Two-tie Spatial Modeling of Base Stations in Cellula Netwoks ifan Zhou, Zhifeng Zhao, Qianlan ing, Rongpeng Li, uan Zhou, and Honggang Zhang ok-zhejiang Lab fo Cognitive Radio and Geen Communications Dept. of Infomation Science and Electonic Engineeing Zhejiang Univesity, Zheda Road 38, Hangzhou 310027, China Email: {zhouyftt, zhaozf, geenjelly, liongpeng, zhouxuan, honggangzhang}@zju.edu.cn Univesité Euopéenne de Betagne & Supélec, Avenue de la Boulaie, CS 47601, 35576 Cesson-Sévigné Cedex, Fance aiv:1404.1142v2 [cs.ni] 1 Aug 2014 Abstact Poisson Point Pocess (PPP) has been widely adopted as an efficient model fo the spatial distibution of base stations (BSs) in cellula netwoks. Howeve, eal BSs deployment ae aely completely andom, due to envionmental impact on actual site planning. Paticulaly, fo multi-tie heteogeneous cellula netwoks, opeatos have to place diffeent BSs accoding to local coveage and capacity equiement, and the divesity of BSs functions may esult in diffeent spatial pattens on each netwoking tie. In this pape, we conside a two-tie scenaio that consists of macocell and micocell BSs in cellula netwoks. By analyzing these two ties sepaately and applying both classical statistics and netwok pefomance as evaluation metics, we obtain accuate spatial model of BSs deployment fo each tie. Basically, we veify the inaccuacy of using PPP in BS locations modeling fo eithe macocells o micocells. Specifically, we find that the fist tie with macocell BSs is dispesed and can be pecisely modelled by Stauss point pocess, while Maten cluste pocess captues the second tie s aggegation natue vey well. These statistical models coincide with the inheent popeties of macocell and micocell BSs espectively, thus poviding a new pespective in undestanding the elationship between spatial stuctue and opeational functions of BSs. I. INTRODUCTION Netwok topology has a geat impact on the pefomance of cellula netwoks, since the powe of eceived signal vaies depending on the distance between tansmitte and eceive. Moeove, intefeence chaacteization become even moe complicated due to path loss and multipath fading effect [1]. Hence, ealistic spatial modeling of base stations (BSs) is essential fo accuate pefomance evaluation in cellula netwoks. In ecent yeas, Poisson point pocess (PPP) has been poposed as an efficient way to model wieless netwok stuctues [2]. The PPP model fo chaacteizing BS locations can povide tactable and useful esults fo key pefomance evaluation in cellula netwoks [3]. Howeve, it may not be a pactically solid model fo BSs distibution because its complete andomness is unealistic in eal deployment [4] [9]. Indeed, BSs ae not independently distibuted in cetain aeas fo pupose of optimizing coveage and capacity pefomance. In [4], the authos discove that the Geye satuated pocess, which takes account of paiwise inteaction between BSs, can accuately epoduce the spatial stuctue fo vaious wieless netwoks. Moe specifically in cellula netwoks, Geye satuation pocess and its special case Stauss pocess ae utilized to model macocell deployment fo diffeent scenaios in [5]. Besides, Poisson had-coe pocess (PHCP) is also poposed to model BSs locations in [6], while it is still shown that Stauss pocess povides the best fit in tems of coveage pobability. Poisson cluste pocess is veified to be able to model uban aea deployment in [7]. Vey ecently, the Ginibe point pocess has been poposed as a suitable model fo wieless netwoks with nodes epulsion [8], in the light of compomise between accuacy and tactability. All these studies above put fowad coesponding spatial models fo BSs deployment, but the types of BSs (i.e. macocell o micocell) have not been taken into account seiously. As we know, cellula netwoks have been undegoing an evolution towads heteogeneous netwoking achitectue. System pefomance evaluation and esouce allocation become moe complicated fo multi-tie scenaio, since each tie may diffe in tansmit powe, coveage aea and suppoted ate [10] [11]. This lasting tend highlights the impotance fo hieachical spatial analysis of multi-tie stuctue. In the typical case of two-tie cellula netwoks, the spatial stuctues of macocell and micocell deployment