Uplink Performance Analysis of Dense Cellular Networks with LoS and NLoS Transmissions

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1 his article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI.9/WC , IEEE Uplink Performance Analysis of Dense Cellular Networks with os and NoS ransmissions ian Ding, Student Member, IEEE, Ming Ding, Member, IEEE, Guoqiang Mao, Senior Member, IEEE, Zihuai in, Senior Member, IEEE, David ópez-pérez, Member, IEEE, Albert Zomaya, Fellow, IEEE Abstract In this paper, we analyse the coverage probability and the area spectral efficiency ASE for the uplink U of dense small cell networks SCNs considering a practical path loss model incorporating both line-ofsight os and non-line-of-sight NoS transmissions. Compared with the existing work, we adopt the following novel approaches in our study: i we assume a practical user association strategy UAS based on the smallest path loss, or equivalently the strongest received signal strength; ii we model the positions of both base stations BSs and the user equipments UEs as two independent Homogeneous Poisson point processes HPPPs; and iii the correlation of BSs and UEs positions is considered, thus making our analytical results more accurate. he performance impact of os and NoS transmissions on the ASE for the U of dense SCNs is shown to be significant, both quantitatively and qualitatively, compared with existing work that does not differentiate os and NoS transmissions. In particular, existing work predicted that a larger U power compensation factor would always result in a better ASE in the practical range of BS density, i.e., 3 BSs/km 2. However, our results show that a smaller U power compensation factor can greatly boost the ASE in the U of dense SCNs, i.e., 2 3 BSs/km 2, ian Ding is with he University of echnology Sydney, Australia ian.ding@student.uts.edu.au. Ming Ding is with Data6, Sydney, Australia Ming.Ding@nicta.com.au. Guoqiang Mao is with the School of Computing and Communication, he University of echnology Sydney, Australia. He also holds adjunct positions at the School of Electronic Information & Communications, Huazhong University of Science & echnology, Wuhan, China, and the School of Information and Communication Engineering, Beijing University of Posts and elecommunications, Beijing, China g.mao@ieee.org. Guoqiang Mao s research is supported by Australian Research Council ARC Discovery project DP538 and Chinese National Science Foundation project Zihuai in is with the School of Electrical and Information Engineering, he University of Sydney, Australia zihuai.lin@sydney.edu.au. David ópez-pérez is with Bell abs, Nokia, Dublin, Ireland dr.david.lopez@ieee.org. Albert Zomaya is with the School of I, the University of Sydney, Australia albert.zomaya@sydney.edu.au. while a larger U power compensation factor is more suitable for sparse SCNs, i.e., 2 BSs/km 2. Index erms dense small cell networks SCNs, Uplink U, ine-of-sight os, Non-ine-of-Sight NoS, coverage probability, area spectral efficiency ASE I. INRODUCION By means of network densification, small cell networks SCNs can achieve a high spatial reuse gain, which further leads to a high network capacity. Particularly, the orthogonal deployment of SCNs within the existing macrocell network, i.e., small cells and macrocells operating on different frequency spectrum Small Cell Scenario #2a defined in 2, is prioritized in the design of the 4th generation 4G ong erm Evolution E networks by the 3rd Generation Partnership oject 3GPP. Furthermore, dense SCNs are envisaged to be the workhorse for capacity enhancement in the 5th generation 5G networks due to its large performance gains and easy deployment, 3, 4. hus, this paper focuses on studying the performance of these orthogonal deployments of dense SCNs. In our previous work 5, we conducted a study on the downlink D of dense SCNs considering a sophisticated path loss model that differentiates line-of-sight os and non-line-of-sight NoS transmissions. os transmission may occur when the distance between a transmitter and a receiver is small, and NoS transmission is more common in office environments and in central business districts. Moreover, the probability that there exists a os path between the transmitter and the receiver increases as their distance decreases. It is observed in 5 that the reduction of the distance between the transmitter and the receiver as the density of small cell base stations BSs increases will cause a transition from NoS transmission to os transmission, c 26 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

2 his article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI.9/WC , IEEE 2 which has a significant impact, both quantitatively and qualitatively, on the performance of D dense SCNs. Motivated by this finding 5, in this paper, we continue to query whether such NoS-to-oS transitions may significantly affect the performance of uplink U dense SCNs. Our work distinguishes from existing work 5, 6, 7 on the performance analysis of U dense SCNs in three major aspects. First, we assume a user association strategy UAS that each UE is associated with the BS with the smallest path loss to the UE, or equivalently each UE is associated with the BS that delivers the strongest received signal strength 5. Note that in our previous work 7 and existing work in the literature 6, the authors assumed that each UE should be associated with the closest BS. Such assumption is not appropriate for the realistic path loss model with os and NoS transmissions, because in practice it is possible for a UE to associate with a BS that is not the closest one but with a os path, instead of the nearest BS with a NoS path. Second, we assume that