Modeling Regular Aggregate Interference by Symmetric Structures

Abstract

In this chapter, downlink co-channel interference statistics in wireless cellular networks with hexagonal grid layout are investigated. The main target is to facilitate the analysis at user locations outside the center of the cell of interest.

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In this chapter, downlink co-channel interference statistics in wireless cellular networks with hexagonal grid layout are investigated. The main target is to facilitate the analysis at user locations outside the center of the cell of interest .

The proposal of a cellular structure for mobile networks dates back to 1947. Two Bell Labs engineers, Douglas H. Ring and W. Rae Young were the first to mention the idea in an internal memorandum [1]. Almost two decades later, in 1966, Richard H. Frenkiel and Philip T. Porter, shaped a hexagonal cellular array of areas to propose the first mobile phone system. Although never proposed as innovative research solution, the hexagon model gained high popularity within the research community and is still extensively utilized nowadays [2–7]. It serves either as the system model itself, or as a reference system for more involved simulation scenarios. On the other hand, its geometric structure renders closed-form analysis of aggregate interference statistics difficult [8]. Hence, simulation results often lack a mathematical back up.

Recently, closed-form results have been reported with system models based on stochastic geometry [9–11]. The stochastic approach is based on an ensemble of network realizations and is therefore not applicable when a fixed structure of the network is given. Since the well-planned deployment of macro-sites is not expected to vanish in the medium term, it is thus considered imperative to make interference analysis in the hexagonal grid model more convenient.

Current work on regular grid models has mainly focused on link-distance statistics [12, 13]. The authors also account for fading and provide closed-form approximations for the co-channel interference of a single link. However, convenient expressions for the moments and the distribution of aggregate co-channel interference are not available yet .

Based on the work in [14, 15], the contributions of this chapter are:

• A circular interference model to facilitate interference analysis in cellular networks with regular grid layout is introduced. Particular focus is placed on the hexagonal grid due to its ubiquity in wireless communication engineering [2–7]. The key idea is to consider the power of the interfering BSs as being uniformly spread along the perimeter of the hexagon .

• It is proposed to model interference statistics in a hexagonal scenario by a single Gamma Random Variable RV. Its shape- and scale parameters are determined in closed form by employing the circular model. The analysis yields key insights on the formative components of the interference distribution. A scenario with regularly arranged macro-sites and randomly distributed small cells demonstrates the model’s expedient application in heterogeneous cellular networks.

The chapter forgoes hexagonal grid setups with more than one ring of interferers as well as further performance analysis, which is enabled by the Gamma approximation. Both are considered straightforward and of no particular relevance for this thesis.

The remainder of this chapter is structured as follows: Sect. 11.1 provides preliminaries on the Gamma distribution. Section 11.2 specifies the hexagonal reference-system model. In Sect. 11.3 the circular interference model and its dual pendant are introduced. Section 11.4 investigates Gamma-distributed interference and its parametrization by the proposed circular interference model. In Sect. 11.5, the accuracy of the Gamma approximation is verified. In Sect. 11.6, the circular model is applied for modeling the interference from the macro BSs in a two-tier heterogeneous cellular network. Section 11.7 provides a comparison of the circular model against Long Term Evolution-Advanced LTE-A system level simulations. Section 11.8 concludes the chapter.

1 Preliminaries on the Gamma Distribution

In the current- as well as the subsequent chapter, particular focus is placed upon the Gamma distribution due to its wide range of useful properties for wireless communication engineering, some of which are outlined in this section .

The Probability Density Function (PDF) of a Gamma distributed RV X with shape parameter k and scale parameter $\theta$, i.e., $G\sim \varGamma [k,\theta ]$, is defined as

$$\begin{aligned} f_G(x) = \frac{1}{\theta ^k \varGamma (k)} x^{k-1}e^{-x/\theta }. \end{aligned}$$

(11.1)

Its mean and variance are given by $\mathbb {E}[G]=k\theta$ and $\mathrm{Var}[G]=k\theta ^2$.

The Gamma distribution exhibits the scaling property, i.e., if $G\sim \varGamma [k,\theta ]$, then $a\,G\sim [k,a\,\theta ],\, \forall a>0$, and the summation property, i.e., if $G_i\sim \varGamma [k_i,\theta ]$ with $i=1,2,\ldots ,N$, then $\sum _{i=1}^N G_i\sim \varGamma [\sum _{i=1}^N k_i, \theta ]$.

Consider an arbitrary distribution with mean $\nu$ and variance $\sigma ^2$. Then, the distribution $\varGamma [k,\theta ]$ with the same first- and second order moments has the parameters

$$\begin{aligned} k = \frac{\nu ^2}{\sigma ^2}, \quad \theta = \frac{\sigma ^2}{\nu }. \end{aligned}$$

(11.2)

These simple moment-matching identities can be exploited for accurately approximating fading distributions [16], such as generalized-K [17, 18] and log-normal [16–19], as well as aggregate interference statistics [11, 20, 21].

Additionally, the Gamma distribution covers the power fading distribution of various single- and multi-antenna schemes under the Rayleigh fading assumption. Conventional Single-Input Single-Output (SISO) yields an exponential distribution $\mathrm{Exp}[1/\theta ]$, which is equivalent to $\varGamma [1,\theta ]$. The power fading of Maximum Ratio Transmission (MRT) with $N_\mathrm{Tx}$ transmit antennas and one receive antenna can be modeled by $\varGamma [N_\mathrm{Tx},\theta ]$, Maximum Ratio Combining (MRC) with one transmit antenna and $N_\mathrm{Rx}$ receive antennas is characterized by $\varGamma [N_\mathrm{Rx},\theta ]$. Furthermore, MRC is often studied in the presence of Nakagami-m fading. Let $Y\sim \mathrm{Nakagami}[m,\varOmega ]$ and $G=Y^2$. Then, $G\sim \varGamma [m,\varOmega /m]$.

