# Optimal Base Station Density for Power Efficiency in Cellular Networks

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2 the moile user at the origin is h x x α SINR = σ 2 P + y Φ \{x} h y y, α where σ 2 = F kt B/(λ u K), and x Φ is the BS closest to the origin. The coverage proaility is defined as (P ) = P(SINR > θ), and the spectral efficiency is defined as τ(p ) = E[ln(1 + SINR)]. B. Prolem Formulation With BSs transmitting at power P, we define the area transmit power consumption as P. In addition to the transmission power, each BS requires a fixed amount of operational power which we denote y P. The power P includes the fixed operational power of the hardware and also the transmit power of the pilot tones. Hence the area power consumption is (P + P ). When the transmit power is infinity, it is shown in [3] that the SINR distriution and hence the coverage proaility and spectral efficiency does not depend on the BS density. The rate demanded y each user is denoted y ; it is independent of the ase station density. The interferencelimited spectral efficiency, corresponding to P =, is τ( ). It is independent of the ase station density and depends only on path loss exponent α. So, irrespective of the transmit power, the maximum achievale rate for a particular density is given y τ( )B u ( ). The user demand rate can e satisfied only if it is less than the maximum achievale rate. But if the user demand exceeds the maximum achievale rate, then the est the operator can do is to support a data rate of τ( )B u ( ). So, the operator must satisfy the target rate or maximum achievale rate, whichever is minimum, with a maximum rate outage δ. The limiting coverage proaility that can e achieved for a given SINR threshold θ and path loss exponent α is given y ( ). So, the operator must provide ( ) coverage with a maximum outage of ɛ. Our goal is to otain the optimal parameters λ and transmit power P so that the area power consumption (P + P ) is minimal while maintaining the QoS constraints of rate and coverage. Formally, λ = arg min { (P + P )} s.t. (P ) (1 ɛ) ( ), (1) R(P ) (1 δ) min{, τ( )B u ( )}. (2) We consider oth the rate and coverage constraints for the following two reasons: Even though B u ( ) ln(1 + SINR) might e high, SINR might e low. This might e ecause of a large BW B. Because of the inclusion of, the lower ound on transmit power otained from the two constraints, may scale differently with the BS density as can e seen in Lemma 1 and Lemma 2 and Fig. 3 and Fig. 6. C. Prior Work Energy efficiency in HetNets has received increasing attention recently. Mostly only interference-limited networks have een considered [4], [5]. Different energy saving techniques in communication networks, have een presented in [6]. Energy efficiency in cellular networks has also een analyzed in [7],[8]. However, the authors have focused on a hexagonal grid model with fixed downlink transmit power. In [4], an optimal ase-station density for oth homogeneous and heterogeneous cellular networks has een otained under the constraint of target rate. However, noise has een ignored. Coverage constraints have not een considered, either.energy efficiency in HetNets has een studied in [5], which provides an optimal macro-pico density ratio that maximizes the overall energy efficiency under fixed noise assumption (i.e., noise power does not vary with the allocated andwidth). In this paper, we consider noise and oth target coverage constraints and target rate constraints. We also show how the transmit power has to e scaled down with increase of ase station density. III. OPTIMAL POWER FOARGET COVERAGE AND RATE A. Minimum transmit power for coverage As the