On the Analytical Tractability of Hexagonal Network Model with Random User Location
11 On the Analytical Tractability of HexagonalNetwork Model with Random User Location
Ridha Nasri and Aymen Jaziri
Abstract —Explicit derivation of interferences in hexagonalwireless networks has been widely considered intractable andrequires extensive computations with system level simulations.In this paper, we fundamentally tackle this problem and ex-plicitly evaluate the downlink Interference-to-Signal Ratio (ISR)for any mobile location m in a hexagonal wireless network,whether composed of omni-directional or tri-sectorized sites. Theexplicit formula of ISR is a very convergent series on m andinvolves the use of Gauss hypergeometric and Hurwitz Riemannzeta functions. Besides, we establish simple identities that wellapproximate this convergent series and turn out quite usefulcompared to other approximations in literature. The derivedexpression of ISR is easily extended to any frequency reusepattern. Moreover, it is also exploited in the derivation of anexplicit form of SINR distribution for any arbitrary distributionof mobile user locations, reflecting the spatial traffic density inthe network. Knowing explicitly about interferences and SINRdistribution is very useful information in capacity and coverageplanning of wireless cellular networks and particularly for macro-cells’ layer that forms almost a regular point pattern. Index Terms —Wireless cellular networks, Interferences,Hexagonal model, SINR distribution, Performance analysis, Ran-dom user locations.
I. I
NTRODUCTION
A. Background and Motivation I NTERFERENCE in wireless networks is the prime concernof telecommunication actors which continuously attemptto minimize it in all the stages of the technology conceptionand deployment (from the standardization to the process ofnetwork design, planning and exploitation). Interferences areoften attributed to co-channel interferences which are theresult of the scarce spectrum reuse in different cells of thenetwork. In the network design phase, planning engineers tryto understand the behavior of the system in terms of radiocapacity and relate it to interferences. Often, they need to havea quick answer on its estimation without having recourse toa simulator. This answer should come up with a closed formexpression of interferences, required to be directly used insome tools essential for capacity and coverage planning, suchas link budget tool.In literature, co-channel interferences are often representedby the Interference-to-Signal Ratio (ISR) metric [1], namedalso interference factor in 3G studies [2], [3]. It is definedas the normalized intercell interferences against the receiveduseful signal (received power from the serving cell). ISRmetric constitutes the chestnut of the performance analysis, in
R. Nasri and A. Jaziri are with Orange Labs, 38/40 avenue GeneralLeclerc, 92794 Issy-les-Moulineaux, France. emails ([email protected],[email protected]). terms of SIR (Signal-to-Interference-Ratio) and SINR (Signal-to-Interference-plus-Noise-Ratio). Its evaluation in wirelessnetworks has been widely addressed but it is still a fresh topicinteresting both academic and industrial communities [1]–[11].Interferences depend on the geometric placement of radio sitesin the network. The two most frequently considered geometrymodels are hexagonal, named also equilateral triangular lattice,and random networks. The first assumes that sites are placedon a regular infinite hexagonal grid whereas the secondsupposes that sites are randomly distributed according to aspatial point process, e.g., Poisson Point Process [6]. Althoughrandom models have recently drawn more attention of theresearch community, hexagonal model is always used as thebasis for coverage dimensioning (estimation of the number ofsites required to cover an area with a given quality of service)and by operational staff for conducting strategic studies of atypical regular network assuming a complete sites’ deploy-ment. Despite its frequent use in network geometry modeling,hexagonal model has been widely considered intractable andthe evaluation of interferences requires extensive numericalcomputations with simulations [12], [13]. This intractabilitywas the key driver in the migration to random networkmodels. Besides, system performances are strongly linkednot only to the network geometry but also to user locationdistributions. With random models, it is hard to analyticallyevaluate the co-channel interference in each user location [1],only its distribution and eventually its average are determined.Moreover, random network models are often taken stationary:distributions of ISR and SINR are independent from the userlocation.In this work, we confine ourselves to mathematically tacklethe tractability issue of the hexagonal model consideringdifferent user location distributions. Besides, this paper findsits motivation from the limit of random network models indealing with non-uniform user location distribution that hasrecently come to prominence [14].
B. Related works
Several works have been made to give approximations ofinterferences in regular hexagonal networks based on observedsimulation results [2]–[4], [11]. To the best of our knowl-edge, no closed exact formula, mathematically and rigorouslyproved, was found.By way of examples, Chan et al. provided in [5] an ap-proximation of the inter-cell interference distribution assumingthat the traffic follows a Poisson model. In [6], Haenggi andGanti provided a good review of interferences in regular and a r X i v : . [ c s . N I] A ug random networks and in particular they gave lower and upperbounds of the cumulated interferences in hexagonal latticesassuming that the receiver is placed at the origin [6, p. 19]. Inthe search of the best approximation that well fits the resultsof simulations, Karray provided in [3] a comparison betweensome approximations of ISR in regular hexagonal networks.To make the hexagonal network model tractable, authors in[2] transformed the hexagonal model to the fluid model:interfering sites are continuously and uniformly distributedin the plane. It was shown that the fluid model is a weakapproximation of the hexagonal network. Almost all worksabout interference evaluation in fluid model assume that sitesin the network are omni-directional.Unlike in the hexagonal model, interference analysis istractable when sites are organized according to a Homoge-neous Poisson Point Process (HPPP), i.e., the number of sitesin a given area follows a Poisson distribution with constantsites’ intensity [6], [8]–[10], [15]. An excellent mathematicaltheory of interferences in Poisson network of interferers canbe found in [6], [8], [9]. Despite its tractability, HPPP model isvery irregular and can not always fit with the geometry of realwireless networks because, in reality, sites exhibit repulsivebehavior while limiting the dispersion of each site positionwithin a small area having fixed radius. This is related tothe fact that radio engineer, while designing sites, places thesite at its theoretical hexagonal position in a planning tool,makes coverage prediction and then looks for real candidatesites within a fixed radius from its theoretical position. Thedeviation of the real network structure from the hexagonalmodel is therefore linked to the availability of candidate sites(within the search radius) that fulfill certain constraints im-posed for example by engineering rules, terrain imperfectionsand government charters.To obtain performances of regular networks, while profitingfrom the tractability of the HPPP model, Haenggi numericallyshowed in [1] and in [16] with Guo that the SIR and SINR dis-tribution for hexagonal (or even perturbed hexagonal) modelwith uniform user location can be approximated by shiftingthe SIR and SINR distribution of the HPPP model. In thesame direction, Deng et al. introduced in [17] the GinibrePoint Process that respects a minimum distance between sitesand thus ensures more the repulsion of sites, feature naturallyidentified in real wireless networks. C. Contributions
