On the Asymptotic Behavior of Ultra-Densification under a Bounded Dual-Slope Path Loss Model
OOn the Asymptotic Behavior of Ultra-Densificationunder a Bounded Dual-Slope Path Loss Model
Yanpeng Yang, Jihong Park † and Ki Won Sung KTH Royal Institute of Technology, Wireless@KTH, Stockholm, SwedenE-mail: [email protected], [email protected] † Dept. of Electronic Systems, Aalborg University, DenmarkE-mail: [email protected]
Abstract —In this paper, we investigate the impact of networkdensification on the performance in terms of downlink signal-to-interference (SIR) coverage probability and network areaspectral efficiency (ASE). A sophisticated bounded dual-slopepath loss model and practical user equipment (UE) densitiesare incorporated in the analysis, which have never been jointlyconsidered before. By using stochastic geometry, we derive anintegral expression along with closed-form bounds of the coverageprobability and ASE, validated by simulation results. Throughthese, we provide the asymptotic behavior of ultra-densification.The coverage probability and ASE have non-zero convergence inasymptotic regions unless UE density goes to infinity (full load).Meanwhile, the effect of UE density on the coverage probabilityis analyzed. The coverage probability will reveal an U-shape forlarge UE densities due to interference fall into the near-field, butit will keep increasing for low UE densites. Furthermore, ourresults indicate that the performance is overestimated withoutapplying the bounded dual-slope path loss model. The derivedexpressions and results in this work pave the way for futurenetwork provisioning.
Index Terms —Network densification, bounded path loss model,dual-slope path loss model, stochastic geometry
I. I
NTRODUCTION
Network densification is considered as a key enabler tocope with the upcoming 5G data tsunami [1] [2]. Deployingmore base stations (BSs) can rapidly increase the networkcapacity by shortening the BS and user equipment (UE)association distance as well as by reducing per-cell trafficload. As densification goes on, BS density may easily exceedUE density, forming an ultra dense network (UDN) [3]. Itssimplest example can be off-peak traffic hours under denseBS deployment. Peak hours can also be suitable cases sinceaverage BS load in practice is only 20% due to networkstability [4]. In the UDN, a large number of UE-void BSswithin their coverages emerge, and the overall network transitsfrom being fully loaded to partially loaded in which not allBSs are active (i.e., transmitting signals to serve the UEswithin their cells). Such a UDN may evaporate the advantageof densification since there is less than one UE per cell onaverage. Therefore, it is crucial to understand the asymptoticbehavior of ultra densification for the purpose of networkdeployment.The pioneering work [5] provides a comprehensive un-derstanding on the impact of BS density in a fully loadeddownlink cellular network with a simple single-slope un- S I R C o v e r a g e P r o b a b ili t y BS Density10 Zero convergenceUnity convergence
Unity-zero interval saturation (a) Bounded path loss [7] (b) Dual-slope path loss [8], [9] (c) Partial load (inactive BS) [10], [11]
Baseline (b) + (c) [6](a) + (b) + (c)
Fig. 1: Impacts of ( a ) bounded path loss, ( b ) dual-slope pathloss, and ( c ) partial load models on asymptotic SIR coverageprobability as BS density increases. bounded path loss model. Illustrated in Fig. 1 as a baseline,it concludes that BS densification does not change the signal-to-interference ratio ( SIR ) of an individual UE, but linearlyimproves the area spectral efficiency (
ASE ) defined as sumrate per unit area. As BS density grows, the desired signal andinterference growths cancel each other, leading to such result.However, it is difficult to apply this conclusion to UDNs dueto its simplified signal propagation and load models [6] [7].In a UDN where BS-UE distance d shrinks, a simpleunbounded path loss model d − α for the path loss exponent α > may amplify the received signals when d < ,which is unrealistic. In addition, a large amount of signalsin a UDN are transmitted from the near-field of a receiver,and these signals experience less attenuation owing to sparseshadowing, i.e. near-field path loss exponent α c < α . Asimple single-slope path loss model cannot capture such adistinction. Furthermore, a full load model forces BSs toalways transmit signals despite non-negligible portion of UE-void BSs, overestimating interference. It is thus important to Minor modification is applied for [8] that considers signal-to-interference-plus-noise ratio instead of