BS density increases, the transmit power of the ase stations may e decreased ecause of the decreasing cell size. However, reducing the transmit power, decreases the coverage proaility ecause of the noise. See Fig. 1. In the next lemma we evaluate the transmit power required to achieve (1 ɛ) ( ) coverage. Lemma 1. The minimum downlink transmit power for which a minimum SINR of θ can e guaranteed for a fraction 1 ɛ of the users is given y where k 1 = P c k 1 θσ2 Γ(α/2+1) ɛπ α/2 [1+ρ(θ,α)] α/2. λ α/2 1, (3) Proof: The coverage proaility without noise (or infinite power) is [3] ( ) = (1 + ρ(θ, α)) 1, (4) where ρ(θ, α) = θ 2/α 1 dx. The coverage proaility 1+x α/2 with noise is [3] (P ) = π e π(1+ρ(θ,α))v θσ 2 v α/2 P 1 dv. (5) Now, using the sustitution θσ 2 P 1 s in (5) and the approximation e svα/2 (1 sv α/2 ) for small s (or, equivalently, small σ 2 ), we get ( θσ 2 ) Γ(α/2 + 1) (P ) ( ) 1, (6) P λ α/2 1 [π(ρ(θ, α) + 1)] α/2 where Γ(x) = t x 1 e t dt is the standard gamma function. This approximation (linearization of the exponential term) is

3 1 Coverage proaility vs Transmit power Min transmit power for target coverage vs ase station density 7 Coverage proaility: (P) =1/1 2 m 2, θ= 15dB =1/5 2 m 2, θ= 15dB =1/4 2 m 2, θ= 15dB ( )(θ= 15dB, α=4) =1/1 2 m 2, θ= db =1/5 2 m 2, θ=db =1/4 2 m 2, θ=db ( )(θ=db, α=4) Transmit power: P (dbm) Fig. 1: Effect of transmit power on coverage proaility: User density λ u =.1 m 2, α = 4 and σ 2 = W. min power for target coverage: P c * (dbm) θ=5db, α=3 θ=2db, α=3 θ=5db, α=4 θ=2db, α= Base station density: λ (in m 2 ) 1 2 Fig. 3: Minimum transmit power for target coverage: User density λ u =.1 m 2, Outage proaility ɛ =.25 and σ 2 = W. Tightness of approximation used to evaluate coverage proaility Rate vs transmit power Coverage Proaility: (P) With Approximation for =1 6 m 2 Without Approximation for =1 6 m 2 With Approximation for =1 2 m 2 Without Approximation for =1 2 m Transmit Power P (dbm) Fig. 2: Tightness of numerical approximation e (θσ2 P 1 )v α/2 (1 θσ 2 P 1 v α/2 ) used to evaluate coverage proaility: SINR threshold θ = -3dB, User density λ u =.1 m 2, α = 3 and σ 2 = W. Rate: R (in ps) = 1/1 2 m 2 = 1/5 2 m 2 = 1/2 2 m 2 = 1/7 2 m 2 R 1 T = 4Kps Transmit power: P (dbm) Fig. 4: Effect of noise on rate: User density λ u =.1 m 2, α = 4.5 and σ 2 = W. very tight, as can e seen from Fig. 2. The lower ound Pc on transmit power P can e otained y comining (4) and (6) with the coverage constraint (1). With increasing ase station density, the operator can scale down the transmit power at a rate proportional to λ 1 α/2 (as shown in Fig. 3), while guaranteeing a certain minimum coverage. B. Minimum transmit power for target data rate The effect of noise on rate is shown in Fig. 4. The transmit power can decrease with increasing BS density. However, a minimum transmit power is required to comat the noise and provide average rate to satisfy the user demand or the maximum achievale rate, whichever is minimum. This constraint imposes a limit on the downlink transmit power, which is provided in the next lemma. Lemma 2. The minimum downlink transmit power that achieves a fraction 1 δ of the minimum of the target rate and maximum supported rate τ( )B u ( ) is given y Pr min{h(), k 2 }, (7) λ α/2 1