In this work, we treat fundamentally the problem of in-terference calculation and give explicit and exact formulasof the downlink ISR in hexagonal wireless networks withomni-directional sites. We prove that the ISR metric, as afunction of the mobile location m = re iθ , admits an absolutelyconvergent Fourier series on θ and an analytic expansion on r . The average of the ISR over the angle θ is rigorouslyderived without any approximation and involves the use of thewell-known Gauss hypergeometric functions [18] and HurwitzRiemann Zeta functions [19]. Different identities and accurateapproximations of ISR are also given and compared withother approximations in [2], [3]. The explicit ISR expression in omni-directional networks is used to derive an accurateformula of ISR in tri-sectorized networks. It is also extendedfor any frequency reuse pattern. All the provided expressionsof the ISR are valid for a propagation parameter b > .The second contribution of this work is the explicit derivationof the network performances in terms of SINR distributionin hexagonal networks with an arbitrary user locations’ dis-tribution reflecting the spatial traffic density [20]. The findingof SINR distribution entails of course the inversion of theSINR, taken as a function of location m . To show the utility ofthe explicit formula of the SINR distribution, we numericallyinvestigate the SINR distribution for two scenarios of userlocations: uniform and heavy-tailed distributions (e.g., Log-normal user location distribution). D. Paper organization
The paper is organized as follows. In Section II, systemmodels, notations and preliminary definitions, including thelocation of omni and tri-sectorized sites, are given. In SectionIII, we derive the explicit formulas of ISR for omni-directionalnetwork and we generalize it for tri-sectorized one. Thisincludes also the presentation of some new approximationsand the sketch of their validities and comparison with otherapproximations in literature. Considering a general frequencypattern and accounting for shadowing effect in the ISR calcu-lation are also provided at the end of section III. In Section IV,analytical expressions of the SINR distributions are given forany user location distribution and numerically presented foruniform and non-uniform user location models. Conclusionssumming up important findings are drawn in Section V.II. S
YSTEM MODELS , NOTATIONS AND DEFINITIONS
A. Location model of omni-directional sites
We consider a hexagonal wireless network composed of aninfinite number of radio sites placed on a regular hexagonalgrid as shown in Fig. 1 left part. In this latter, each siteis located at the center of the hexagon and composed ofonly one sector (named also cell). Antenna in each site isassumed to have an omni-directional radiation pattern (uniformantenna gain in all directions) and covers a geographical area,named Voronoi cell. A network with such omni-directionalsites is called regular omni-directional network. As largelyconsidered, we assume also that all cells transmit with thesame power level P .We identify R with the complex plane C and we divide itinto equally 6 regions Ω l (0 ≤ l ≤ , defined by Ω l = { m ∈ C | lπ ≤ arg ( m ) < ( l + 1) π } (1)Interfering sites are arranged into rings of increasing radii.Rings of sites, surrounding the central site located at the originof R , are hexagons. We denote by k , k ≥ , the index ofeach ring in the network. The number of sites in the ring k and belonging to region Ω l is k . We give so the notation S l,k,j to the site indexed by j in ring k and located in region Ω l .The site located at the origin of R is denoted formally by S , , but, to simplify notations, we omit the lower index in Fig. 1: Hexagonal network model with omni-directional sites(Left side) and tri-sectorized sites (Right side). S , , and use simply S . We shall use the same label S l,k,j torefer to the geographical position of the site. Lemma In a regular hexagonal network, the site position S l,k,j , for k ≥ , is given by S l,k,j = D k,j e i ( θ k,j + lπ ) (2)where D k,j = δ p k + j − jk , θ k,j = atan ( j √ k − j ) , δ is theinter-site distance and i = √− is the imaginary unit of thecomplex plane C .The proof of the Lemma is simple using the geometry of thenetwork and it is not provided here.A mobile location in the plane is denoted by the random vari-able m . Likewise, label m is used to denote the geographicallocation of the mobile. In the complex plane, m = re iθ ; where r = | m | is the absolute value of m representing its distanceto the central site and θ is the angle coordinate of the mobilerelative to the horizontal axis.For hexagonal omni-directional network, we assume that lo-cation m is connected to central cell S covering a disk ofradius R < δ so that location variable m ≤ R . The choice of R is problematic because the disk circumscribing the hexagonover-estimates interferences whereas the disk inscribed in thehexagon under-estimates interferences and generates coverageholes. For all the theoretical study, we take an arbitrary R , suchthat R < δ , but for the numerical results we use R = δ q √ π which is the radius of the disk having the same area of thehexagon representing the cell. Actually, the real use of R appears only in section IV where we provide the distributionof SINR assuming that m is a random variable having dt ( m ) as its probability measure. B. Location model of tri-sectorized sites
Unlike for omni-directional sites, tri-sectorized sites arelocated at the corner of the hexagon and composed of sectorscovering each one a cell. Sites are arranged to form alwaysa hexagonal grid (see Fig. 1). This model is called regulartri-sectorized network. Since each site is divided into threesectors, we shall use S l,k,j,c to identify the sector c , ≤ c ≤ ,of the site j in the ring k and located in region Ω l . The sector c in the central site is simply denoted S c . The azimuth of theantenna, in which the radiation is at its maximum, is takenrelative to the geographical North. Let φ l,k,j = arg ( m − S l,k,j ) be the angle between site S l,k,j and mobile m , then the angle φ l,k,j,c between the azimuth of sector c in the site S l,k,j andthe mobile m is given by φ l,k,j,c = π c −
3) + φ l,k,j (3)We denote by G ( . ) the antenna mask of the sector (relative toits azimuth) and we define the antenna mask of a site, denoted G s , by the sum of all antenna masks of sectors belonging tothe same site G s ( φ l,k,j ) = X c =1 G ( φ l,k,j,c ) (4) C. Propagation model