the BSs and the UEs are deployed according to two independent Homogeneous Poisson point processes HPPPs, which is more practical and realistic compared with the previous work 6, 7. hird, we consider the correlation of BS and UE positions explained later in the paper, thus making our numerical results more accurate than the previous work 6, which ignored such correlation. he main contributions of this paper are as follows: Numerically tractable results are obtained for the U coverage probability and the U area spectral efficiency ASE performance using a piecewise path loss model, incorporating both os and NoS transmissions. Our theoretical analysis of the U of dense SCNs shows a similar performance trend that was found for the D of dense SCNs in our previous work 5, i.e., when the density of UEs is larger than a threshold, the ASE may suffer from a slow growth or even a decrease. hen, the ASE will grow almost linearly as the UE density increases above another larger threshold. his finding is in stark contrast with previous results using a simplistic path loss model that does not differentiate os and NoS transmissions 6. Our theoretical analysis also indicates that the performance impact of os and NoS transmissions on the U of SCNs with U power compensation is significant both quantitatively and qualitatively compared with existing work in the literature that does not differentiate os and NoS transmissions. he details of the U power compensation scheme will be introduced in Section III. In particular, the previous work 6 showed that a larger U power compensation factor should always deliver a better ASE performance in the practical range of BS density, i.e., 3 BSs/km 2. However, our results show that a smaller U power compensation factor can greatly boost the ASE performance in dense SCNs, i.e., 2 3 BSs/km 2, while a larger U power compensation factor is more suitable for sparse SCNs, i.e., 2 BSs/km 2. Our new finding indicates that it is possible to save UE battery and meanwhile obtain a high ASE in the U of dense SCNs in 5G, if the U power compensation factor is optimized. he remainder of this paper is structured as follows. Section II compares the closest related work to our work. Section III describes the system model. Section IV presents our main analytical results on the U coverage probability and the U ASE. Section V presents the application of our main analytical results on the U coverage probability and the U ASE in a 3GPP special case, followed by a more efficient computation method to evaluate the results using the Gauss-aguerre quadrature. he numerical results and simulation results are discussed in Section VI. Finally, the conclusions are drawn in Section VII. II. REAED WORK In the D performance analysis of cellular networks based on stochastic geometry, BS positions are typically modeled as a Homogeneous Poisson point process HPPP on the plane 8, and in this case, the coverage probability can be expressed in a closedform. Furthermore, an important and novel capacity model was proposed for HPPP random cellular networks, where the impact of random interference on the cooperative communications is analyzed by a closed-form expression 9. In the U performance analysis of cellular networks based on stochastic geometry, UE positions are typically modeled as a HPPP on the plane 6, and BS positions are c 26 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

3 his article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI.9/WC , IEEE 3 assumed to be uniformly and randomly deployed in the Voronoi cell of each UE. he difficulty of modeling both BSs and UEs as a HPPP is that the BS and UE positions are coupled 6,, and the dependence of UE positions is therefore hard to analyse, 2, 3. Such dependence occurs because if a UE is associated with a BS that delivers the strongest received signal or is closest to the UE, it implies that there are no other BSs that can be located in positions that deliver the strongest received signal or in a closer distance than the aforementioned BS. o derive tractable and closed-form results, previous work ignored this dependence and modeled the distance between a UE and its associated BS as an independent identical distributed i.i.d. random variable. In greater detail, in 6, the authors assumed that the UEs are randomly distributed following a HPPP, and exactly one BS is randomly and uniformly located in each UE s Voronoi cell, i.e., each BS associates with its nearest UE. It is also assumed that the distance between each BS and its serving UE is i.i.d. Rayleigh distributed. he system model of only deploying UEs as a HPPP 6 makes it difficult to conduct network performance analysis for U of SCNs. Furthermore, the association strategy that each BS associates with its nearest UE 6 is impractical, and the assumption that all of the BS-UE association distances are i.i.d. Rayleigh distributed 6 is unrealistic. In 4, the authors considered UE spatial blocking, which is referred to as the outage caused by limited number of usable channels, and derived approximate expressions for the U blocking probability and the U coverage probability. In 5, the authors proposed a tractable model to characterize the U rate distribution in a K-tier heterogeneous cellular networks HCNs considering power control and load balancing. In 6, the authors considered the maximum power limitation for UEs and obtained approximate expressions for the U outage probability and U spectral efficiency. However, none of the aforementioned U related work considered a realistic path loss model with line-of-sight os and non-line-of-sight NoS transmissions. In contrast, in this paper, we consider a sophisticated path loss model incorporating