Outage Probability for Joint Processing Coordinated Multi-Point (JP-CoMP) Performance Analysis in LTE-Advanced Networks

Transcription

1 Outage Probability for Joint Processing Coordinated Multi-Point (JP-CoMP) Performance Analysis in LTE-Advanced Networks Marceau Coupechoux TELECOM ParisTech, INFRES/RMS joint work with D. Ben Cheikh (Orange Labs/TPT), J.M. Kelif (Orange Labs) and Ph. Godlewski (TPT) Indian Institute of Science, ECE Seminar 09 Mar M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

2 LTE and LTE-Advanced Cellular Networks Evolution (3GPP) I UMTS R99 UMTS R4 HSDPA R5 HSUPA R6 HSPA+ R7 HSPA+ R8 WCDMA IP core network DL improvements: 16 QAM AMC/HARQ Fast scheduling MAC in Node-B UL improvements: DL improvements: UL improvements: AMC/HARQ Fast scheduling MAC in Node-B MBMS (DL) MIMO 64 QAM L3 improvements L2 improvements 16 QAM L2 improvements Voice over HSPA LTE R8 LTE R9 LTE-A R10 LTE-A R11 OFDMA (DL) SC-FDMA (UL) MIMO AMC/HARQ 64 QAM (DL) 16 QAM (UL) MBSFN MIMO improv. MIMO improv. Relays Carrier aggregation Het Nets CoMP Figure: Evolution of the 3GPP Standards. M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

3 LTE and LTE-Advanced Cellular Networks Evolution (3GPP) II M. Coupechoux and Ph. Martins, Vers la quatrième génération (Towards Fourth Generation), Springer Verlag, 315 pp, to appear (hopefully) mid M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

4 LTE LTE and LTE-Advanced LTE Requirements: Peak data rates: 300 (DL) / 75 Mbps (UL) Peak spectrum efficiency: 15 (DL) / 3.75 bps/hz (UL) Average spectrum efficiency: 2.67 (4x4 DL) / 0.74 (UL 1x2) bps/hz/cell U-plane latency < 5 ms Spectrum flexibility: from 1.4 to 20 MHz LTE Key Features: OFDMA (DL) / SC-FDMA (UL) Adaptive Modulation (up to 64-QAM) and Coding / HARQ MIMO (up to 4x4): Spatial Multiplexing, Transmit Diversity, Beamforming, MU-MIMO Frequency selective fast scheduling IP-based flat architecture M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

5 LTE and LTE-Advanced LTE-Advanced Release 10 LTE-Advanced Requirements: Peak data rates: 1 Gbps (DL) / 500 Mbps (UL) Peak spectrum efficiency: 30 (DL) / 15 bps/hz (UL) Average spectrum efficiency: 3.7 (4x4 DL) / 2.0 (UL 2x4) bps/hz/cell LTE-Advanced Rel. 10 Features: Carrier aggregation (up to 100 MHz) Clustered SC-FDMA (UL) MIMO: 8x8 (DL), 4x8 (UL), enhanced MU-MIMO Relaying Heterogeneous Networks (HetNets) M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

6 LTE and LTE-Advanced Coordinated Multi-Point for LTE-A Rel. 11 Goal: coordinate BS transmissions in order to reduce interference and improve cell-edge throughput. CS/CB: scheduling and beamforming are coordinated among clustered BS. JP: data is transmitted from several BS to the same UE coherently (tight synchronization is needed) or non-coherently (gain obtained from power boost). Figure: Coordinated Multi-Point (CoMP). M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

7 LTE and LTE-Advanced Outage Probability and SIR CDF SINR/SIR CDF is a crucial input for cellular network dimensioning: Coverage studies: P[γ < γ th ], where γ th is the the common control channel SIR threshold provides coverage; γ th s.t. P[γ < γ th ] = 10% provides cell-edge throughput. Capacity studies: channel state distribution + M/G/1-PS (or other Markov chain [25]) approach provides capacity figures. probability probability SIR threshold Coverage studies SIR threshold cell edge throughput coverage probability Capacity studies Figure: Outage probability and dimensioning issues. = M/G/1-PS queue M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

8 LTE and LTE-Advanced Outlines Related Work Maximum Ratio Transmission System Model Outage Probability Numerical Results M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