SIR . a r X i v : . [ c s . N I] A p r ncorporate such propagation and load characteristics in detailso as to examine the impact of ultra densification. For this endour system model considers the tri-fold aspect: ( a ) boundedpath loss, ( b ) dual-slope path loss and ( c ) partial load models ,as illustrated in Figs. 1 and 2.In preceding works, the impacts of ( a ) and ( b ) on SIR coverage probability, defined as
Pr (
SIR > t ) for a targetthreshold t > , are respectively investigated by [7] and [9],[10]. Both models leads to a conclusion that SIR coverageprobability asymptotically converges to (i.e. SIR → )as BS density increases. The reason is as follows: as BSdensity grows, interference keeps increasing while the desiredsignal increase is saturated under (a); the number of near-field interferer increases under (b), dominating the increase inthe desired signal from ‘a single’ BS. On the other hand, theimpact of ( c ) is clarified in [11], [12], showing SIR coverageprobability asymptotically converges to , i.e. SIR → ∞ .It comes from the fact that UEs’ neighboring BSs under anearest association rule are always active while the rest ofBSs become inactive. This results in making interfering BSdensity converge to UE density while the desired signal keepsincreasing. A recent work [8] combines both ( b ) and ( c ) ,and interestingly concludes that SIR coverage probability stillconverges to asymptotically since ( c ) dominates ( b ) .Motivated by the discussions, we aim to combine ( a ) , ( b ) ,and ( c ) altogether, and investigate their aggregate impact onthe asymptotic SIR coverage behavior. The main contributionsof this paper are listed below. • Asymptotic unity-zero interval
SIR coverage saturationis derived, which also leads to the same ASE saturation.This verifies combining ( a ) and ( b ) exactly cancels out ( c ) (Prop. 3 and 4). • Numerically tractable integral-form of coverage proba-bility under a bounded dual-slope path loss model arederived (1). Moreover, closed-form bounds of coverageprobability and ASE are provided (Prop. 2 and 5). • The impact of UE density on coverage probability andASE are analyzed (Fig. 3). Meanwhile, the trend ofcoverage probability and ASE are interpreted (Fig. 4 andFig. 5). The scaling trend of ASE in terms of BS densityis derived (Prop. 5).II. S
YSTEM M ODEL
We consider a downlink cellular network where BSs andUEs are distributed according to two independent homo-geneous Poisson Point Processes (PPPs) Φ b and Φ u . Thedensities of BS and UE are denoted as λ b and λ u respectively.We assume each UE is associated with its closest BS whosecoverage area comprises a Voronoi tessellation as shown inFig. 2. Each BS becomes inactive without transmitting anysignal when its coverage area, the Voronoi cell, is empty ofactive UEs. Correspondingly, each active BS has at least oneUE in its cell and will randomly choose one of them to serve. Near-field environment in [10] is confined to α c < α in this article, whichoriginally considers a general α c > . BS active probability(c)
Partial load model p a UEActive BSInactive BS
Baseline. Unbounded single-slope path loss model + Full load modelbounded region radius(a)