In this study, we use simplified Okumura-Hata propagationmodel [21], i.e., the path-loss between site S l,k,j and mobileuser m is expressed as L l,k,j ( m ) = a | m − S l,k,j | b (5)where b > is the amplitude loss exponent, (i.e., b is thepathloss exponent), a is a constant dependent on the type ofthe environment (indoor, outdoor, rural, urban,...). We omitalso the lower index in L , , and simply use the notation L .With this pathloss model and considering the antenna radiationpattern G , the received power from sector S l,k,j,c at location m is P G ( φ l,k,j,c ) L l,k,j ( m ) , where φ l,k,j,c is given as in (3).Since we perform system level analysis, fast fading effectis taken from link level results. This latter feeds the formerwith link level curve that relates SINR to the throughput or tothe packet/bit error rate (see for example [12], [13]). Whenit is constructed, link level curve accounts for fast fadingconsidering different types of propagation environment anduser equipment speeds and categories. Also, the use of fadingeffect in the link performances finds its motivation from thelink adaptation, performed in real networks, in which fastfading is naturally taken in the relation between throughputand SINR. In fact, mobile user sends periodically the channelfeedback report in terms of CQI (Channel Quality Indicatorreflecting the SINR and including the fading effect) to theenodeB (taking LTE technology for example), and when it isscheduled, the latter chooses the Transport Block size and theModulation and channel Coding Scheme (MCS) according tothe received CQI. So, the throughput is given knowing the CQI(or SINR) and the instantaneous imperfection of the channeldue, inter alia, to fast fading.Moreover, it is known that, for perfect channel (additive WhiteGaussian Noise Channel), the relation between throughput th (normalized with the bandwidth) and SINR ξ is provided bythe famous Shannon’s formula th = log (1 + ξ ) . To accountfor the imperfection of the channel due to fast fading, one canmodify Shannon’s formula to have th = K log (1 + K ξ ) ,with some constants K and K calibrated from practicalsystems as stated in [22].Shadowing effect is not extensively investigated in thiswork but we give in section III-D the way of how it can be The antenna mask, G ( θ ) ≤ , should not be confused with the antennagain. In logarithmic (dB) scale, antenna gain is the sum of the maximumantenna gain and the mask. considered in the ISR calculation. Besides, since the analysisis carried out for arbitrary user location distribution, randomvariability of the signal, due to shadowing and fast fading,can be aggregated in the user location variable by changingthe distribution of user locations accordingly. D. Definition of ISR
Definition
1: Let m = re iθ be a location in the plane C andlet S l,k,j , as defined in Lemma 1, be the location of a givensite in a regular hexagonal network. We denote by f l,k,j ( m ) the individual ISR received from site S l,k,j in location m . Itis calculated relatively to the first sector of central site S by f l,k,j ( m ) = L ( m ) L l,k,j ( m ) G s ( φ l,k,j ) G ( θ − π ) (6)Note that for omni-directional site, there is only one sectorand hence G s = G = 1 .With the homogeneity hypothesis that all sectors transmit withthe same power, f l,k,j ( m ) is the normalized received powerfrom site S l,k,j , against the useful received power at location m served by the first sector of central site S . Definition
2: The ISR, denoted by f ( m, b ) , is a function oflocation m and propagation parameter b . It is defined as thesum of all individual ISRs f ( m, b ) = X ≤ l ≤ ,k ≥ , ≤ j ≤ k − f l,k,j ( m ) (7)III. M ATHEMATICAL ANALYSIS AND EXPLICITDERIVATION OF THE
ISRIn this section, we use mathematical tools to derive theFourier and Maclaurin series expansion of f ( m, b ) . We firstanalyze the ISR for omni-directional and tri-sectorized net-work and we point out, at the end of the section, the possibilityto explicitly account for higher frequency reuse pattern andshadowing effect in the ISR calculation. A. ISR formulas in hexagonal omni-directional networks1) Explicit and exact formulas of ISR:
Using the definition of the individual ISR in (6) andsubstituting L l,k,j by its form in (5), we shall establish thefollowing mathematical results. Proposition For a given location m = re iθ connected tothe central cell S , the individual ISR, defined in (6), is givenfor omni-directional sites by f l,k,j ( m ) = + ∞ X n = −∞ (cid:18) rD k,j (cid:19) b + | n | Γ( b + | n | ) e in ( θ − θ k,j − l π ) Γ( b )Γ(1 + | n | ) × F b, b + | n | , | n | , (cid:18) rD k,j (cid:19) ! (8)where F ( ., ., ., . ) and Γ( . ) are respectively the Gauss hyper-geometric and Euler Gamma functions [18, p. 561]. Proof:
The proof of proposition 1 is in Appendix A. (cid:4)
Theorem Let f , defined as in (7), be the ISR function of alocation m = re iθ and a propagation parameter b in a regularhexagonal network with infinite number of omni-directionalsites. Let δ be the inter-site distance. Take x = rδ , then for b > and x < , f admits an absolutely convergent Fourierseries expansion on θ and an analytic expansion on xf ( m, b ) = H ( x, b ) + 2 + ∞ X n =1 H n ( x, b ) cos (6 nθ ) (9)where H ( x, b ) = 6 x b Γ( b ) ∞ X h =0 Γ( b + h ) Γ( h + 1) ω ( b + h ) x h (10) H n ( x, b ) ≈ b + 6 n )Γ( b )Γ(1 + 6 n ) x b +6 n (1 − x ) b (11)and ω ( b ) = 3 − b ζ ( b ) (cid:18) ζ ( b,
13 ) − ζ ( b,
23 ) (cid:19) (12)with ζ ( . ) and ζ ( ., . ) are respectively the Riemann Zeta andHurwitz Riemann Zeta functions [19, p. 1036].To prove theorem 1, we first establish the following lemma,proved in Appendix B. This Lemma explicitly provides thenormalized interferences received at the origin of C in termsof known functions. Lemma Let z be any complex number such that < ( z ) > , where < ( z ) is the real part of the complex number z , then ω ( z ) = + ∞ X k =1 k − X j =0 k + j − jk ) z =3 − z ζ ( z ) (cid:18) ζ ( z,
13 ) − ζ ( z,
23 ) (cid:19) (13)where ζ ( . ) and ζ ( ., . ) are respectively the Riemann Zeta andHurwitz Riemann Zeta functions. Proof of theorem 1:
Using the definition of f ( m, b ) in (7) and the explicit form of f l,k,j ( m ) in (8), we obtain f ( m, b ) = X ≤ l ≤ ,k ≥ , ≤ j ≤ k − ∞ X n = −∞ (cid:18) rD k,j (cid:19) b + | n | e in ( θ − θ k,j − l π ) × Γ( b + | n | )Γ( b )Γ(1 + | n | ) F b, b + | n | , | n | , (cid:18) rD k,j (cid:19) ! (14)The sum over l is simple to evaluate because it vanishesfor every integer n not multiple of 6. In addition, the infinitehexagonal network is symmetric with respect to the real axis,then f ( m, b ) stays unchanged when we substitute the location m by its complex conjugate. It follows that f is an evenfunction on θ and can be written in a closed form f ( m, b ) = H ( x, b ) + 2 + ∞ X n =1 H n ( x, b ) cos (6 nθ ) (15) where H n ( x, b ) = 6Γ( b + 6 n )Γ( b )Γ(1 + 6 n ) + ∞ X k =1 k − X j =0 (cid:18) xD k,j (cid:19) b +6 n cos (6 nθ k,j ) F b, b + 6 n, n, (cid:18) xD k,j (cid:19) ! (16)To prove that the Fourier series expansion of f is absolutelyconvergent, we shall explicitly evaluate H n and thus show (10)and (11).We first substitute D k,j with δ p k + j − jk and r with xδ .Expanding then Gauss hypergeometric function F in (16)and using lemma 2 yield H ( x, b ) = 6 x b Γ( b ) ∞ X h =0 Γ( b + h ) Γ( h + 1) ω ( b + h ) x h (17)and for n ≥ H n ( x, b ) = 6 x b +6 n Γ( b ) ∞ X h =0 Γ( b + h )Γ( b + 6 n + h )Γ( h + 1)Γ(1 + 6 n + h ) x h × + ∞ X k =1 k − X j =0 cos (6 nθ k,j )( k + j − jk ) b + h +3 n (18)Now when n is sufficiently high, we have the followingequivalences [18, p. 262]: Γ( b + 6 n + h )Γ(1 + 6 n + h ) ∼ Γ( b + 6 n )Γ(1 + 6 n ) ∼ (6 n ) b − (19)and + ∞ X k =1 k − X j =0 cos (6 nθ k,j )( k + j − jk ) b + h +3 n ∼ (20)Replacing the previous relations (19) and (20) in the expres-sion (18) of H n ( x, b ) yields H n ( x, b ) ∼ b + 6 n )Γ( b ) Γ(1 + 6 n ) x b +6 n + ∞ X h =0 Γ( b + h )Γ( h + 1) x h (21)The last sum in (21) converges uniformly in the unit disk to