According to [22], the quotient $\gamma = S / I$ of two RVs $S\sim \varGamma [k_S,\theta _S]$ and $I\sim \varGamma [k_{I},\theta _{I}]$ is distributed as

$$\begin{aligned} f_\gamma (x) = \frac{(\theta _{I}/\theta _S)^{k_S}}{B(k_S,k_{I})} \left( 1+\frac{\theta _{I}}{\theta _S} x\right) ^{-k_S-k_{I}}x^{k_S-1},\ x>0 \end{aligned}$$

(11.3)

with $B(\cdot ,\cdot )$ denoting the Beta function. Interpreting $\gamma$ as a Signal-to-Interference Ratio (SIR) allows to determine the success probability $\mathbb {P}[\gamma >\delta ]$ for a given threshold $\delta$ as

$$\begin{aligned} \mathbb {P}[\gamma >\delta ] = \frac{\varGamma (k_S + k_{I})}{\varGamma (k_S)}\left( \frac{\theta _S}{\delta \, \theta _{I}}\right) ^{k_{I}} \, _2\bar{F}_1\left( k_{I}, k_S+k_{I}, 1+k_{I}, -\frac{\theta _S}{\delta \, \theta _{I}}\right) , \end{aligned}$$

(11.4)

where $_2\bar{F}_1(\cdot ,\cdot ,\cdot ,\cdot )$ is a regularized hypergeometric function [11].

These observations motivate the application of the Gamma RV as a sensible compromise between accuracy and tractability. Further properties of Gamma RVs will be discussed as needed.

2 Hexagonal Reference Model

The reference hexagonal setup is composed of a central cell and six interfering BSs, as shown in Fig. 11.1. The interferers are equipped with omnidirectional antennas and are located at the edges of a hexagon with radius R (marked as ‘$+$’ in Fig. 11.1). All BSs are assumed to transmit with the same power . The signal from the ith interfering BS with polar coordinates $\left( R,\varPsi _i\right)$ to a user with polar coordinates $(r,\phi )$ experiences macroscopic path loss and fading. It is assumed that $0<r\le R/2$, so as to assure that the user is associated with the central BS. The path loss is modeled by the exponential law

$$\begin{aligned} \ell \left( d_{r, \varDelta _i}^{(M)}\right) =\min \left( b_\mathrm{P}, \frac{1}{c_\mathrm{P}} \left( d_{r, \varDelta _i}^{(M)}\right) ^{-\alpha }\right) , \end{aligned}$$

(11.5)

where $b_\mathrm{P}$ denotes the intercept, $c_\mathrm{P}$ is a constant, $\alpha$ refers to the path loss exponent and

$$\begin{aligned} d_{r, \varDelta _i}^{(M)} = \sqrt{R^2 + r^2 - 2Rr\cos \left( \varDelta _i\right) }, \end{aligned}$$

(11.6)

with $\varDelta _i = \phi -\varPsi _i$ and $\varPsi _i = 2\pi \,i /M,\ i=1,\ldots , M$. In the remainder of this chapter, it is assumed that $d_{r,\varDelta _i}^{(M)} > (b_\mathrm{P}c_\mathrm{P})^{-1/\alpha }$. Exemplifying from [23], a minimum coupling loss of $70\,\,\mathrm{dB}$ and free space propagation at an LTE-A frequency of $f_\mathrm{c} = 2.14\,\,\mathrm{GHz}$ yield $b_\mathrm{P}=10^{-7}$, $c_\mathrm{P}= 8.05\times 10^{-3}$ and $(b_\mathrm{P}c_\mathrm{P})^{-1/\alpha } = 0.028\,\,\mathrm{m}$, hence justifying this assumption.

In the hexagonal scenario, $M=6$. The terms r and $\varDelta _i$ denote the user’s distance to the center and its angle-difference to the ith interfering BS, respectively. Motivated by Sect. 11.1, fading is modeled by an independent and Identically Distributed (I.I.D.) Gamma RV $G_i\sim \varGamma [k_0, \theta _0]$, where $k_0$ and $\theta _0$ refer to shape- and scale parameter, respectively.

Fig. 11.1

System model. Center cell with user at $(r,\phi )$. Interfering BSs are located at $(R,\varPsi _i$), where $\varPsi _i = 2\pi \,i /M,\ i=1,\ldots , M$

3 Circular Interference Model

In a one-tier hexagonal grid scenario, as presented in Sect. 11.2, the experienced aggregate interference power at position $(r,\phi )$ can be expressed as

$$\begin{aligned} I_{6}(r,\phi ) = \sum \limits _{i=1}^{6} P_\mathrm{M}\, G_i\, \ell \left( d_{r, \varDelta _i}^{(6)}\right) , \end{aligned}$$

(11.7)

where $P_\mathrm{M}$ denotes the transmit power, $G_i$ is the fading and $\ell (d_{r, \varDelta _i}^{(6)})$ refers to the path loss at distance $d_{r,\varDelta _i}^{(6)}$, as specified in Eqs. (11.5) and (11.6), respectively. Each sum term can be regarded as a RV $G_i$, which is weighted by the factor $P_\mathrm{M}\, \ell (d_{r,\varDelta _i}^{(6)})$. Hence, the statistics of $I_{6}(r,\phi )$ outside the cell-center, i.e., $r> 0$, are accessible via a sum of differently weighted RVs. Since, in general, this does not lead to closed-form results, as detailed in Chap. 12, in this chapter a circular interference model is proposed in order to facilitate the statistical analysis.