Bounded path loss model R c : near-field region radius ↵ c : near-field path loss exponent(b) Dual-slope path loss modelnear-field R b Fig. 2: Network layout of traditional fully loaded network (top)and partially loaded UDN (bottom) as well as illustration ofthe bounded dual-slope path loss model (middle).According to [13], the probability that a BS becomes activeis given as p a ≈ − (cid:18) λ u . λ b (cid:19) − . (1)which allows us to incorporate the partial load model.Both BS and UE are equipped with a single antenna and BSstransmit with unit power. Rayleigh fading is used to model thechannel gain, with the fading coefficients h are i.i.d zero meanunit variance complex normal distributed random variables.Since we will focus on the asymptotic behavior and the systemis interference-limited in dense networks, we will neglect thenoise power and examine SIR throughout the paper.We consider that path loss attenuation from a BS to a UEis distinguished under two different regions, near-field and far-field at the UE as illustrated in Fig. 2. In a near-field withinradius R c , a transmitted signal experiences less absorption anddiffraction, so the near-field path loss exponent α c becomesless than the far-field exponent α > . In the innermostnear field, the transmitted signal becomes no longer attenuatedwithin radius R b because of the physical volume of the UE.This dual-slope path loss (cid:96) ( α, α c , d ) can be formulated as apiece-wise function of the propagation distance d , shown as (cid:96) ( α, α c , d ) = , ≤ d ≤ R b ; d − α c , R b < d ≤ R c ; τ d − α , d > R c (2)here R b > is the radius of bounded path loss region, i.e.the path loss in the range of [0 , R b ] is assumed constant; τ (cid:44) R cα − α c ; R c ≥ R b is the critical distance to divide the near-and far-field; and α c and α are the near- and far-field path lossexponents for < α c < α , respectively. A. Performance Metrics
In this paper, we will focus on two performance metricsfrom both user and network perspectives:
SIR coverage prob-ability of a typical UE and the
ASE of the network. Weanalyze the performance of a typical user located at the origin o randomly selected by the BS, which is permissible in ahomogeneous PPP by Slivnyak’s theorem [14]. The SIR ofa typical user denoted as 0 can be expressed as SIR = | h , | (cid:96) ( d , ) (cid:88) i ∈ Φ ∗ b \{ } | h i, | (cid:96) ( d i, ) . (3)where d i,j and h i,j denote the distance and channel betweenBS i and UE j , | h i,j | ∼ exp(1) . Φ ∗ b represents the set ofactive BSs which is not a homogeneous PPP. Nevertheless,we can assume Φ ∗ b as a homogeneous PPP with density λ ∗ b = λ b p a , which has been shown to be accurate according to [15][16]. Given the downlink SIR of the typical user, the coverageprobability is defined as: P c ( λ b , λ u , T ) (cid:44) P [SIR > T ] (4)where T is the target SIR level.The network
ASE Γ is defined as the sum average spectralefficiency of all active BSs achieving the target threshold in aunit area [16] and is given by Γ( λ b , λ u , T ) (cid:44) p a λ b P c ( λ b , λ u , T ) log (1 + T ) (5)where p a λ b P c can be interpreted as the density of the BSsthat successfully transmit the symbols to their users. III. C OVERAGE P ROBABILITY AND
ASE
ANALYSIS
In this section, we derive the coverage probability andASE expressions under a bounded dual-slope path loss modeland provide closed-form bounds of them. Furthermore, wedemonstrate the convergence of them in asymptotic regionswhere λ b → ∞ . Proposition 1: (Coverage probability expression) In a cellu-lar network with BS active probability p a , the coverage proba-bility under a bounded dual-slope path loss model is expressedin (6) at the bottom of this page, where the supplementaryequations are listed in (7)-(10) and F ( b, z ) = F (1 , b, b, − z ) with F ( a, b, c, z ) being the Gauss hypergeometricfunction. Proof
See Appendix A.Despite the complicated form of (6), the first and thirdintegrals can be calculated into exponential expressions. Inthis case, by applying transforms to the second integral, wecan derive closed-form bounds of (6) with only exponentialand hypergeometric functions as shown in the followingproposition.
Proposition 2: (Coverage probability bounds)
SIR coverageprobability’s lower bound P LB c and upper bound P UB c are givenas (11) and (12) at the bottom of this page, where H =1 − p a T T , H l = 1 + p a G ( R b ) , H u = 1 + p a G ( R c ) , H = 1 + p a G ( T ) . Proof
See Appendix B.Applying Proposition 2 in asymptotic regions leads to thefollowing proposition.