both os and NoS transmissions to study their performance impact on dense SCNs and show that os and NoS transmissions have a significant impact on the performance of U dense SCNs. os and NoS transmissions have been previously investigated in the D performance analysis of dense SCNs 5, 7. One major conclusion of 5 is that the ASE performance will slowly increase or even decrease in certain BS density regions. It is interesting to see whether this conclusion holds for U dense SCNs. In our previous work on the U performance analysis of dense SCNs 7, we assume that each UE is associated with its nearest BS, which may not be a practical assumption when considering os and NoS transmissions. Compared with 7, in this work we consider a more realistic user association strategy, in which a UE associates with the BS that has the smallest path loss, or equivalently delivers the strongest received signal strength. his user association strategy is more realistic and is particularly important when considering both os and NoS transmissions that are present in realistic radio environment, because the closest BS may possibly have only a NoS path to the UE and therefore may offer a weaker signal than a BS that is further away but has a os path to the UE. III. SYSEM MODE Different from the assumption that only UEs deployment follows HPPP distribution 6, in this paper, we assume that both BSs and UEs are distributed following HPPPs with densities λ BSs/km 2 and λ UE UEs/km 2, respectively. Here, we assume that λ UE λ so that all the BSs are activated to serve at least one UE. Each UE is assumed to associate with the BS with the smallest path loss. We focus on U and consider a randomly tagged BS, which is denoted as the typical BS located at the origin. With the assumption of λ UE λ, on each time-frequency resource block, each BS has one active UE in its coverage area. he UE associated with the typical BS is denoted as the typical UE, and the other UEs using the same time-frequency resource block are denoted as the interfering UEs. he distance from the typical UE to the typical BS is denoted by R, which is a random variable whose distribution will be analyzed later. hroughout the paper, we use the upper case letters, e.g., R, to denote a random variable and use the lower case letters, e.g., r, to denote specific instance of the random variable c 26 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

4 his article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI.9/WC , IEEE 4 he link from the typical UE to the typical BS has a os path or a NoS path with probability r and r, respectively, where such probability can be computed by the following piecewise function 5, r, < r d 2 r, d < r < d 2 r =... N r, r > d N he distance dependent path loss is expressed as ζ r with r being the distance, and the path loss gain is ζ r, where the path loss of each link is modeled as 2, which is shown on the top of next page. In 2, for n {, 2,, N}, A n is the path loss of os path at a reference distance of r =, A N n is the path loss of NoS path at a reference distance of r =, αn is the path loss exponent of os link, and αn N is the path loss exponent of NoS link. he U transmission power of UE k located at a distance of r is denoted by P k, and is subject to a semi-static power control PC mechanism, i.e., the fractional path loss compensation FPC scheme 8. Based on this FPC scheme, P k is modeled as P k = P ζ r ɛ, 3 where P is the baseline power on the considered RB at the UE, ɛ, is the FPC factor, and ζ r is expressed in 2. In 3, the distance-based fractional power compensation term ζ r ɛ is denoted by β r and written as β r = ζ r ɛ. 4 herefore, the received signal power at the typical BS can be written as P sig = P β R ζ R g = P ζ R ɛ 5 g, where g denotes the channel gain of the multipath fading channel and is an i.i.d. exponential distributed random variable. Hence, g follows an exponential distribution with unit mean. As a result, the SINR at the typical BS of the typical UE can be expressed as SINR = P sig, 6 σ 2 + I Z where σ 2 is the noise power, Z is the set of interfering UEs, and I Z is the interference given by I Z = Z P β R z ζ D z g z, 7 where g z denotes the channel gain of the multipath fading channel of interferer z Z, and is an i.i.d. exponential distributed random variable, which follows an exponential distribution with unit mean. he distance of interferer z Z to its serving BS is denoted by R z, and the distance of interferer z Z to the typical BS is denoted by D z. he details of the distribution of R z and R are given in Section V. Since D z R z, D z can be approximated by the distance from the serving BS of interferer z to the typical BS. IV. ANAYSIS BASED ON HE PROPOSED PAH OSS MODE he U coverage probability for the typical BS can be formulated as P cov λ, = SINR, 8 where is the SINR threshold. he area spectral efficiency ASE in bps/hz/km 2 for a given λ can be formulated as 5 A ASE λ, = λ log 2 + x f X λ, x dx, 9 where is the minimum working SINR for the considered SCN, and f X λ, x is the PDF of the SINR observed at the typical BS for a particular value of λ. Based on the definition of P cov λ,, which is the complementary cumulative distribution function CCDF of SINR, f X λ, x can be computed as f X λ, x = P cov λ, x. x Based on the system model presented in Section III, we can calculate P cov λ, and present it in the following theorem. heorem. P cov λ, can be derived as N P cov λ, = n + n N, where n = d n d n N n = d n d n n= P g A r α ɛ P g A N r αn ɛ os fr,n rdr, NoS fr,n N r dr, c 26 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