3.1 Proposed Model

In the proposed circular interference model, the power of the six reference BSs is spread uniformly along a circle of radius R. This is achieved by evenly distributing the total transmit power $6\, P_\mathrm{M}$ among M equally spaced BSs and considering the limiting case $M\rightarrow \infty$. By generalizing Eq. (11.7), this is expressed as

$$\begin{aligned} I_{C}(r) = \lim \limits _{M\rightarrow \infty } \frac{6\, P_\mathrm{M}}{M} \sum \limits _{i=1}^{M} G_i\,\ell \left( d_{r, \varDelta _i}^{(M)}\right) =\frac{6\, P_\mathrm{M}\,}{2\pi }\mathbb {E}\left[ {G_i}\right] \int \limits _{-\pi }^{\pi } \ell \left( d_{r,\varDelta }\right) d\varDelta , \end{aligned}$$

(11.8)

with $\ell (\cdot )$ from Eq. (11.5) and $d_{r,\varDelta _i}^{(M)}$ from Eq. (11.6). The terms $d_{r, \varDelta }$ and $\varDelta$ denote distance and angle-difference between the user and an infinitesimal interfering circular segment, as illustrated in Fig. 11.1.

Assuming a path loss exponent $\alpha = 2$, i.e., free space propagation, Eq. (11.8) can explicitly be evaluated as

$$\begin{aligned} I_{C}(r)=6\, P_\mathrm{M}\, \mathbb {E}\left[ G_i\right] \frac{1}{c_\mathrm{P}}\, \frac{1}{R^2-r^2}. \end{aligned}$$

(11.9)

An intuitive interpretation of this result is provided in the next section by the model’s pendant.

In the remainder of this chapter, $\alpha =2$ is employed. It represents the worst case of low interference attenuation. However, previously- as well as all subsequently presented analysis can be carried out in closed-form for $\alpha =2n$ with $n\in \mathbb {N}$. Values $\alpha$ other than these require the evaluation of elliptic integrals (see, e.g., [24]). Thus, a practical first order estimate for arbitrary values of $\alpha$ is achieved by evaluating the performance with 2n and $2(n+1)$, where $2n < \alpha <2(n+1)$.

3.2 The Dual Model

Consider a user in a hexagonal scenario, which is moved along a circle of radius r from $-\pi$ to $\pi$, as indicated in Fig. 11.1. The average expected interference along the circle can be calulated as

$$\begin{aligned} I'_{C}(r)&=\frac{1}{2\pi } \int \limits _{-\pi }^{\pi } \mathbb {E}\left[ I_{6}(r,\phi ')\right] d\phi ' \end{aligned}$$

(11.10)

$$\begin{aligned}&= \sum \limits _{i=1}^{6} P_\mathrm{M}\, \mathbb {E}\left[ G_i\right] \, \frac{1}{2\pi }\int \limits _{-\pi }^\pi \ell (d_{r, \varDelta }) d\phi '. \end{aligned}$$

(11.11)

The result is obtained by plugging Eq. (11.7) into Eq. (11.10), exchanging sum and integral, and exploiting the linearity of the expectation. It is equivalent to $I_{C}(r)$ in Eq. (11.8) and, consequently, also yields Eq. (11.9). Thus, the result is independent of the user’s angle-position. It can be interpreted as the average expected interference, i.e., the interference experienced by a typical user in a hexagonal scenario at distance r.

From Eq. (11.9) it is observed that the average expected interference increases by either increasing the transmit power $P_\mathrm{M}$, decreasing the distance of the interferers R, or moving the user further away from the origin, which is reflected by the parameter r. The fading enters the equation only via the expectation, i.e., Eqs. (11.8) and (11.11) hold for arbitrary fading distributions with finite mean. Finally, note that the circular interference model is not restricted to hexagons. By replacing ‘6’ by ‘N’ in Eqs. (11.7)–(11.11), it can generally be applied for substituting any convex regular N-polygonal model, as validated in Sect. 11.5.1.

4 Statistics of Aggregate Interference

In this section, aggregate interference in a hexagonal scenario with I.I.D. Gamma fading is investigated. Motivated by the findings in Sect. 11.1, it is proposed to approximate its statistics by a single Gamma RV. The corresponding shape- and scale parameters are dependent on the distance and can be determined by applying the previously presented circular model.

4.1 Interference Statistics at the Center

Assume I.I.D. Gamma fading with $G_i\sim \varGamma \left[ k_0, \theta _0\right]$. Then, according to Sect. 11.3, interference can be considered as a sum of Gamma RVs, which are weighted by the factors $P_\mathrm{M}\,\ell (d_{r, \varDelta _i}^{(6)})$, i.e., the received power without fading.

At the center of a hexagonal scenario (i.e., at $r=0$), all weighting factors are equal, i.e., $P_\mathrm{M}\,\ell (d_{r,\varDelta _i}^{(6)}) = P_\mathrm{M}\,\ell (R)$. By virtue of the scaling- and summation property of a Gamma RV (cf. Sect. 11.1), the resulting interference is distributed as

$$\begin{aligned} I_{6}(0,\phi ) \sim \varGamma \left[ 6\, k_0, \theta _0P_\mathrm{M}\ell (R) \right] . \end{aligned}$$

(11.12)

4.2 Interference Statistics Outside the Center

Outside the center (i.e., at $r>0$), the distances $d_{r, \varDelta _i}^{(6)}$ and, thus, also the weighting factors ${P_\mathrm{M}}\,{\ell (d_{r,\varDelta _i}^{(6)})}$ generally differ from each other. Hence, a non-uniform impact of the interferers is observed. Then, the interference statistics are only accessible via evaluating the distribution of a sum of Gamma RVs with varying scale parameter. This method is particularized in Chap. 12.

The current chapter resorts to the following first order estimate. It is proposed to approximate the typically experienced interference distribution at distance r, $0<r\le R/2$, by

$$\begin{aligned} \hat{I}(r)\sim \varGamma [\hat{k}(r), \hat{\theta }(r)]. \end{aligned}$$

(11.13)

The rationale for this model are findings in prior work, where out-of-cell interference in stochastic networks is appropriately assessed by a Gamma distribution [11]. If it can be proven as accurate, it considerably facilitates further performance analysis by applying the methods in Sect. 11.1.