Γ( b )(1 − x ) b . We obtain then H n ( x, b ) ∼ b + 6 n )Γ( b )Γ(1 + 6 n ) x b +6 n (1 − x ) b , as n → + ∞ (22)It follows that the Fourier series expansion (9) of the ISR f converges like the series H ( x, b ) + 12 x b Γ( b )(1 − x ) b + ∞ X n =1 Γ( b + 6 n )Γ(1 + 6 n ) x n cos (6 nθ ) (23)which converges absolutely for | x | < . In addition, the ISR f can be very well approximated by the series in (23) for every x < and b > . (cid:4) It is interesting to note from theorem 1 the followingremarks: • The explicit formulas of the ISR f ( m, b ) in (9) andits average over θ in (10) are rapidly convergent series on m , for | m | < δ , and were unknown before thiswork. The first elements of the series are sufficient tocapture with a high precision the ISR value at location m . Therefore, using these exact expressions is better, interms of computation time, than performing simulationswith high number of sites. • The received interference at location m is I ( m, b ) = f ( m, b ) Par b . At the origin of R , i.e. m = 0 , it is simplyreduced to the quantity I (0 , b ) = P ω ( b ) aδ b . • The Mean Interference-to-Signal Ratio (MISR) for thehexagonal model can be simply deduced by averaging f ( m, b ) over m . When m follows a uniform randomvariable on the disk of radius R = κδ (where κ could befor example / , q √ π or / √ ), the MISR is ¯ f ( b ) = 6Γ( b ) ∞ X h =0 Γ( b + h ) ω ( b + h )( b + h + 1)Γ( h + 1) κ b +2 h (24)which is independent from the intersite distance δ . Corollary Let f , as defined in (7), be the ISR functionof location m and parameter b . Let x be as in theorem 1, thenwe have for b > and x < f ( m, b ) ≈ H ( x, b ) − x b (1 − x ) b +2 x b (1 − x ) b X l =0 < (cid:20)(cid:16) − xe i ( θ + lπ/ (cid:17) − b (cid:21) (25)The proof of corollary 1 is simple. We have just to replace H n ( x, b ) in the expression of f ( m, b ) by the approximationof (11), valid for n ≥ , and then evaluate the sum in (23). Remark: • As can be observed from equation (11), H n ( x, b ) decaysvery fast with n and becomes very small for n ≥ . Thus,the coefficients H n ( x, b ) , for n ≥ , contribute very littleon the value of f ( m, b ) . The ISR f is then a very slowlyvarying function on θ . This observation was pointed outin [3] based on the curve of f . • The ISR f ( m, b ) can be well approximated only with H ( x, b ) with an error equal to O ( x b +6 ) ; where O isthe Knuth’s notation. Corollary For b > and x < , the followingapproximations hold. f ( m, b ) ≈ H ( x, b ) (26) H ( x, b ) = 6 x b (cid:0) ω ( b ) + ω ( b + 1) b x + O ( x ) (cid:1) (27) H ( x, b ) ≈ x b (cid:0) F ( b, b, , x ) + ω ( b ) − (cid:1) (28) H ( x, b ) ≈ x b (cid:18) − b ) x (1 − x ) b − + ω ( b ) − (cid:19) (29) Proof:
Approximation (26) arises directly from the previousremark. Approximation (27) results from the Maclaurin seriesexpansions of H provided in (10) and we take only thedevelopment to order 2 with respect to x . δ I n t e r f e r en c e − t o − S i gna l R a t i o ( I S R ) Simulated ISR with θ = π /6Explicit form of f in (25) with θ = π /6Simulated ISR with θ =0Explicit form of f in (25) with θ =0 b=1.25b=1.4b=2 Fig. 2: Explicit formula of f in (25) versus simulated ISR ateach location m .To prove (28), let ω be the function in (13), take the expansionof H in (10) and use the approximation ω ( b + h ) ≈ , for h ≥ .To prove (29), we start from (28) and use Euler transformationformula of the Gauss hypergeometric function [18, p. 564]: F ( b, b, , x ) = (1 − x ) − b F (1 − b, − b, , x ) (30)To complete the proof, take only the Maclaurin series devel-opment of order 2 in F (1 − b, − b, , x ) because, for x < and b > , the other terms of higher order are negligible. (cid:4)
2) Validation with numerical examples:
In Fig. 2, we plot the explicit formula (25) of the ISR andits simulated value in each location m considering differentamplitude exponent coefficients b ∈ { . , . , } ; and for twocases of angle θ ∈ { , π/ } . For each value of location m ,the simulated ISR is performed with 1000 rings of cells. Wenotice that the explicit simple form (25) of f is exactly similarto the one obtained by simulation following its definition in(7), for all x < and b > , but in practice we focus onlyon the part where the location m is served by the central cell,i.e., x = rδ < √ .It is worth noting that the ISR undergoes small variations with θ except at cell edge ( x ≈ √ ) where its impact becomessignificant. Moreover, the influence of θ on the ISR increaseswith b .Fig. 3 represents the exact expression of H in (10) comparedwith their development of order , namely ω ( b ) x b , and the nd order given in (27) for b = 1 . , b = 1 . and b = 2 .For b close to , both the order and the nd order matchwell with the exact form of H . However, for high value of b ,the exact value of H begins to distance from its first orders’developments and mainly from order . This is explained bythe fact that when b increases, the coefficient Γ( b + h ) Γ( h +1) of theseries expansion of H also increases. Hence, we need to gofor higher order and notably at the cell edge area. To sum up,The nd order development (27) is a good approximation of H in the covered area of the cell for < b ≤ . .
3) Comparison with other approximations in literature:
In addition to the explicit calculation (10) of H , this papergives a simple approximation of H in (29) that would matchvery well with H for every b > . To show the effectivenessof this approximation, we numerically compare it with other δ I n t e r f e r en c e − t o − S i gna l R a t i o ( I S R ) Exact formula of H in (10)Approximation with the order 0Approximation with the 2 nd order in (27) b=1.3 b=2b=1.1 Fig. 3: H and its developments of order 0 and nd order. δ I n t e r f e r en c e − t o − S i gna l R a t i o ( I S R ) Explicit formula of H in (10)Approximation in (29)Approximation with f in (31)Approximation with f in (32) b=1.4b=1.2b=2 Fig. 4: Comparison of the ISR formula (29) with otherapproximations for b ∈ { . , . , } .approximations provided in [2] and [3]. The approximation ofthe ISR (provided for the fluid model) given in [2, Eq.15] is f ( m ) = 2 πx b √ b −
1) (1 − x ) − b (31)Whereas, the provided one in [3, Eq.7] is f ( m ) = ζ (2 b − x b (1 − x ) b (cid:18) − x ) b + (1 − x ) b (1 + x ) b (cid:19) (32)In Fig. 4, numerical results show that, for any value of x ≤ √ ,the average over θ of the ISR, H , is well captured andprecisely estimated by the proposed simple approximation(29). The approximation f , in [3], is valid for low values of x but is limited at the cell edge (with error higher than )for all b > . The approximation f , in [2], is good only when b approaches . For higher b (e.g., close to ), f moves awayfrom H and becomes inefficient to estimate the interferencesin hexagonal network. To reduce the gap between f and theexact value of H , Kelif et al. introduced in [2, Eq.16, p. 7]a corrective term obtained by simulation. B. ISR expression in hexagonal tri-sectorized networks
Sectorization effect on interferences was largely investigatedbut mostly with simulations [23], [24]. In regular hexago-nal networks with tri-sectorized sites, interferences obviouslydepend on the antenna radiation patterns of the sectors inboth serving and interfering sites. In general, both horizontaland vertical radiation patterns influence interferences, but herewe give an approximation of the ISR considering only thehorizontal pattern. The vertical one is neglected and we assumethat its effect is taken in the constant term of the antenna gain.
