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The Stochastic Geometry Analyses of Cellular Networks with {\alpha}-Stable Self-Similarity

To understand the spatial deployment of base stations (BSs) is the first step to facilitate the performance analyses of cellular networks, as well as the design of efficient networking protocols. Poisson point process (PPP) has been widely adopted to characterize the deployment of BSs and established the reputation to give tractable results in the stochastic geometry analyses. However, given the total randomness assumption in PPP, its accuracy has been recently questioned. On the other hand, the actual deployment of BSs during the past long evolution is highly correlated with heavy-tailed human activities. The {\alpha}-stable distribution, one kind of heavy-tailed distributions, has demonstrated superior accuracy to statistically model the spatial density of BSs. In this paper, we start with the new findings on {\alpha}-stable distributed BSs deployment and investigate the intrinsic feature (i.e., the spatial self-similarity) embedded in the BSs. Based on these findings, we perform the coverage probability analyses and provide the relevant theoretical results. In particular, we show that for some special cases, our work could reduce to the fundamental work by J. G. Andrews. We also examine the network performance under extensive simulation settings and validate that the simulations results are consistent with our theoretical derivations.


I. INTRODUCTION
As a key enabler in the information and communications technology (ICT) industry, cellular networks play a decisive role in delivering communication messages and entertainment content [2]. In order to meet the increasing traffic demand, cellular network operators gradually deploy different types of necessary infrastructure including lots of base stations (BSs). To understand the spatial deployment of BSs is the first step to facilitate the performance analyses of cellular networks, as well as the design of efficient networking protocols. In the earliest stages, a two-dimensional hexagonal grid model was used, implying that BSs were deployed regularly, which deviates from the real scenarios. In the recent stages, Poisson point process (PPP) [1], [3]- [5] was assumed, which could roughly model the randomness in the realistic deployment of BSs in cellular networks, meanwhile leading to tractable results. However, given the total randomness assumption in PPP, its practical accuracy has been recently questioned [6], [7]. Consequently, in order to reduce the modeling gap between the PPP model and the realistic spatial deployment of R. Li, Z. Zhao, C. Qi  BSs, other kinds of point processes have been exploited to obtain more precise analytical results [8]. In particular, Poisson cluster processes [9], [10] and Gauss-Poisson process [11] are exploited to model the attractive behavior between BSs, while Matérn hard core point process (MHCP) [12], Ginibre point process (GPP) [8], [13], [14], and determinantal point process [15] are studied to characterize the repulsive feature between BSs [7], [16]- [18].
On the other hand, the actual deployment of BSs during the past long evolution is highly correlated with human activities [19]. Humans tend to live together with societal relationship, and their social behaviors lead to mobile traffic hotspots, thus causing BSs to be more densely deployed in certain areas. Interestingly, according to an observation named "preferential attachment" [20], Barabási et al. argued that many large networks should grow to be heavy-tailed. As its names implies, a heavy-tailed distribution has non-exponential bounded tail. Mathematically, for a heavy-tailed random variable X, the probability Pr(X > x) satisfies lim x→∞ e κx Pr(X > x) = ∞, for all κ > 0. Many well-known statistical distributions including power-law distribution (also named as generalized Pareto distribution), Weibull distribution, log-normal distribution, and α-stable distribution [6], [21], [22] belong to the heavy-tailed family. Therefore, heavy-tailed distributions appear to be more suitable to capture the societal feature and clustering property for the practical BSs.
Based on the practical BS deployment information, we have validated that the α-stable distributions demonstrate superior accuracy to statistically model the spatial density of BSs and also outperform other aforementioned heavy-tailed distributions [7], [19], [23]- [25]. But the spatial density alone cannot reach some intuitive conclusions. Fortunately, the α-stable distributions, which have been widely adopted to characterize the distribution of aggregated traffic at the BS level in cellular networks and at the switch level in wired broadband networks, often imply the self-similarity of traffic [22], [26]. As a popular property in complex networks [27], the self-similarity of traffic in the temporal domain means that the distribution of traffic remains invariant under different temporal scales. Similar to the case of traffic, as the practically deployed BSs are αstable distributed, it is natural to ask whether the spatial selfsimilarity holds in cellular networks. Furthermore, compared to other point process models, the inhomogeneous PPP with αstable distributed density becomes applicable with significant merit. The system model based on multiple tiers of PPPs may lead to some tractable results, but it lacks the reasonable explanation to divide the gradually deployed and increasingly denser heterogeneous cells [28] into different tiers. On the contrary, α-stable distributions contribute to understanding the spatial self-similarity and bridging the gap between cellular networks and other social behavior-based complex networks.

