On the Asymptotic Behavior of UltraDensification under a Bounded DualSlope Path Loss Model
Abstract
In this paper, we investigate the impact of network densification on the performance in terms of downlink signaltointerference (SIR) coverage probability and network area spectral efficiency (ASE). A sophisticated bounded dualslope path loss model and practical user equipment (UE) densities are incorporated in the analysis, which have never been jointly considered before. By using stochastic geometry, we derive an integral expression along with closedform bounds of the coverage probability and ASE, validated by simulation results. Through these, we provide the asymptotic behavior of ultradensification. The coverage probability and ASE have nonzero convergence in asymptotic regions unless UE density goes to infinity (full load). Meanwhile, the effect of UE density on the coverage probability is analyzed. The coverage probability will reveal an Ushape for large UE densities due to interference fall into the nearfield, but it will keep increasing for low UE densites. Furthermore, our results indicate that the performance is overestimated without applying the bounded dualslope path loss model. The derived expressions and results in this work pave the way for future network provisioning.
I Introduction
Network densification is considered as a key enabler to cope with the upcoming 5G data tsunami [1][2]. Deploying more base stations (BSs) can rapidly increase the network capacity by shortening the BS and user equipment (UE) association distance as well as by reducing percell traffic load. As densification goes on, BS density may easily exceed UE density, forming an ultra dense network (UDN) [3]. Its simplest example can be offpeak traffic hours under dense BS deployment. Peak hours can also be suitable cases since average BS load in practice is only 20% due to network stability [4]. In the UDN, a large number of UEvoid BSs within their coverages emerge, and the overall network transits from being fully loaded to partially loaded in which not all BSs are active (i.e., transmitting signals to serve the UEs within their cells). Such a UDN may evaporate the advantage of densification since there is less than one UE per cell on average. Therefore, it is crucial to understand the asymptotic behavior of ultra densification for the purpose of network deployment.
The pioneering work [5] provides a comprehensive understanding on the impact of BS density in a fully loaded downlink cellular network with a simple singleslope unbounded path loss model. Illustrated in Fig. 1 as a baseline, it concludes that BS densification does not change the signaltointerference ratio (SIR) of an individual UE, but linearly improves the area spectral efficiency (ASE) defined as sum rate per unit area. As BS density grows, the desired signal and interference growths cancel each other, leading to such result. However, it is difficult to apply this conclusion to UDNs due to its simplified signal propagation and load models [6][7].
In a UDN where BSUE distance shrinks, a simple unbounded path loss model for the path loss exponent may amplify the received signals when , which is unrealistic. In addition, a large amount of signals in a UDN are transmitted from the nearfield of a receiver, and these signals experience less attenuation owing to sparse shadowing, i.e. nearfield path loss exponent . A simple singleslope path loss model cannot capture such a distinction. Furthermore, a full load model forces BSs to always transmit signals despite nonnegligible portion of UEvoid BSs, overestimating interference. It is thus important to incorporate such propagation and load characteristics in detail so as to examine the impact of ultra densification. For this end our system model considers the trifold aspect: bounded path loss, dualslope path loss and partial load models, as illustrated in Figs. 1 and 2.
In preceding works, the impacts of and on SIR coverage probability, defined as for a target threshold , are respectively investigated by [7] and [9, 10]. Both models leads to a conclusion that SIR coverage probability asymptotically converges to (i.e. ) as BS density increases.^{3}^{3}3Nearfield environment in [10] is confined to in this article, which originally considers a general . The reason is as follows: as BS density grows, interference keeps increasing while the desired signal increase is saturated under (a); the number of nearfield interferer increases under (b), dominating the increase in the desired signal from ‘a single’ BS. On the other hand, the impact of is clarified in [11, 12], showing SIR coverage probability asymptotically converges to , i.e. . It comes from the fact that UEs’ neighboring BSs under a nearest association rule are always active while the rest of BSs become inactive. This results in making interfering BS density converge to UE density while the desired signal keeps increasing. A recent work [8] combines both and , and interestingly concludes that SIR coverage probability still converges to asymptotically since dominates .
Motivated by the discussions, we aim to combine , , and altogether, and investigate their aggregate impact on the asymptotic SIR coverage behavior. The main contributions of this paper are listed below.
Ii System Model
We consider a downlink cellular network where BSs and UEs are distributed according to two independent homogeneous Poisson Point Processes (PPPs) and . The densities of BS and UE are denoted as and respectively. We assume each UE is associated with its closest BS whose coverage area comprises a Voronoi tessellation as shown in Fig. 2. Each BS becomes inactive without transmitting any signal when its coverage area, the Voronoi cell, is empty of active UEs. Correspondingly, each active BS has at least one UE in its cell and will randomly choose one of them to serve. According to [13], the probability that a BS becomes active is given as
(1) 
which allows us to incorporate the partial load model.
Both BS and UE are equipped with a single antenna and BSs transmit with unit power. Rayleigh fading is used to model the channel gain, with the fading coefficients are i.i.d zero mean unit variance complex normal distributed random variables. Since we will focus on the asymptotic behavior and the system is interferencelimited in dense networks, we will neglect the noise power and examine SIR throughout the paper.