5 his article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI.9/WC , IEEE { A r α, os with probability r A N r αn, NoS with probability r, < r d { A 2r α 2, os with probability 2 r ζ r = A N 2 r αn 2, NoS with probability 2 r, d < r < d 2, 2. {. A N rα N, os with probability N r A N N rαn N, NoS with probability N r, r > d N 5 and d and d N are respectively defined as and. Moreover, fr,n N r and fr,n r can be respectively derived as fr,n r = exp r n u 2πuλdu exp r n u 2πuλdu 3 n r 2πrλ, d n < r < d n, and f N R,n r = exp r 2 n u 2πuλdu exp r n u 2πuλdu n r 2πrλ, d n < r < d n, 4 where r and r 2 are determined respectively by /α N r = A r α /A N, 5 and /α r 2 = A N r αn /A. 6 P g A Furthermore, r α ɛ os and P g A N r αn ɛ NoS are respectively computed by P g A r α ɛ os IZ = exp and σ 2 P A r α ɛ P g A N r αn ɛ = exp σ 2 P A N r αn ɛ NoS IZ P A r α ɛ, 7 P A N r αn ɛ 8, where IZ s is the aplace transform of RV I Z evaluated at s. oof: See Appendix A. As can be observed from heorem, the piece-wise path loss function for os transmission, the piecewise path loss function for NoS transmission, and the piece-wise os probability function play active roles in determining the final result of P cov λ,. We will investigate their impacts on network performance in detail in the following sections. Plugging P cov λ, obtained from into, we can get the result of the ASE using 9. V. SUDY OF A 3GPP SPECIA CASE As a special case for heorem, we consider a path loss function adopted in the 3GPP as 8 { A r α, os with probability r ζ r = A N r αn, NoS with probability r, 9 together with a linear os probability function of r, defined in the 3GPP as 9 { r r = d, < r d, 2, r > d where d is the cut-off distance of the os link. For the 3GPP special case, according to heorem, P cov λ, γ can then be computed by P cov λ, = 2 n= n + n N. 2 In the following subsections, we will investigate the results of, N, 2, and 2 N, respectively c 26 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

6 his article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI.9/WC , IEEE 6 A. he Result of Regarding the result of, which is the coverage probability when the typical UE is associated with the typical BS with a os link of distance less than d, we present emma 2 in the following. and f N R z,2 u = exp πλu 2 2πuλ, 29 where /α N emma 2. When the typical UE is associated with u = A u α /A N, 3 a os BS of a distance less than d, the coverage probability can be computed by and σ d 2 /α P = e A r α ɛ IZ fr,rdr, u 2 = A N u αn /A, 3 P A r α ɛ 22 Specifically, when the interference comes from a os where fr, r path, fr z u can be derived as { = exp πλr 2 r + 2πλ 3 3d r3 3d r d 2πrλ, f fr R 23 z u = u, os, < u x z, fr N u, NoS, < u x, 32 z, and the aplace transform IZ s is expressed as IZ s { = exp 2πλ where d xd /α N r } x = A x α /A N. 33 f +s P βu ζx R z u du os xdx { exp 2πλ d x Conditioned on x d, when the interference path r d } is NoS, fr N f N z u can be derived as +s P βu ζx R z u du NoS xdx { exp 2πλ fr N z { u d fr, } u, os, < u x 2 f 2N +s P βu ζx R z u du NoS xdx. f R, N u, NoS, < u x, r < x y 24 = fr, u, os, < u d, According to the HPPP system model, the distribution of R z is the same as R, but bounded by x. he fr, N u, NoS, < u y, y < x d fr, 2N PDF of R z can be written as u, NoS, y < u x fr u, os, < u x 34 z, where fr N f Rz u = u, NoS, < u x z, fr 2N u, NoS, y, z, < u d y = A d α /A N /αn, 35 fr N u, NoS, d z,2 < u x 25 and where /α f u x 2 = A N x αn /A. 36 Rz, = exp πλu 2 u + 2πλ 3 3d u3 3d 26 Conditioned on x > d, when the interference path ud 2πuλ, is NoS, fr 2N z u can be derived as fr N u z, fr = exp πλu 2 u πλ 2 u, os, < u d z, 3d u3 3d 27 f 2N fr N R z u = u, NoS, < u y z,. u d 2πuλ, fr 2N z, u, NoS, y < u d f fr 2N z, u = exp 2πλ d2 6 u3 R N u, NoS, d z,2 < u x u 37 2πuλ, 3d d 28 oof: See Appendix B c 26 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

7 his article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI.9/WC , IEEE 7 B. he Result of N Regarding the result of N, which is the coverage probability when the typical UE is associated with the typical BS with a NoS link of distance less than d, we propose emma 3 in the following. emma 3. N can be derived as d N = e σ 2 P r αn ɛ IZ f N P r αn ɛ R, r dr, 38 where fr, N r exp πλr 2 r πλ 2 3d r3 3d r d = 2πrλ, < r y, exp πλd2 2πλr3 3 3d r d 2πrλ, y < r d 39 and the aplace transform IZ s for < r y and y < r d are respectively expressed as IZ s = exp 2πλ d r 2 x d f +s P βu ζx R z u du os xdx exp 2πλ d x r d f N +s P βu ζx R z u du NoS xdx exp 2πλ d f 2N +s P βu ζx R z u du NoS xdx, 4 and IZ s = exp 2πλ d r +s P exp 2πλ where r 2 = +s P x d f N βu ζx R z d f 2N βu ζx R z A r αn /A N /α. u du NoS u du NoS xdx xdx, 4 oof: he proof is very similar to that in Appendix B. hus it is omitted for brevity. C. he Result of 2 he result of 2 is the coverage probability when the typical UE is associated with the typical BS with a os link of distance larger than d. From heorem, 2 can be derived as ɛ 2 P g A r α = os d σ 2 fr,2 rdr. + I Z 42 According to heorem and 2, fr,2 r can be calculated by fr,2 r = exp r u 2πuλdu exp r u 2πuλdu 2πrλ =, r > d. 43 Plugging 43 into 42, yields 2 =. 44 D. he Result of N 2 Regarding the result of N 2, which is the coverage probability when the typical UE is associated with the typical BS with a NoS link of distance larger than d, we propose emma 4 in the following. emma 4. 2 N can be derived as 2 N σ 2 P = r αn ɛ IZ P r αn ɛ where d e f N R,2r dr, 45 f N R,2 r = exp πλr 2 2πrλ, 46 and the aplace transform IZ s is expressed as IZ s = exp 2πλ r f 2N +s P βu ζx R z u du NoS xdx. 47 oof: he proof is very similar to that in Appendix B. hus it is omitted for brevity.. E. Evaluation Using the Gauss-aguerre Quadrature o improve the tractability of the derived results, we propose to approximate the infinite integral of outer-most integrals in 45 by the Gauss-aguerre quadrature 2, expressed as f u e u du n ω i f u i, 48 i= where n is the degree of aguerre polynomial, and u i and ω i are the i-th abscissas and weight c 26 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