have not been analyzed sepaately eve, despite of thei inheent diffeences in functionality and netwoking featue. Actually, macocell BSs ae neithe too close no too fa away fom each othe in ode to satisfy coveage equiement and decease inte-cell intefeence. Theefoe, thee is epulsion between macocell BSs. On the othe hand, micocells ae usually deployed to diminish coveage hole and offload netwok taffic, which always exhibits aggegation featue. So micocell BSs would be clusteed. These facts povide easonable basis to adopt Gibbs and Neyman-Scott pocesses [12] fo modeling macocell and micocell deployment, espectively. This pape aims to find ealistic spatial models fo macocell and micocell BSs sepaately. Based on massive and detailed eal data fom one of the lagest opeatos in China, we obtain enough location ecods fo fitting and testing models. By applying statistical metics in point pocess field such as L- function and neaest neighbo distance distibution [13], along with pefomance metics in cellula netwok analysis such as coveage pobability and cell coveage aea, diffeent hypothe-
ses of BSs deployment on each tie ae tested. Accodingly, the main contibutions of this pape ae summaized as follows: 1) Veification of the inaccuacy of PPP: Diffeent with the elated woks, the validity of PPP fo BS locations on diffeent tie is tested and we veify its inaccuacy in tems of classical spatial statistical metic fo both macocells and micocells. 2) Realistic spatial model fo macocells: We find that macocell BSs ae egulaly deployed and can be well modeled by Stauss point pocess. The fitting esult coincides with the opeational function of macocells which is tageted at poviding oveall coveage in cellula netwoks. 3) Realistic spatial model fo micocells: Diffeent with macocells, we discove that micocell BSs tend to be moe clusteed, and can be accuately epoduced by Maten cluste pocess (MCP), which is consistent with micocells main functionality of diminishing coveage hole and taffic offloading. II. BACKGROUND-POINT PROCESS THEOR Gibbs model [12] is a kind of geneal statistical model that can be chaacteized by pobability density, which is helpful in fitting and simulation using Monte Calo method. Without loss of geneality, we conside a point patten z = {z 1,z 2,...,z n(z) } placed in a bounded windoww, wheen(z) is the numbe of points in z. Fo simplicity, only paiwise inteaction is consideed hee, and its pobability density function (PDF) can be defined as: n(z) f(z) = α [ i=1 µ(z i )] [ i<j ρ(z i,z j )], (1) whee α is a nomalizing facto to ensue the integal to unity, µ(z i ) ae functions modeling the fist ode popety, and ρ(z i,z j ) ae functions modeling paiwise inteaction. By setting µ(z) a constant β, and defining ρ(z i,z j ) as follows: { 1, zi z ρ(z i,z j ) = j > γ, z i z j, (2) the PDF is simplified to f(z) = αβ n(z) γ p(z), (3) whee p(z) is the numbe of point pais that ae less than units apat. If γ = 1, thee is no inteaction between points, and it can be simplified to PPP. By adding a constaint that no distinct points ae allowed to come close than distance hc, the PPP futhe educes to Poisson hadcoe pocess (PHCP) with had coe distance hc. Clealy, γ < 1 indicates the epulsive case, and the density deceases with p(z), Stauss point pocess is one of the epesentative examples. Othewise, when γ > 1, in ode to make the PDF integable and capable of modeling clusteing effect, a satuation theshold is added in the exponent and the PDF is: f(z) = αβ n(z) γ min(p(z),sat). (4) Fig. 1. (left): Macocells as point patten x exhibits inhibition. (ight): Micocells in the same aea as point patten y appeas to be clusteed. It s temed as Geye satuation pocess, a genealization of Stauss point pocess. Moeove, it would educe to a PPP fo sat = 0, o a Stauss point pocess fo sat. The Neyman-Scott pocess [12] consists of the set of clustes of offsping points, centeed aound an unobseved set of paent points. Maten cluste pocess (MCP) is a special case of the Neyman-Scott pocess, whee the numbe of offsping points pe cluste is Poisson distibuted with intensity λ c, and thei positions ae placed unifomly inside a disc of adius R cented on the paent points. We assume that the cluste