Density Analysis of Small Cell Networks: From NoiseLimited to Dense InterferenceLimited
Abstract
Considering both nonlineofsight (NLOS) and lineofsight (LOS) transmissions, the transitional behaviors from noiselimited regime to dense interferencelimited regime have been elaborated for the fifth generation (5G) small cell networks (SCNs). Besides, we identify four performance regimes based on base station (BS) density, i.e., (i) the noiselimited regime, (ii) the signal NLOStoLOStransition regime, (iii) the interference NLOStoLOStransition regime, and (iv) the dense interferencelimited regime. To analytically illustrate the performance regime, we propose a unified framework characterizing the future 5G wireless networks over generalized shadowing/fading channels. Simulation results indicate that different factors, i.e., noise, desired signal and interference, successively and separately dominate the network performance with the increase of BS density. Hence, our results shed light on the design and management of SCNs in urban and rural areas with different BS deployment density.
I Introduction
According to the study of Prof.Webb [1], the wireless capacity has increased about 1 million fold from 1950 to 2000. Data shows that around improvement was achieved by cell splitting and network densification, while the rest of the gain, was mainly obtained from the use of a wider spectrum, better coding techniques and modulation schemes. In this context, network densification has been and will still be the main force to achieve the fold increase of data rates in the future fifth generation (5G) wireless networks [2, 3], due to its large spectrum reuse as well as its easy management. In this paper, we focus on the analysis of transitional behaviors for small cell networks (SCNs) using an orthogonal deployment with the existing macrocells, i.e., small cells and macrocells are operating on different frequency spectrum [4, 5, 6].
Regarding the network performance of SCNs, a fundamental question is: What is the performance trend of SCNs as the base station (BS) density increases? In this paper, we answer this question and identify four performance regimes based on BS density with considerations of nonlineofsight (NLOS) and lineofsight (LOS) transmissions. These four performance regimes are: (i) the noiselimited regime, (ii) the signal NLOStoLOStransition regime, (iii) the interference NLOStoLOStransition regime, and (iv) the dense interferencelimited regime. To analytically illustrate the performance regime, we propose a unified framework characterizing the future 5G wireless networks over generalized shadowing/fading channels. The main contributions of this paper are listed as follows:

We reveal the transitional behaviors from noiselimited regime to dense interferencelimited regime in SCNs and analyze in detail the factors that affect the performance trend. The analysis results will benefit the design and management of SCNs in urban and rural areas with different BS deployment density.

We identify four performance regimes based on BS density. For the discovered regimes, we present tractable definitions for the regime boundaries. More specifically,

The boundary between the noiselimited regime and the signal NLOStoLOStransition regime;

The boundary between the signal NLOStoLOStransition regime and the interference NLOStoLOStransition regime;

The boundary between the interference NLOStoLOStransition regime and the dense interferencelimited regime.


An accurate SCN model and generalized theoretical analysis: For characterizing the NLOStoLOS transitional behaviors in SCNs, we propose a unified framework, which is applicable to analyze a SCN utilizing the strongest received signal power association, assuming generalized shadowing/fading channels and incorporating both NLOS and LOS transmissions.
The reminder of this paper is organized as follows. In Section II, motivations and some recent work closely related to ours are presented. Section III introduces the system model and network assumptions. An important theorem used in the analysis on transforming the original network into an equivalent distancedependent network, i.e., the Equivalence Theorem, is presented and proven in Section IV. Section V studies the coverage probability and the ASE of SCNs. In Section VI, a comparison with other cell association scheme is provided. In Section VII, the analytical results are validated via Monte Carlo simulations. Besides, the transitional behaviors are elaborated and tractable definitions for the regime boundaries are presented. Finally, Section VIII concludes this paper and discusses possible future work.
