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SIR distribution analysis in cellular networks considering the joint impact of pathloss, shadowing and fast fading
EURASIP Journal on Wireless Communications and Networking volume 2011, Article number: 137 (2011)
Abstract
In this paper, we propose an analysis of the joint impact of pathloss, shadowing and fast fading on cellular networks. Two analytical methods are developed to express the outage probability. The first one based on the FentonWilkinson approach, approximates a sum of lognormal random variables by a lognormal random variable and approximates fast fading coefficients in interference terms by their average value. We denote it FWBM for FentonWilkinson based method. The second one is based on the central limit theorem for causal functions. It allows to approximate a sum of positive random variables by a Gamma distribution. We denote it CLCFM for central limit theorem for causal functions method. Each method allows to establish a simple and easily computable outage probability formula, which jointly takes into account pathloss, shadowing and fast fading. We compute the outage probability, for mobile stations located at any distance from their serving BS, by using a fluid model network that considers the cellular network as a continuum of BS. We validate our approach by comparing all results to extensive Monte Carlo simulations performed in a traditional hexagonal network and we provide the limits of the two methods in terms of system parameters. The proposed framework is a powerful tool to study performances of cellular networks, e.g., OFDMA systems (WiMAX, LTE).
1 Introduction
In this paper, we are interested in characterizing the signal to interference ratio (SIR) distribution on the downlink of a cellular network. The SIR outage probability and the SIR cumulative distribution function (CDF) are important metrics for the performance evaluation of wireless communication systems. In this paper, we define the outage probability as the probability that the SIR at the input of the receiver chain is falling below a given threshold value. This performance parameter is crucial for both coverage and capacity studies. In terms of coverage, mobile stations should be able to decode common control channels (like pilots or broadcast channels) and thus to attain a certain SIR threshold on these channels with high probability. In this case, we are interested in the low SIR region of the SIR distribution in order to evaluate the cell coverage. In terms of capacity and for systems implementing link adaptation on shared downlink channels (such as HSPA or LTE), the whole SIR distribution is needed for performance evaluation. The ergodic capacity at a certain distance from the base station is indeed evaluated as an expectation of the Shannon classical formula over the channel variations. The cell capacity is obtained by integration over the cell area.
The issue of expressing outage probability in cellular networks has been extensively addressed in the literature. For this study, a difficult task is to take into account the joint impact of pathloss, shadowing and fast fading. There are two classical assumptions: (1) considering only the shadowing effect, (2) considering both shadowing and fast fading effects. In the former case, authors mainly face the problem of expressing the distribution of the sum of lognormally random variables; several classical methods can be applied to solve this issue (see e.g. [1, 2]). In the latter case, formulas usually consist in many infinite integrals, which are uneasy to handle in a practical way (see e.g. [3]). In both cases, outage probability is always an explicit function of all distances from the user to interferers.
As the need for easytouse formulas for outage probability is clear, approximations need to be done. Working on the uplink [4], derived the distribution function of a ratio of pathlosses with shadowing, which is essential for the evaluation of external interference. For that, authors approximate the hexagonal cell with a disk of same area. Authors of [5] assume perfect power control on the uplink, while neglecting fast fading. On the downlink, Chan and Hanly [6] precisely approximate the distribution of the othercell interference. They, however, provide formulas that are difficult to handle in practice and do not consider fast fading. Immovilli and Merani [7] take into account both channel effects and make several assumptions in order to obtain simplified formulas. In particular, they approximate interference by its mean value. Outage probability is however an explicit function of all distances from receiver to every interferer. Zorzi [8], proposes a formula essentially valid for packet radio networks rather than for cellular systems. Authors of [9] provide some interesting characterizations and upper bounds of the outage probability but neglects the slowly varying pathgains. In [10], authors consider both shadowing and fast fading but assume a single interferer.