The distribution in Eq. (11.13) is fully determined by the distance-dependent shape- and scale parameters $\hat{k}(r)$ and $\hat{\theta }(r)$, respectively. In order to evaluate the two parameters, firstly the proposed circular interference model is employed to determine expectation and variance of $\hat{I}(r)$. Then, it is exploited that $\mathbb {E}[\hat{I}(r)] = \hat{k}(r)\, \hat{\theta }(r)$ and $\mathrm{Var}[\hat{I}(r)] = \hat{k}(r)\, \hat{\theta }^2(r)$.

As discussed in Sect. 11.3.1, the distinct received powers from the interfering BSs can be averaged along a circle of radius r. Thus, the typical impact of one interferer is calculated as

$$\begin{aligned} P_\mathrm{M}\frac{1}{2\pi }\int \limits _{-\pi }^{\pi } \ell \left( d_{r, \varDelta }\right) d\varDelta = \frac{P_\mathrm{M}}{c_\mathrm{P}}\frac{1}{R^2-r^2}, \end{aligned}$$

(11.14)

and yields the average expected interference at distance r as

$$\begin{aligned} \mathbb {E}\left[ \hat{I}(r)\right] = 6\, k_0\theta _0\frac{P_\mathrm{M}}{c_\mathrm{P}}\frac{1}{R^2-r^2}. \end{aligned}$$

(11.15)

The variance of the aggregate interference comprises two components :

• 1. The variance of the fading, which calculates as

  $$\begin{aligned} \mathrm{Var}_f\left[ \hat{I}(r)\right] = 6\, k_0 \left( \theta _0 \frac{P_\mathrm{M}}{c_\mathrm{P}}\frac{1}{R^2-r^2}\right) ^2. \end{aligned}$$

  (11.16)

• 2. The variance of the received power without fading, which is caused by the unequal distances $d_{r, \varDelta _i}$. With

  $$\begin{aligned} \frac{1}{2\pi }\int \limits _{-\pi }^{\pi } \left( P_\mathrm{M}\ell \left( d_{r, \varDelta }\right) - P_\mathrm{M}\ell (R)\right) ^2d\varDelta = \left( \frac{P_\mathrm{M}}{c_\mathrm{P}R^2}\right) ^2 \frac{2r^2R^4 + r^4R^2-r^6}{\left( R^2-r^2\right) ^3}, \end{aligned}$$

  (11.17)

  the second variance component is obtained as

  $$\begin{aligned} \mathrm{Var}_d\left[ \hat{I}(r)\right] = 6 k_0 \left( \theta _0 \frac{P_\mathrm{M}}{c_\mathrm{P}R^2}\right) ^2 \frac{2r^2R^4 + r^4R^2-r^6}{\left( R^2-r^2\right) ^3}. \end{aligned}$$

  (11.18)

Since the two components are statistically independent, the overall variance is calculated as

$$\begin{aligned} \mathrm{Var}\left[ \hat{I}(r)\right]&= \mathrm{Var}_f\left[ \hat{I}(r)\right] + \mathrm{Var}_d\left[ \hat{I}(r)\right] \nonumber \\&= 6\, k_0\left( \theta _0 \frac{P_\mathrm{M}}{c_\mathrm{P}}\frac{1}{R^2-r^2}\right) ^2\left( 1+\frac{2r^2R^4 + r^4R^2-r^6}{R^6-r^2R^4}\right) \end{aligned}$$

(11.19)

where $\mathrm{Var}_f[\hat{I}(r)]$ and $\mathrm{Var}_d[\hat{I}(r)]$ refer to Eqs. (11.16) and (11.18), respectively.

Finally, the distance-dependent shape- and scale parameter are derived from Eqs. (11.15) and (11.19) as

$$\begin{aligned} \hat{k}(r)&= 6\, k_0\, \frac{R^4(R^2-r^2)}{R^6+r^2R^4+r^4R^2-r^6}, \end{aligned}$$

(11.20)

$$\begin{aligned} \hat{\theta }(r)&= \theta _0 \frac{P_\mathrm{M}}{c_\mathrm{P}}\, \frac{1}{R^2-r^2}\left( 1 + \frac{2r^2R^4 + r^4R^2-r^6}{R^6-r^2R^4}\right) . \end{aligned}$$

(11.21)

5 Numerical Results and Discussion

In this section, the accuracy of the circular model and the proposed Gamma approximation are verified by numerical evaluation.

5.1 Validation of Expected Aggregate Interference

First, the expected interference powers in the hexagonal reference scenario and the proposed circular interference setup are compared to each other. The transmit power and inter-site distance are specified as $P_\mathrm{M}=40\,\,\mathrm{W}$ and $R=500\,\,\mathrm{m}$, based on the standard 3rd Generation Partnership Project (3GPP) macro cell scenario from [23]. Intercept and constant of the path loss $\ell (\cdot )$ are set $b_\mathrm{P}=1$ and $c_\mathrm{P}= 1$ for simplicity . Fading is assumed to be distributed as $G_i\sim \varGamma [1,1]$. The parameters are summarized in Table 11.1.

Consider a user which is moved along a semi circle $\left\{ \left. \left( r,\phi \right) \right| \phi \in [0,\pi ] \right\}$, as indicated in Fig. 11.1. The expected interference in the hexagonal scenario is calculated as

$$\begin{aligned} \mathbb {E}\left[ I_{6}(r,\phi )\right] = \sum \limits _{i=1}^{6} {P_\mathrm{M}\, \mathbb {E}[G_i]\, \ell \left( d_{r, \varDelta _i}^{(6)}\right) }, \end{aligned}$$

(11.22)

with $I_{6}(r,\phi )$ from Eq. (11.7) and $\mathbb {E}[G_i]=1$. For the circular model, $\mathbb {E}\left[ I_{C}(r) \right] = I_{C}(r)$, with $I_C(r)$ from Eq. (11.9).