Ii Motivations and Related Work
The modeling of the spatial distribution of SCNs using stochastic geometry has resulted in significant progress in understanding the performance of cellular networks [7, 8, 9]. Random spatial point processes, especially the homogeneous Poisson point process (PPP), have now been widely used to model the locations of small cell BSs in various scenarios. Existing results are likely to analyze the performance assuming that the networks operate in the noiselimited regime or the interferencelimited regime. However, the transitional behaviors from noiselimited regime to interferencelimited regime were rarely mentioned in their work. Some assumptions in the system model were even conflicted with each other, e.g., in [10] and [11], the millimeter wave networks were assumed to be noiselimited and interferencelimited, respectively. Besides, most work is usually based on certain simplified assumptions, e.g., Rayleigh fading, a single path loss exponent with no thermal noise, etc, for analytical tractability, which may not hold in a more realistic scenario. For instance, consider a SCN in urban areas, the path loss model may not follow a single power law relationship in the near filed and thus nonsingular [12, 13] or multipleslop path loss model [14] should be applied. Besides, signal transmissions between BSs and MUs are frequently affected by reflection, diffraction, and even blockage due to highrise buildings in urban areas, and thus NLOS/LOS transmissions should also be considered [11]. As a consequence, the detailed analysis of transitional behaviors are needed, with considerations of a more generalized propagation model incorporating both NLOS and LOS transmissions, to cope with these new characteristics in SCNs.
A number of more recent work had a new look at dense SCNs considering more practical propagation models. The closest system model to the one in this paper are in [15, 16, 11, 17, 10, 18, 12, 19]. In [15], the transitional behaviors of interference in millimeter wave networks was analyzed, but it focused on the medium access control. In [11] and [16], the coverage probability and capacity were calculated based on the smallest path loss cell association model assuming multipath fading modeled as Rayleigh fading and Nakagami fading, respectively. However, shadowing was ignored in their models, which may not be very practical for a SCN. The authors of [10] and [17] analyzed the coverage and capacity performance in millimeter wave cellular networks. In [10], selfbackhauled millimeter wave cellular networks were analyzed assuming a cell association scheme based on the smallest path loss. In [17], a threestate statistical model for each link was assumed, in which a link can either be in a NLOS, LOS or in an outage state. Besides, both [10] and [17] assumed a noiselimited network ignoring intercell interference, which may not be very practical since modern wireless networks generally work in an interferencelimited region. In [18], the authors assumed Rayleigh fading for NLOS transmissions and Nakagami fading for LOS transmissions which is more practical than work in [16]. However, the cell association scheme in [18] is only applicable to the scenario where the SINR threshold is greater than 0 dB. Besides, the ASE performance was not analyzed in [18]. In [12], a nearfiled path loss model with bounded pathloss was studied. In [19], a tractable performance evaluation method, i.e., the intensity matching, was proposed to model and optimize the networks.
To summarize, in this paper, we propose a more generalized framework to analyze the transitional behaviors for SCNs compared with the work in [15, 16, 11, 17, 10, 18, 12, 19]. Our framework takes into account a cell association scheme based on the strongest received signal power, probabilistic NLOS and LOS transmissions, multipath fading and/or shadowing. Furthermore, the proposed framework can also be applied to analyze dense SCNs, where BSs are distributed according to nonhomogeneous PPPs, i.e., the BS density is spatially varying.
Iii System Model
We consider a homogeneous SCN in urban areas and focus on the analysis of downlink performance. Assume that BSs are spatially distributed on an infinite plane and the locations of BSs follow a homogeneous PPP denoted by with an density of , where is the BS index. MUs are deployed according to another independent homogeneous PPP denoted by with an density of . All BSs in the network operate at the same power and share the same bandwidth. Within a cell, MUs use orthogonal frequencies for downlink transmissions and therefore intracell interference is not considered in our analysis. However, adjacent BSs may generate intercell interference to MUs, which is the main focus of our work.