In this paper, we propose two methods to analyze the outage probability of mobile stations located at any distance r from their serving base station (BS). The first method, based on the FentonWilkinson approach [11], approximates a sum of lognormal random variables as a lognormal random variable and approximates fast fading coefficients in interference terms by their average value. We denote it FWBM (for FentonWilkinson Based Method). The second one is based on the central limit theorem for causal functions [12]. It allows to approximate a sum of positive random variables by a gamma distribution. We denote it CLCFM (for central limit theorem for causal functions method). Each method allows to establish a simple and easily computable outage probability formula, which jointly takes into account pathloss, shadowing and fast fading. We compare our proposed formulas with results obtained with extensive Monte Carlo simulations in a classical hexagonal network. At last, we rely on fluid model proposed in [13, 14] in order to express the outage probability as a simple analytical expression depending only on the distance to the serving BS. Such an expression allows further integrations much more easily than with existing formulas. Note that part of the presented results are included in [15].
The paper is organized as follows. FWBM is explained in sect. 3. We derive the outage probability while considering first only pathloss and shadowing and then pathloss, shadowing and fast fading jointly. Sect. 4 develops the CLCF method. Outage probability is calculated while considering first only pathloss and fast fading and then pathloss, shadowing and fast fading jointly. The computation is based on the fluid model (sect. 5). In sect. 6, we validate our approach and compare analytical expressions with results obtained through Monte Carlo simulations.
2 Interference model
We first define the interference model assumed in this paper. We consider a hexagonal cellular network with frequency reuse one and we focus on the downlink. We are interested in evaluating the SIR at a mobile station u, served by base station BS_{0} and interfered by N interfering base stations. We assume that mobile stations are attached to their closest BS.
2.0.1 Propagation Model
The power received by u depends on the radio channel state and varies with time due to fading effects (shadowing and fast fading). Let P_{ j } be the transmission power of base station j, the power p_{j,u}received by u can be written as:
The term {P}_{j}K{r}_{j,u}^{\eta}, where K is a constant, represents the mean value of the received power at distance r_{j,u}from the transmitter (BS_{ j } ). X_{j,u}is a random variable (RV) representing the Rayleigh fading effects, whose pdf is p_{ X } (x) = e^{x}. The term {Y}_{j,u}=1{0}^{{\xi}_{j,u}\u221510} is a lognormal RV characterizing shadowing. ξ_{j,u}is a Normal distributed RV, with zero mean and standard deviation, σ, which is typically between 0 and 8 dB. Parameter η, which is typically between 2.7 and 3.5, is the pathloss exponent.
2.0.2 SIR Calculation
The interference power received by a mobile u can be written as:
We consider in this paper that the thermal noise is negligible (in a urban environment) and so we focus on the SIR rather than on the signal to interference plus noise ratio (SINR). Considering moreover that all BS transmit with the same power P_{0}, we can express the SIR expression (dropping index u and setting r_{0,u}= r):
2.0.3 Outage Probability
In this paper, we define the outage probability as the probability that the SIR at u falls below a given threshold δ. Note that while varying δ, we obtain the definition of the SIR CDF. In this paper, we indifferently speak of outage probability or CDF.
3 FentonWilkinson based method
In this section, we propose a first method based on the FentonWilkinson approximation. We first analyze the pathloss and shadowing impact. We then extend the result to the joint influence of pathloss, shadowing and fast fading based on the previous obtained result.
3.1 Pathloss and shadowing impact
The power p_{ j } received by u can be written in this case:
The probability density function (PDF) of this slowly varying received power is given by
Where a=\frac{ln10}{10}, m=\frac{1}{a}ln\left(K{P}_{j}{r}_{j}^{\eta}\right) is the (logarithmic) received mean power expressed in decibels (dB), which is related to the pathloss and σ is the (logarithmic) standard deviation of the mean received signal due to the shadowing.
The SIR at user u is now given by:
We see that the SIR can be written γ = 1/F with
The factor F is defined for any mobile u and it is thus location dependent. The numerator of this factor is a sum of lognormally distributed RV, which can be approximated by a lognormally distributed RV [2]. The denominator of the factor is a lognormally distributed RV. F can thus be approximated by a lognormal RV. Using the FentonWilkinson [11] method, we can calculate the logarithmic mean and standard deviation, m_{ f } and s_{ f } of F for any mobile at the distance r from its serving BS, BS_{0} (see Appendix 1):
where
From Eqs. (4) and (9), we notice that f (r, η) represents the factor F without shadowing.