9 Related Work I Related Work Performance of CoMP has been asserted in several works: The JP strategy offers larger performance gain than the CS/CB [1, 2]. Multicell cooperation attain larger capacity than an isolated cell (measurement study [3]). A field trial [4] confirms the throughput enhancement introduced by JP-CoMP. In [5], different JP schemes are proposed and compared numerically. In [6], zero forcing (ZF) and maximum ratio transmission (MRT) schemes are studied for femto cell coordination. Two power allocation algorithms are proposed and compared. In [3, 4, 5, 6], authors performed simulation, measurement or field study but no theoretical studies were conducted. M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

10 Related Work Related Work II There are few analytical works: In [7], an analytical expression of the capacity outage probability was derived for an open-loop Alamouti-like CoMP downlink transmission in Rayleigh fading. The proposed SINR expression can only be achieved when using a distributed Alamouti for two cooperating BSs. In [8], an analytical study of a multi-cell multi-antenna cooperative MRT/MRC scheme was conducted. An analytical expression of the PDF of the SIR was derived considering path-loss, shadowing and Rayleigh fading. Limitations of the study are: User is supposed to be at equal distances from cooperative BSs, Shadowing is modeled with a Gamma distribution, Transmitters are supposed to be Poisson distributed, Significant differences bw. simulations and theoretical results. Several works on feedback constraints. Several works on BS selection algorithms. M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

11 Maximum Ratio Transmission Maximum Ratio Transmission I Assume n t Tx antennas and n r Rx antennas without interference, then the received signal is: y = Hws + n, where H is the n r n t channel gain matrix, w is the n t 1 precoding vector with w 2 1, s is the information symbol, n is AWGN. n t transmit antennas n r receive antennas Figure: MRT Model. M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

12 Maximum Ratio Transmission Maximum Ratio Transmission II This equation can be interpreted as an AWGN channel with SNR: γ = Hw 2 σn 2 E[ s 2 ]. The scheme that maximizes Hw 2 under the constraint w 2 1 is called MRT. In general: Hw 2 w 2 λ max (H H H), and the optimal precoder is the normalized eigenvector associated to the largest eigenvalue of H H H. M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

13 Maximum Ratio Transmission Maximum Ratio Transmission III If n r = 1, then H = h and using Cauchy-Schwartz inequality: hw 2 h 2 w 2 = h 2, with equality if w is proportional to h. The MRT precoder can thus be written: w MRT = h h. MRT maximizes the received signal power. Drawbacks of the scheme: Interference is not taken into account. Perfect channel information is required at transmitter. M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

14 System Model System Model I We consider the following system model: Interfering cluster M Tx antennas N BSs/cluster K users B BSs in the network user k Cluster B k Figure: System Model. M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

15 System Model II System Model We assume that JP-CoMP is based on MRT [14]. Received signal can be written: where y k = b B k pb,k h b,k x b,k + K i=1,i k j B i pj,k h j,k x j,i + n, (1) p b,k is the power received by user k from BS b, n is the AWGN, h b,k C 1 M is the complex Gaussian channel bw. k and b, x b,k C M 1 is the MRT data vector: x b,k = h b,k h b,k s k. (2) M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

16 System Model III System Model The received power p b,k includes path-loss and shadowing: p b,k = P Tx Cr η b,k 10 ξ b,k 10, (3) where P Tx is the total transmit power of BS b, C is a constant, r b,k is the distance bw. k and b, η is the path-loss exponent and ξ b,k is N(0,σ). Shadowing is now assumed to be constant. The output SIR perceived by a user k is given by: γ k = ( 2 b Bk pb,k h b,k ) j B pj,k h h j,i j,k i K i=1,i k h j,i 2 = X Y. (4) M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

17 Outage Probability I Useful signal CDF Outage Probability We can write: X = U 2, U = b B k pb,k h b,k. (5) U can also be written as: U = pb,k M h b,k,i 2. (6) b B k Closed form expression of the PDF of U is not known, we thus rely on the central limit theorem for causal functions. i=1 M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

18 Outage Probability Outage Probability II Theorem (CLCF) Consider n RV X i 0 with density f i, means η i and variances σ 2 i. Let X = X X n with density f (t) = f 1 (t) f 2 (t) f n (t). Denote Under certain conditions, f (t) where α + 1 = η2 σ 2 and β = σ2 η. Proof. see [13]. η = η 1 + η η n, σ 2 = σ σ σ 2 n. tα e t/β β α+1 Γ(α+1) as n, M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

19 Outage Probability Outage Probability III The use of the CLCF provides: f U (u) = uν 1 e u θ Γ(ν)θ ν. (7) where ν = E[U]2 var(u) M var(u) and θ = E[U]. Denoting V = i=1 h b,k,i 2, V is a square root of a Gamma RV and thus we can write the PDF of V as: f V (v) = E[V ] can be derived using (8) and is given by: E[V ] = 0 2 (M 1)! v2m 1 e v2, (8) 2 (M 1)! v2m e v2 dv = (2M 1)!! π, (9) 2 M (M 1)! M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