Iiia Path Loss Model
We incorporate both NLOS and LOS transmissions into the path loss model, whose performance impact is attracting growing interest among researchers recently. In reality, the occurrence of NLOS or LOS transmissions depends on various environmental factors, including geographical structure, distance and clusters, etc. The following definition gives a simplified oneparameter model of NLOS and LOS transmissions.
The occurrence of NLOS and LOS transmissions can be modeled using probabilities and , respectively. The probabilities are functions of the distance between a BS and a MU which satisfy
(1) 
where denotes the Euclidean distance between a BS at and the typical MU (aka the probe MU or the tagged MU) located at the origin .
Regarding the mathematical form of (or ), N. Blaunstein [20] formulated as a negative exponential function, i.e., , where is a parameter determined by the density and the mean length of the blockages lying in the visual path between the typical MU and BSs. Bai [21] extended N. Blaunstein’s work by using random shape theory which shows that is not only determined by the mean length but also the mean width of the blockages. The authors of [17] and [21] approximated by using piecewise functions and step functions, respectively. The authors of [16] considered to be a linear function and a twopiece exponential function, respectively; both are recommended by the 3GPP.
It should be noted that the occurrence of NLOS (or LOS) transmissions is assumed to be independent for different BSMU pairs. Though such assumption might not be entirely realistic, e.g., NLOS transmissions caused by a large obstacle may be spatially correlated, the authors of [21] showed that the impact of the independence assumption on the SINR analysis is negligible.
Note that from the viewpoint of the typical MU, each BS in the infinite plane is either a NLOS BS or a LOS BS to the typical MU. Accordingly, we perform a thinning procedure on points in the PPP to model the distributions of NLOS BSs and LOS BSs, respectively. That is, each BS in will be kept if a BS has a NLOS transmission with the typical MU, thus forming a new point process denoted by . While BSs in form another point process denoted by , representing the set of BSs with LOS path to the typical MU. As a consequence of the independence assumption between LOS and NLOS transmissions mentioned above, and are two independent nonhomogeneous PPPs with intensity^{1}^{1}1In this article, density and intensity have the same meaning. and , respectively.
In general, NLOS and LOS transmissions incur different path losses captured by^{2}^{2}2As the derivations in scenarios with lognormal shadowing is much more complicated than that with Rayleigh fading, we choose to take the former as an example. It is found in Eq. (4) and Eq. (5) that the model can also be applied to Rayleigh fading and other generalized shadowing/fading models.
(2) 
and
(3) 
where the path loss is expressed in dB unit, and are the path losses at the reference distance (usually at 1 meter), and are respective path loss exponents for NLOS and LOS transmissions, and are independent Gaussian random variables with zero means, i.e., and , reflecting the signal attenuation caused by shadow fading. The corresponding model parameters can be found in [22, 23, 24, 25].
Accordingly, the received signal power for NLOS and LOS transmissions in W (watt) can be expressed by
(4) 
and
(5) 
respectively, where (or ) denotes lognormal shadowing for NLOS (or LOS) transmission, and , and are all constants.
Therefore, the received power by the typical MU from BS is given by
(6) 
where is a random indicator variable, which equals to 1 for NLOS transmission and 0 for LOS transmission, and the corresponding probabilities are and , respectively, i.e.,
(7) 
Based on the path loss model discussed above, for downlink transmissions, the SINR experienced by the typical MU associated with BS can be written by
(8) 
where is the Palm point process [26] representing the set of interfering BSs in the network to the typical MU and denotes the noise power at the MU side, which is assumed to be the additive white Gaussian noise (AWGN).
IiiB Cell Association Scheme
Considering NLOS and LOS transmissions, two cell association schemes can be chosen, i.e., the maximum average received power and the maximum instantaneous SINR. In our work, we assume the typical MU should connect with the BS that provides the highest SINR [8, 27]. More specifically, the typical MU associates itself to the BS given by
(9) 
Intuitively, the highest SINR association is equivalent to the strongest received signal power association. Such intuition is formally presented and proved in Lemma 1.