Proposition 3: (Asymptotic
SIR coverage probability) As λ b → ∞ , SIR coverage probability P c ( λ b , λ u , T ) convergesto a finite value as follows. lim λ b →∞ P c ( λ b , λ u , T ) = e − λ u πc T ( α,α c ,R c ,R b ) (13) Proof
From (11) and (12) in Proposition 1, we have lim λ b →∞ P LB c = lim λ b →∞ P UB c = H e − λ u πc T ( α,α c ,R c ,R b ) P c ( λ b , λ u , T ) = λ b π (cid:32)(cid:90) R b e − λ b πr (1+ p a G ( r,T )) d r + (cid:90) R c R b e − λ b πr (1+ p a G ( r,T )) d r + (cid:90) ∞ R c e − λ b πr (1+ p a G ( T )) d r (cid:33) (6) G ( r, T ) = c T ( α, α c , R c , R b ) r − − T (1 + T ) − (7) G ( r, T ) = (cid:20) c T (cid:0) α, α c , R c , √ r (cid:1) + R b (cid:18) F (cid:18) α , T (cid:19) − T T (cid:19)(cid:21) r − − F (cid:18) α , T − (cid:19) (8) G ( T ) = T (cid:16) α c − (cid:17) − F (cid:18) − α c , T (cid:19) (9) c T ( α, α c , R c , x ) := R c F (cid:18) α , T (cid:20) R c x (cid:21) α (cid:19) − R b (cid:20) F (cid:18) α , T (cid:19) − T T (cid:21) + 2 T R bα R c − α α c − F (cid:18) − α c , T (cid:20) xR c (cid:21) α (cid:19) (10) P LB c = 1 H (cid:16) e − λ b p a πc T ( α,α c ,R c ,R b ) − e − λ b π ( R b H + p a c T ( α,α c ,R c ,R b ) ) (cid:17) + 1 H l (cid:16) e − λ b πR b H l − e − λ b πR c H l (cid:17) + 1 H (cid:16) e − λ b πR c H (cid:17) (11) P UB c = 1 H (cid:16) e − λ b p a πc T ( α,α c ,R c ,R b ) − e − λ b π ( R b H + p a c T ( α,α c ,R c ,R b ) ) (cid:17) + 1 H u (cid:16) e − λ b πR b H u − e − λ b πR c H u (cid:17) + 1 H (cid:16) e − λ b πR c H (cid:17) (12) ince all the other terms tend to 0 as λ b → ∞ . According tothe Squeeze theorem, lim λ b →∞ P c = H e − λ u πm . Meanwhile, lim λ b →∞ H = 1 since p a → . Thus lim λ b →∞ P c = e − λ u πc T ( α,α c ,R c ,R b ) .Proposition 3 emphasizes the importance of consideringUE density in a UDN. The converged value is a decreasingfunction of λ u and it tends to 0 when λ u → ∞ , i.e., ina fully loaded network. Thus, deploying infinite number ofBSs will not bring the UE performance to a unprecedentedlevel. In contrast, extreme densification will put the coverageprobability into the danger of decreasing to 0, as shown in Fig.3 in the next section. The converged result also depends onenvironmental parameters ( α, α c , R c , R b ) . It will increase asboth path loss exponents grow since the coverage probabilityis a increasing function of path loss exponents [9]. A larger R b or R c will decline the performance since it either reducethe signal power or amplify the interference in the near-field .We now turn to the network perspective and study theasymptotic behavior of the ASE. Combining the definition in(5) with Proposition 3, we can easily obtain the followingproposition. Proposition 4: (Asymptotic
ASE ) As λ b → ∞ , ASE Γ converges to a finite value as follows. lim λ b →∞ Γ ( λ b , λ u , T ) = λ u e − λ u πc T ( α,α c ,R c ,R b ) log (1 + T ) (14) Proposition 4 shows that the asymptotic ASE will increasewith λ u when λ u is small and tends to 0 as λ u → ∞ .Unlike the coverage probability, the asymptotic ASE will bebeneficial from ultra-desification to some extent and but stillhighly depends on the UE density.Returning from the asymptotic regions, we demonstrate howthe ASE scales with BS density in the next proposition. Proposition 5: ( ASE scaling) The
ASE scales with λ b p a e − λ b p a πc T ( α,α c ,R c ,R b ) and is bounded by Γ LB = λ b p a P LB c log (1 + T ) (15) Γ UB = λ b p a P UB c log (1 + T ) . (16) Proof