The outage probability is now defined as the probability for the SIR γ to be lower than a threshold value δ and can be expressed as:
where Q is the complementary error function:Q\left(u\right)=\frac{1}{2}er\phantom{\rule{2.77695pt}{0ex}}fc\left(\frac{u}{\sqrt{2}}\right).
3.2 Pathloss, shadowing and fast fading impact
In this case, the outage probability can be expressed as:
The interference power received by a mobile u due to fast fading effects varies with time. As a consequence, the fast fading can increase or decrease the power received by u. We consider that the increase of interfering power due to fast fading coming from some base stations are compensated by the decrease of interfering powers coming from other base stations. As a consequence, for the interfering power, only the slow fading effect has a significant impact on the SIR. We thus assume that ∀j ≠ 0, X_{ j } ≈ E[X_{ j } ] = 1 (this assumption will be validated by simulations in the next section), and we can write \mathbb{P}\left({r}^{\eta}{X}_{0}{Y}_{0}>\delta \left(\sum _{j=1}^{N}{r}_{j}^{\eta}{X}_{j}{Y}_{j}\right)\right)\approx \mathbb{P}\left({r}^{\eta}{X}_{0}{Y}_{0}>\delta \left({\sum}_{j=1}^{N}{r}_{j}^{\eta}{Y}_{j}\right)\right).
So we have:
As a consequence, the outage probability for a mobile located at a distance r from its serving BS, taking into account pathloss, shadowing and fast fading can be written as:
4 Central limit theorem for causal functions method
In this section, we adopt a different path for deriving the SIR CDF. We first express the outage probability by considering the pathloss and fast fading. Afterwards, we use this result in order to introduce the shadowing impact.
4.1 Pathloss and fast fading impact
Assuming only fast fading channel, the SIR is given by:
where:
are two independent RV. To calculate the outage probability, we need to calculate first, the probability distribution function (PDF) f_{ S } (x)of S and the PDF f_{ I } (x) of I. The PDF of the useful power is given by [16]:
We now approximate the interference PDF using the central limit theorem for causal functions [12] by a Gamma distribution given by:
where \nu =\frac{E{\left[I\right]}^{2}}{var\left(I\right)} and \lambda =\frac{var\left(I\right)}{E\left[I\right]}. Since E[X_{ j } ] = 1 and var(X_{ j } ) = 1 for j = 1,..., N, the mean of the interference power is given by:
The variance of I can be expressed as:
So we have:
The outage probability can now be derived as follows:
4.2 Pathloss, shadowing and fast fading impact
In this section we will consider that the shadowing follows a lognormal distribution. We can again write γ = S/I with now
We again approximate the interference PDF using the Central Limit Theorem for Causal Functions. We thus need to compute the two quantities {\nu}_{s}=\frac{E{\left[I\right]}^{2}}{var\left(I\right)} and {\lambda}_{s}=\frac{var\left(I\right)}{E\left[I\right]}. Since involved RV are independent, the average value of I is given by:
In the same way, the variance of I is given by:
The two parameters ν _{ s } and λ_{ s } can now be obtained:
Recall that Y_{0} follows a lognormal distribution with logarithmic mean 0 and standard deviation σ. We can thus average the outage probability over the variations of Y_{0}:
5 Analytical fluid model
With the two proposed methods, we obtain expressions of the SIR CDF at a given distance r from the serving basestation. We see however that expressions depend also on the distances r_{ i } between the considered mobile terminal and all interfering BS. With FWBM, parameters m_{ f } and s_{ f } in (11) depend on the r_{ i } (see Eqs. (5) and (6)). With CLCFM, parameters ν_{ s } and λ_{ s } in (17) depend also on the r_{ i } (see Eqs. (15) and (16)). The presence of all distances r_{ i } make proposed formulas sometimes uneasy to use for dimensioning purposes. In this section, we thus express the parameters m_{ f } , s_{ f } , ν_{ s } and λ_{ s } as functions dependent only on the distance r using the fluid model.