8 his article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI.9/WC , IEEE 8 of the quadrature. For practical use, n should be set to a value above to ensure good numerical accuracy 2. o utilize the Gauss-aguerre quadrature, the outermost integral in 45 is rewritten by using the change of variable r = πλr 2. o evaluate 45 by means of the Gauss-aguerre quadrature, we propose emma 5 in the following. emma 5. By using the Gauss-aguerre quadrature as shown in 48, 45 can be approximated and simplified as N 2 n σ ω i exp 2 i= πλ d 2 IZ P u i +πλd 2 πλ α N ɛ P u i +πλd 2 πλ α N ɛ. 49 oof: See Appendix C. hanks to emma 5, the 3-fold integral computation in 45 can now be simplified as a 2-fold integral computation, which improves the tractability of our results. VI. SIMUAION AND DISCUSSION In this section, we present numerical and simulation results to establish the accuracy of our analysis and further study the performance of the U of dense SCNs. We adopt the following parameters according to the 3GPP recommendations 8, 2: d =.3 km, α = 2.9, α N = 3.75, P = -76 dbm, σ 2 = -99 dbm with a noise figure of 5 db at each BS. We first consider a sparse network in subsection VI-A, and then we analyze a dense network in the subsections VI-B and VI-C. A. Validation of the Analytical Results of P cov λ, For comparison, we first compute analytical results using a single-slope path loss model that does not differentiate os and NoS transmissions 6. Note that in 6, only one path loss exponent is defined and denoted by α, the value of which is α = α N = he results of P cov λ, in a sparse network obability of SINR> he analysis in 4 he proposed analysis using the single-slop path loss model in 4 Simulation SINR hreshold db Fig.. he coverage probability P cov λ, vs. the SINR threshold in 6 with λ = BSs/km 2, α = 3.75, and ɛ =.7. scenario with λ = BSs/km 2, α = 3.75, and ɛ =.7 are plotted in Fig.. In the case of the single-slope path loss model 4, as can be observed from Fig., our analytical result is much more accurate than that in 6 because our system model assumptions are more reasonable than those in 6: first, the distributions of BSs and UEs are modeled as two independent HPPPs, instead of the assumption that only UEs are distributed according to a HPPP 6; second, the dependence of BS and UE positions are discussed, instead of being ignored 6. In the case of the 3GPP path loss model 8, the results of P cov λ, in a sparse network scenario with λ = BSs/km 2 and in a dense network scenario with λ = 3 BSs/km 2 are plotted in Fig. 2. As can be observed from Fig. 2, our analytical results match the simulation results very well, and thus we will only use analytical results of P cov λ, in our discussion hereafter. As can be seen from Fig. 2, for the case of λ = BSs/km 2, when the SINR threshold is small e.g., < 4dB, the analytical result of the coverage probability is larger than the simulation result. his is because in our analysis, the approximation of replacing the location of UE by that of its serving BS, may exclude the cases of strong interfering UEs located at the proximity of the typical BS, thus underestimating the total interference, and overestimating the coverage probability. However, as the SINR threshold increases e.g., > 4dB, c 26 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

9 his article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI.9/WC , IEEE he proposed analysis λ=, ǫ=.7 Analytical he proposed analysis λ=, ǫ=.7 Simulation he proposed analysis λ=, ǫ=.7 Analytical he proposed analysis λ=, ǫ=.7 Simulation.9.8 he proposed analysis ǫ=.6 Analytical he proposed analysis ǫ=.7 Analytical he proposed analysis ǫ=.8 Analytical he analysis in 4 α=3.75, ǫ=.7 Analytical obability of SINR> obability of SINR> SINR hreshold db Fig. 2. he coverage probability P cov λ, vs. the SINR threshold with λ = BSs/km 2 and λ = 3 BSs/km 2. the impact of the overestimation of the coverage probability will decrease, and our analytical result matches the simulation result well. Another interesting finding as can be observed from Fig. 2 is that the analytical result with a larger BS density is more accurate than that with a smaller BS density. his is because in denser networks, the distance between a UE and its serving BS is smaller, and the approximation of replacing the location of a UE by that of its serving BS has less impact on the estimation of the total interference, thus making the analytical result more accurate. B. he Results of P cov λ, vs. λ he results of P cov λ, against the BS density for = db are plotted in Fig. 3. From Fig. 3, we can observe that when considering both os and NoS transmissions, the coverage probability presents a significantly different behavior. When the SCN is sparse and thus noise-limited, the coverage probability given by the proposed analysis grows as λ increases, similarly as that observed in 6. However, when the network is dense enough, the coverage probability decreases as λ increases, due to the transition of a large number of interference paths from NoS to os, which is not captured in 6. Particularly, during this region, interference increases at a faster rate than the signal due to the transition from mostly NoS interference to os interference, thereby causing a drop in the SINR hence the coverage probability. In more detail, the coverage probability given by the proposed analysis. λ 2 3 BS density λ BSs/km 2 Fig. 3. he coverage