A. Related Works
Stochastic geometry tools have successfully established the reputation to model and analyze wireless networks, as they are able to capture the topological randomness in the network geometry and lead to tractable analytical results [1], [3]- [5], [9]- [18]. There is no doubt that the PPP belongs to the most popular point process used in the literature because of its tractability and has led to a lot of meaningful research results for cellular networks with different kinds of cutting-edge techniques. The baseline operations in singletier and/or multiple-tier downlink cases are examined in [1], [29]- [32] (and references therein) while their counterparts in uplink cases are considered in references like [33]- [38]. User association and load balancing are examined in [39]- [42]. Cognitive, cooperative and intelligent cellular networks are taken into account in [18], [43]- [47]. Energy efficiency, energy harvesting, and BS sleeping for green cellular networks are investigated in [48]- [52]. Besides, multiple-input multipleoutput (MIMO) antenna and in-band full-duplex (FD) systems are studied in [53]- [62]. Also, the PPP-related works have been extended to mobile ad hoc networks [63]. Along with the continuous progress in techniques for physical-layer and network management in wireless networks, the PPP model offers a useful theoretical platform for the initial performance calibration, so as to save economic expenditure and avoid timeconsuming large-scale setup for realistic tests. For example, FD communication, which benefits from the advance in signal processing and self-interference cancellation techniques, is optimistically promoted to double the spectral efficiency for wireless networks [64]. However, given the more complicated interference in FD networks, it is essential to conduct a careful re-examination. Consistent with the system-level simulation results in [64], [65] has taken advantage of the PPP model and demonstrated the potential negative effect that FD communications might impose on the uplink transmission.
Although the PPP model provides many useful performance trends, the concern about total independence between nodes (e.g., BSs) has never stopped. Hence, in order to reduce the modeling gap between the PPP model and the practical BS deployment, point processes with either attractive or repulsive spatial correlations have been explored as well.
• For attractive point processes, [66] has shown that Poisson cluster process is more suitable to model the spatial distribution of BSs in urban areas. The general clustering nature of deployed BSs in highly populated urban areas clearly reflects the aggregation property of ever-growing traffic demands in cellular networks [7]. [10] has further verified the aggregated interference when the transmitting nodes are modeled by Poisson cluster process and compared the corresponding results with that in the classical PPP model. • On the contrary, [29] and [67] have argued that BSs, in particular macro BSs, in cellular networks tend to be deployed systematically, such that any two BSs are not too close. Thus, a spatial model based on a point process with the repulsive nature seems to be more desirable [29], [67]. Furthermore, the MHCP model [12] possesses limited tractability and leaves many open challenges to be addressed. The GPP model, one typical example of determinantal point processes on the complex plane, has been widely adopted and shows a promising tradeoff between accuracy and tractability [8], [13], [14]. In particular, Deng et al. [8] have shown that the GPP leads to the same trend curve of coverage probability 1 as the PPP. Hence, we can come to the following conclusion that the spatial BS density of large-scale cellular networks simultaneously possesses two conflicting features, that is, the density is very large in some clustering regions while being significantly smaller in others. The conclusion is also consistent with our common sense. Furthermore, the heavy-tailed distributions, which generate small values with the high probability but still allow comparatively larger values, fit well to model the spatial BS density. Our previous works [6], [7], [23]- [25] have shown that the α-stable distribution, one kind of heavy-tailed distributions, could be used to perfectly model the spatial BS density in China and Italy, especially in urban areas.
In fact, α-stable distributions are not totally new to researchers in the communications field. On one hand, a distribution is called stable if a linear combination of two independent identically distributed random variables has the same distribution [21]. In particular, the Gaussian distribution, which is one specific kind of α-stable distributions with α = 2, has been widely adopted to model the distribution of channel noise. On the other hand, α-stable distributions have been observed for the distribution of wired broadband network traffic [68] since two decades ago, and have recently been found in modeling the cellular network traffic [22], [26]. αstable distributions also owe their importance in both theory and practice to the generalization of the central limit theorem [21] and the accompanying self-similarity [27], [69]- [71] of the stable family. Furthermore, the well-known self-similarity lays the very foundation for the traffic predictability, thus making it possible to perform some actions beforehand. For example, [70] has highlighted the practicability to perform resource reservation schemes for the broadband network traffic with self-similarity. Hence, as the spatial BS density could be better modeled by the α-stable distributions, it is essential to theoretically examine the spatial self-similarity and understand its impact on coverage probability. In other words, we will take the very first step to investigate the spatial self-similarity in the BS deployment and perform stochastic geometry analyses of cellular networks when the BS density is α-stable distributed.