centes fom the point patten c which is Poisson distibuted with intensity λ p > 0, while conditional on c = {c 1,c 2,...,c n } associate eachc i with a Poisson point pocessx i with intensity λ c > 0 and these offsping point pocesses ae independent with each othe. The density function of Maten cluste pocess can be witten as: f(ξ c i ) = 2 R 2, fo = ξ c i R. (5) All these models mentioned above can be fitted using maximum pseudolikelihood method [14] povided in Spatstat, an R package [15]. III. EVALUATION STATISTICS AND FITTING METHOD In ode to obtain the sepaate spatial models of two-tie BSs, we investigate a dense uban aea in a pospeous city in China. BSs of GSM cellula netwoks ae consideed hee only fo epesentativeness and consistency. The selected 3 3 km 2 squae aea contains 266 BSs including 77 macocells and 189 micocells, wheeas the high BS density implies stong coveage and capacity demands. The macocell BSs ae efeed to as point patten x, while point patten y is fo micocell BSs. Both of them have been mapped by the same scale onto a unit squae as seen in Fig.1. Fo hypothesis testing, we use the following fou evaluation statistics including two classical metics and two netwok pefomance metics: 1) L-function: L-function is a tansfomation of the Ripley s K-function (K()), which is widely used to test the validity of a point pocess in stochastic geomety [13]. It eflects egulaity o clusteing popety of a point patten and is defined as: K() L() = π. (6)
L() Theoetical (PPP) (macocells) (micocells) Fig. 2. L-function of point patten x and y, compaed with the theoetical cuve fo PPP. Fo a completely andom (unifom Poisson) point patten, the theoetical value is L() =, which is used as a baseline to judge a point patten s spatial chaacte [13]. If L() <, then thee is dispesion on this scale and should be modeled by an epulsive pocess; othewise it is aggegated if L() > and should be modeled by a clusteing point pocess. Due to its explicitness and impotance, L-function is used as the fist-step metic in this pape. 2) Neaest Neighbo Distance Distibution: Neaest neighbo distance distibution function of a point pocess z is the cumulative distibution function (CDF) G() of the distance fom a typical andom point of z to the neaest othe points of z [13]. The estimate of G() is a useful statistic summaising the clusteing popety of point patten by compaing with the theoetical G() of a PPP which is G() = 1 exp( πλ 2 ). (7) This statistic is utilized as a useful evaluation metic in micocells spatial modeling, since it epesents the intenal stuctue of clustes in eal BSs deployment. 3) SIR Distibution: In ode to find a ealistic model, we choose signal-to-intefeence-atio (SIR) as an evaluation metic to bidge the modeling validity and actual netwok pefomance. Assuming each mobile use connects to the BS at location y that offes the highest eceived SIR. The esulting SIR in position z is defined as: SIR(s,z) = P y h y d(s,y) α x z\y P xh x d(s,x) α, (8) whee Rayleigh fading is adopted as h x, h y exp(1), and the path loss exponent α is assumed to be 4 consideing dense uban scenaio. P x and P y ae tansmit powes of the coesponding BSs. 4) Voonoi Cell Aea Distibution: The Voonoi cell of a node z z is defined as {y R 2 : d(y,x) > d(y,z), x z\z}. It is poposed as an evaluation metic due to its similaity with the coveage egion of BSs [5], which is a valuable paamete in pactical netwok opeations. As mentioned in Section I, macocell BSs ae esponsible fo coveage-centic equiement, theefoe we use this metic as an evaluation statistic in the hypothesis testing of point patten x. IV. PROPOSED MODELS AND FITTING RESULTS In ode to obtain the accuate point pocess model fo each tie, we fist fit the candidate models to the BSs data set using maximum pseudolikelihood method and get the coesponding paametes. To test models validity, we geneate 600 ealisations fo each fitted model, and fo each ealization the evaluation statistics ae computed to obtain the simulation envelope. Afte that, we thow out the 30 highest and 30 lowest values to ceate 90% confidence intevals fo judgement. TABLE I PARAMETERS OF FITTED MODELS. Point Patten Fitted Model Paametes PPP λ = 20.306 x PHCP h c = 