Table 11.1 System parameters for validation. Transmit power and circle radius are referred from [23]

Fig. 11.2

Expected aggregate interference experienced at position $(r,\phi )$ in circular- ($I_{C}(r)$) and hexagonal model ($\mathbb {E}[I_{6}(r,\phi )]$), respectively. Receiver distances $r = \{0,125,250\}\,\,\mathrm{m}$ refer to cell-center, middle of cell and cell-edge, respectively

Figure 11.2 depicts the evaluated results of Eqs. (11.9) and (11.22) for various distances r. It is observed that

• At cell-center, i.e., at $r = 0\,\,\mathrm{m}$, the expected interference powers in the hexagonal- and circular scenario ($\mathbb {E}[I_{6}(0,\phi )]$ and $I_{C}(0)$) are equal.

• Outside the center, i.e., at $r>0\,\,\mathrm{m}$, $\mathbb {E}[I_{6}(r,\phi )]$ fluctuates around $I_{C}(r)$. The deviation is weak in the middle of the cell ($r=125\,\,\mathrm{m}$), and strong at cell-edge ($r = 250\,\,\mathrm{m}$). Note that $\mathbb {E}[I_{6}(r,\phi )]$ is not symmetric about $I_{C}(r)$ due to the concavity of the path loss model.

The relative error of the circular interference model is calculated as

$$\begin{aligned} \varepsilon \, (r, \phi ) = \left| \frac{\mathbb {E}\left[ I_{6}(r,\phi )\right] - I_{C}(r)}{\mathbb {E}\left[ I_{6}(r,\phi )\right] }\right| , \end{aligned}$$

(11.23)

with $I_{C}(0)$ and $\mathbb {E}[I_{6}(0,\phi )]$ from Eqs. (11.9) to (11.22), respectively. The largest error occurs at cell-edge, i.e.,

$$\begin{aligned} \max \limits _{r, \phi } \epsilon \, (r, \phi ) = \max \limits _{\phi }\epsilon \, (250, \phi ). \end{aligned}$$

(11.24)

In the specified scenario, $\max _{\phi }\epsilon \, (250, \phi ) = 3.2\,\%$, as shown in Fig. 11.3.

Fig. 11.3

Maximum deviation of circular interference model from expected interference in convex regular N-polygonal models. The labeled cell-shapes can be arranged in a grid without overlapping areas

5.2 Validation of Gamma Approximation

In this section, the accuracy of the Gamma approximation in Eq. (11.13) and its parameterization by Eqs. (11.20) and (11.21) are verified. The exact position-dependent distributions of $I_6(r,\phi )$ are obtained by evaluating Theorem 12.1.

In order to capture a representative profile of distributions, three user distances $r=\{0,125,250\}\,\,\mathrm{m}$ and three angle-positions $\phi = \{0, \frac{\pi }{12}, \frac{\pi }{6} \}$ are considered, as illustrated by bold dots in Fig. 11.4. The distances correspond to cell-center, middle of cell and cell-edge, respectively. The angle $\phi = 0$ represents a user, which is moved directly towards its strongest interferer, $\phi =\frac{\pi }{6}$ refers to the path with two equidistant dominant interferers, and $\phi = \frac{\pi }{12}$ is a variation thereof .

Fading is specified as $G_i\sim \varGamma [2,1]$. This corresponds to a $1\times 2$ Single-Input Multiple-Output SIMO system with Rayleigh-fading and Multiple-Input Single-Output (MISO) at the user, or, equivalently, a $2\times 1$ MISO system with MRT at the BS.

Fig. 11.4

Setup for evaluation. Cutout of Fig. 11.1 (upper right quadrant)

The Cumulative Distribution Function (CDF) of the Gamma approximation, $F_{\hat{I}}(x;\hat{k}(r), \hat{\theta }(r))$ and the CDF $F_6(x; r, \phi )$ of $I_6(r,\theta )$ are evaluated at each distance r and angle $\phi$, respectively. The accuracy of the Gamma approximation is quantified by Kolmogorov–Smirnov KS statistics, which formulate as

$$\begin{aligned} D_\mathrm{KS}(r,\phi _m) = \sup _x\left| F_{\hat{I}}\left( x;\hat{k}(r), \hat{\theta }(r)\right) - F_{6}\left( x;r,\phi \right) \right| . \end{aligned}$$

(11.25)

Results are depicted in Fig. 11.5. The Gamma approximation most closely resembles the experienced interference distributions at $\phi =\frac{\pi }{12}$. In this case, the difference between exact- and approximated CDFs is less than $1\,\%$ for $r<159\,\,\mathrm{m}$ and $2.75\,\%$ at cell-edge ($r=250\,\,\mathrm{m}$). The largest deviation occurs at $\phi =\frac{\pi }{6}$, where the user is moved centrally in between its two dominant interferers (upper curve). Then, the distributions differ by less than $1\,\%$ for $r<155\,\mathrm{m}$ and by 3.7 % at cell-edge.

Fig. 11.5

KS statistics at position $(r,\phi _m)$, comparing Gamma approximation and exact distribution. Receiver distances $r\,\,\!{=}\!\,\,\{0,125,250\}$ m refer to cell-center, middle of cell and cell-edge, respectively

For qualitative evaluation, Fig. 11.6 depicts the exact CDFs and the corresponding Gamma approximations at the specified representative user positions, which are denoted by bold dots in Figs. 11.4 and 11.5, respectively.