20 Outage Probability Outage Probability IV So that: E[U] = (2M 1)!! π pb,k 2 M. (10) (M 1)! b B k On the other hand: var(u) = p b,k E[ h b,k 2 ] p b,k E[ h b,k ] 2 b B k b B k (2M 1)!! = (M π( 2 M (M 1)! )2 ) p b,k. (11) b B k Note: (2N)!! = 2 4 (2N) and (2N + 1)!! = 3 5 (2N + 1). M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

21 Outage Probability V Outage Probability The parameters ν and θ are thus given by: ν = (2M 1)!! 2 ( b B pb,k k ) 2 M2 2M (M 1)! 2 π(2m 1)!! 2, b B k p b,k θ = M22M (M 1)! 2 π(2m 1)!! 2 2 M (M 1)!(2M 1)!! π From (10) and (11): Mean useful signal power increases with N. Variability also increases with N. (12) b B k p b,k. (13) b B pb,k k It can be shown that mean and variance increase with M. M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

22 Outage Probability Outage Probability VI Using the Gamma approximation of the PDF of U, we can derive the CDF of X as follows: F X (v) = v 0 = γ(ν, v Γ(ν) u ν 1 e u θ du, (14) Γ(ν)θν θ ) where γ(.,.) is the lower incomplete gamma function: γ(a,x) = x 0, (15) t a 1 e t dt. M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

23 Outage Probability Outage Probability VII Interference PDF We can write now: K Y = i=1,i k pj,k g j,k,i j B i 2, (16) where g j,k,i = h j,k h j,i h j,i. g j,k,i is complex Gaussian independent of h j,i. M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

24 Outage Probability Outage Probability VIII Claim ([15]) Consider zero-mean complex Gaussian vectors h 0 = [h 1,0,h 2,0,,h N,0 ] H and h j = [h 1,j,h 2,j,,h N,j ] H. Let g j = hh 0 h j h 0. Then, g j is independent on h 0. Proof. g j h 0 is zero-mean complex Gaussian and: E[g j h 0 ] = hh 0 h 0 E[h j] = 0 E[ g j 2 h 0 ] = hh 0 E[h j h H j ]h 0 h 0 2 = hh 0 I N h 0 h 0 2 = 1 M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

25 Outage Probability Outage Probability IX Let c k,i = j B i pj,k g j,k,i, it is the sum of independent complex Gaussian random variables. Y can be written as: Y = K i=1,i k c k,i 2, (17) {c k,i } (i=1...k,i k) being independent zero-mean complex Gaussian elements with variances: λ i,k = var(c k,i ) = j B i p j,k. (18) M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

26 Outage Probability X Outage Probability The PDF of Y is, hence, given by [16]: f Y (y) = K i=1,i k λ p,k λ p,k λ i,k. Π i exp( y ), (19) λ i,k λ i,k where Π i = p=1..k,p i,p k Having the PDF of X and the PDF of Y, and since they are independent RV, we derive the outage probability as follows: P(γ k < γ th ) = F X (γ th y)f Y (y)dy, (20) = K i=1,i k Π i λ i,k Γ(ν) exp( y γth y )γ(ν, )dy. (21) λ i θ M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

27 Outage Probability Outage Probability XI The outage probability is given by [17] [18]: P(γ k < γ th )= K i=1,i k ( ) ν γth λ i,k Π i U 2θ ( ν 2, 1 2, γ ) thλ i,k 4θ 2 (CoMP) where U(.,.,.) is the confluent hypergeometric function of second kind: U(a,b,z) = 1 Γ(a) 0 e zt t a 1 (1 + t) b a 1 dt. M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

28 Numerical Results I Numerical Results Figure: Comparison between simulated and analytical results of the outage probability without CoMP and with CoMP MRT strategy for 3, 4 or 5 cooperating BSs. M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

29 Numerical Results II Numerical Results Figure: Influence of the number of transmit antennas. M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

30 Numerical Results III Numerical Results Figure: Impact of the noise power on the outage probability. M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

31 Numerical Results IV Numerical Results Figure: Comparison with an isolated cell. M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