4 Spectral efficiency: τ (in ps/hz) Tightness of approximation used to evaluate spectral efficiency without approx, =1 3 m 2, α=4 with approx, =1 3 m 2, α=4 without approx, =1 4 m 2, α=4 with approx, =1 4 m 2, α=4 without approx, =1 4 m 2, α=3 with approx, =1 4 m 2, α= Transmit power: P (in dbm) Fig. 5: Tightness of numerical approximation e σ2 P rα (e t 1) [1 σ2 P rα (e t 1)] used to evaluate spectral efficiency: User density λ u =.1m 2, noise power σ 2 = W. Min power for target rate: P r * ( in dbm ) Min transmit power target rate vs ase station density =1Kps, α=3 =1Mps, α=3 =1Gps, α=3 =1Kps, α=4 =1Mps, α=4 =1Gps, α= Base station density λ (m 2 ) Fig. 6: Minimum transmit power for target rate: User density λ u =.1 m 2, total andwidth B = 2 MHz, Outage proaility δ =.25 and noise power σ 2 = W. where k 2 = σ2 g(α) δτ( ), h() = g(α) = π α/2 t> z> e z z α/2 σ 2 g(α) [τ( ) (1 δ){ λu B }], and (e t 1)e z{(et 1) 2/α (e t 1) 2/α 1 1+x α/2 dx} dtdz. Proof: The spectral efficiency τ(p ) has een derived in [3] as τ(p ) E[ln(1 + SINR)] = e πr 2 P r> t> e σ 2 λ rα (e t 1) L Ir (r α (e t 1))dt2π rdr, where L Ir (r α (e t 1)) is the Laplace transform of the interference I r evaluated at r α (e t 1), which is given y ( ) exp π r 2 (e t 1) 2/α 1 dx. (e t 1) 1 + x 2/α α/2 Since σ 2 is very small, we use the approximation e σ2 P rα (e t 1) (1 σ2 P rα (e t 1)) in (8). This approximation is tight as shown in Fig. 5. Sustituting π r 2 z in (8), we otain τ(p ) = τ( ) σ 2 g(α)p 1 λ α/2+1 + o(σ 2 ), σ 2, (9) where τ( ) is the spectral efficiency at zero noise or high power, which is independent of the BS density [3]. A fraction 1 δ of the users must achieve a data rate at least equal to the target rate or maximum achievale rate, whichever is (8) Rate ( in Mps ) Target rate and maximum acheivale rate Vs ase station density Maximum achievale rate Target rate or user demand Base station density: ( in m 2 ) Fig. 7: Maximum achievale rate and target rate (user demand): User density λ u =.1 m 2, α = 4, total andwidth B = 2 MHz and target rate (user demand) = 5 kps. minimum. This, in turn, imposes a constraint on the spectral efficiency τ(p ): { } RT τ(p ) (1 δ) min B u ( ), τ( ). (1) Comining (1) and (9) we get a lower ound P r on the transmit power P. Depending on the target data rate, the transmit power of the ase stations can e scaled down (under target data rate constraint) with the increase of small cell density, at a rate proportional to λ 1 α 2 or faster than that as shown in Fig. 6. IV. OPTIMUM BASE STATION DENSITY To ensure QoS (oth coverage and user demand rate), the minimum downlink transmit power should e P m=max(p c, P r ). So, the total power consumption y a ase station is P m + P. The area power consumption follows as Λ 1 Λ 2

5 P A ( ) = (P m + P ) = (max(p c, P r ) + P ). Using (3) and (7), we get P A ( ) = max{k 1, min{h( ), k 2 }} λ α/2 2 + P. (11) The optimal λ is then otained y setting dp A d =. Theorem 1. When α > 4, the optimum ase station density λ that minimizes the area power consumption under coverage and rate constraints is given y where and [ (α 4) max{k1, k 2 } ] 2 λ = 2P [ (α 4)k1 ] 2 2P R 2, R 1, [ (α 4)(Bτ( )) α/2 1 ] 2 k 1 R 1 =, 2P λ α/2 1 u [ (α 4)(Bτ( )) α/2 1 ] 2 max{k 1, k 2 } R 2 =. 2P λ α/2 1 u Proof: We egin the proof y considering 3 different ranges of. We define Λ 1 = λ u Bτ( ) and Λ k 2 = 1c 2 k 1τ( ) c 1. λ We also re-write h( ) as h( ) = c 1 τ( ) c 2. The demand rate is at least equal to the maximum achievale rate B u ( )τ( ) only when (, Λ 1 ], as can e seen in Fig. 7. Re-arranging the expressions of h( ), Λ 1, Λ 2 and k 2 we get k 2 (, Λ 1 ], min{h( ), k 2 } = h( ) (Λ 1, Λ 2 ), h( ) [Λ 2, ). Re-arranging the expressions of Λ 2, k 1 and h( ), it can e shown that, k 1 h( ) only for [Λ 2, ). Therefore, the area power consumption given in (11) can e re-written as max{k 1, k 2 } + P λ α/2 2 (, Λ 1 ], h( ) P A = + P (Λ 1, Λ 2 ), (12) λ α/2 2 k 1 + P λ α/2 2 [Λ 2, ). Area power consumption: P A (in W/m 2 ) Area power consumption: analytical and simulation results α=6 α=5 α=4 α= Base station density: λ (in m 2 ) 1 