Lemma 1.
For a nonnegative set , , if and only if , .
Proof:
For a nonnegative set , , if and only if , thus if and only if , which completes the proof. ∎
Lemma 1 states that providing the highest SINR is equivalent to providing the strongest received power to the typical MU. It follows from Eq. (9) and Lemma 1 that the BS associated with the typical MU can also be written as
(10) 
where , and the set . In the following, we mainly use Eq. (10) to characterize the considered cell association scheme.
Iv The Equivalence of SCNs and the Distribution of the Strongest Received Signal Power
Before presenting our main analytical results, first we introduce the Equivalence Theorem that will be used throughout the paper. The purpose of introducing the Equivalence Theorem is to unify the analysis considering different multipath fading and/or shadowing, and to reduce the complexity of our theoretical analysis. Then based on this theorem, we derive the cumulative distribution function (CDF) of the strongest received signal power.
Iva The Equivalence of SCNs
In this subsection, an equivalent SCN to the one being analyzed will be introduced, which specifies how the intensity measure and the intensity are changed after a transformation of original PPPs. More specifically, denoting by
(11) 
and
(12) 
the received signal power in Eq. (4) and Eq. (5) can be written as
(13) 
and
(14) 
From the discussion in Subsection IIIA, the BS’s location can be viewed as a nonhomogeneous PPP with an equivalent intensity of . Through the above transformation which scales the distances between the typical MU and all other BSs using Eq. (11) and (12), the scaled point process for NLOS BSs (or LOS BSs) still remains a PPP denoted by (or ) according to the displacement theorem [28]. The intuition is that in the equivalent networks, the received signal power and cell association scheme are only dependent on the new equivalent distance (or ) between the BSs and the typical MU, while the effects of transmit power, multipath fading and shadowing are incorporated into the equivalent intensity (or the equivalent intensity measure) of the transformed point process. Besides, and are mutually independent because of the independence between and . As a result, the performance analysis involving path loss, multipath fading, shadowing, etc, can be handled in a unified framework. This motivates the following theorem.
Theorem 2 (The Equivalence Theorem).
Assume that a general fading or shadowing satisfy . The system which consists of two nonhomogeneous PPPs with intensities and respectively, representing the sets of NLOS and LOS BSs, and in which each MU is associated with the BS providing the strongest received signal power is equivalent, in terms of performance to the typical MU located at the origin, to another system consisting of two nonhomogeneous PPPs with intensities (functions) and respectively, representing the sets of NLOS and LOS BSs, and in which the typical MU is associated with the nearest BS. Moreover, intensities (functions) and are given by
(15) 
and
(16) 
respectively, where
(17) 
and
(18) 
Proof:
See Appendix A. ∎
In [29], a similar theorem which was also extended from Blaszczyszyn’s work [8] was proposed to analyze a dimensional network, in which NLOS and LOS transmissions are not considered. By utilizing the Equivalence theorem above, the transformed cellular network has the exactly same performance for the typical MU with respect to the coverage probability and the ASE compared with the original network, which is proved in Appendix A and validated by Monte Carlo simulations in Section VII. After transformation, the received signal power and cell association scheme are only dependent on the equivalent distance between the BSs and the typical MU, i.e., and , while the effects of transmit power, multipath fading and shadowing are incorporated into the equivalent intensity shown in Eq. (15) and Eq. (16). Therefore, the complexity of theoretical analysis can be significantly reduced.
Remark 3.
Remark 4.
In our system model, lognormal shadowing is assumed. Thus the condition of is satisfied. For a general case of shadowing and/or fading model, can be easily met due to the bounded fading in practice.
In the next subsection, we will provide an application of the Equivalence theorem, i.e., using the equivalence theorem to derive the distribution of the strongest received signal power.