32 Other Obtained Results Other Obtained Results I OFDMA Cellular networks [19]: P ( [ log2 (1 + δ) µ MIC SIR eff < δ) = 1 Q σ MIC ]. (OFDMA) µ MIC = E[C n ], (22) σmic 2 = 1 (E[C N n] 2 E[C n ] 2 ). (23) sc E[C n ] = (1 Q f (x,t)) e x dxdt, (24) E[Cn 2 ] = 2t (1 Q f (x,t)) e x dxdt. (25) [ 10log10 ( x 2 Q f (x,t) = Q t 1 ) m ] f. (26) s f M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

33 Other Obtained Results Other Obtained Results II analysis simulation Outage probability r = R c r = R c / r = R c r = R c / δ [db] δ [db] Figure: Outage probability at cell edge (r = R c ) and inside the cell (r = R c /2); comparison between analysis (solid curves) and simulations (dotted curves) on a hexagonal network (N sc = 48, η = 3, σ = 6 db). M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

34 Other Obtained Results Other Obtained Results III 10 5 multiple carrier analysis simulation SIR threshold [db] for 2% outage single carrier Shadowing standard deviation (σ ) [db] Figure: SIR threshold δ in the cell (r = R c /2) for 2% outage probability as a function of the shadowing standard deviation; comparison between analysis (solid curves) and simulations (dotted curves) on a hexagonal network (N sc = 48, η = 3). M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

35 Other Obtained Results Other Obtained Results IV MIMO Alamouti transmission [20]: ( P out (γ th ) = 1 p 0 2γ th β+p 0 ) α ( 1+ 2γ ) thβ Γ(α+1). (ALAM) 2γ th β+p 0 Γ(α) α = 2 ( B j=1 p j) 2, (27) 1 + ρ z B j=1 p2 j β = 1 + ρ z 2 B j=1 p2 j B j=1 p. (28) j ρ z = (29) M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

36 Other Obtained Results Other Obtained Results V Figure: P out versus SINR threshold for 2 1 MISO Alamouti and 2 2 MIMO Alamouti with MRC receiver systems in case of equal (a) and unequal (b) received interference power (EP and UEP resp. in legends). M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

37 Other Obtained Results Other Obtained Results VI MU-MIMO Zero-Forcing/Beamforming [21]: P out (γ th ) = (ZF) Γ(ν + α) Γ(ν)µ ν α! p ν 0 (Kγ th) α (γ th K + p 0 γ th K µ ) ν+α 2 F 1 (1,ν+ α;α + 1; γ th K + p ) 0 µ where 2 F 1 (.,.;.;.) is the Gauss hypergeometric function [17], ν = (1 2(2α 1)θ2 (α 1) 2 (α 2) (5α 4)θ 2 +2(α 1)(α 2)θ ) ( B j=1 p j) 2 B j=1 p2 j (30) µ = (5α 4)θ 2 +2(α 1)(α 2)θ (α 1)(α 2)θ+(α 1) 2 (α 2) B j=1 p2 j B j=1 p j (31) α = M K + 1, θ = (1 ρ 2 ), ρ = (32) M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

38 Other Obtained Results Other Obtained Results VII Figure: Outage probability of a zero forcing multicellular precoded system (4 users, 6 antennas at the BS). M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

39 Conclusion Conclusion We have derived an approximate closed form expression of the SIR CDF. Further work: Consider more realistic scenarios Compare with other beamforming schemes Consider dynamic studies Outage probabilities can be efficiently used for solving various dimensioning issues: Effect of network densification [22] Limiting transmit power in green cellular networks [23] Perf. evaluation of frequency reuse schemes in OFDMA networks [24] Dynamic performance study of WiMAX networks [25] M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

40 References I Conclusion [1] J. Li, E. Lu, and I-Tai Lu. Performance Benchmark for Network MIMO Systems: A Unified Approach for MMSE Transceiver Design and Performance Analysis. In Proc. IEEE GLOBECOM, pages 1 6, Dec [2] Y. Rui, M. Li, P. Chengo, Y. Luo, and A. Guo. Achievable Rates of Coordinated Multi-Point Transmission Schemes under Imperfect CSI. In Proc. IEEE ICC, pages 1 6, [3] V. Jungnickel and al. Capacity Measurements in a Cooperative MIMO Network. IEEE Trans. on Veh. Tech., 58(5): , Jun M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

41 References II Conclusion [4] P. Marsch, M. Grieger, and G. Fettweis. Large Scale Field Trial Results on Different Uplink Coordinated Multi-Point (CoMP) Concepts in an Urban Environment. In Proc. IEEE WCNC, pages , Mar [5] H. Zhang, H. Dai, and Q. Zhou. Base Station Cooperation for Multiuser MIMO: Joint Transmission and BS Selection. In Proc. Conf. Inform. Sciences and Sys., Mar [6] S. Ben Halima, M. Helard, and D.T. Phan-Huy. New Coordination and Resource Allocation Schemes for Uniform Rate in Femtocell Networks. In Proc. IEEE VTC-Spring, pages 1 5, May M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