Fig. 11.6

Aggregate interference at particular user positions (see bold dots in Fig. 11.4): Exact CDFs, as obtained by numerically evaluating [25] for a hexagon scenario dashed lines and corresponding Gamma approximations solid lines

The Gamma CDFs perfectly fit at cell center ($r=0\,\,\mathrm{m}$) and in the middle of the cell ($r=125\,\mathrm{m}$). At cell-edge ($r=250\,\,\mathrm{m}$), the Gamma approximation closely resembles the experienced interference of a user at $\phi =\frac{\pi }{12}$. The probability of high interference values at $\phi =\frac{\pi }{6}$ is slightly underestimated by at most $3.7\,\%$ (cf. Fig. 11.5).

6 Application in Heterogeneous Networks

In this section, aggregate interference statistics in a two-tier heterogeneous cellular network with regularly placed macro-BSs and randomly distributed small cell BSs are investigated. The interference contribution from each tier is approximated by a single Gamma RV and the total interference is calculated as the sum of the two. The accuracy of the approximations is verified by extensive Monte Carlo simulations .

Fig. 11.7

Snapshot of a heterogeneous network deployment. Macro-BSs are arranged on a hexagon. Small cell BSs are randomly distributed around a user at $(r,\phi )$ and excluded from a ball of radius $R_\mathrm{Ex}$

The macro-tier comprises six hexagonally arranged BSs at distance $R=500\,\,\mathrm{m}$, each transmitting with $P_\mathrm{M}=40\,\,\mathrm{W}$. Small cell BSs are distributed according to a Poisson Point Process PPP of density $\mu _\mathrm{S}=10^{-4}\,\,\mathrm{m}^{-2}$ and transmit with a power of $P_\mathrm{S}= 0.4\,\,\mathrm{W}$. As indicated in Fig. 11.7, they are excluded from a ball (in fact, it is a disc, but ball is the more common term in literature, e.g., in [11]) of radius R$_\mathrm{Ex} = (P_\mathrm{M}/P_\mathrm{S})^{-1/\alpha }R/2$ around the user so as to ensure user association to the central macro-BS at cell-edge. In both tiers, the path loss $\ell (\cdot )$ is modeled according to Eq. (11.5), with intercept $b_\mathrm{P}= 1$, constant $c_\mathrm{P}=1$ and exponent $\alpha =4$. Fading is assumed to be distributed as $G_i\sim \varGamma [1,1]$. The parameters are summarized in Table 11.2 .

Table 11.2 Parameters for numerical evaluation of heterogeneous scenario

In the first step, the interference contribution from the macro-tier is approximated by a Gamma RV ${\hat{I}}_\mathrm{M}(r) \sim \varGamma [{\hat{k}}_\mathrm{M}(r),{\hat{\theta }}_\mathrm{M}(r)]$. According to Sect. 11.4, it can be parameterized by the circular interference model. Recalculating Eqs. (11.15), (11.16), (11.18) and (11.19) for $\alpha =4$ yields

$$\begin{aligned} {\hat{k}}_\mathrm{M}(r)&= \frac{6\, k_0 R^8\left( r ^2-R^2\right) \left( r ^2+R^2\right) ^2}{r ^{14}-7 r ^{12} R^2 +23 r ^{10}R^4 -41 r ^8 R^6 + 39 r ^6 R^8 - 25 r ^4 R^{10} - 9 r ^2 R^{12}-R^{14}},\end{aligned}$$

(11.26)

$$\begin{aligned} \hat{\theta }_\mathrm{M}(r)&= { \frac{\theta _0 P_\mathrm{M}\left( -r ^{14}+7 r ^{12}R^2-23 r ^{10}R^4+41 r ^8 R^6 - 39 r ^6 R^8 + 25 r ^4 R^{10}+9 r ^2 R^{12}+R^{14}\right) }{c_\mathrm{P}\left( r ^2-R^2\right) ^4 \left( r ^2+R^2\right) R^8}.} \end{aligned}$$

(11.27)

Secondly, the contribution of the small cell tier is also approximated by a Gamma RV $\hat{I}_\mathrm{S}\sim \varGamma [\hat{k}_\mathrm{S}, \hat{\theta }_\mathrm{S}]$. Along the lines of [26, Eqs. (2.19) and (2.21)], mean and variance of the actual interference $I_\mathrm{A,S}$ from the PPP model are determined as

$$\begin{aligned} \mathbb {E}[I_\mathrm{A,S}]&= P_\mathrm{S}\, \mathbb {E}[G_i] 2\pi \mu _\mathrm{S}\int _{R_\mathrm{Ex}}^{\infty }r\, \ell (r)dr \nonumber \\&= P_\mathrm{S}\, k_0\,\theta _0 \frac{1}{c_\mathrm{P}} \pi \mu _\mathrm{S}\,R_\mathrm{Ex}^{-2},\end{aligned}$$

(11.28)

$$\begin{aligned} \mathrm{Var}[I_\mathrm{A,S}]&= \mathbb {E}[G_i^2] 2\pi \mu _\mathrm{S}\int _{R_\mathrm{Ex}}^{\infty }r\,\ell (r)^2dr \nonumber \\&= (1+k_0)k_0\,\theta _0^2\, P_\mathrm{S}^2 \frac{1}{c_\mathrm{P}^2} \frac{\mu _\mathrm{S} \pi }{3}R_\mathrm{Ex}^{-6}. \end{aligned}$$

(11.29)

Then, exploiting the identities $\mathbb {E}[\hat{I}_\mathrm{S}]=\hat{k}_\mathrm{S}\, \hat{\theta }_\mathrm{S}$ and $\mathrm{Var}[\hat{I}_\mathrm{S}]=\hat{k}_\mathrm{S}\, \hat{\theta }_\mathrm{S}^2$ yields

$$\begin{aligned} \hat{k}_\mathrm{S}&= \frac{3 R_\mathrm{Ex}^2\, k_0\,\theta _0\, \mu _\mathrm{S}}{(1+k_0)}, \end{aligned}$$

