been found also by Ku cera et al. in [7], as it can be seen as a special case of their Theorem 4. However, we point out that it can only be used for calculation of ρ(f) and not to detect the second type of infeasibility, which is due to P p P max. We propose to calculate the Rayleigh-quotient not with the noiseless version of P(k + 1) but the version including the normalized noise u. That is, we introduce a feasibility test: T F ( P(k)) = P T (k) ( ) FP(k)+u P(k) 2. (10) We will now show how this test behaves for feasible and infeasible situations. Proposition 3: lim T F( P(k)) = k with c (1,ρ(F)] greater than one. Proof: See Appendix. { 1, if ρ(f) 1 and P p P max c, else IV. PROPOSED PROTOCOLS We will now develop testing protocols based on the discussion in Section III. For this, we consider a cellular wireless system with LTE-like transmission resource structure, with separate transmission- and control-channels that can be used

4 Protocol 1 Direct Measurement Test (DMT) Protocol 1: enb broadcasts P T, S and T S. 2: All senders report P max,i to enb. 3: for i = 1 to N do 4: Sender i sends with P T at TTI T S + TTI i (S). 5: All receivers calculate and store h ij. 6: end for 7: All receivers report their h ij s and η i to enb. 8: enb calculates ρ(f), P p and checks for P p P max. 9: If ρ(f) < 1 and P p P max, reuse is feasible. differently at each TTI. Compared to the discussed references, where arriving calls seek admission to a channel, we assume that a controller, e.g., the enb, wants to check the reuse feasibility of a predefined set of links at a certain time instant. We imagine this check to happen on a regular basis and the output to be used for scheduling decisions of the enb. In set-ups with a high spatial density of D2D-links reusing resources, control channels might form a bottleneck as they cannot be reused. Because of this we are interested in the control overhead produced by our protocols and include it in our evaluations. We assume control traffic to be collision free and one message to carrybbyte per transported numeric value. A. Direct Measurement Test (DMT) Protocol The Direct Measurement Test Protocol implements the brute-force reference mentioned in Section III. The enb defines a transmit power P T, a probing order S and a start time T S. TTI i (S) is the relative TTI offset of link i w.r.t. T S, based on S. The protocol is described as Protocol 1. Assuming that all probing- and control-messages take one TTI for transmission, the algorithm will make a feasibility statement N +2 TTI s after T S. Control messages are sent in lines 1, 2 and 7 carrying different quantities of numeric values. We assume S = N and consider that every receiver reports N channel gains, so the number control bytes sent is CB DMT (N) = (N+2)b+Nb+N(N+1)b = Ω(bN 2 ). (11) Ω( ) here is the landau-symbol, indicating thatcb DMT grows at least as fast as( ). Although some optimization can be done, e.g., S or P max,i might be pre-defined, the main contributor to channel usage, which is the channel report by the receivers, cannot be circumvented. B. Rayleigh-Quotient Test (RQT) Protocol The following protocol is based on Proposition 3. We use that (FP +u) i = P i γ i /SINR i and hence: N T F ( P(k)) i=1 = P i(k) 2 γ i /SINR i (k) N i=1 P. (12) i(k) 2 Alternatively, R F ( P(k)) can be used, in which case SINR i becomes. All links use PCA (4) and receivers feed Protocol 2 Rayleigh Quotient Test (RQT) Protocol 1: enb broadcasts P T, T S and I. 2: for k = 1 to I do 3: All senders transmit with P i (k). 4: All receivers report γ i /SINR i (k) back to sender. 5: if k < I then 6: Senders set P i (k +1) = min{ Pi(k)γi SINR i(k),p max,i} 7: end if 8: end for 9: Senders report P i (I) and γ i /SINR i (I) back to enb. 10: enb calculates T F ( P(k)). 11: If T F ( P(k)) 1 reuse is feasible, else infeasible. information back to their senders using control channels. The enb defines an initial transmission power P T that all senders use, a starting time T S and a number of power-control iterations I 1. The protocol is described as Protocol 2. Again assuming that all probing- and control-messages (including feedback from receivers to transmitters) take one TTI, the algorithm will make a feasibility statement 2I + 1 TTI s after T S. As Proposition 3 holds in the limit, the feasibility statement made after I iterations is only true with a probability, depending on how well the PCA has converged. Control messages are sent at lines 1, 4 and 9. So the number of control bytes sent is: CB RQT (N,I) = 3b+NIb+2Nb = Ω(bIN). (13) We conclude that the RQT Protocol will cause less control traffic than the DMT Protocol, as long as I < N. However, when I > N/2, the DMT Protocol will finish faster. C. SI(N)R-Bound Test (SBT) Protocol The third protocol uses Proposition 1 and 2. All links use PCA (4) and report status updates to the enb, waiting for the algorithm to converge. In particular, user equipments (UEs) report to the enb whenever their SIR or SINR is lower than their SINR requirement. According to Proposition 1, if all UEs report their SIR at one iteration, reuse is assumed infeasible. If no UE reports at all, it is assumed feasible. Following Proposition 2, if the UEs of a link observe no further SINR changes but SINR i < γ i ( converged infeasible ), this is also reported and interpreted as infeasible. The PCA (4) is adapted such that powers are not updated if their SINR is within a threshold γ above γ i, which enforces an earlier convergence behavior. Again, the enb defines an initial transmit power P T and a starting time T S. The protocol is described as Protocol 3. Because neither the duration of the SBT Protocol, nor the number of messages per iteration are deterministic, an analytical derivation of convergence speed and control overhead of the SBT Protocol is out of scope of this paper.