IvB The Distribution of the Strongest Received Signal Power
In this subsection, we use stochastic geometry and Theorem 2 to obtain the distribution of the strongest received signal power. Then we will use simulation results to validate our theoretical analysis.
Lemma 5.
Proof:
See Appendix B. ∎
If a specific NLOS/LOS transmission model is given, the distribution of the strongest received signal power can be easily derived using Lemma 5. The following is an example assuming that the LOS transmission probability follows a negative exponential distribution.
Assume that and , where is a constant determined by the density and the mean length of blockages lying in the visual path between the typical MU and the connected BS [11], then the CDF of the strongest received signal power is given by Eq. (19). Fig. 1 illustrates the CDF of the strongest received signal power and it can be seen that the simulation results perfectly match the analytical results. From Fig. 1, we can find that over 50% of the strongest received signal power is larger than 51 dBm when and this value increases by approximately 16 dB when , which indicates that the strongest received signal power improves as the BS density increases.
V The Coverage Probability and ASE Analysis
In downlink performance evaluation, for networks where BSs are random distributed according to a homogeneous PPP, it is sufficient to study the performance of the typical MU located at the origin to characterize the performance of a SCN using the Palm theory [26, Eq. (4.71)]. In this section, the coverage probability is firstly investigated and then the ASE will be derived from the results of coverage probability.
The coverage probability is generally defined as the probability that the typical MU’s measured SINR is greater than a designated threshold , i.e.,
(20) 
where the definition of SINR is given by Eq. (8) and the subscript is omitted here for simplicity. Now, we present a main result in this section on the coverage probability as follows.
Theorem 6 (Coverage Probability).
Proof:
See Appendix C. ∎
The coverage probability evaluated by Eq. (21) in Theorem 6 is at least a 3fold integral which is complicated for numerical computation. However, Theorem 6 gives general results that can be applied to various multipath fading or shadowing models, e.g., Rayleigh fading, Nakagami fading, etc, and various NLOS/LOS transmission models as well.
In the following, we focus on studying a special scenario in which a simplified NLOS/LOS transmission model is adopted for ease of numerical evaluations, which is expressed as follows
(24) 
where is a constant distance below which all BSs connect with the typical MU with LOS transmissions. This model has been used in some recent work [11, 10]. With assumptions above, the intensity measure for NLOS transmissions, i.e., , is expressed as follows
(25) 
where is the complementary error function, , and are all constants. After obtaining , the density of NLOS BSs, i.e., , can be readily derived as follows
(26) 
Similarly, the intensity measure and density for LOS BSs are
(27) 
(28) 
respectively, where , and are all constants. By substituting and above into Eq. (22) and Eq. (23), the coverage probability can be obtained in this specific scenario, followed by results in Section VII.
In the above scenario, the shadowing follows lognormal distributions. However, Theorem 6 can also be applied to a generalized shadowing/fading model and the coverage probability with Rayleigh fading will be derived in Section VI.
In the following, an asymptotic analysis will be given for the situation where BS deployment becomes ultra dense, i.e., , which helps to analyze the performance with a concise form.
Corollary 7.
If , the coverage probability of in Eq. (21) when converges as follows
(29) 
Proof:
A sketch of the proof of Corollary 7 is given here. In Eq. (29), is due to the reason that when , the network is interferencelimited and noise can be ignored compared with the aggregate interference, which is also validated by results in Section VII. The proof of can can be found in [8, Remark 9] and [11, Theorem 4] and are omitted here. ∎
From Corollary 7, it can be concluded that for dense SCNs the coverage probability is invariant with respect to BS density and even the distribution of shadowing/fading. However, when the BS density is not dense enough, the coverage probability reveals an interesting performance, which will be fully studied in Section VII.