The fluid model approach has been developed, e.g., in [17]. It consists in replacing on the downlink a given fixed finite number of transmitters (base stations) by an equivalent continuum of transmitters which are distributed according to some distribution function. For a homogeneous and regular cellular network, inteferers are now characterized by the interfering BS density ρ_{BS}. Let denote:
Assuming an infinite network, the fluid model allows us to approximate g by the following function [17] (see Appendix 2 for more details). Introducing the dependence of g on r, we obtain:
where R_{ c } is the half interBS distance.
In the FWBM method, parameters f (r, η) and G(r, η) given by Eqs. (9) and (8), respectively, can be expressed as follows:
Parameters m_{ f } and s_{ f } can thus be written as functions only on the distance r to the serving BS. In the same way, for the CLCFM, we have:
6 Performance evaluation
In this section, we compare the figures obtained with analytical expressions (11) and (17) to those obtained by Monte Carlo simulations. It is clear that several approximations have been done in order to obtain easytouse closedform formulas: (1) the FentonWilkinson method is known to be accurate for low standard deviations; (2) the Central Limit Theorem for Causal Functions is an approximation; (3) the fluid model also. It is thus important to know to what extent approximations are acceptable.
6.1 Monte Carlo simulator
The simulator assumes an homogeneous hexagonal network as the one shown in Figure 1, made of fifteen rings around a central cell. The cell range is denoted R, the halfdistance between BS is set to R_{ c } = 1 km.
The simulation consists in computing at each snapshot the SIR for a uniformly random location in the central cell. This computation can be done independently of the BS output power because noise is supposed to be negligible both in simulations and analytical study. At each snapshot, shadowing (lognormal distribution with standard deviation σ) and fast fading (exponential distribution of mean 1) RV are independently drawn between the MS and the serving BS and between MS and interfering BS. We do not consider correlation between shadowing coefficient. SIR samples at a given distance from the central BS are recorded in order to compute the outage probability. Five thousand (5,000) snapshots are considered.
6.2 Results
In this section, we compare obtained formulas with results obtained by Monte Carlo simulations in a hexagonal network. We study the robustness of our approaches while varying three important parameters: σ, the standard deviation of the shadowing, η, the pathloss exponent, and r, the distance to the serving BS.
Moreover, we compare the results for the following approaches:

SIM: results obtained from Monte Carlo simulations;

FWBM: the FentonWilkinson Based Method in conjunction with the fluid model;

CLCFM: the Central Limit Theorem for Causal Functions Method in conjunction with the fluid model;
In Figure 2, we compare SIR CDF obtained with SIM, FWBM and CLCFM at r = 0.2 Km, for η = 3.0 and while varying σ from 3 to 8 dB. Parameter σ is definitely the coefficient that influences the most the difference between analysis and simulations. It is clear that the highest is σ, the highest is the error induced by approximations. For σ = 8 dB, the CLCFM is not valid anymore if we consider the whole CDF, but remains accurate for low SIR region. For σ = 3 dB, both methods provide very accurate results.
In Figure 3, we study the influence of the distance to the serving BS for σ = 4 dB and η = 3.0. This distance has a small influence on the accuracy of the proposed methods, all analytical CDF fit well with the CDF obtained by simulations.
In Figure 4, we study the pathloss exponent η with fixed σ = 4 dB and distance r = 0.2 km. Here again, the parameter has a small influence on the globally good accuracy of the methods. Error seems, however, to increase with η.
For the sake of completeness, we present in Tables 1 and 2 extensive results for the comparison of SIM, FWBM, and CLCFM. We set three probability thresholds (5, 50, and 90%) and we obtain the corresponding SIR thresholds in dB from the different CDF. Reported figures represent the difference in dB between the threshold obtained with SIM on the one side and the threshold obtained with FWBM or CLCFM on the other side. Excessive differences (more than 3 dB) are marked with a star. From these tables, we can draw some conclusions:

CLCFM provides accurate results for σ ≤ 6 dB and η ≤ 3.0. For all σ ≤ 8 dB, results are still accurate in the low SIR region; this is an interesting result for coverage issues where outage computations are involved.

FWBM provides accurate results for σ ≤ 8 dB and η ≤ 3.0, η can be greater if σ is strictly less than 8 dB. At σ = 8 dB and for η ≤ 3.5, results are still accurate in the low SIR region.
Theses results show that proposed methods, especially FWBM, can provide accurate results for typical values of parameters r, σ and η usually considered in cellular networks.