B. Contributions
Different from the previous works, this paper aims at explaining the implications to statistically model the spatial BS density by α-stable distributions and to build the theory to derive the coverage probability mathematically. Belonging to one of the precursor works, we take advantage of a large amount of practical BS deployment records and provide the following key insights: • Firstly, this paper visits and confirms the self-similarity in the spatial deployment of BSs. To our best knowledge, it is the first work in the literature to validate the spatial selfsimilarity among BSs in cellular networks. Therefore, this paper lays the foundation to apply the advance in complex network theory to examine cellular networks. • Secondly, this paper provides the approach to characterize the coverage probability and derive the computable representation when the spatial BS deployment satisfies the self-similarity and the density of the practical BSs obeys α-stable distributions. • Thirdly, this paper theoretically proves that for some special cases, our result could be reduced to the fundamental work by J. G. Andrews in [1]. Besides, this paper studies the coverage probability performance under extensive simulation settings and validates that the simulations results are consistent with our theoretical derivations. The paper also evaluates the impact of self-similarity on the coverage probability performance. The remainder of the paper is organized as follows. In Section II, we present some necessary mathematical background, introduce the realistic dataset of spatial BS deployment and validate the spatial self-similarity. In Section III, we provide the analyses of coverage probability. Section IV evaluates the computable representation obtained by the theoretical analysis and compares it with the classical PPP model. Finally, we conclude this paper in Section V.

A. Mathematical Background
Beforehand, Table I summarizes the most used notations in this paper.
and µ ∈ R such that its characteristic function is of the following form: (1) The function E(·) represents the expectation operation with respect to a random variable. α is called the characteristic exponent and indicates the index of stability, while β is identified as the skewness parameter. α and β together determine the shape of the distribution. Moreover, σ and µ are called scale and location shift parameters, respectively. In particular, if α = 2, the α-stable distribution reduce to the Gaussian distribution.
Our previous works [6], [23] have validated that β = 1 holds for the fitting results of actual spatial BS density in both Hangzhou, China and Rome, Italy. For simplicity of representation, we use the operator b ∼ c to denote that b and c have the same distribution and further have λ ∼ S(α, σ, µ) to indicate that the spatial BS density λ follows the α-stable distributions with β = 1.
2) The self-similarity: The self-similarity has been mostly applied to model time series [69]. Given zero-mean, stationary is the Hurst parameter indicating the decay rate of statistical dependence of two points with increasing time interval. In other words, the self-similarity implies that the time series is exactly or approximately similar to a part of itself.
The concept of self-similarity is also applicable to spatial scenarios. For example, H. ElSawy et al. [16] have shown the aggregated interference from distributed nodes to a receiver obey α-stable distributions, while X. Ge et al. [72] have verified that the statistical characteristics of the wireless cellular coverage boundary possess the self-similarity or fractal property.
However, no exact self-similar phenomenon exists in the real world. Most of the self-similar phenomena observed in the real world only have the statistical characteristic. Moreover, the self-similarity of random processes is usually evaluated by the Hurst parameter, which can be estimated using two typical methods and should be in the interval (0, 1).