probability P cov λ, vs. the BS density with different ɛ and SINR threshold = db. peaks at a certain density λ. When λ increases above λ, interfering UEs become closer to the typical BS and their interfering signals start reaching the typical BS via strong os paths. When λ is further increased far above λ, the coverage probability decreases at a slower pace because both the signal power and the interference power are os dominated and increase at approximately the same rate. here are still more and more interferers whose signal reach the typical BS via os paths but their effect is smaller than the dominating interferers. It should also be noted that the coverage probability with different FPC factor ɛ exhibits different trends. Specifically, when the SCN is sparse, adopting a higher ɛ e.g., ɛ =.8 leads to a higher coverage probability. his is because the sparse SCN is noiselimited and hence increasing the transmission power provides better coverage performance. However, when the SCN is dense, adopting a lower ɛ e.g., ɛ =.6 leads to higher coverage probability. his is because the dense SCN is interference-limited, and the network experiences a surplus of strong os interference instead of shortage of U transmission power, and hence decreasing the transmission power provides better coverage performance. herefore, our results suggest that in dense SCNs, increasing the U transmission power may degrade the coverage probability. Such observation is further investigated in terms of ASE in the following subsection c 26 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

10 his article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI.9/WC , IEEE Area spectral efficiency bps/hz/km 2 3 he proposed analysis ǫ=.6 Analytical he proposed analysis ǫ=.7 Analytical he proposed analysis ǫ=.8 Analytical he analysis in 4 α=3.75, ǫ=.6 Analytical he analysis in 4 α=3.75, ǫ=.7 Analytical he analysis in 4 α=3.75, ǫ=.8 Analytical 2 λ 2 λ 3 BS density λ BSs/km 2 Fig. 4. Area spectral efficiency A ASE λ, vs. the BS density with different ɛ and SINR threshold = db. λ and λ correspond to the BS density when the ASE given by the proposed analysis starts to suffer from a slow growth and when it starts to pick up the growth, respectively. C. he Results of A ASE λ, vs. λ In this subsection, we investigate the ASE with = db based on the analytical results of P cov λ,. he results of A ASE λ, obtained by comparing the proposed analysis with the analysis from 6 are plotted in Fig. 4. As can be seen from Fig. 4, the analysis from 6 indicates that when the SCN is dense enough, the ASE increases linearly with λ. In contrast, our proposed analysis reveals a more complicated ASE trend. Specifically, when the SCN is relatively sparse, i.e., BSs/km 2, the ASE quickly increases with λ since the network is generally noise-limited, and thus having UEs closer to their serving BSs improves performance. When the SCN is extremely dense, i.e., around 3 BSs/km 2, the ASE increases linearly with λ because both the signal power and the interference power are os dominated. As for the practical range of λ for the existing and the future cellular networks, i.e., 3 BSs/km 2, the ASE trend is interesting. First, when λ λ, λ, where λ is around 2 and λ λ > λ is around 25 in Fig. 4, the ASE exhibits a slow-down in the rate of growth due to the fast decrease of coverage probability shown in Fig. 3. hereafter, when λ λ, the ASE exhibits an acceleration in the growth rate due to the slowdown in the decrease of coverage probability also shown in Fig. 3. Our finding, the ASE may exhibits a slow-down in the rate of growth as the BS density increases, is similar to our results reported for the D of SCNs 5, which indicates the significant impact of the path loss model incorporating both NoS and os transmissions. Such impact makes a difference for dense SCNs in terms of the ASE both quantitatively and qualitatively, comparing to that with a simplistic path loss model that does not differentiate os and NoS transmissions. Our proposed analysis also shows another important finding. A smaller U power compensation factor ɛ e.g., ɛ =.6 can greatly boost the ASE performance in 5G dense SCNs, i.e., 2 3 BSs/km 2, while a larger ɛ e.g., ɛ =.8 is more suitable for sparse SCNs, i.e., 2 BSs/km 2. his contradicts the results in 6 where a larger U power compensation factor was predicted to always result in a better ASE in the practical range of BS density, i.e., 3 BSs/km 2, as shown in Fig. 4. herefore, our theoretical analysis indicates that the performance impact of os and NoS transmissions on U SCNs with U power compensation is also significant both quantitatively and qualitatively, compared with the previous work in 6 that does not differentiate os and NoS transmissions. Interestingly, our new finding implies that its is possible to save UE battery and meanwhile achieve a high ASE in the U of 5G dense SCNs, if ɛ is optimized. he intuition is that in dense SCNs, the network experiences a surplus of strong os interference instead of shortage of U transmission power, and thus reducing the transmission powers of a large number of interferers turns out to be a good strategy that enhances the ASE. Note that our conclusion is made from the investigated set of parameters, and it is of significant interest to further study the generality of this conclusion in other network models and with other parameter sets. D. Discussion on Various Values of α In this subsection, we change the value of α from 2.9 to.9 and 3.9, respectively, to investigate the performance impact of α. In Fig. 5, the analytical results of P cov λ, with = db and with various α and various ɛ are compared. As can be seen from Fig. 5, the smaller the α, the larger the difference between the NoS path loss exponent α N and α. As a result, performance c 26 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