(11.30)

$$\begin{aligned} \hat{\theta }_\mathrm{S}&= \frac{P_\mathrm{S}(1+ k_0)}{3\,c_\mathrm{P}\, R_\mathrm{Ex}^{\alpha }}. \end{aligned}$$

(11.31)

Finally, the PDF of the total aggregate interference, $\hat{I}_\mathrm{A}(r)=\hat{I}_\mathrm{M}(r) + \hat{I}_\mathrm{S}$, at user distance r is calculated as

$$\begin{aligned} f_{\hat{I}_\mathrm{A}}(x;r)&= {\hat{\theta }_\mathrm{M}\left( r\right) }^{-{\hat{k}_\mathrm{M}\left( r\right) }} {\hat{\theta }_\mathrm{S}}^{-{\hat{k}_\mathrm{S}}} e^{-\frac{x}{{\hat{\theta }_\mathrm{S}}}} x^{{\hat{k}_\mathrm{M}\left( r\right) }+{\hat{k}_\mathrm{S}}-1} \nonumber \\&\quad \times \, _1\tilde{F}_1\left( {\hat{k}_\mathrm{M}\left( r\right) };{\hat{k}_\mathrm{M}\left( r\right) }+{\hat{k}_\mathrm{S}};\left( \frac{1}{{\hat{\theta }_\mathrm{S}}}-\frac{1}{{\hat{\theta }_\mathrm{M}\left( r\right) }}\right) x\right) , \end{aligned}$$

(11.32)

where $_1\tilde{F}_1(\cdot ;\cdot ;\cdot )$ denotes the regularized confluent hypergeometric function.

In order to verify the accuracy of this approximation, Monte Carlo simulations are carried out. The results for a typical user at distance r are obtained by averaging over $10^6$ uniformly distributed angle-positions on $[0,2\pi ]$. For each position, $10^5$ fading- and $10^4$ spatial realizations of the small cell deployment are generated. The small cell BSs are distributed over a circular area of radius 10 R.

Fig. 11.8

CDFs of interference from macro- and small cell tier a and aggregate interference from both tiers b. Solid lines indicate results as obtained by approximating the contribution of each tier by a Gamma RV. Dashed lines show results from Monte Carlo simulations. User distances $r = \{0,125,250\} \,\mathrm{m}$ refer to cell-center, middle of cell and cell-edge, respectively

Figure 11.8a depicts the individual interference contributions from the macro- and the small cell tier at various user distances r. It is observed that the CDFs for the macro tier, which correspond to the approximation in Eqs. (11.26), and (11.27), show an accurate fit with the Monte Carlo simulations. This corroborates the claim in Sect. 11.3.1 that the circular model is also applicable for path loss exponents other than $\alpha =2$. The interference CDF of the small cell tier, which refers to the approximation in Eqs. (11.30) and (11.31), is independent of the user distance r due to the fixed exclusion radius $R_\mathrm{Ex}$. It is also in close agreement with the simulations. Figure 11.8b shows the CDFs of the aggregate interference from both macro- and small cell tier. It is found that the approximation by a sum of two independently parameterized Gamma RVs almost perfectly captures the actual interference characteristics at the cell center ($r=0\,\,\mathrm{m}$) and in the middle of the cell ($r=125\,\,\mathrm{m}$). It even provides an accurate fit at cell-edge ($r=250\,\,\mathrm{m}$).

7 LTE-Advanced System Level Simulations

In this section, the validity of the Gamma distribution for approximating aggregate interference in symmetric interference scenarios is evaluated. In the first part, the corresponding system model is introduced.

Fig. 11.9

Hexagonal grid setup with central cell and six interfering eNodeBs. UEs are equidistantly distributed along circles of radius r = {50, 120, 210} m. Bold dots indicate representative UE positions. The corresponding angles are given by $\phi =\{0,\frac{\pi }{12},\frac{\pi }{6}\}$, respectively. In the case of BS collaboration, eNodeB 7 does not contribute to the aggregate interference

Table 11.3 Specific parameters for simulations of homogeneous macro cellular network

7.1 System Model

The system model is composed of a central macro site and six neighboring nodes, which are arranged according to a hexagonal grid, as illustrated in Fig. 11.9. Each site employs a single eNodeB, which is equipped with an omni-directional antenna. For systematic investigations, the UEs are equidistantly distributed along concentric circles of radius $r = \{50, 120, 210\}\,\,\mathrm{m}$, referring to cell-center, middle of cell and cell-edge, respectively. Each circle encompasses 24 UEs, which are uniquely identified by the tuple $(r,\phi )$, where $\phi$ denotes the angle position. The signal experiences free-space path loss, fast fading according to a time-correlated Rayleigh channel, and spatially-correlated log-normal shadowing with $8\,\,\mathrm{dB}$ standard deviation. Hereinafter, the combination of these three mechanisms is termed composite fading. The free-space path loss law is defined as $\min (b_\mathrm{P}, 1/c_\mathrm{P}d^{-2})$. In this section, $b_\mathrm{P}=10^{-7}$ and $f_c = 2.14\,\,\mathrm{GHz}$, yielding $c_\mathrm{P}= (4\pi \, f_c/ c_0)^2 = 8.0465\times 10^{3}$, where $c_0$ is the speed of light. The shadow fading maps are computed by applying the method in [27]. The results in this section are obtained by averaging over 100 channel realizations and 100 TTIs. The simulation parameters are summarized in Tables 9.6 and 11.3, respectively.

7.2 Validation of Gamma Approximation

In this section, the introduced circular interference model is validated. The particular focus lies on the accuracy of the Gamma distribution as an approximation for both composite fading and aggregate interference. The system model is referred from Sect. 11.7.1 .