0.0047 (macocells) Stauss = 0.085, γ = 0.3547 Geye = 0.12,sat = 3,γ = 0.2448 y PPP λ = 37.837 (micocells) Geye = 0.05,sat = 5,γ = 1.4011 MCP λ p = 71.552,λ c = 2.641,R = 0.087 A. Macocell Point Patten Modeling The macocell point patten x consists of 77 points. Fistly, we usel-function to detemine whethe the point patten is epulsive o clusteed. As shown in Fig.2, the L() of x is unde the L() = (PPP) baseline, which indicates that macocell BSs tend to be dispesively distibuted. Secondly, given the epulsive popety, we adopt the Stauss pocess, PHCP, and Geye satuation pocess fo the following modeling and use the PPP as a benchmak fo compaison. Applying maximum pseudolikelihood method [14], we obtain these fitted models and thei coesponding paametes in Table I. In ode to test the validity of each model, we utilize the confidence inteval of evaluation statistics fo futhe veification. Explicitly, the L-function of x is pesented in Fig.3 along with the fitted PPP and Geye pocess envelope, while the PHCP and Stauss envelope ae depicted in Fig.4. We obseve that the eall() is mostly below the PPP envelope, especially in the ange of 0.05 < < 0.15 which veifies the iationality of PPP s complete andomness. Similaly, Geye pocess is unable to captue the chaacteistics of x since its L-function envelope can t always suound that of the data set. In Fig.4, the hypothesis that x is PHCP can also be ejected, but Stauss pocess gives a pefect fitting accoding to the confidence inteval. Conclusively, we can eject the PPP, PHCP and Geye pocess hypothesis by the L-function statistic evaluation, and the Stauss pocess is shown to be a suitable model fo macocell deployment. Futhemoe, Voonoi cell aea and coveage pobability distibutions ae applied to einfoce this conclusion, and the elated esults ae illustated in Fig.5 and 6. The esults show that the Stauss pocess povides an accuate modeling fo x not only in tems of classical statistic metics but also in pactical netwok pefomance metics. Given these
L() PPP envelope Geye envelope Coveage Pobability 0.2 0.4 0.6 0.8 1.0 Stauss envelope 10 5 0 5 10 15 20 SINR theshold(db) Fig. 3. Rejection of PPP and Geye pocess fo x by L-function. Fig. 6. x s SIR distibution lies within the fitted Stauss envelope. L() Stauss envelope PHCP envelope L() PPP envelope Geye envelope Fig. 4. PHCP. x s L-function lies within the fitted Stauss envelope but ejects Fig. 7. Rejection of PPP and Geye pocess fo y by L-function. P(aea<=A) 0.0 0.2 0.4 0.6 0.8 1.0 Stauss envelope L() 0.00 0.01 0.02 0.03 0.04 Voonoi cell aea(a) Fig. 5. x s Voonoi aea distibution lies within the fitted Stauss envelope. Fig. 8. y s L-function lies within the fitted Maten envelope. hypothesis testing esults above, we can claim that the Stauss point pocess gives the best fit fo macocell deployment. B. Micocell Point Patten Modeling The micocell point patten y contains 189 points, and thus it has fa moe points than x as expected. The L() cuve of y in Fig.2 indicates that micocell BSs ae aggegately deployed in this aea. Hence, in addition to using PPP fo inaccuacy veification, the Geye satuation pocess and Maten cluste pocess ae consideed cedible candidates fo accuate chaacteization. The fitted models and coesponding paametes ae listed in Table I. Fistly, we use L-function to test each hypothesis. The envelope of PPP and Geye simulations ae pesented in Fig.7. As we can see, the L() of y is totally above the PPP s envelope which veifies the inaccuacy of PPP fimly.
Fig. 10. G() 0.0 0.2 0.4 0.6 0.8 1.0 Fig. 9. Coveage Pobability 0.2 0.4 0.6 0.8 1.0 0.00 0.05 0.10 0.15 y s G() lies within Maten pocess envelope. 