42 References III Conclusion [7] V. Garcia, N. Lebedev, and J.-M. Gorce. Capacity Outage Probability for Multi-Cell Processing Under Rayleigh Fading. IEEE Commun. Lett., 15(8): , Aug [8] X. Ge, K. Huang, C-X. Wang, X. Hong, and X. Yang. Capacity Analysis of a Multi-Cell Multi-Antenna Cooperative Cellular Network with Co-Channel Interference. IEEE Trans. on Wireless Commun., 10(10): , Oct [9] J.-M. Kélif, M. Coupechoux, and Ph. Godlewski. A fluid model for performance analysis in cellular networks. EURASIP Journal on Wireless Communications and Networking, 2010, July M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

43 References IV Conclusion [10] J. M. Kélif, M. Coupechoux, and Ph. Godlewski. Effect of shadowing on outage probability in fluid cellular radio networks. In Intl. Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks, WiOpt, Berlin, Germany, March [11] J. M. Kélif and M. Coupechoux. Impact of topology and shadowing on the outage probability of cellular networks. In IEEE International Conference on Communications, ICC, Dresden, Germany, June M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

44 References V Conclusion [12] D. Ben Cheikh, J.-M. Kélif, M. Coupechoux, and Ph. Godlewski. Sir distribution analysis in cellular networks considering the joint impact of path-loss, shadowing and fast fading. EURASIP Journal on Wireless Communications and Networking, 2011(137), October [13] A. Papoulis. The Fourier Integral and Its Applications. McGraw-Hill, New York, NY, [14] Lo, T.K.Y. Maximum Ratio Transmission. In Proc. IEEE ICC, M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

45 References VI Conclusion [15] A. Shah and A. Haimovich. Performance Analysis of Maximum Ratio Combining and Comparison with Optimum Combining for Mobile Radio Communications with Cochannel Interference. IEEE Tran. on Veh. Tech., 49(4): , Jul [16] D. Ben Cheikh Battikh, J.-M. Kelif, F.A. Abdallah, and Dinh-Thuy Phan-Huy. Time Reversal Outage Probability for Wideband Indoor Wireless Communications. In Proc. IEEE PIMRC, pages , Sept [17] I.S. Gradshteyn and I.W. Ryzhik. Table of integrals series and products. Academic Press, Moscow, M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

46 References VII Conclusion [18] D. Ben Cheikh, J.-M. Kélif, M. Coupechoux, and Ph. Godlewski. Analytical Joint Processing Multi-Point Cooperation Performance in Rayleigh Fading. submitted, [19] J. M. Kélif, M. Coupechoux, and Ph. Godlewski. On the Dimensioning of Cellular OFDMA Networks. Elsevier Physical Communication, 2012 (in press). [20] D. Ben Cheikh, J.-M. Kélif, M. Coupechoux, and Ph. Godlewski. Outage probability in a multi-cellular network using alamouti scheme. In IEEE Sarnoff Symposium, Princeton, USA, April M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

47 References VIII Conclusion [21] D. Ben Cheikh, J.-M. Kélif, M. Coupechoux, and Ph. Godlewski. Multicellular zero forcing precoding performance in rayleigh and shadow fading. In IEEE Vehicular Technology Conference, VTC Spring, Budapest, Hungary, May [22] J. M. Kélif and M. Coupechoux. Cell breathing, sectorization and densification in cellular networks. In Intl. Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks, WiOpt, Seoul, Korea, June [23] J.-M. Kélif, M. Coupechoux, and F. Marache. Limiting power transmission of green cellular networks: Impact on coverage and capacity. In IEEE International Conference on Communications, ICC, Cape Town, South Africa, May M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48

48 References IX Conclusion [24] M. Maqbool, Ph. Godlewski, M. Coupechoux, and J.-M. Kélif. Analytical performance evaluation of various frequency reuse and scheduling schemes in cellular ofdma networks. Performance Evaluation, 67(4): , April [25] S. Doirieux, B. Baynat, M. Maqbool, and M. Coupechoux. An efficient analytical model for the dimensioning of wimax networks supporting multi-profile best effort traffic. Elsevier Computer Communications, 33(10): , June M. Coupechoux (TPT) Outage Prob. for JP-CoMP 09 Mar / 48