As mentioned earlier, antenna pattern (in dB scale) is the sumof a constant gain and a mask depending on the angle betweenthe mobile location and the antenna’s boresight.The antenna mask G s of a site, as defined in (4), is a periodicfunction with period π if all sectors forming the site have thesame antenna pattern. In tri-sectorized network, the relativeinterference F , received in location m , is the sum over allinterferences received from each site S l,k,j (for ≤ l ≤ , k ≥ and ≤ j ≤ k − ) and from all collocated sectors ofthe same site. F ( m, b ) = − G s ( θ ) G ( θ − π ) + f s ( m, b ) (33)where the quantity − G s ( θ ) G ( θ − π ) corresponds to the intra-site ISR generated by sectors belonging to the same site and f s ( m, b ) is the inter-site ISR as defined in sub-section II-D. Agood approximation of F ( m, b ) is provided in the followingproposition. Proposition The ISR F ( m, b ) , dependent on b andreceived at location m connected to the first sector of site S in a regular hexagonal network with infinite number of tri-sectorized sites, has the following approximation: F ( m, b ) ≈ − G s ( θ ) G ( θ − π ) + α f ( m, b ) G ( θ − π ) − α x b G ( θ − π ) X l =0 < h(cid:0) e il π − xe iθ (cid:1) i(cid:12)(cid:12) − xe i ( θ − l π ) (cid:12)(cid:12) b +3 (34)where f is recalled the ISR function for omni-directionalnetwork, given in theorem 1, and x = rδ . The coefficients α and α are given by α p = 3 π Z π G s ( θ ) cos (3 pθ ) dθ, p ∈ { , } (35) Proof:
The proof is provided in Appendix C. (cid:4)
It is important to note that the approach, used to give the closedformula (34) of the ISR in tri-sectorized network, is applicableto any sectorization level such as quadri or hexa-sectorization.In order to assess the validity of ISR approximation in tri-sectorized network, we use an antenna mask from a real patternhaving a beamwidth equal to ◦ . The coefficients of the sitemask calculated in (35) for the considered antenna are α =0 . and α = − . .In Fig. 5a, we draw both simulated results and the explicitform (34) of the ISR in a hexagonal tri-sectorized network fordifferent values of b and for the extreme values of θ : and π . Simulations is performed using 1000 rings of tri-sectorizedsites. For θ = 0 , r ≤ δ whereas for θ = π , r varies up to δ .The simple explicit formula (34) approximates very well theISR for all locations in the cell except with small differencewhen m approaches δ e i π and for high value of b . In addition,for θ = 0 , the signal coming from the serving sector is almostequal to the signal from the collocated one and consequentlyISR is higher than 1 even for locations close to site center.Furthermore, we observe a high gap between ISR in θ = 0 and in θ = π . Hence, the impact of θ is no longer negligible x=r/ δ I n t e r f e r en c e − t o − S i gna l R a t i o ( I S R ) Simulated ISR Explicit formula in (34) −0.7 −0.4 −0.1 0.2 0.5 0.7−0.7−0.4−0.10.20.50.70510 x=(r/ δ ).cos( θ )y=(r/ δ ).sin( θ ) G ( θ − π / ) .f s ( m ) b=1.25b=1.4b=2 b=1.25 θ =0 θ = π /3 b=2b=1.4 (a) ISR in tri−sectorized network for θ =0 and θ = π /3. (b) Illustration of the intersite ISRG( θ − π /3)f s (m) for b=1.4. Fig. 5: ISR in tri-sectorized network.and it plays a crucial role in the calculation of the ISR.In Fig. 5b, the inter-site ISR G ( θ − π ) f s ( m, b ) , in (59),is represented as a function of θ and r . In contrast tothe small impact of θ in the omni-directional network, wenotice further that the inter-site ISR in tri-sectorized networkdepends on θ as much as on r . In fact, the antenna maskdefines the envelope curve of the ISR calculated in omni-directional network. Furthermore, Fig. 5b shows that the inter-site interference is periodic with period π . This result is easyto be mathematically proved. C. ISR expression for higher frequency reuse patterns
Frequency reuse is the core concept of wireless cellularnetworks and consists in using the same frequency in differentgeographical areas, spaced far enough apart from each other.To avoid excessive interferences, fixed frequency allocation toeach cell is performed in the planning process according to afrequency reuse pattern, denoted here by υ . According to thispattern, υ different frequency bands of the operator are usedfor each cluster of υ adjacent cells.Co-channel interfering cells are arranged to form again ahexagonal lattice with first ring distant from the cell at theorigin by ∆ = √ υδ (see [25, p. 8]), where δ is the inter-sitedistance and defined as in section II-A. Let ˆ f ( m, b, υ ) be theISR of a hexagonal network having a reuse pattern υ , it isobtained by changing δ in the ISR of reuse 1 by ∆ . Since theISR of reuse 1, f ( m, b ) = ˆ f ( m, b, , depends on the ratio m/δ , it follows that ˆ f ( m, b, υ ) = f ( m/ √ υ, b ) (36)It is clear that ˆ f ( m, b, υ ) converges faster when υ increases.Consequently, for high value of υ , the first term of the seriesof f ( m/ √ υ, b ) is enough to capture the essential value ofthe ISR. It means that for high value of υ , ˆ f ( m, b, υ ) ≈ ω ( b ) x b /υ b , where again x = | m | /δ .It is known that for high frequency reuse patterns, theoperator is obliged to allocate a low bandwidth to each cellresulting in a decrease of cell capacity. In order to maximizethe spatial spectrum reuse while minimizing interferences,Fractional Frequency Reuse (FFR) pattern has been proposedin GSM networks for the concentric-cells’ deployment (calledalso Underlay-Overlay cells) and in LTE for the Inter-Cell In-terference Coordination (ICIC) scheme. The fundamental idea of FFR is to apply two frequency reuse patterns in the samecell: tight frequency reuse with short reuse distance, restrictedto users close to the cell, and regular frequency reuse with longreuse distance, useful for users at cell edge. For FFR, the ISRprovided in (36) is still valid but the frequency reuse pattern υ depends indeed on the location m . A well known exampleof FFR pattern is υ ( m ) = ( | m | ≤ r ) + 3 × ( | m | > r ) ,where ( | m | ≤ r ) is the indicator function that takes ifthe condition " | m | ≤ r " is verified and else and r is athreshold separating cell edge users from those close to thecell. This means that a frequency reuse pattern υ = 3 is usedat cell edge whereas at cell center υ = 1 . D. Inclusion of Log-normal shadowing in the ISR calculation
To take into account Log-normal shadowing effect on thecalculation of the ISR, the pathloss L l,k,j ( m ) between location m and site S l,k,j becomes a Log-normal random variablemultiplied by its expression in (5). In each location m , L l,k,j ( m ) and L l ,k ,j ( m ) are also supposed to be independentand identically distributed whenever ( l, k, j ) = ( l , k , j ) .Consequently, the individual ISR, defined in (6), is also a Log-normal random variable since it is the ratio of two Log-normalvariables. Let f ( sh ) ( m, b ) be the ISR when accounting for theeffect of shadowing, its definition in (7) is changed for omni-directional network by f ( sh ) ( m, b ) = X ≤ l ≤ ,k ≥ , ≤ j ≤ k − | m | b | m − S l,k,j | b χ l,k,j (37)where { χ l,k,j } is a sequence of independent and identicallydistributed Log-normal variables with mean E [ χ ] and variance V ar [ χ ] .The distribution of the Log-normal sum is not exactly knownand always constitutes an open problem in probability theory.Nevertheless, it has been recognized that the Log-normal sumcan be well approximated by a new Log-normal random vari-able using Fenton-Wilkinson method [26]. This is achieved bymatching the mean and the variance of the new resulted Log-normal variable with those of the Log-normal sum. The choiceof Fenton-Wilkinson approach is motivated by its conveniencefor the present study in contrast with other approaches in [27],[28]. Applying Fenton-Wilkinson method to the Log-normalsum in (37), it follows that f ( sh ) ( m, b ) is approximated by aLog-normal random variable with mean E [ f ( sh ) ( m, b )] = f ( m, b ) E [ χ ] and variance V ar [ f ( sh ) ( m, b )] = f ( m, b ) V ar [ χ ] where f ( m, b ) is the ISR without the inclusion of shadowingand given explicitly in (9) or in (25).It is important to note that, with the inclusion of shadowing,the Voronoi cell associated to the central cell is no longer thehexagon or the disk having the same area as the hexagon.It becomes the set { m ∈ C | L ( m ) ≤ L l,k,j ( m ) , k ≥ } ,instead. IV. A PPLICATION TO PERFORMANCE ANALYSIS : SINR
DISTRIBUTION
The performance analysis of wireless networks definitelyentails the evaluation of the SINR distribution. Its knowledgeallows to estimate the throughput which is related to the SINRby a continuous, differentiable, and strictly monotonicallyincreasing function. As mentioned in section II-C, this functionis called the link level curve of the system.In this section, we give the explicit formula of the Comple-mentary Cumulative Distribution Function (CCDF) of SINRfor a general location distribution. Considering an arbitraryuser location distribution is of great interest for capacitydimensioning because it was shown that the spatial userdistribution is practically very heterogeneous between cells ofthe network and even inside cells [20], [29].