10 5 0 5 10 15 20 SINR theshold(db) y s SIR distibution lies within the fitted Maten envelope. The same esult applies fo Geye pocess in the ange of 0.10 < < 0.20. So we can eject these two hypotheses accoding to L() in the fist place. The hypothesis of MCP can be accepted in tem of L-function metic in Fig.8. In ode to sustain this claim, the neaest neighbo distance distibution is applied as an evaluation metic in Fig.9, indicating y s G() lies within the envelope exactly. Moeove, the coveage pobability distibution of y falls into the envelope of fitted MCP in Fig.10, which veifies the MCP s accuacy in tems of netwok pefomance metic as well. Fom these esults above, it clealy confims that MCP can epoduce y s eal deployment accuately, so we conclude that micocell BSs tend to be clusteed deployed and can be well modelled by Maten cluste pocess. V. CONCLUSIONS In this pape, we poposed a new method fo spatial modeling of BS locations and obtained accuate models fo both macocell and micocell deployment in a two-tie cellula netwok. Specifically, by applying both classical statistics and netwok pefomance as fitting metics, we found that macocell BSs ae dispesedly distibuted and can be well modeled by Stauss point pocess, while micocell BSs pesent aggegation popety and can be accuately epoduced by Maten cluste pocess. To the best of ou knowledge, it is the fist time that diffeent types of BSs (i.e. macocell o micocell) ae spatially modeled sepaately, hence we povide a new pespective in undestanding the fundamental elationship between spatial stuctue and opeational functions of BSs in heteogeneous cellula netwok. et, lage-scale veification in diffeent places is still necessay to einfoce this conclusion moe geneally. Besides, unifying these two sepaate spatial models in one integated theoetical expessions can be left as futue wok, given the potential usefulness fo moe ealistic netwok pefomance analysis. ACKNOWLEDGMENT This pape is patially suppoted by the National Basic Reseach Pogam of China (973Geen, No. 2012CB316000), the Key Poject of Chinese Ministy of Education (No. 313053), the Key Technologies R&D Pogam of China (No. 2012BAH75F01), and the gant of Investing fo the Futue pogam of Fance ANR to the CominLabs excellence laboatoy (ANR-10-LAB-07-01). REFERENCES [1] J. G. Andews, R. K. Ganti, M. Haenggi, N. Jindal, and S. Webe, A pime on spatial modeling and analysis in wieless netwoks, IEEE Commun. Mag., vol. 48, pp. 156 163, Nov. 2010. [2] M. Haenggi, J. G. Andews, F. Baccelli, O. Dousse, and M. Fanceschetti, Stochastic geomety and andom gaphs fo the analysis and design of wieless netwoks, IEEE J. Sel. Aea. Commun., vol. 27, pp. 1029 1046, Sep. 2009. [3] J. G. Andews, F. Baccelli, and R. K. Ganti, A tactable appoach to coveage and ate in cellula netwoks, IEEE Tans. Commun., vol. 59, pp. 3122 3134, Nov. 2011. [4] J. Riihijavi and P. Mahonen, Modeling spatial stuctue of wieless communication netwoks, in Poc. IEEE INFOCOM Comput. Commun. Wokshops, San Diego, CA, Ma. 2010. [5] D. B. Taylo, H. S. Dhillon, T. D. Novlan, and J. G. Andews, Paiwise inteaction pocesses fo modeling cellula netwok topology, in Poc. IEEE GLOBECOM, Anaheim, CA, Dec. 2012. [6] A. Guo and M. Haenggi, Spatial stochastic models and metics fo the stuctue of base stations in cellula netwoks, IEEE Tans. Wieless. Commun., vol. 12, pp. 5800 5812, Nov. 2013. [7] C.-H. Lee, C.-. Shih, and.-s. Chen, Stochastic geomety based models fo modeling cellula netwoks in uban aeas, Wieless Netwoks, vol. 19, pp. 1063 1072, Aug. 2013. [8] N. Deng, W. Zhou, and M. Haenggi, The Ginibe Point Pocess as a model fo wieless netwoks with epulsion, aiv pepint aiv:1401.3677, 2014. [9] L. Wu,. Zhong, and W. Zhang, Spatial statistical modeling fo heteogeneous cellula netwoks an empiical study, IEEE Vehicula Technology Confeence, Seoul, Koea, May. 2014. [10] H. S. Dhillon, R. K. Ganti, F. Baccelli, and J. G. Andews, Modeling and analysis of K-tie downlink heteogeneous cellula netwoks, IEEE J. Sel. Aea. Commun., vol. 30, pp. 550 560, Ap. 2012. [11] W. C. Cheung, T. Q. Quek, and M. Kountouis, Thoughput optimization, spectum allocation, and access contol in two-tie femtocell netwoks, IEEE J. Sel. Aea. Commun., vol. 30, pp. 561 574, Ap. 2012. [12] S. N. Chiu, D. Stoyan, W. S. Kendall, and J. Mecke, Stochastic Geomety and Its Applications. John Wiley & Sons, 2013. [13] B. D. Ripley, Statistical Infeence fo Spatial Pocesses. Cambidge univesity pess, 1991. [14] A. Baddeley and R. Tune, Pactical maximum pseudolikelihood fo spatial point pattens, Austalian & New Zealand J. Stat., vol. 42, pp. 283 322, Sep. 2000. [15] A. Baddeley and R. Tune, Spatstat: an R package fo analyzing spatial point pattens, Jounal of statistical softwae, vol. 12, no. 6, pp. 1 42, 2005.