We consider that path loss attenuation from a BS to a UE is distinguished under two different regions, nearfield and farfield at the UE as illustrated in Fig. 2. In a nearfield within radius , a transmitted signal experiences less absorption and diffraction, so the nearfield path loss exponent becomes less than the farfield exponent . In the innermost near field, the transmitted signal becomes no longer attenuated within radius because of the physical volume of the UE. This dualslope path loss can be formulated as a piecewise function of the propagation distance , shown as
(2) 
where is the radius of bounded path loss region, i.e. the path loss in the range of is assumed constant; ; is the critical distance to divide the near and farfield; and and are the near and farfield path loss exponents for , respectively.
(6)  
(7)  
(8)  
(9)  
(10) 
(11)  
(12) 
Iia Performance Metrics
In this paper, we will focus on two performance metrics from both user and network perspectives: SIR coverage probability of a typical UE and the ASE of the network. We analyze the performance of a typical user located at the origin randomly selected by the BS, which is permissible in a homogeneous PPP by Slivnyak’s theorem [14]. The SIR of a typical user denoted as 0 can be expressed as
(3) 
where and denote the distance and channel between BS and UE , . represents the set of active BSs which is not a homogeneous PPP. Nevertheless, we can assume as a homogeneous PPP with density , which has been shown to be accurate according to [15][16]. Given the downlink SIR of the typical user, the coverage probability is defined as:
(4) 
where T is the target SIR level.
The network ASE is defined as the sum average spectral efficiency of all active BSs achieving the target threshold in a unit area [16] and is given by
(5) 
where can be interpreted as the density of the BSs that successfully transmit the symbols to their users.
Iii Coverage Probability and ASE analysis
In this section, we derive the coverage probability and ASE expressions under a bounded dualslope path loss model and provide closedform bounds of them. Furthermore, we demonstrate the convergence of them in asymptotic regions where .
Proposition 1
(Coverage probability expression) In a cellular network with BS active probability , the coverage probability under a bounded dualslope path loss model is expressed in (6) at the bottom of this page, where the supplementary equations are listed in (7)(10) and with being the Gauss hypergeometric function.

See Appendix A.
Despite the complicated form of (6), the first and third integrals can be calculated into exponential expressions. In this case, by applying transforms to the second integral, we can derive closedform bounds of (6) with only exponential and hypergeometric functions as shown in the following proposition.
Proposition 2
(Coverage probability bounds) SIR coverage probability’s lower bound and upper bound are given as (11) and (12) at the bottom of this page, where , , , .

See Appendix B.
Applying Proposition 2 in asymptotic regions leads to the following proposition.
Proposition 3
(Asymptotic SIR coverage probability) As , SIR coverage probability converges to a finite value as follows.
(13) 
From (11) and (12) in Proposition 1, we have P_missingc^LB = lim_λ_b →∞ P_c^UB = 1H1e^λ_u πc_T(α, α_c, R_c, R_b)λ_b →∞lim_λ_b →∞ P_c = 1H1e^λ_u πmlim_λ_b →∞ H_1 = 1p_a →0lim_λ_b →∞ P_c = e^λ_u πc_T(α, α_c, R_c, R_b)
Proposition 3 emphasizes the importance of considering UE density in a UDN. The converged value is a decreasing function of and it tends to 0 when , i.e., in a fully loaded network. Thus, deploying infinite number of BSs will not bring the UE performance to a unprecedented level. In contrast, extreme densification will put the coverage probability into the danger of decreasing to 0, as shown in Fig. 3 in the next section. The converged result also depends on environmental parameters . It will increase as both path loss exponents grow since the coverage probability is a increasing function of path loss exponents [9]. A larger or will decline the performance since it either reduce the signal power or amplify the interference in the nearfield.
We now turn to the network perspective and study the asymptotic behavior of the ASE. Combining the definition in (5) with Proposition 3, we can easily obtain the following proposition.
Proposition 4
(Asymptotic ASE) As , ASE converges to a finite value as follows.
(14) Proposition 4 shows that the asymptotic ASE will increase with when is small and tends to 0 as . Unlike the coverage probability, the asymptotic ASE will be beneficial from ultradesification to some extent and but still highly depends on the UE density.
Returning from the asymptotic regions, we demonstrate how the ASE scales with BS density in the next proposition.
Proposition 5
(ASE scaling) The ASE scales with and is bounded by
(15) (16) 
See Appendix C.
Similar with its asymptotic behavior, the ASE will increase with BS density when is small and finally converge to as shown in Proposition 4.
Iv Numerical Results
In this section, we present the numerical results to study the performance of network densification and validate our theoretical analysis. We assume , , , and set the SIR threshold T = 10dB in all of our results. To calculate or simulate ‘fully loaded network’, we set which is a sufficiently large value so that .
Iva Effect of UE density
Fig. 3 shows the effect of UE density on coverage probability. An exact match between simulation and analysis is observed. Meanwhile, we find that coverage probabilities show completely different trends among different UE densities.