• The rescaled adjusted range statistic (R/S) method:
For the random process Y t , the partial sum is denoted by Z(n) = n t=1 Y t , and then the sample variance could be denoted by S 2 (n) = n t=1 (Yt− Z(n) n ) 2 n . The R/S statistic could be formulated as R(n) . The log-log relationship between the R/S statistic and n should also be linear with a slope of H. with respect to m should follow a straight line with the slope ι if the data are statistical self-similar. In this case, H = 1 − ι 2 . Moreover, a larger H ∈ (0.5, 1) corresponds to stronger selfsimilarity.

B. Statistical Modeling
As validated in [6], [23], [25], when we divide the region into several parts and calculate the corresponding probability density function (PDF) of the spatial BS density in each part, we observe that the PDF follows the α-stable distributions. For example, Fig. 1(a) depicts that a region is divided into four parts. The spatial BS density in each part λ 1 , λ 2 , λ 3 and λ 4 are mutually dependent and obey the α-stable distributions The Rayleigh fading gain for the signal link g The fading exponent for the interfering links Assumed to follow Rayleigh fading. ζ Rayleigh fading factor The Hurst parameter The self-similarity zooming parameter a = 2 R The predefined radius of inner circle in Fig. 1 The spatial density of BSs -N (r) The number of BSs within an r-radius circle - 1. An illustration of the self-similarity for the spatial BS deployment.
with the same values α, σ and µ. The correlation between BS density in different parts makes it challenging to directly derive the coverage probability. Instead, it is essential to take the statistical modeling of the correlation into consideration. In that regards, self-similarity emerges as a promising technique to characterize the correlation of the spatial density, as it has manifested its importance and effectiveness by modeling the correlation in different scales. In order to show the self-similarity in spatial BS deployment, we take advantage of the collected BS deployment records in Hangzhou, China and Rome, Italy 2 . Fig. 2(a) and 2(d) illustrate the corresponding deployment situation in both cities. Similar to the common validation process in temporal dimension [69], we verify the accuracy of self-similarity in spatial BS deployment as follows. As depicted in Fig. 1(b), we randomly select a point as the starting "origin point" and thus get some concentric circles with increasingly larger radii (e.g., from R to 2R, 3R, · · · ). Then, we could get a series, each value corresponding to the number of BSs in one circle. Afterwards, we could apply the aforementioned R/S method and V-T method in Section II-A2. Fig. 2(b)(c) and Fig. 2(e)(f) provide the corresponding log-log plots for both cities. From these figures, it can be observed that all the estimated Hurst parameters are very close to 1. Therefore, we could boldly argue that the spatially deployed BSs possess the spatial selfsimilarity. Then, according to the definition of self-similarity, the number of BSs N (aR) within a circle of radius aR should satisfy that N (aR) ∼ a H N (R), ∀R, a > 0, 0 ≤ H < 1, where a indicates the self-similarity zooming parameter and H denotes the Hurst parameter.
Remark 1. For cellular networks, the number of BSs exhibit the spatial self-similarity.

III. THE COVERAGE PROBABILITY ANALYSES
In this part, we derive the coverage probability in a downlink cellular network. Specifically, the coverage probability is defined as the probability that the SINR for a mobile user achieves a target threshold. Mathematically, the coverage probability is formulated as where δ denotes the pathloss exponent factor of the standard propagation channel. N 0 is the additive white Gaussian noise (AWGN) factor and T is the target threshold. Besides, assuming that the fading follows the Rayleigh fading, we have h ∼ exp(ζ). Without loss of generality, we assume the mobile user is located at the "origin point". When the distance from a mobile user to its serving BS b 0 is r, I r denotes the cumulative interference from all other BSs i (except BS b 0 ) to the mobile user.

A. Impact of α-Stable Distributions and Self-similarity
Before delving into the coverage probability, we first talk about how the α-stable distributions and the self-similarity will mathematically shape the BS deployment?
As depicted in Fig. 1(c), we assume that there exists a specific R (i.e., R > r) to divide the whole region of interest into two parts. Besides, the spatial BS density λ for the region within the radius R satisfies λ ∼ S(α, σ, µ). Moreover, we can have the following lemma to characterize the spatial BS density λ S for the region outside the radius R.
Then, the Laplace transform of λ S could be derived as By merging Eq. (6) and Eq. (7), we obtain the result.