Finally, the ASE in units of bps/Hz/k for a given BS density can be derived as follows
(30) 
Vi Comparisons with Other Cell Association Schemes
In this section, we will apply Theorem 6 to a SCN with Rayleigh fading. Moreover, the strongest received signal power association scheme is compared with the nearest BS association scheme to evaluate the performance impact of the two different cell association schemes. For brevity, we denote SPAS and NBAS by the Strongest Power Association Scheme and the Nearest BS Association Scheme, respectively.
Via Rayleigh Fading Assuming SPAS
As the majority of previous work considered Rayleigh fading and ignored lognormal shadowing, in this part we will apply our proposed model to Rayleigh fading scenario for the sake of a fair comparison. The main difference for theoretical analysis when replacing lognormal shadowing with Rayleigh fading lies in the intensity measure and the intensity. Assuming that and follow exponential distributions with rates and , respectively, then , , , can be calculated based on Theorem 2 as follows
(31) 
(32) 
(33) 
and
(34) 
where and denote the upper and the lower incomplete gamma functions, respectively, is the gamma function.
ViB Rayleigh Fading Assuming NBAS
In this part, the coverage probability will be provided by applying NBAS. Two scenarios will be considered, i.e., with consideration of NLOS and LOS transmissions and without considering the coexistence of NLOS and LOS transmissions, for comparisons with SPAS.
With consideration of NLOS and LOS transmissions: The derivations are very similar to the work in [30] and we just present the results as follows
(35) 
Without considering the coexistence of NLOS and LOS transmissions: If we do not differentiate NLOS and LOS transmissions, the coverage probability using NBAS is given in [7, Theorem 2] as follows
(36) 
where and
Vii Simulations and Discussions
This section presents numerical results to validate our analysis, followed by discussions to shed new light on the performance of SCNs. We use the following parameter values, , , , , , , ^{3}^{3}3The mean of Rayleigh fading for NLOS (or LOS) transmissions is equal to the mean of lognormal shadowing for NLOS (or LOS) transmissions., , , ^{4}^{4}4Including a noise figure of 9 dB at the typical MU., and [11, 10, 22].
Viia Validation of the Analytical Results of with Monte Carlo Simulations
Fig. 2 illustrates the coverage probability with respect to different SINR thresholds. We can observe that the analytical results (solid lines or dash lines) match well with the simulation results (markers), which validate our analytical analysis. The coverage probability decrease with the increase of SINR threshold, which is intuitively correct as a higher SINR threshold requires a higher signal quality. We also plot figures adopting different shadowing/fading models, i.e., lognormal shadowing and Rayleigh fading.
To fully study the SINR coverage probability with respect to BS density, the results of configured with and assuming SPAS are plotted in Fig. 3 and Fig. 4. As can be observed from Fig. 3, the analytical results match the simulation results well with respect to various BS intensities from to . With the assistance of Fig. 4, we conclude that the performance of small cell networks can be divided into four different regimes according to the density of small cell BSs, where in each regime, the performance is dominated by different factors.

NoiseLimited Regime (NLR): ( in Fig. (3) and using the parameters in the simulation). In this regime, the typical MU is likely to have a NLOS path with the serving BS. The network in the NLR regime is very sparse and thus the interference can be ignored compared with the thermal noise if we use SINR for performance metric. In this case, and the coverage probability will increase with the increase of as the strongest received power () will grow and noise power () will remain the same. While if we use SIR for performance metric, the SIR coverage probability remain almost stable in this regime as increases. This is because the increase in the received signal power is counterbalanced by the increase in the aggregate interference power. Besides, as the aggregate interference power is smaller than noise power, the SIR coverage probability is larger than the SINR coverage probability.

Signal NLOStoLOSTransition Regime (SN2LTR): ( in Fig. (3) and using the parameters in the simulation). In this regime, when is small, the typical MU has a higher probability to connect to a NLOS BS; while when becomes larger, the typical MU has an increasingly higher probability to connect to a LOS BS. That is to say, with the increase of , the typical MU is more likely to be in LOS with the associated BS, i.e., the received signal transforms from NLOS to LOS path. Even though the associated BS is LOS, the majority of interfering BSs are still NLOS in this regime and thus the SINR (or SIR) coverage probability keeps growing. Besides, from this regime on, noise power has a negligible impact on coverage performance, i.e., the SCN is interferencelimited.