Firstly, the average aggregate interference is measured along each of the three UE circles. The results are depicted as solid lines in Fig. 11.10. In accordance with Sect. 11.5.1, it is observed that average aggregate interference is almost constant at the cell-center and in the middle of the cell. At cell-edge, the curves exhibit fluctuations due to the vicinity of the dominant interferers. Results from the circular interference model accurately assess the average behavior, as shown by the dashed lines.

Fig. 11.10

Average aggregate interference power along the three UE circles in Fig. 11.9. Solid curves refer to simulation results, dashed curves denote results from circular interference model

In the next step, the Empirical Cumulative Distribution Function (ECDF) of the aggregate interference is computed at nine representative UE locations, which are marked by bold dots in Fig. 11.9. Similar to Sect. 11.5.2, the angle positions $\phi = \{0, \frac{\pi }{6}, \frac{\pi }{12}\}$ refer to UEs with one dominant interferer (eNodeB 7), two equidistant dominant interferers (eNodeBs 6 and 7) and a variation thereof. Solid lines in Fig. 11.11 depict the results. In accordance with Sect. 11.5.2, the interference distributions are dominated by the UEs’ distances to the origin while their angle positions have only minor impact. The latter is illustrated by the enlarged section in Fig. 11.11.

Fig. 11.11

Aggregate interference distributions at representative UE locations $r=\{50,120,210\}$ m and $\phi =\{0,\frac{\pi }{12},\frac{\pi }{6}\}$, as marked by bold dots in Fig. 11.9. Solid lines refer to ECDF curves from simulations, dashed lines denote Gamma approximations as obtained with circular model

Finally, the introduced circular model is applied to approximate the aggregate interference distribution at a certain distance r by a Gamma RV. The first step consists in determining the parameters $k_0$ and $\theta _0$ of the Gamma distribution $\varGamma [k_0,\theta _0]$ that models the composite fading (cf. Sect. 11.4). This is achieved by applying Algorithm 2. The intital values $k_0'$ and $\theta _0'$ are obtained from Maximum Likelihood Estimation (MLE). MLE maximizes the likelihood $\mathcal {L}(k_0',\theta _0'|x) = f(x|k_0',\theta _0')$, where $f(\cdot )$ denotes a Gamma PDF with shape $k_0'$ and scale $\theta _0'$, and x are the given outcomes. Using a step size of $\varDelta ~=~0.001$ and $N_\mathrm{iterations} = 100$ yields a KS distance of 0.0512 between simulated- and approximated composite fading distribution. For comparison, employing only MLE achieves a KS distance of 0.0917 .

Then, for each UE distance, the parameters of the aggregate interference distribution, $\hat{k}(r)$ and $\hat{\theta }(r)$, are calculated with Eqs. (11.20) and (11.21), respectively. The corresponding CDF curves are depicted as dashed lines in Fig. 11.11. It is observed that the approximated distributions slightly underestimate the occurrence of high interference values. In order to quantify the deviation from the simulated curves, the first row in Table 11.4 provides the KS distance for each UE location $(r,\phi )$. The values range from 0.05 at $r=210\,\,\mathrm{m}$ to 0.08 at $r=50\,\,\mathrm{m}$.

Table 11.4 KS distances between Gamma approximations and simulated ECDF curves at representative UE locations

For comparison, each simulated curve is also approximated by two further Gamma distributions. The first distribution adapts the circular model and estimates the composite fading by MLE, i.e., it employs the parameters $k_0$ and $\theta _0$ that were used above to initialize Algorithm 2. The second distribution is computed by applying MLE directly to the simulated aggregate interference. The corresponding KS distances are likewise listed in the second- and third row of Table 11.4 for each UE location $(r,\phi )$. The first observation is that Algorithm 2 considerably improves the performance of the circular model, such that it even exceeds pure MLE of the aggregate interference. Hence, the accuracy of the circular model crucially depends on the precision of the composite fading approximation. Secondly, the results of the MLE range from 0.07 to 0.08, indicating that the assumption of Gamma-distributed interference itself induces a systematic error.

In summary, the circular model achieves a remarkable accuracy of fit despite its simplicity, thus corroborating its applicability.

8 Conclusion

In this chapter, a circular interference model for aggregate interference analysis in regular grid deployments is introduced. Particular focus is placed on characterizing a user at an eccentric location. The expected interference from the circular model is identified as the interference that is experienced by a typical user in a hexagonal grid at a certain distance from the origin. At cell-edge, it deviates by at most $3.2\,\%$ from the actual values.

In a second step, the corresponding interference statistics are approximated by a single Gamma RV. By means of the circular model, the distance-dependent shape- and scale parameters are determined in closed form and unveil the two key formative components of the distribution as the variance of the fading and the variance of the path loss due to the eccentric user location, respectively. Qualitative- and quantitative comparisons with the exact distributions confirm the accuracy of the Gamma approximation, yielding KS statistics no higher than $3.7\,\%$.

The circular model’s expedient adaption for representing the well-planned part of a two-tier heterogeneous cellular network is demonstrated. The example merges a fully regular macro-deployment with completely randomly distributed small cells and models the interference contribution from each tier by a single Gamma RV. The resulting aggregate interference distribution shows a remarkably good fit with Monte Carlo simulations. Hence, the model enables to accurately capture the impact of both user eccentricity and heterogeneity of the network with only few key parameters.

In the last part of the chapter, it is shown that the circular model enables an accurate prediction of the interference statistics in an LTE-A hexagonal grid scenario. Deviations from the simulation results mainly stem from the inaccurate approximation of the composite fading. The remainder of the approximation error is caused by the assumption of Gamma distributed aggregate interference itself.

The presented circular model does not allow to account for power control and coordination among BSs. This is a major motivation for the next chapter, which extends the model by non-uniform power profiles.