7 Conclusion
In this paper, we establish simple formulas of the outage probability in cellular networks, while considering pathloss, shadowing and fast fading. Using FWBM approach, we take into account pathloss and shadowing to first express the inverse of the SIR of a mobile located at a given distance of its serving BS as a lognormal random variable. We then consider both pathloss, shadowing and fast fading and give an analytical expression of the outage probability at a given distance of the serving BS. Using CLCFM approach, we take into account pathloss and fast fading to express the SIR of a mobile located at a given distance of its serving BS. We then consider pathloss, fast fading and shadowing and give an analytical expression of the outage probability at a given distance of the serving BS. The fluid model allows us to obtain formulas that only depend on the distance to the serving BS. The analytical model that we propose is validated by comparisons with Monte Carlo simulations. The formulas derived in this paper allow to obtain performances results instantaneously.
The proposed framework is a powerful tool to study performances of cellular networks and to design fine algorithms taking into account the distance to the serving BS, shadowing and fast fading. It can particularly easily be used to study frequency reuse schemes in OFDMA systems.
Appendix 1
In this section, we derive the logarithmic mean and standard deviation of F (see equation (4)) using the FentonWilkinson method. Several classical techniques exist in the literature in order to approximate a sum of lognormal RV, e.g., SchwartzYeh [18] and Farley [19]. The former needs complex recursive calculations, the latter assumes identical mean and standard deviation. On the contrary, the FentonWilkinson method provides a closedform formula for non identical distributed lognormal RV.
Each term of the sum in the numerator of F is a RV following a lognormal distribution. We can write ln\left({r}_{j}^{\eta}{Y}_{j}\right)\propto \mathcal{N}\left(a{m}_{j},{a}^{2}{\sigma}^{2}\right), where a = ln(10)/10 and {m}_{j}=\frac{1}{a}ln\left({r}_{j}^{\eta}\right). The sum S={\sum}_{j=1}^{N}{r}_{j}^{\eta}{Y}_{j} can itself be approximated by a lognormal RV:ln\left(S\right)\propto \mathcal{N}\left(am,{a}^{2}{\sigma}_{s}^{2}\right)with:
These two expressions simplify to:
Taking now into account the denominator, the logarithmic mean value of F is simply am_{ f } = am  (η ln(r)) and the logarithmic standard deviation of F is {a}^{2}{s}_{f}^{2}={a}^{2}{\sigma}_{s}^{2}+{a}^{2}{\sigma}^{2}. The ratio of two lognormal RV is indeed a lognormal RV with mean, the difference of the means, and variance, the sum of the variances. With the definitions of f, G and H, we obtain Eqs. (5) and (6). Note that m_{ f } and s_{ f } are expressed in dB.
Appendix 2
In this section, we shortly recall the fluid model approach, that will allow us to simplify Eq. 18. The fluid model approach consists of replacing a given fixed finite number of interfering BS by an equivalent continuum of transmitters which are uniformly distributed with density ρ_{ BS } .
We consider a central cell and a round shaped network around this cell with radius R_{ nw } . The intersite distance is 2R_{ c } (see Figure 5). Let's consider a mobile u at a distance r_{ u } from its serving BS. In the fluid model, each elementary surface zdzdθ at a distance z from u contains ρ_{ BS }zdzdθ BS which contribute to function g. Their contribution to the sum over j is ρ_{ BS }zdzdθz^{η} (η > 2). We approximate the integration surface by a ring with centre u, inner radius 2R_{ c } r_{ u } , and outer radius R_{ nw }  r_{ u } (see Figure 6):
If the network is large, i.e., R_{ nw } is big in front of R_{ c } , g can be further approximated by (dropping subscript u):
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Ben Cheikh, D., Kelif, JM., Coupechoux, M. et al. SIR distribution analysis in cellular networks considering the joint impact of pathloss, shadowing and fast fading. J Wireless Com Network 2011, 137 (2011). https://doi.org/10.1186/168714992011137
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DOI: https://doi.org/10.1186/168714992011137
Keywords
 Central Limit Theorem
 Mobile Station
 Cellular Network
 Outage Probability
 Fluid Model