A. Explicit form of SINR distribution
Before the explicit calculation of the SINR CCDF, we givethe following definitions.
Definition
3: The SINR, experienced at location m connectedto central cell S in a regular omni-directional hexagonalnetwork, is the ratio between the useful received power andthe total received interferences including thermal noise power SIN R = ( ηf ( m, b ) + y x b ) − (38)where y = aP N P δ b , P is recalled the transmitted power ofthe cell, including the antenna gain, a and b are the path lossparameters given in (5), δ is recalled the intersite distance, x = rδ , η is the average load of the interfering cells and P N is the thermal noise power calculated in the same spectrumbandwidth as for the cell transmit power P .Equation (38) is obtained by firstly considering the classicaldefinition of SINR at location mSIN R = P/L ( m ) X l,k =0 ,j η l,k,j P/L l,k,j ( m ) + P N (39)where η l,k,j is the percentage of occupied resources (schedul-ing slots or physical resource blocks) in the interfering celland reflects its load averaged over a period corresponding tothe averaging window of the measurement reports of mobiles;and secondly assuming that all interfering cells have the sameaverage load η . Definition
4: The CCDF of SINR for any scenario of userlocations’ distribution is given by the following integral Ψ( y ) = P ( SIN R > y ) = Z S ( SIN R > y ) dt ( m ) (40)where S is the area of the central cell assumed to be a diskof radius R < δ , ( . ) is the indicator function. t ( m ) is theprobability measure of the location variable m and reflects thespatial traffic distribution in cell S .We assume that t is a probability measure on S and conse-quently Z S dt ( m ) = 1 Remark: • The CCDF of SINR is widely called coverage probabilitybecause it gives the percentage of locations in which thecondition "
SIN R > y " is guaranteed. • When the thermal noise power is neglected with respectto interferences, y ≈ and the SINR might be similarto SIR. • When the SINR in each location m is a random variable(e.g., through the shadowing effect), definition (40) of theSINR CCDF is replaced by Ψ( y ) = Z S P ( SIN R > y | m ) dt ( m ) In the calculation of the SINR distribution, we consider theexpression of the ISR in (26) since we showed earlier that, foran omni-directional network, the angle θ has a little impact.This latter would be further smoothed when looking for thedistribution of any transformation of the ISR function.Let g be the function defined on [0 , by g ( x ) = ηH ( x ) + y x b (41)where again x = rδ , it is clear that g ( x ) = 1 /SIN R and thecalculation of SINR CCDF requires the inverse function of g which realizes a continuous bijection from [0 , ρ ] to [0 , g ( ρ )] for every real number ρ satisfying ≤ ρ < . The inversionof g can be explicitly given by series reversion method. Inthe following proposition, we provide an approximation of g − ( y ) . Proposition Let g be defined as in (41). Let y = g ( x ) forevery x ≤ ρ = √ , then the following approximation holdsfor x = g − ( y ) g − ( y ) ≈ C ( y, b ) r + q + β ( b ) C ( y, b ) (42)where C ( y, b ) = ( y ηω ( b ) + y ) b and β ( b ) = 6 bηω ( b + 1)6 ηω ( b ) + y Proof:
The proof is given in Appendix D. (cid:4)
Considering now an arbitrary traffic distribution, we shall statethe explicit form of SINR CCDF involving the inverse function g − in the following theorem. Theorem Let g be defined as in (41). Recall δ be theinter-site distance and R be the radius of the central cell S .Then the SINR CCDF is explicitly given by Ψ( y ) = T (Λ( y )) , ∀ y > (43)with Λ( y ) = min (cid:18) δ × g − ( 1 y ) , R (cid:19) (44)The function T ( . ) is the marginal Cumulative DistributionFunction of the location random variable m and is obtainedfrom the probability measure t ( m ) = t ( r, θ ) by dT ( r ) = R π dt ( r, θ ) . The function min( u, v ) gives the smallest oneamong the real numbers u and v . Proof:
Taking the definition of SINR and Ψ in respectively(38) and (40) and with the assumption that the impact of θ onthe ISR is neglected, we have Ψ( y ) = Z π Z R ( 1 g ( rδ ) > y ) dt ( r, θ )= Z min ( δ × g − ( y ) ,R ) Z π dt ( r, θ )= Z Λ( y )0 dT ( r ) B. Numerical results for different user locations’ distributions
In order to evaluate and validate the analytical expressionof SINR distribution Ψ for different scenarios of user loca-tions’ distributions, we consider a uniform and non-uniformlocations’ distributions in the cell. Note that in practice, theestimation of the spatial traffic density (the measure t ( m ) )in each location can be based on processing probes or onthe manipulation of some key performance indicators of thenetwork as in [20]. Furthermore, authors in [29] showedwith real data measurements that the spatial traffic densitycan be approximated by a Log-normal distribution. Therefore,numerical results of this work are provided for both uniformand Log-normal user locations’ distribution scenarios.Besides, for each location scenario, we simulateinterferences in a hexagonal omni-directional networkusing Monte-Carlo method and compare it to the theoreticalresults. The considered network is composed of 1000rings of cells operating in the band M hz of the LTEtechnology. Only M hz of spectrum bandwidth is availableand reused everywhere in the network. i.e., frequencyreuse pattern equals 1. Each cell of the network transmitswith a power equal to P = 60 dBm and distant from itsadjacent cells by δ = 1 Km . We set the radius of the cell to R = δ q √ π = 0 . δ and we assume that interfering cellsare fully loaded, i.e., η = 1 . The downlink thermal noise isset to − dBm . It is worth noting that parameters in dB scale are transformed to linear when used for the calculationof the simulated metrics ISR and SINR.For each scenario of simulation, 20000 users are generatedaccording to the considered location distribution in subsectionsIV-B1 and IV-B2. For each generated user location,interferences, ISR and SINR are calculated.The comparison between theoretical and simulation resultsare provided for 2 types of environment, different by the valueof the propagation parameter a : Outdoor environment with a = 130 dB and deep indoor environment with a = 166 dB .Furthermore, We investigate results for 3 different values of b : . , . and . The Log-normal user location distribution should not be confused withthe Log-normal shadowing. P = 60 dBm corresponds to dBm from the transmitter poweramplifier and dB for the antenna gain of the transmitter. P N = E n + 10 log ( W ) + NF ; where E n is the energy of the thermalnoise equal to − dBm/Hz , W is the spectrum bandwidth equal to Mhz and NF is the noise figure of the transmitter, set here to dB . −10 −5 0 5 10 15 20 25 3000.10.20.30.40.50.60.70.80.91 SINR in dB
SINR CCDF withMonte Carlo simulationsCalculated SINR CCDF in (45)−10 −5 0 5 10 15 20 25 3000.10.20.30.40.50.60.70.80.91
SINR in dB S I NR CCD F b=2b=1.5b=1.25 b=2b=1.5b=1.25 (a) In outdoor environment (b) In deep indoor environment Fig. 6: CCDF of the SINR for uniform location distributionscenario with different values of b .