B. The PDF of the Distance from Serving BS to UE
In this paper, we adopt the minimum distance as the user association metric. Therefore, when the distance from a mobile user to its serving BS b 0 is r, all distances from the interference BSs to the target mobile user must be larger than r. The derivation methodology of the PDF of r could basically follow the well-established lines in [1]. However, as the BS density varies in space and obeys the α-stable distributions, the derivations should be re-considered. Therefore, we give the following the theorem. Theorem 1. The PDF of the distance from the closest BS r to the target mobile user in a cellular network with α-stable distributed BS density is expressed in Eq. (8).
Proof. Firstly, we take advantage of the simple fact that the null probability of a 2-D Poisson process in an area A with the given density λ is exp(−λA). Since λ ∼ S(α, σ, µ), we have This corollary could be achieved by directly applying Theorem 1 with a static density λ = µ and thus σ = 0. It is also consistent with the conclusions in [1].

C. Main Result
We now state our main result for the coverage probability analysis. Generally, the coverage probability p c is mainly determined by the SINR threshold T , the spatial BS density distribution S(α, σ, µ), and the pathloss exponent δ. In some sense, p c (T, S(α, σ, µ), δ) is tightly coupled with the cumulative interference I r for the α-stable distributed and selfsimilar BS deployment. The following lemma characterizes the relationship between p c (T, S(α, σ, µ), δ) and I r .
Lemma 2. The coverage probability p c (T, S(α, σ, µ), δ) could be obtained from the following formula, that is, where L Ir (s) is the Laplace transformation of random variable I r evaluated at s conditioned on the distance to the closest BS from the origin. The Rayleigh fading coefficient is assumed to satisfy h ∼ exp(ζ).
We leave the proof of Lemma 2 in Appendix A, as the proof could be simply obtained by calculating the condition probability on the distance r from the serving BS to the mobile user.
Next, we focus on how to calculate the Laplace transform of the cumulative interference L Ir (s), when the BS deployment obey α-stable distributions and self-similarity. In Section III-A, we divide the region of interest into two concentric circles. For the inner circle ∆ with a radius R, the impact of the spatial BS density λ is similar to the impact of r and explicitly represented by the latter's PDF f (r). Meanwhile, the outer circle ∆ S with the radius spanning from R to aR has a spatial BS density λ S ∼ λa H−2 and understanding the impact of the interference from BSs in the outer circle is one of the core contributions of this paper. For simplicity of representation, we let I r = i∈∆/b0 g iR −δ i + i∈∆S g iR −δ i to denote the cumulative interference from all the other BSs i (except the serving BS b 0 ) to the mobile user with the distancê R i and pathloss g i . We could obtain the following lemma, Lemma 3. The Laplace transform of the cumulative interference I r for a cellular network with α-stable distributed BSs could be formulated as Eq. (12), where Θ(s, b, c) Proof. According to the definition of Laplace transform, we have The equation (a) comes from the identical, independent distribution of g i and its further independence from the point process.
Based on Corollary 2, we have the following remark.
Remark 2. When we expand the interference to the whole region, the cumulative interference from the outer circle could be negligible and the coverage probability is mainly contingent on the SINR within the inner circle.
When R → ∞, we can further simplify the results to get more interesting insight.
Proof. Similar to the lines to prove Corollary 2, we have lim Then lim R→∞ Ξ(ζT r δ , R, aR) = 0. On the other hand, as R → ∞, where the Equation (a) comes from Γ(s, There, we obtain the result in the theorem after simple mathematical manipulations. Next, we state the coverage probability when extra constraints are imposed, that is, the spatial density in the inner circle is fixed (i.e., σ = 0 and λ = µ in Theorem 3). We get the following corollary.
Remark 3. When the spatial density is fixed and equals λ, the self-similarity patterns no longer take effect and our result could be reduced to the well-recognized conclusions obtained by J. G. Andrews et al. in [1].