Interference NLOStoLOSTransition Regime (IN2LTR): ( in Fig. (3) and using the parameters in the simulation). In this regime, the typical MU is connected to a LOS BS with a high probability. However, different from the situation in the SN2LTR, the majority of interfering BSs experience transitions from NLOS to LOS path, which causes much more severe interference to the typical MU compared with NLOS interfering BSs. As a result, the SINR (or SIR) coverage probability decreases as the increase of because the transition of interference from NLOS path to LOS path causes a larger increase in interference compared with that in signal.

Dense InterferenceLimited Regime (DILR): ( in Fig. (3) and using the parameters in the simulation). In this regime, the network is extremely dense and grow close to the LOSBSonly scenario as the increase of . The SINR (or SIR) coverage probability will become stable with the increase in BS density as any increase in the received LOS BS signal power is counterbalanced by the increase in the aggregate LOS BS interference power, which is also illuminated by Corollary 7.
Another interesting observation in Fig. 3 is that the network experiencing Rayleigh fading outperforms that experiencing lognormal shadowing when the network is sparse, while the network experiencing lognormal shadowing outperforms that experiencing Rayleigh fading when SCNs becomes dense. However, this phenomenon is highly related to the assumed SINR threshold, which is shown in Fig. 2. If the SCN becomes ultra dense, the coverage probability approaches the same asymptotic value regardless of shadowing or fading model.
In Fig. 5, we compare the performance of different cell association schemes using NBAS as a baseline. To guarantee the fairness of comparison, Rayleigh fading is assumed for all studied scenarios. It is observed that by assuming SPAS, the coverage probability have a considerable gain compared with that assuming NBAS, with the peak coverage probability rising from 0.6 to 0.8. Besides, we also plot figures which are exclusive of NLOS/LOS transmissions, i.e., signalslope path loss model [7] is adopted. It is found that the coverage probability firstly increase with the increase of BS density and then becomes stable and independent of when the SCN is dense, if we adopt parameters of NLOS transmissions to the signalslope path loss model. In comparison, the coverage probability is stable even when the BS density is rather sparse, if we adopt parameters of LOS transmissions to the signalslope path loss model.
ViiB Boundary Definitions
Based on the qualitative results above, it is interesting to develop a qualitative definition of the boundaries among adjacent regimes. In this subsection, we propose the following definition to characterize three BS density boundaries, which makes the analysis of SCNS more formal.
Definition 8.
The boundary between NLR and SN2LTR is which is defined as follows
(37) 
The intuition of this definition is when , the aggregate interference has a greater impact on network performance than that caused by noise.
Definition 9.
The The boundary between SN2LTR and IN2LTR is , which is defined as follows
(38) 
The definition above reveals that is the maximum coverage probability if other parameters are fixed. When , LOS interference will degrade the coverage performance.
Definition 10.
The The boundary between IN2LTR and DILR is , which is defined as follows
(39) 
where .
When becomes larger and larger, the SCNs fall into the DILR, i.e., the aggregate interference might be extremely large, which is shown by Eq. (39). In the following, we will analyze the ASE performance in the four defined regimes.
ViiC Discussion on the Analytical Results of
In this part, the ASE with is evaluated analytically only, as is a function of shown in Eq. (30).