11 his article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI.9/WC , IEEE.8.7 he proposed analysis ǫ=.6 Analytical he proposed analysis ǫ=.8 Analytical 4 he proposed analysis ǫ=.6 Simulation he proposed analysis ǫ=.7 Simulation he proposed analysis ǫ=.8 Simulation obability of SINR> α =2.9 α =3.9 Area spectral efficiency bps/hz/km α = BS density λ BSs/km 2 Fig. 5. he coverage probability P cov λ, vs. the BS density with different ɛ and α. SINR threshold = db BS density λ BSs/km 2 Fig. 6. Area spectral efficiency A ASE λ, vs. the BS density with the exponential os probability model, different ɛ and SINR threshold = db. impact of the transition of interference from the NoS transmission to the os transmission becomes more drastic as λ increases. In other words, the slow growth of the P cov λ, is more obvious to observe. For example, when α takes a near-field path loss exponent such as.9, the decrease of the P cov λ, at λ λ, λ BSs/km 2 is substantial and it hardly recovers after λ. As has been discussed in the subsection VI-B, when the SCN is sparse, adopting a higher ɛ leads to a higher coverage probability. However, as λ increases, adopting a lower ɛ leads to a higher coverage probability. he BS density around which the coverage probability with smaller ɛ surpasses that with larger ɛ is defined as the transition point of ɛ. As can be seen from Fig. 5, the transition point of various ɛ increases as α increases. It indicates that in dense SCNs with smaller α, the coverage probability using a smaller ɛ can soon outperform that using a larger ɛ as the SCN becomes denser. E. Investigation of a Different Path oss Model In this subsection, we investigate the U ASE performance assuming a more complicated path loss model, in which the os probability is defined as follows 8 { 5 exp R r, < r d r = 5 exp r, 5 R 2, r > d where R =.56 km, R 2 =.3 km, and d = R. he simulation results of the area spectral ln efficiency A ASE λ, vs. the BS density is shown in Fig. 6. As can be seen from Fig. 6, the area spectral efficiency with the exponential os probability model exhibits a slow-down in the rate of growth in certain BS density regions, which qualitatively confirms our observations in subsection VI-C with the linear os probability model. Specifically, in Fig. 6, the numerical result for λ is around 2 BSs/km 2. Furthermore, the area spectral efficiency with the exponential os probability model exhibits a similar trend as discussed in subsection VI-C with the linear os probability model, i.e., using a smaller U power compensation factor ɛ can outperform that using a larger ɛ as the SCN becomes denser. F. Investigation of the Performance Impact of Ricean Fading In this subsection, we investigate the U ASE performance assuming a linear path loss model including the Ricean fading. Here we adopt a practical model of Ricean fading 7 with K factor K = 5 db. he simulation results of the area spectral efficiency A ASE λ, vs. the BS density is shown in Fig. 7. As can be seen from Fig. 7, the area spectral efficiency with the linear os probability model and the Ricean fading exhibits a slow-down in the rate of growth as the BS density increases, which qualitatively confirms our observations in subsection VI-C for the linear os probability model and the Rayleigh fading. Furthermore, the area spectral c 26 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

12 his article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI.9/WC , IEEE 2 Area spectral efficiency bps/hz/km 2 4 he proposed analysis ǫ=.6 Simulation he proposed analysis ǫ=.7 Simulation he proposed analysis ǫ=.8 Simulation BS density λ BSs/km 2 Figure 7. Area spectral efficiency A ASE λ, vs. the BS density with the linear os probability model, different ɛ and SINR threshold = db, including the Ricean fading. efficiency with the Ricean fading exhibits a similar trend as discussed in subsection VI-C with the Rayleigh fading, i.e., using a smaller U power compensation factor ɛ can outperform that using a larger ɛ as the SCN becomes denser. Since the simulation results of Ricean fading and Rayleigh fading are not qualitatively different, we suggest to use a simplified model with the Rayleigh fading in theoretical analysis. VII. CONCUSION In this paper, we have investigated the impact of a piecewise linear path loss model incorporating both os and NoS transmissions in the performance of the U of dense SCNs. Analytical results were obtained for the coverage probability and the ASE performance. he results show that os and NoS transmissions have a significant impact in the ASE of the U of dense SCNs, both quantitatively and qualitatively, compared with previous works that does not differentiate os and NoS transmissions. Specifically, we found that he ASE may suffer from a slow growth as the UE density increases in the U of dense SCNs. he ASE with a smaller U power compensation factor considerably outperforms that with a larger U power compensation factor in dense SCNs. he reverse is true for sparse SCNs. As our future work, we will consider other factors of realistic networks in the theoretical analysis for SCNs, such as the introduction of Ricean fading or Nakagami fading, because the multi-path fading model is also affected by the os and NoS transmissions. APPENDIX A: PROOF OF HEOREM Given the piecewise path loss model presented in Section III, P cov λ, can be derived as P cov λ, = SINR r f R r dr = = d + d d N + d N N n= P gζr ɛ P g A r α ɛ P g A N r αn ɛ P g A r α ɛ P g A N r αn ɛ n + n N. f R r dr os fr, rdr NoS fr, N r dr os fr,n rdr NoS fr,n N r dr 5 In the following, we show how to compute fr,n r and fr,n N r. o compute fr,n r, we define two events as follows Event B : he nearest BS with a os path to the UE is located at distance X. he CCDF of X is written as F X x = exp x u 2πuλdu 5. aking the derivative of F X x with regard to x, we can get the PDF of X as x fx x = exp u 2πuλdu x 2πxλ. 52 Event C N conditioned on the value of X : Given that X = x, the nearest BS with a NoS path to the UE is located farther than distance x, where /α N A x α = A N x αn, and x = A x α /A N. he conditional probability of C N on condition of X = x can be computed by C N X = x = exp x u 2πuλdu 53. hen, we consider the event that the UE is associated with a BS with a os path and such BS is located at distance Rn. fr,n r can be derived as c 26 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