5 Protocol 3 SI(N)R-Bound Test (SBT) Protocol 1: enb broadcasts P T and T S. 2: loop 3: All senders transmit { with P i (k). 4: Receivers set β i = 1,γ i SINR i (k) < γ i + γ γ i /SINR i (k), else 5: Receivers report β i back to sender. 6: Receivers report SINR i < γ i, < γ i and converged infeasible to enb 7: Senders set P i (k +1) = min{p i (k) β i,p max,i } 8: end loop 9: enb receives no message within an iteration: Feasible 10: < γ i i or converged infeasible: Infeasible Parameter Cell Radius Maximum Transmission Power Maximum D2D Link Distance Channel Model Noise Power SINR constraint γ i γ TABLE I SIMULATION PARAMETERS Value 500m 26 dbm 50m ITU-R M Urban Macro dbm 10 db i L 0.5 in linear domain Percentage TTI SBT RQT RF( P) RQT TF( P) User Density in [Users/ha] Fig. 2. Percentage of Wrong Results Produced by the Protocols 5 DMT SBT RQT RF( P) RQT TF( P) User Density in [Users/ha] Fig. 3. Convergence Speed of the Protocols V. SIMULATIVE EVALUATION We now present simulation results for the proposed protocols. The results were generated using the SimuLTEframework [13] of the OMNeT++ simulator. As the framework abstracts intracell-interference, we implemented it ourselves. We assume a single cell D2D setup of an urban macro cell, containing one randomly positioned cellular uplink user and a set of D2D links. The number and position of D2D senders follow a Poisson Point Process of defined density and D2D receivers are placed uniformly distributed within a radius of 50m of their sender. Further used simulation parameters are summarized in Table I. At a predefined time instant, the enb tests the reuse feasibility of all links with the proposed protocols. The protocols are compared against the DMT Protocol and algorithm A1 from [7], which we implement like the RQT Protocol using R F ( P(k)). In Figure 2 we compare the percentage of wrong results produced by the tests, after convergence, for different user densities. Convergence is here assumed at the first TTI after which the feasibility result of the respective protocol does not change anymore. The DMT Protocol was used as reference for the true result. It can be seen that the SBT Protocol and the RQT Protocol with T F ( P(k)) always produce the correct result after enough iterations, whereas the RQT Protocol with R F ( P(k)) can produce a significant amount of wrong results. A further investigation shows that all wrong results of R F ( P(k)) are false positive and in particular due to maximum power violations. This proves that the second type of infeasibility is not evaluated correctly by R F ( P). Figure 3 shows the mean number of TTIs needed for convergence by the protocols. It can be seen that the DMT Protocol behaves deterministic according to our analysis from Section IV. The RQT Protocols converge after 3-4 TTIs, i.e., 1-2 PCA iterations, for all densities. For high densities the RQT Protocols converge slightly faster because the set-ups become clearly infeasible. The SBT Protocol takes longest to converge for low D2D-link densities but saturates for higher densities. The number of bytes sent on control channels until convergence is shown in Figure 4. Here, b = 1 was assumed. The control traffic produced by the DMT Protocol increases quadratically, as predicted in section IV. The traffic produced by the RQT Protocols grows linearly, which is consistent with Figure 3, as the average required iterations stay nearly constant but the average number of links increases linearly. The SBT Protocol produces the least control traffic for large parts of the considered parameter range. This is notable because it also takes longest to converge. The reason for the low control traffic produced by the SBT Protocol is that UEs mainly report SINR/SIR outages. So only a subset of the UEs actually sends