Fig. 6 illustrates the ASE with different cell association schemes vs. BS density . It is found that the ASE of the sparse SCN has a similar growth tendency with that of the SCN which employs NLOS transmission configurations, while the ASE of dense SCN approaches the performance of the SCN which employs LOS transmission configurations. Specifically, when the SCN is in the NLR and SN2LTR, e.g., , the ASE quickly increases with because the network is generally noiselimited or the aggregate interference power is relatively low, and thus adding more small cells immensely benefits the ASE of the SCN. And when the network is in the front section of IN2LTR , i.e., , the ASE exhibits a slowingdown in the rate of growth due to the fast decrease of the coverage probability at , which is shown in Fig. 3, Fig. 4 and Fig. 5. Second, when , the ASE will pick up the growth rate since the decrease of the coverage probability becomes a minor factor compared with the increase of BS density . Finally, when the SCN becomes extremely dense, i.e., in the DILR, the ASE exhibits a nearly linear trajectory with regard to because both the signal power and the interference power are now LOS dominated, and thus statistically stable as explained before.
ViiD Guidance on Network Design
Based on the findings of NLOStoLOStransition, in this subsection we will introduce some guidance on how to design and manage the cellular networks in order to optimize the network performance as we evolve into dense SCNs.
As described in section VIIA and VIIC, the ASE increases almost for sure as SCNs becomes denser due to the gain of frequency reuse. In contrast, the coverage probability of SCNs will firstly increase and then decrease with the increase of BS density . In this context, there is a trade off between the coverage probability and the ASE in the future 5G SCNs incorporating both NLOS and LOS transmissions. While in [7], denser SCNs always provide better network performance with respect to the ASE as well as the coverage probability. According to the data, the current 4G network is operating in the SN2LTR. As we deploy more and more BSs in the future to meet the skyrocketing demands on wireless data, the network will fall into the IN2LTR. In this regime, we need elaborately design the network system including transmission techniques, medium access control (MAC) protocols and coding techniques, etc, to compensate the impair of network coverage caused by strong LOS interference. The most common MAC protocols are interference cancellation, interference avoidance and interference control. By jointly utilizing advanced transmission techniques like beamforming techniques, multipleinput multipleoutput (MIMO), multiantenna, coordinated multipoint (CoMP) transmissions and better coding techniques, the interference will be mitigated to a acceptable level, which benefits both the coverage probability and the ASE a lot.
Viii Conclusions and Future Work
In this paper, we illustrated the transition behaviors in SCNs incorporating both NLOS and LOS transmissions. Based on our analysis, the network can be divided into four regimes, i.e., the NLR, the SN2LTR, the IN2LTR and the DILR, where in each regime the performance is dominated by different factors. The analysis helps to understand as the BS density grows continually, which dominant factor that determines the cellular network performance and therefore provide guidance on the design and management of the cellular networks as we evolve into dense SCNs. Moreover, our work adopt a generalized shadowing/fading model, in which lognormal shadowing and/or Rayleigh fading can be treated in a unified framework.
In our future work, shadowing and multipath fading model will be considered simultaneously which is more practical for the real network. Furthermore, heterogeneous networks (HetNets) incorporating both NLOS and LOS transmissions will also be investigated.
Appendix A: Proof of Theorem 2
Firstly, we will obtain the intensity measure of ; and then the intensity will be easily acquired by taking a derivation of . By using displacement theorem [8, 28], the point process is Poisson with intensity measure
(40) 
where is a ball centered at the origin with radius and results by converting from Cartesian to polar coordinates. Then the intensity of denoted by can be given by
(41) 
Note that to ensure the intensity measure is finite for any bounded set (a set is bounded if it can be contained in a ball with a finite radius), has to satisfy a certain condition. As , from Eq. (40), we get an inequality as follows
(42) 
If the expectation , then . Using similar approach, the intensity measure and intensity of the PPP are obtained by Eq. (18) and Eq. (16), respectively.
As for the cell association scheme, it is obvious that the original scheme is equivalent to the scheme which actually corresponds to the nearest BS association scheme. Thus the proof is completed.
Appendix B: Proof of Lemma 5
Denote the strongest NLOS received signal power and the strongest LOS received signal power by and , respectively. That is, and . Then the probability can be derived as