13 his article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI.9/WC , IEEE 3 fr,n r = fx r CN X = r = exp r u 2πuλdu r 2πrλ exp r u 2πuλdu, d n < r < d n. 54 Having obtained fr,n r, we move on to evaluate P g A r α ɛ os in 7 as P g A r α ɛ os = g σ 2 +I Z P A r α ɛ os } = E IZ {exp = exp σ 2 +I Z P A r α ɛ σ 2 P A r α ɛ E IZ {exp = exp σ 2 P A r α ɛ I Z P A r α ɛ IZ } P A r α ɛ, 55 where IZ s is the aplace transform of RV I Z evaluated at s. o compute fr,n N r, we define two events as follows Event B N : he nearest BS with a NoS path to the UE is located at distance X N. he CCDF of X N is written as F X N x = exp x u 2πuλdu. aking the derivative of F X x with regard to x, we can get the PDF of X N as fx N x = exp x u 2πuλdu x 56 2πxλ. Event C conditioned on the value of X N : Given that X N = x, the nearest BS with a os path to the UE is located farther than distance x 2, where /α A x α 2 = A N x αn, and x 2 = A N x αn /A. he conditional probability of C on condition of X N = x can be computed by { C X N = x x2 exp u, < x y = exp d u. 2πuλdu, x > y 57 hen, we consider the event that the UE is associated with a BS with a NoS path and such BS is located at distance Rn N. f N r can be derived as R,n fr,n N r = fx N r C X N = r = exp r u 2πuλdu r 2πrλ exp r 2 u 2πuλdu, d n < r < d n. Similar to 55, can be computed by P g A N r αn ɛ = E IZ {exp = exp P g A N r αn ɛ NoS } σ 2 +I Z P A N r αn ɛ σ 2 P A N r αn ɛ IZ 58 NoS P A N r αn ɛ 59 Our proof is completed by applying the definition of n and n N in. APPENDIX B: PROOF OF EMMA 2 Based on 2, = d = d exp IZ can be obtained as P g A r α ɛ σ 2 P A r α ɛ P A r α ɛ os fr, rdr f R, r dr. 6 he aplace transform IZ s is expressed as c 26 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

14 his article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI.9/WC , IEEE IZ s = E IZ exp si Z = E rz,dz,g z exp s P β r z ζ d z g z Z = E rz,dz E gz exp sp β r z ζ d z g z Z = E rz,dz Z = exp 2πλ E r rz +sp βr zζx = exp 2πλ E r rz xdx +s P βr z ζx = exp 2πλ d xd r E rz +s P βr z os xdx ζx exp 2πλ d x r d E rz +s P βr z NoS xdx ζx exp 2πλ d E rz +s P βr z NoS xdx, ζx 6 where the expectation function averaged over r z is derived as follows E rz +s os P βr z ζx 62 = +sp βr zζd z +s P βu ζx f R z u du By plugging 62 into 6, we can obtain 24. APPENDIX C: PROOF OF EMMA 5. xdx By using the change of variable πλr 2 r, 45 can be rewritten as N 2 = πλd 2 IZ exp σ 2 α N ɛ P rπλ P rπλ α N ɛ e r d r. 63 By using the change of variable r πλ d 2 v, 63 can be rewritten as 2 N = σ exp 2 IZ P v+πλd 2 πλ α N ɛ P v+πλd 2 πλ α N ɛ 4 e πλd 2 e v dv. 64 By using the method of Gauss-aguerre quadrature as shown in 48, we complete the proof. REFERENCES D. ópez-pérez, M. Ding, H. Claussen, and A. H. Jafari, owards Gbps/UE in cellular systems: Understanding ultradense small cell deployments, IEEE Commun. Surveys and utorials, vol. 7, no. 4, pp , Fourthquarter GPP, R , Small cell enhancements for E-URA and E-URAN - Physical layer aspects, Dec X. Ge, S. u, G. Mao, C.-X. Wang, and. Han, 5G Ultra- Dense Cellular Networks, IEEE Wireless Commun., vol. 23, no., pp , Feb Xiaohu Ge, Song u, ao Han, Qiang i and Guoqiang Mao, Energy Efficiency of Small Cell Backhaul Networks Based on Gauss-Markov Mobile Models, IE Networks, vol. 4, no. 2, pp , M. Ding, P. Wang, D. ópez-pérez, G. Mao, and Z. in, Performance Impact of os and NoS ransmissions in Dense Cellular Networks, IEEE rans. Wireless Commun., vol. 5, no. 3, pp , Mar D. Novlan, H. S. Dhillon, and J. G. Andrews, Analytical modeling of uplink cellular networks, IEEE rans. Wireless Commun., vol. 2, no. 6, pp , Jun Ding, M. Ding, G. Mao, Z. in, and D. ópez-pérez, Uplink Performance Analysis of Dense Cellular Networks with os and NoS ransmissions, IEEE Int. Conf. Commun. ICC, Kuala umpur, Malaysia, May J. G. Andrews, F. Baccelli, and R. K. Ganti, A tractable approach to coverage and rate in cellular networks, IEEE rans. Commun., vol. 59, no., pp , Nov 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 rans. Wireless Commun., vol., no., pp , Oct. 2. B. Yu,. Yang, H. Ishii, and S. Mukherjee, Dynamic DD Support in Macrocell-Assisted Small Cell Architecture, IEEE J. Sel. Areas Commun., vol. 33, no. 6, pp. 2-23, Jun. 25. Anushiya Kannan, Baris Fidan and Guoqiang Mao, Robust Distributed Sensor Network ocalization Based on Analysis of Flip Ambiguities, IEEE Global elecommunications Conference Globecom, pp. - 6, Ruixue Mao and Guoqiang Mao, Road raffic Density Estimation in Vehicular Networks, IEEE WCNC, pp , Guoqiang Mao and Brian D.O. Anderson, Graph heoretic Models and ools for the Analysis of Dynamic Wireless Multihop Networks, IEEE WCNC, pp. - 6, Y. Hu, Y. Hong, and J. Evans, Uplink Coverage and Spatial Blocking in Poisson Cellular Networks, IEEE Int. Conf. Commun. ICC, Sydney, Australia, pp , Jun c 26 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.