See Appendix C.Similar with its asymptotic behavior, the ASE will increasewith BS density when λ b is small and finally converge to λ u e − λ u πc T log (1 + T ) as shown in Proposition 4.IV. N UMERICAL R ESULTS
In this section, we present the numerical results to studythe performance of network densification and validate ourtheoretical analysis. We assume R b = 1 m , R c = 70 m , α c = 2 . , α = 4 and set the SIR threshold T = 10dB in allof our results. To calculate or simulate ‘fully loaded network’,we set λ u = 2 × / km which is a sufficiently large valueso that p a ≈ . λ b (BSs/km ) C o v e r age p r obab ili t y Expression in (6)Simulation
Full load λ u =20 λ u =200 λ u =2000 Fig. 3: Effect of λ u on coverage probability under boundeddual-slope model. A. Effect of UE density
Fig. 3 shows the effect of UE density on coverage prob-ability. An exact match between simulation and analysis isobserved. Meanwhile, we find that coverage probabilities showcompletely different trends among different UE densities.In full load model, the diminishing of coverage probabilitystarts when interfering BSs fall into the near-field of thetypical UE and keeps decreasing since interference will con-tinue increasing. When λ u is finite, the interferer coordinatesconverge to UE coordinates in a UDN regime [11]. Thus thedistance from the typical UE to its closest interferer can beapproximated as the distance to the its closest neighbor UE, which has an expected value of √ λ u . When UE densityis low (e.g. λ u = 20 ), the expected value is larger than thecritical distance, which means the probability of no interfererinside the near-field of the typical UE is very high. Hence,the coverage probability is a non-decreasing function of λ b asin a single-slope model. In contrast, higher UE density (e.g. λ u = 200 or ) leads to more potential interferers withincritical distance. Thus coverage probability will decrease forthe same reason as in fully loaded network. Nevertheless, whenall the UEs in the near-field get service, coverage probabilitywill start increasing again since the interference are saturatedand no longer increase. Therefore, it is important to estimatethe active UE density for efficient network deployment oroperation in order to avoid the decreasing region of coverageprobability. B. Coverage probability analysis
In Fig. 4, we compare the coverage probability under ourmodel with the previous models. We observe that the perfor-mance is overestimated with unbounded or single-slope mod-els in highly densified regions. The reason is that those modelseither exaggerate the received power inside the bounded regionor underestimate the interference in the near-field. λ b (BSs/km ) C o v e r age P r obab ili t y Bounded dual−slope (BD)Lower bound of BDUpper bound of BDUnbounded single−slope (US)Unbounded dual−slope (UD) (a) Coverage Probability when network is fully loaded λ b (BSs/km ) C o v e r age p r obab ili t y Bounded dual−slope (BD)Lower bound of BDUpper bound of BDUnbounded single−slope (US)Unbounded dual−slope (UD) (b) Coverage Probability when λ u = 200 Fig. 4: Coverage probability bounds and comparison withprevious modelsThe inaccuracy of path loss models may mislead the pre-diction of asymptotic behavior. For instance, the coverageprobability will converge to 1 with unbounded models but to e − λ u πc T ( α,α c ,R c ,R b ) which is smaller than 1 (assume λ u > )when applying a bounded model. Consistent with the result inProposition 3, the converged value will decrease as UE densitygrows and finally falls to zero in the full load case as shownin Fig. 3. This is because the signal is limited by the boundeffect and the overall performance will be dominated by theinterference which depends on UE density. C. ASE analysis