1) SINR distribution for uniform user location variable:
When the user location distribution is uniform in the disk(representing the cell) of radius R , we have dt ( m ) = rdrdθπR .The distribution Ψ of SINR in (43) becomes Ψ( y ) = Λ( y ) R (45)Since the ISR is a function of the polar coordinates of m ,approximating the central Voronoi cell with a disk of radius R gives a simple and closed form of SINR distribution. Thisassumption has been widely considered, see for example [30],[31].In Fig. 6, we present the curve of Ψ in (45) and the SINRCCDF obtained by Monte Carlo simulations for both outdoorenvironment (Fig.6a) and deep indoor environment (Fig. 6b).The results show that the expression of Ψ calculated in (45)fits with simulations results of the SINR CCDF quite well.Moreover, as expected the impact of θ on Ψ is smoothedand thus neglected. In addition, it is important to note, fromthe comparison between Fig. 6a and Fig. 6b, that in the deepindoor environment, the impact of the value y on the SINRis clear and tends to decrease it by to dB depending on b . This is explained by the high value of the deep indoorpenetration margin that makes the path loss very high and thusthe signal arrives very low at indoor locations. Conversely, foroutdoor environment, y is low and SINR CCDF should bequite similar to SIR CCDF.
2) SINR distribution for Log-normal user location variable:
For a Log-normal location distribution, we assume that vari-able r follows a Log-normal distribution ∼ ln N ( µ, σ ) withmean µ and standard deviation σ . Variable θ is always uniform ∼ U (0 , π ) . The probability measure of location m is writtenby dt ( m ) = e − (ln( r ) − µ )22 σ drσ √ πQ (cid:2) σ (ln( R ) − µ ) (cid:3) r dθ π and consequently the expression (43) of Ψ becomes Ψ( y ) = Q (cid:2) σ (ln(Λ( y )) − µ ) (cid:3) Q (cid:2) σ (ln( R ) − µ ) (cid:3) (46)where Q is the cumulative distribution function of the standardnormal distribution and is given by the following integral. Q ( z ) = 1 √ π Z z −∞ e − u du, ∀ z ∈ R −0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.6−0.4−0.200.20.40.6 x=(r/ δ ).cos( θ ) y = (r / δ ) . s i n ( θ ) −0.6 −0.4 −0.2 0 0.2 0.4 0.6 x=(r/ δ ).cos( θ ) (b) Traffic hotspot at cell edge (a) Traffic hotspot at cell center Fig. 7: Snapshots of the two traffic hotspots following Log-normal distribution: The first is close to cell center (a) with ( µ, σ ) = ( − , . and the second is at cell edge (b) with ( µ, σ ) = ( − . , . .The parameters µ and σ of the Log-normal distribution allowto deal with different scenarios of traffic hotspots. Besides, µ and σ can be tuned to precisely approximate the distributionof SINR in each cell of a real network.Here, we show the SINR distribution for two cases of userlocations distributions. The first case is a Log-normal hotspotclose to cell center with parameters ( µ, σ ) = ( − , . ,whereas the second one is a Log-normal traffic hotspot locatedat cell edge and having parameters ( µ, σ ) = ( − . , . .Both hotspots are depicted in Fig. 7.In Fig. 8, we show the CCDF of the SINR for the previouslycited environments (Fig. 8a for outdoor and Fig. 8b for deepindoor environment) considering a hotspot close to the cellcenter (Fig. 7a). Simulation Results confirm once again theprecision brought by the analytical form of SINR CCDF in(43). We notice indeed that numerical and analytical curvesof SINR CCDF are very close. This consolidates also thedifferent observations from the study of uniform locationdistribution, notably regarding the negligible influence of θ onthe calculation of the SINR distribution and the importance ofthe penetration margin that deteriorates more or less the SINRin deep indoor environment.In Fig. 9, we present the SINR CCDF for a Log-normaltraffic hotspot located at cell edge (see Fig.7b). Even if θ influences the ISR value in cell edge (see Fig. 2), the curve ofSINR CCDF shows to be still insensitive to θ for traffic hotspotat cell border because the simulation result considering θ isalmost similar to the analytical expression (43) which doesnot account for θ variation. All locations in a hotspot at celledge suffer from degraded SINR. We note also that the SINRCCDF curve of a hotspot close to cell center is more scatteredthan that of a hotspot at cell edge. This is related to the valueof σ which is higher for the first hotspot.The same previous conclusions regarding the impact of thepenetration margin in indoor environment are still valid forhotspot at cell edge. We could say that the SINR CCDF fordeep indoor is obtained from that of outdoor environment bya translation of almost dB to low values.V. C ONCLUSIONS
This paper has analytically and fundamentally investigateddownlink interferences in hexagonal wireless networks. Sev-eral results have been established for the ISR function f −5 0 5 10 15 20 25 3000.10.20.30.40.50.60.70.80.91 SINR in dB S I NR CCD F SINR CCDF with Monte Carlo simulationsCalculated SINRCCDF in (46)−5 0 5 10 15 20 25 3000.10.20.30.40.50.60.70.80.91
SINR in dB S I NR CCD F b=1.5b=1.5 b=2b=1.25 b=1.25 b=2 (a) In outdoor environment (b) In deep indoor environment Fig. 8: CCDF of the SINR in a cell having traffic hotspot closeto its center:( µ = − and σ = 0 . ) for different values of b . −10 −5 0 500.10.20.30.40.50.60.70.80.91 SINR in dB
SINR CCDF with MonteCarlo simulationsCalculated SINR CCDF in (46)−10 −5 0 5 1000.10.20.30.40.50.60.70.80.91