6 Control Bytes DMT SBT RQT RF( P) RQT TF( P) User Density in [Users/ha] Fig. 4. Bytes Sent on Control Channels Until Convergence reports and in overall, the reports sum up to be less than the regular reports produced by the other protocols. We conclude that both proposed protocols, RQT and SBT, meet the target of reliable feasibility statements. The RQT Protocol is very fast and makes reliable statements after few TTI but causes control traffic in amount linear to the number of participating links. The SBT Protocol, on the other hand, is rather slow in convergence but produces less control traffic in overall and significantly less per TTI. The right choice depends on the considered scenario. VI. CONCLUSION In this paper, we introduced two new protocols to test for reuse feasibility in D2D scenarios, the SI(N)R Bound Test Protocol and the Rayleigh Quotient Test Protocol. The latter extends the existing, rayleigh-quotient based protocol from [7] to cope with maximum transmission powers. We evaluated all protocols by means of simulation in terms of reliability, convergence speed and sent control bytes. The results show that both, SBT Protocol and RQT Protocol can achieve reliable feasibility results for all types of infeasibility. The RQT Protocol converges fast and with linearly growing control traffic, whereas the SBT Protocol is slower in general but produces less control traffic. APPENDIX Proof of Proposition 3 Proof: We will first consider the case without maximum transmission powers and then discuss their influence. We can rewrite P(k) = P p + P(k) and consider that P p = F P p +u by definition. Because F( P p + P(k))+u = P p +F P(k), limit (8) applies to P(k) and (10) turns into: lim T F( P(k)) = k ( P p + P(k)) (F( T P p + P(k))+u = lim k P p + P(k) 2 ( P = lim p +λ k 1v 1 ) T ( P p +λ k+1 1 v 1 ) k P p +λ k 1 v 1 2 ) P p 2 +(1+λ 1 )λ k 1P p T = lim v 1 +λ 2k+1 1 v 1 2 k P p 2 +2λ k 1Pp T v 1 +λ 2k 1 v (14) 1 2 It can be seen from (14) that if λ 1 1, then lim k T F ( P(k)) = 1 and if λ 1 > 1, then lim k T F ( P(k)) = λ 1. When λ 1 1 but P p P max, the PCA will converge to some vector that does not fulfill (2) [5]. However, as it does converge at all, P(k +1) = min{f P(k)+u, P max } P(k) (15) will hold for some sufficient large k, meaning that all links will either achieve their requirement with equality or reach their transmit power limit P max,i. P(k) will hence satisfy P(k) F P(k)+u element-wise. As all vector elements are nonnegative, lim T F( P T (k)(fp(k)+u) P(k)) = lim k k P T (k) P(k) > 1. (16) We could show that if λ 1 1 and P p P max, T F ( P(k)) converges to 1, that if λ 1 1 and P p P max it converges to a value larger than one and if λ 1 > 1 it converges to λ 1, which was stated. REFERENCES [1] K. Doppler, M. Rinne et al., Device-to-device communication as an underlay to lte-advanced networks, IEEE Communications Magazine, vol. 47, no. 12, pp , Dec [2] A. Asadi, Q. Wang, and V. Mancuso, A survey on device-to-device communication in cellular networks, IEEE Communications Surveys & Tutorials, no. 99, [3] T. L. Mung Chiang, Prashanth Hande and C. W. Tan, Power control in wireless cellular networks, Foundations and Trends in Networking, vol. 2, no. 4, pp , [4] N. Bambos, S. Chen, and D. Mitra, Channel probing for distributed access control in wireless communication networks, in GLOBECOM Proceedings, vol. 1, Nov 1995, pp vol.1. [5] M. Xiao, N. Shroff, and E. Chong, Distributed admission control for power-controlled cellular wireless systems, IEEE/ACM Transactions on Networking, vol. 9, no. 6, pp , Dec [6] C. Zhu and M. Corson, A distributed channel probing scheme for wireless networks, in INFOCOM Proceedings, vol. 1, 2001, pp vol.1. [7] S. Kucera, L. Kucera, and B. Zhang, Efficient distributed algorithms for dynamic access to shared multiuser channels in sinr-constrained wireless networks, IEEE Transactions on Mobile Computing, vol. 11, no. 12, pp , Dec [8] P.-C. Lin, Feasibility problem of channel spatial reuse in powercontrolled wireless communication networks, in WCNC Proceedings, 2013, pp [9] G. Foschini and Z. Miljanic, A simple distributed autonomous power control algorithm and its convergence, IEEE Transactions on Vehicular Technology, vol. 42, no. 4, pp , Nov [10] G. Fodor, D. Della Penda et al., A comparative study of power control approaches for device-to-device communications, in ICC Proceedings, June 2013, pp [11] A. Berman and R. Plemmons, Nonnegative Matrices in the Mathematical Sciences. Society for Industrial and Applied Mathematics, [12] A. Quarteroni, R. Sacco, and F. Saleri, Numerical Mathematics, ser. Texts in Applied Mathematics. Springer Berlin Heidelberg, [13] A. Virdis, G. Stea, and G. Nardini, Simulte: A modular system-level simulator for lte/lte-a networks based on omnet++, proc. of SimulTech, pp , 2014.