IV. NUMERICAL ANALYSES
In this part, we provide numerical evaluations through which we compare the coverage probability of the cellular network with α-stable distributed BS density and the classical PPP model. Our simulation parameters are configured according to Table I. In particular, the default spatial BS density for the PPP model is consistent with that in [1]. As [21] states that when α ∈ (0, 1), the mean value of S(α, σ, µ) equals µ. Therefore, it is fair to compare the network with α-stable distributed BS density and the classical PPP model.
Firstly, we leverage the theoretical analysis results and give the coverage probability comparison under different AWGN environment in Fig. 3. From the figure, the network with αstable distributed BS density exhibits significant performance improvement than the PPP model in low SNR environment while the performance gap becomes gradually narrow along with the increase of SNR. On the other hand, the performance gap between SNR = 0 dB and SNR = 20 dB is only 4 dB and 6 dB for the α-stable model and the PPP, respectively, which demonstrates that the cellular network is interferencelimited rather than noise-limited. Fig. 3 also depicts the results by Monte Carlo simulations (see the dashed or dotted curves) and each curve is averaged under 15000 iterations. It can be observed that the results obtained by Monte Carlo simulations match well with those obtained by the theoretical analyses, which further verifies the correctness of our theoretical derivations. Fig. 4 illustrates interesting coverage probability comparison under various (average) spatial BS densities. It shows that µ = 0.25 has lead to remarkable coverage probability and  also made the networks to be interference-limited. Therefore, further increasing the spatial density would add more to the aggregated interference and impose negative impact on the coverage probability. It also implies that the influence of the spatial BS density on the coverage probability is rather sophisticated. Fig. 5 and Fig. 6 further examine the impact of α and σ. They show that when α becomes larger or σ gets smaller, the network coverage probability increases. In some cases, the network with α-stable self-similarity could produce interior performance than the PPP model. Next, we focus on the impact of the spatial self-similarity on the coverage probability. Recalling the statements in Section II-A2, the spatial self-similarity is examplified by the relationship N (aR) ∼ a H N (R) or λ S ∼ a H−2 λ. Therefore, we examine the coverage probability when a and H differs and provide the corresponding results in Table II. From the table, we observe increasing H or decreasing a will result in a slight reduction of the coverage probability. The trivial impact of a and H on the network performance can be explained as that a variation of H or a will make the spatial BS density  λ S of the outer circle larger from a probabilistic sense, thus making BSs over-crowded and generating huge interference. This phenomenon that stronger self-similarity incurs negative impact is also consistent with its counterpart of self-similar Ethernet traffic. As stated in [73], stronger self-similarity in Ethernet traffic will make heavier traffic prone to arrive in a sequel and congest the network. We continue the performance analyses in Fig. 7(a) and Fig.  7(b). Fig. 7(a) shows reducing R will cause a reduction of coverage probability for low SINR threshold but an increment for high SINR threshold. On the other hand, consistent with our intuition, Fig. 7(b) depicts that a smaller pathloss exponent will bring a lower coverage probability. However, the network with α-stable distributed BSs always demonstrates relatively larger coverage probability than the PPP model.

V. CONCLUSION
In this paper, we have performed the stochastic geometry analyses in cellular networks with α-stable self-similarly deployed BSs. Based on the practical BS deployment records in China and Europe (Italy), we have validated the spatial self-similarity among BSs. Afterwards, we have theoretically derived the coverage probability for cellular networks with self-similarity and provided a tractable result. In particular, we have demonstrated that our analytical results could be reduced to the works achieved by J. G. Andrews [1]. We have also simulated the coverage probability performance under extensive parameter settings and verified the consistence between the theoretical and simulation results. Our results  have shown that compared to the classical PPP model, the modeling with α-stable self-similarity could contribute to analyze the performance under more sophisticated network configurations, thus making it easier to understand the actual network variations.
There still exist some open questions to be addressed. For example, our work has shown that instead of improving the coverage performance, simple yet stubborn deployment of BSs incurs significant interference and degrades the coverage performance. Therefore, frequency reuse has been applied in practical cellular communication. In this case, it is still mean-ingful to combine α-stable self-similarity with more realistic network configurations, so as to produce more valuable results.

B. Derivations of Eq. (14)
In this part, we provide the details about the equation (a) in the Eq. (14).