Figure 5 depicts the scaling of ASE with regard to BSdensity. Aligning with proposition 5, the ASE first increasewith BS density and then converge to a constant. The constantis larger than 0 in partially loaded network and decreases to0 when the network is full load ( λ u → ∞ ) as proved inProposition 4.By comparing Fig. 5 with Fig. 4, we can observe a trade-off between UE and network performance during BS densifi- −10 −5 BS density λ b (BSs/km ) ASE ( bp s / H z / k m ) Bounded dual−slope (BD)Lower bound of BDUpper bound of BDUnbounded single−slope (US)Unbounded dual−slope (UD) (a) ASE when network is fully loaded −1 BS density λ b (BSs/km ) ASE ( bp s / H z / k m ) Bounded dual−slope (BD)Lower bound of BDUpper bound of BDUnbounded single−slope (US)Unbounded dual−slope (UD) (b) ASE when λ u = 200 Fig. 5: ASE bounds and comparison with previous models.cation. In full load case, there exists an BS density thresholdaround λ b = 10 . Before the threshold, although individualperformance gets worse, densification is still beneficial fromthe network perspective. For partially loaded network, thetrade-off appears approximately between λ b = 10 . to λ b =10 . The phenomenon further demonstrates the necessity ofapplying a dual-slope model because the coverage probabilityis a non-decreasing function of BS density in a single-slopemodel.Our bounds in Proposition 2 and 5 are compared with theintegral expression and shown in Fig. 4 and 5. The figuresverify the asymptotic tightness of the bounds for λ → ∞ . Theupper bound is tighter in small UE density scenarios while thelower bound fits better for large UE densities. This is becausethe upper bound is close to single-slope model which is similarwith small UE density scenario. The bounds can be used asapproximations in large BS density regions.V. C ONCLUSION
In this paper, we investigate the asymptotic behavior ofultra-densification of base stations. To our best knowledge,his is the first work incorporating two key aspects of UDNmodeling: a partially loaded network due to a finite active UEdensity and a dual-slope path loss model with a bounded losswithin a unit distance. With such models, we find that theasymptotic behavior of ultra-dense base station deployment isdifferent from what was known with simpler assumptions, e.g.unit or zero convergence of coverage probability. Dependingon the UE density, both UE coverage probability and ASEconverge to either zero or a constant value. Even before theasymptotic regions, our results suggest that the densificationcannot always improve the individual UE performance orboost the network throughput as well. The increment areprevented by introducing extra interference in the near-fielduntil all the UEs in the near-field are served. Our workprovides insights into the scaling of the network densification,and thus gives a guideline for the network deployment.A
PPENDIX AP ROOF OF P ROPOSITION P lc = P [SIR > T ] = P (cid:20) | h | (cid:96) ( r ) I > T (cid:21) ( a ) = (cid:90) r> P (cid:20) | h | > T I(cid:96) ( r ) | r (cid:21) f r ( r )d r ( b ) = (cid:90) r> L I (cid:18) T(cid:96) ( r ) (cid:19) f r ( r )d r (17)where (a) follows from BS distribution and (b) is due to thefact that | h | ∼ exp(1) , L I ( s ) is the Laplace transform ofinterference which can be derived as L I ( s ) = E I [ e − sI ] = E Φ ∗ b ,g i [ exp ( − s ( (cid:88) x ∈ Φ ∗ b g i (cid:96) ( d i )))] ( a ) = E Φ ∗ b (cid:89) x ∈ Φ ∗ b