SINR in dB S I NR CCD F b=1.25b=1.25 b=1.5 b=1.5 b=2b=2 (a) In outdoor environment (b) In deep indoor environment Fig. 9: CCDF of the SINR in a cell having traffic hotspot atcell edge: ( µ = − . and σ = 0 . ) for different values of b .and the SINR distribution. Given a location m = re iθ , wehave proved that the ISR can be written as an absolutelyconvergent Fourier series on θ with explicit coefficients. Thefirst coefficient, which is the average of the ISR over θ , hasbeen explicitly evaluated without any approximation. Besides,we have proved for omni-directional network that the ISR isa slowly varying function on θ . Its average, as a function of r , is enough to capture the essential information of f wheninvestigating the SINR distribution. In addition to the providedclosed formulas of ISR, this paper has also given very simpleapproximations, rigorously proved and shown to be better thanother approximations in literature. Subsequently, the derivedexpressions have been extended for tri-sectorized hexagonalnetworks and for the use of high frequency reuse patterns.Moreover, SINR distribution has been explicitly given forevery location distribution and entails, of course, the inversionof the function g : x ηH ( x, b ) + y x b . Numerical resultshave been carried out for two types of user locations: uniformand Log-normal. The exact formula of SINR distributionapproximates quite well the simulated SINR even when userlocations form a traffic hotspot at cell edge.For future works, we will extend the results of this paper tomathematically analyze interferences in perturbed hexagonalnetwork model: site location is perturbed by a random variable.Apart from the fact that any new result for such a networkmodel is scientifically interesting, there are additional motiva-tions for looking on it: • Perturbed hexagonal network model approximates morethan any other model the geometry of the real network[10]. Consequently, performances of perturbed hexagonal model would be closer to the performances of a given realnetwork, • In addition to the parameters of the location randomvariable m , considered in this paper, the perturbed modelintroduces a new parameter, named the average pertur-bation. SINR or throughput distribution for a given realcell can be precisely determined with tuning the averageperturbation and the first moments of the user locationvariable. A PPENDIX AP ROOF OF PROPOSITION L l,k,j ( m ) in (6) by its expression in (5)and use (2), then we can write down f l,k,j ( m ) = τ ( m ) τ ( m ) ;where τ ( m ) = (cid:18) rD k,j (cid:19) b (cid:18) − rD k,j e i ( θ − θ k,j − l π ) (cid:19) − b (47)and τ ( m ) is its complex conjugate.Since rD k,j < , τ ( m ) is expanded to an absolutely convergentseries τ ( m ) = + ∞ X n =0 (cid:18) rD k,j (cid:19) n + b e in ( θ − θ k,j − l π ) Γ( b + n )Γ( b )Γ(1 + n ) (48)Arranging the product of the two series τ ( m ) τ ( m ) in orderto get a Fourier series expansion on θ and using the definitionof the Gauss hypergeometric function [18, p. 561], we arriveat (8). A PPENDIX BP ROOF OF LEMMA z be any complex number such that < ( z ) > , define ω ( z ) = + ∞ X k =1 k − X j =0 k + j − jk ) z (49)Observe that ω ( z )Γ( z ) is the Mellin transform of the func-tion ϑ defined for all complex number t , such that < ( t ) > ,by ϑ ( t ) = + ∞ X k =1 k − X j =0 e − t ( k + j − jk ) = + ∞ X k =1 + ∞ X j =0 e − t ( k + j + jk ) (50)Now using the Borwein cubic theta function defined by thebilateral sum [32], we have X k,j ∈ Z e − t ( k + j + jk ) = 1 + 2 ϑ ( t ) + 2 + ∞ X k =1 + ∞ X j =0 e − t ( k + j − jk ) (51)Dividing the last term in the right hand side of (51) intotwo parts and manipulating the indexes of the sum give thefollowing identity + ∞ X k =1 + ∞ X j =0 e − t ( k + j − jk ) = 2 ϑ ( t ) (52) It follows, from the last identity and (51), that X k,j ∈ Z e − t ( k + j + jk ) = 1 + 6 ϑ ( t ) (53)Using the Lambert Series form of Borwein cubic thetafunction [32], we obtain ϑ ( t ) = + ∞ X k =0 (cid:18) e t (3 k +1) − − e t (3 k +2) − (cid:19) (54)Finally, applying the Mellin transform to both right and lefthand sides of the last equation yields ω ( z )Γ( z ) = 3 − z ζ ( z ) (cid:18) ζ ( z,
13 ) − ζ ( z,
23 ) (cid:19) Γ( z ) (55)which completes the proof of lemma 2.A PPENDIX CP ROOF OF PROPOSITION G s be defined as in (4). Given that G s is a periodicfunction on θ with period π , we assume that it has aconvergent Fourier series expansion G s ( θ ) = α + 2 + ∞ X p =1 α p cos (3 pθ ) (56)where α p = 3 π Z π G s ( θ ) cos (3 pθ ) dθ, ∀ p ∈ Z Let again m be an arbitrary location in S and recall that e iφ l,k,j = ( m − S l,k,j ) / | m − S l,k,j | . Using the expression ofpathloss in (5), we have L ( m ) G s ( φ l,k,j ) L l,k,j ( m ) = r b X p ∈ Z α p ( m − S l,k,j ) p | m − S l,k,j | b +3 p (57)Thanks to the convergence of the Fourier series (56), thecoefficient α p can be neglected for p ≥ . Consequently, wehave L ( m ) G s ( φ l,k,j ) L l,k,j ( m ) ≈ α L ( m ) L l,k,j ( m ) + 2 α r b < (cid:2) ( m − S l,k,j ) (cid:3) | m − S l,k,j | b +3 (58)We now sum all individual ISRs arriving at location m . Forthe second term having the coefficient α , we limit the sumonly to the first ring k = 1 because it behaves like an ISRwith path loss exponent equals b + 3 . To prove the latter, wedevelop r b < (cid:2) ( m − S l,k,j ) (cid:3) | m − S l,k,j | − b − as we did inproposition 1. It follows that G ( θ − π f s ( m, b ) ≈ X ≤ l ≤ ,k ≥ , ≤ j ≤ k − L ( m ) L l,k,j ( m ) G s ( φ l,k,j ) ≈ α f ( m, b ) + 2 α r b X l =0 < (cid:2) ( m − S l, , ) (cid:3) | m − S l, , | b +3 (59)Replacing S l, , by δe il π and m by re iθ , we complete theproof. A PPENDIX DP ROOF OF PROPOSITION g be the function defined on [0 , by (41). Using theseries expansion of H ( x ) given in (10), it is clear that g admits an analytic expansion on x = r/δ by g ( x ) = (6 ηω ( b ) + y ) x b + ∞ X n =1 γ n x n ! (60)where γ n = 6 η Γ( b + n ) ω ( b + n )(6 ηω ( b ) + y )Γ( b ) Γ( n + 1) (61)Let y = g ( x ) , equation (60) can be transformed to (cid:18) y ηω ( b ) + y (cid:19) b = x + ∞ X n =1 γ n x n ! b = x + γ b x + O ( x ) (62)where O is the Knuth’s notation. The second approximationis justified for x sufficiently small. e.g., valid for x < √ .Omitting the error terms in equation (62) gives a second-orderequation, in x , which admits two solutions: x ± = C ( y, b ) ± q + γ b C ( y, b ) (63)where C ( y, b ) = (cid:16) y ηω ( b )+ y (cid:17) b .Since x = r/δ is a positive real number, only the positivesolution x is valid. Now, replacing γ by its expressionin (61), we obtain the approximation of x = g − ( y ) inproposition 3. A CKNOWLEDGMENT
The authors thank the editor and the anonymous reviewersfor their constructive comments. They also gratefully Ac-knowledge Prof. Martin Haenggi for his useful suggestionsimproving the quality of the paper.R
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