Fig. 4: Coverage probability bounds and comparison with previous models In full load model, the diminishing of coverage probability starts when interfering BSs fall into the nearfield of the typical UE and keeps decreasing since interference will continue increasing. When is finite, the interferer coordinates converge to UE coordinates in a UDN regime [11]. Thus the distance from the typical UE to its closest interferer can be approximated as the distance to the its closest neighbor UE , which has an expected value of . When UE density is low (e.g. ), the expected value is larger than the critical distance, which means the probability of no interferer inside the nearfield of the typical UE is very high. Hence, the coverage probability is a nondecreasing function of as in a singleslope model. In contrast, higher UE density (e.g. or ) leads to more potential interferers within critical distance. Thus coverage probability will decrease for the same reason as in fully loaded network. Nevertheless, when all the UEs in the nearfield get service, coverage probability will start increasing again since the interference are saturated and no longer increase. Therefore, it is important to estimate the active UE density for efficient network deployment or operation in order to avoid the decreasing region of coverage probability.
IvB Coverage probability analysis
In Fig. 4, we compare the coverage probability under our model with the previous models. We observe that the performance is overestimated with unbounded or singleslope models in highly densified regions. The reason is that those models either exaggerate the received power inside the bounded region or underestimate the interference in the nearfield.
The inaccuracy of path loss models may mislead the prediction of asymptotic behavior. For instance, the coverage probability will converge to 1 with unbounded models but to which is smaller than 1 (assume ) when applying a bounded model. Consistent with the result in Proposition 3, the converged value will decrease as UE density grows and finally falls to zero in the full load case as shown in Fig. 3. This is because the signal is limited by the bound effect and the overall performance will be dominated by the interference which depends on UE density.
Fig. 5: ASE bounds and comparison with previous models. IvC ASE analysis
Figure 5 depicts the scaling of ASE with regard to BS density. Aligning with proposition 5, the ASE first increase with BS density and then converge to a constant. The constant is larger than 0 in partially loaded network and decreases to 0 when the network is full load () as proved in Proposition 4.
By comparing Fig. 5 with Fig. 4, we can observe a tradeoff between UE and network performance during BS densification. In full load case, there exists an BS density threshold around . Before the threshold, although individual performance gets worse, densification is still beneficial from the network perspective. For partially loaded network, the tradeoff appears approximately between to . The phenomenon further demonstrates the necessity of applying a dualslope model because the coverage probability is a nondecreasing function of BS density in a singleslope model.
Our bounds in Proposition 2 and 5 are compared with the integral expression and shown in Fig. 4 and 5. The figures verify the asymptotic tightness of the bounds for . The upper bound is tighter in small UE density scenarios while the lower bound fits better for large UE densities. This is because the upper bound is close to singleslope model which is similar with small UE density scenario. The bounds can be used as approximations in large BS density regions.
V Conclusion
In this paper, we investigate the asymptotic behavior of ultradensification of base stations. To our best knowledge, this is the first work incorporating two key aspects of UDN modeling: a partially loaded network due to a finite active UE density and a dualslope path loss model with a bounded loss within a unit distance. With such models, we find that the asymptotic behavior of ultradense base station deployment is different from what was known with simpler assumptions, e.g. unit or zero convergence of coverage probability. Depending on the UE density, both UE coverage probability and ASE converge to either zero or a constant value. Even before the asymptotic regions, our results suggest that the densification cannot always improve the individual UE performance or boost the network throughput as well. The increment are prevented by introducing extra interference in the nearfield until all the UEs in the nearfield are served. Our work provides insights into the scaling of the network densification, and thus gives a guideline for the network deployment.
Appendix A Proof of Proposition 1
We start from the coverage probability expression under general path loss model and then plug in with our bounded dualslope model. According to the definition, the coverage probability can be expressed as:
(17) where (a) follows from BS distribution and (b) is due to the fact that , is the Laplace transform of interference which can be derived as
(18) where (a) is because and (b) follows the probability generating functional (PGFL) of the PPP. Plugging in and employing a change of variables results in
(19) Plugging (19) into (17) with gives the coverage probability under a general path loss fucntion in (20) as:
(20) Based on (20), we can substitute our bounded dualslope model (2) into it and get the expression in (6).
Appendix B Proof of Proposition 2
According to the expression of (7) and (9), and are linear functions of . Thus we can rewrite the first and third integral in (6) as follows:
(21) (22) In the second integral, from we can get . With the inequality, we can provide bounds for the second integral as:
(23) (24) Replacing the integrals in (6) with the exponential expressions above completes the proof.
Appendix C Proof of Proposition 3
Notation: Let and be two functions defined on some subset of the real numbers. One writes if and only if there exists a positive real number and a real number such that for all .
We omit the proof of the bounds since they come directly from Proposition 2. To prove the ASE scales with is equivalent with showing and . Denote as and from (12) we have:
(25) Then we can show and since and the other two parts 0 as . Thus .
For , from (11) we have
(26) which can be rephrased as:
(27) Thus, thus . Therefore and we complete the proof.
Acknowledgment
Part of this work has been supported by the H2020 project METISII cofunded by the EU. The views expressed are those of the authors and do not necessarily represent the project. The consortium is not liable for any use that may be made of any of the information contained therein.
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