11 + s(cid:96) ( d i ) ( b ) = exp (cid:18) − πλ ∗ b (cid:90) ∞ r (cid:18) −
11 + s(cid:96) ( v ) (cid:19) d v (cid:19) (18)where (a) is because g ∼ exp(1) and (b) follows the prob-ability generating functional (PGFL) of the PPP. Plugging in s = (cid:16) T(cid:96) ( r ) (cid:17) and employing a change of variables v = √ tr results in L I (cid:18) T(cid:96) ( r ) (cid:19) = exp − πλ b P a (cid:90) ∞ TT + (cid:96) ( r ) (cid:96) ( √ tr ) d t . (19)Plugging (19) into (17) with z → r gives the coverageprobability under a general path loss fucntion in (20) as: P lc ( λ b , λ u , T ) = λ b π (cid:90) ∞ exp − λ b πz p a (cid:90) ∞
11 + (cid:96) ( √ z ) T(cid:96) ( √ tz ) d t d z (20) Based on (20), we can substitute our bounded dual-slopemodel (2) into it and get the expression in (6).A
PPENDIX BP ROOF OF P ROPOSITION rG ( r, T ) and rG ( T ) are linear functions of r . Thus we can rewrite the firstand third integral in (6) as follows: (cid:90) R b e − λ b πr (1+ p a G ( r )) d r = 1 H (cid:16) e − λ b p a πc T − e − λ b π ( R b H + p a c T ) (cid:17) (21) (cid:90) ∞ R c e − λ b πr (1+ p a G ( r )) d r = 1 H (cid:16) e − λ b πR c H (cid:17) . (22) In the second integral, from G (cid:48) ( r ) < we can get G ( R b ) ≥ G ( r ) ≥ G ( R c ) . With the inequality, we can provide boundsfor the second integral as: (cid:90) R c R b e − λ b πr (1+ p a G ( r )) d r ≤ (cid:90) R c R b e − λ b πr (1+ p a G ( R b )) d r = 1 H u ( e − λ b πH u R b − e − λ b πH u R c ) (23) (cid:90) R c R b e − λ b πr (1+ p a G ( r )) d r ≥ (cid:90) R c R b e − λ b πr (1+ p a G ( R c )) d r = 1 H l ( e − λ b πH l R b − e − λ b πH l R c ) . (24) Replacing the integrals in (6) with the exponential expressionsabove completes the proof.A
PPENDIX CP ROOF OF P ROPOSITION Notation : Let f and g be two functions defined on somesubset of the real numbers. One writes f ( x ) = O ( g ( x )) ifand only if there exists a positive real number M and a realnumber x such that f ( x ) ≤ M g ( x ) for all x ≥ x .We omit the proof of the bounds since they come di-rectly from Proposition 2. To prove the ASE scales with λ ∗ b e − λ ∗ b πc T is equivalent with showing Γ UB = O ( λ ∗ b e − λ ∗ b πc T ) and λ ∗ b e − λ ∗ b πc T = O (Γ LB ) . Denote log (1+ T ) as τ and from(12) we have: Γ UB ≤ λ ∗ b ( 1 H e − λ b p a πc T + 1 H u e − λ b πH u R b + 1 H e − λ b πH R c ) τ. (25) Then we can show ∃ λ > , ∀ λ b > λ , H e − λ b p a πc T > H u e − λ b πH u R b and H e − λ b p a πc T > H e − λ b πH R c since H e − λ b p a πc T → H e − λ u πc T and the other two parts → λ b → ∞ . Thus ∃ λ > , ∀ λ b > λ , Γ UB ≤ H λ ∗ b e − λ ∗ b πc T = ⇒ Γ UB = O ( λ ∗ b e − λ ∗ b πc T ) .For Γ LB , from (11) we have Γ LB ≥ k λ ∗ b (cid:16) e − λ b p a πc T − e − λ b π ( H R b + p a c T ) (cid:17) τ, (26)hich can be rephrased as: λ ∗ b e − λ b p a πc T ≤ H (1 − e − λ b πH R b ) τ Γ LB . (27)Thus, ∃ λ > , k > , ∀ λ b > λ , e − λ b πH R b < k thus λ ∗ b e − λ b p a πc T ≤ H (1 − k ) τ Γ LB . Therefore λ ∗ b e − λ ∗ b πc T = O (Γ LB ) and we complete the proof.A CKNOWLEDGMENT
Part of this work has been supported by the H2020 projectMETIS-II co-funded by the EU. The views expressed are thoseof the authors and do not necessarily represent the project. Theconsortium is not liable for any use that may be made of anyof the information contained therein.R
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