Fig. 8: Area power consumption: Target rate = 5Mps, SINR threshold θ = 12 db, User density λ u =.1 m 2, Outage proaility ɛ =.25, Rate outage δ =.25, P = 1.5 W and noise power σ 2 = W. The dashed lines correspond to our analysis while the old lines are otained y Monte Carlo simulation. Re-arranging the expressions of R 1, R 2, Λ 1, Λ 2 and using the When R 1 < < R 2, the optimal ase station density λ is derivative of (12), for α > 4, we otain the solution to the following equation dp A (Λ 1 ) λ α/2 2 (2P c 2 2) λ α/2 1 (4P τ( )c 2 ) + 2λ α/2 P (τ( )) 2 R 2, d dp A (Λ 1 ) λ <, dp A(Λ 2 ) > R ((α 4)c 1 τ( )) + ((α 6)c 1 c 2 ) =, 1 < < R 2, d d dp where c 1 = σ 2 g(α), c 2 = (1 δ) λ u A (Λ 2 ) R B. There exists no T R 1. d optimum density for α 4. Therefore for α > 4, (, Λ 1 ] R 2, λ (Λ 1, Λ 2 ) R 1 < < R 2, (13) [Λ 2, ) R 1. Now, for α > 4, optimal ase station density λ can e otained y setting the derivative of (12) to zero. When α 4, the total area transmit power consumption Pm as well as the total area fixed power consumption P increase with ase station density, for (, Λ 1 ] and [Λ 2, ), see (12). Hence, in this case, there exists no optimal density. The area power consumption always increases with ase station density for α = 3 and α = 4 as can e seen in Fig. 8. The simulation curves are otained using the Monte Carlo method. This result indicates that cell size shrinking or dense small cell deployment is not power-efficient for these values of the path loss exponent. Fig. 8 also shows the area power consumption and the optimal ase station densities for α = 5 and α = 6. This result indicates that small cell deployment can e power efficient in uran areas (where it is possile that α > 4).

6 V. CONCLUSION In this paper, we have derived a lower ound on downlink transmit power in a homogeneous Poisson networks under target coverage and target data rate constraints. For a path loss exponent α > 4, we have proven the existence of an optimal deployment density after which further reduction of the cell size is not efficient from an energy viewpoint. Our results indicate that, for α 4, counter-intuitively, smaller BS densities lead to improved area power consumption. In other words, increasing the BS density is not green for α 4. REFERENCES [1] M. Haenggi, Stochastic geometry for wireless networks. Camridge University Press, 212. [2] V. Chandrasekhar, J. Andrews, and A. Gatherer, Femtocell networks: a survey, Communications Magazine, IEEE, vol. 46, no. 9, pp , 28. [3] J. G. Andrews, F. Baccelli, and R. K. Ganti, A tractale approach to coverage and rate in cellular networks, IEEE Trans. on Communications, vol. 59, no. 11, pp , Nov [4] D. Cao, S. Zhou, and Z. Niu, Optimal ase station density for energyefficient heterogeneous cellular networks, International Conference on Communications (ICC), IEEE, pp , 212. [5] T. Q. Quek, W. C. Cheung, and M. Kountouris, Energy efficiency analysis of two-tier heterogeneous networks, in 11th European Wireless Conference 211-Sustainale Wireless Technologies (European Wireless). VDE, 211, pp [6] X. Wang, A. V. Vasilakos, M. Chen, Y. Liu, and T. T. Kwon, A survey of green moile networks: Opportunities and challenges, Moile Networks and Applications, vol. 17, no. 1, pp. 4 2, 212. [7] F. Richter and G. Fettweis, Cellular moile network densification utilizing micro ase stations, in International Conference on Communications (ICC). IEEE, 21, pp [8] F. Richter, A. J. Fehske, P. Marsch, and G. P. Fettweis, Traffic demand and energy efficiency in heterogeneous cellular moile radio networks, in 71st Vehicular Technology Conference (VTC). IEEE, 21-Spring, pp. 1 6.

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