A Tractable Framework for Coverage and Outage in Heterogeneous Cellular Networks

Transcription

1 A Tractable Framework for Coverage and Outage n Heterogeneous Cellular Networks Harpreet S. Dhllon, Radha Krshna Gant and Jeffrey G. Andrews Wreless Networkng and Communcatons Group WNCG, Department of Electrcal and Computer Engneerng, The Unversty of Texas at Austn, Unversty Staton C0803, Austn, TX. Emal: {dhllon@utexas.edu, rgant@austn.utexas.edu, jandrews@ece.utexas.edu} Abstract We develop a tractable, flexble, and accurate model for downlnk heterogeneous cellular networks. It conssts of K ters of randomly-located base statons BSs, where each ter may dffer n terms of the average transmt power, the supported data rate and the BS densty. Ths allows elements spannng tradtonal, mcro, pco, and femtocell BSs to be smultaneously consdered. Assumng a moble user connects to ts strongest BS, we derve ts Sgnal-to-Interference-Rato SIR dstrbuton and use that to fnd the coverage equvalently outage probablty over the entre network. We verfy the accuracy of these analytcal results through emprcal comparsons wth an actual 4G macrocell network. I. INTRODUCTION Cellular networks are n the process of a major transton, and wll become ncreasngly complex over the next decade due to the co-deployment n space and frequency of very dfferent classes of BSs. Tradtonal BSs wll be joned by mcro, pco, and femtocell BSs, as well as dstrbuted antennas and out-of-band relays []-[6]. These heterogeneous cellular networks HCNs have BSs that dffer by a few orders of magntude n terms of ther transmt power and hence range and the densty n whch they wll be deployed. For example, t s easy to magne a tradtonal cellular network havng perhaps 0 pco-cells and 00 low-power femtocells n each hghpower macro-cell, sharng the same lcensed spectrum. Varous plausble scenaros for HCNs are graphcally llustrated n Fgs. -4. Clearly, the coverage, rate, and relablty that moble users can acheve n such networks would seem to be qute dfferent versus the famlar one-ter cellular networks of yesterday. There are two man challenges n understandng these multter heterogeneous networks. Frst and foremost s to develop system models that capture the heterogenety of these networks wth enough accuracy to be realstc but enough smplcty to be useful. Second, s to be able to quantfy mportant performance metrcs lke outage probablty versus an arbtrary Sgnal-to-Interference-plus-Nose-Rato SINR, wth the end goal of better understandng system desgn prncples for HCNs. Solvng these two problems s the goal of ths paper. Those famlar wth cellular network analyss are lkely to mmedately recognze that these two goals are farly ambtous, snce even for tradtonal cellular networks, very smplfed models must be used to gan tractablty. For example, even to the present day the most popular analytcal model s the ncredbly smplstc Wyner model, whch assumes channel gans from all the nterferng BSs to be equal [8]. Although ths model has a mert of beng tractable, t s not accurate n most cases and does not even have a noton of outage snce SIR s determnstc [9]. The most popular but largely ntractable model s the two-dmensonal hexagonal grd model, whch s ubqutous n textbooks and research alke. Typcally, the grd model s used as the bass of system-level smulatons and analyss s not possble [0]-[2]. However, the accuracy of such models n the case of random BS locatons, such as those of femtocells, s questonable. A thrd way s to model the locatons of BSs by a pont process and then use technques wth foundatons n stochastc geometry [3]-[6] to develop a tractable analytcal model [7]. In [7], we showed perhaps surprsngly that not only does the added randomness result n a much more tractable framework, but even wth the smple Posson Pont Process PPP t s about as accurate as the grd model. The results n ths paper can be vewed as an extenson of [7] to a general downlnk mult-ter network. Ths random model for the BS placements s lkely even more sensble for a K-ter network snce the BSs n lower ters are more lkely to mmc a random spatal dstrbuton than the macro base statons. The model thus conssts of K ters of PPP dstrbuted BSs, where each ter may dffer n terms of the average transmt power, the supported data rate, and the BS densty the average number of BSs per unt area. We assume each moble connects to the strongest BS open access, or the strongest one t s authorzed to access closed access. The man result of the paper s a smple closed-form expresson for the coverage probablty, whch s bascally the probablty that a randomly located moble can acheve a target SIR β wth at least one of the ters. Ths coverage probablty can be easly vsualzed as the complementary cumulatve dstrbuton functon CCDF of the effectve receved SIR when all the ters have the same SIR threshold β. We confrm the accuracy of our results and assumptons wth smulatons aganst an actual 4G macro-cell network wth lower ters agan modeled as Posson dstrbuted.

2 II. SYSTEM MODEL A. Heterogeneous Cellular Network Model We model a HCN as a K-ter cellular network where each ter models the BSs of a partcular type, such as those of femtocells or pco-cells. The BSs across ters may dffer n terms of the transmt power, the supported data rate and ther spatal densty. We assume that the BSs n the -th ter are spatally dstrbuted as a PPP Φ of densty λ, transmt at a power P, and have a Sgnal-to-Interference-plus-Nose-Rato SINR target of β. More precsely a moble can relably communcate wth a BS x n the -th ter only f ts downlnk SINR wth respect to that BS s greater than β. Thus, each ter can be unquely defned by the tuple {P, β, λ }. The mobles are also modeled by an ndependent PPP Φ m of densty λ m. Wthout loss of generalty, we conduct analyss on a typcal moble user located at the orgn. The fadng power between a BS located at pont x and the typcal moble s denoted by h x and s assumed to be..d exponental Raylegh fadng. More complex channel dstrbutons can be consdered, at the prce of decreased tractablty, and we leave such generalzatons to future work. The standard path loss functon s gven by lx x α, where α > 2 s the path loss exponent. Hence, the receved power at a typcal moble user from a BS located at pont x belongng to th ter s P h x x α, where h x exp. The resultng SIR expresson assumng the user connects to ths BS s: SIRx P h x x α. j x Φ j\x P j h x x α the SIR of the th BS, at most m b s can be greater than /m for any postve nteger m. As a specal case of nterest, only one b can be greater than and hence each randomly chosen moble user has at most one BS wth whch t can successfully communcate. Proof: See Appendx. We wll comment on the applcablty of the derved results n the β < 0 db regme n the Numercal Results secton. Fg.. Coverage regons n two-ter network. Both macro red crcles and femto black squares BSs are dstrbuted as ndependent PPPs. Also P femto P macro/000, λ femto 5λ macro. Snce self-nterference domnates nose n all cellular networks of even modest densty, we neglect thermal nose n ths work, whch further mproves tractablty. It was shown that addng nose has no affect on these results n the regmes of most nterest n the -ter case [7], and we would expect that to hold even more strongly for a mult-ter network. We consder the maxmum SIR connectvty model, where each moble user connects to ts strongest BS,.e., the BS that offers the hghest receved SIR. Mathematcally the typcal node at the orgn s n coverage f: max SIRx > β, x Φ for some K. An mportant assumpton we make s that the thresholds β > 0 db. The followng Lemma shows that under ths assumpton, at most one BS n the entre network can provde SIR greater than the requred threshold, whch admts a smple form for the resultng coverage probablty. Although some users n commercal cellular networks ndeed have operatng SIR below 0 db, they are n a dstnct mnorty cell edge users and ndeed later we show that ths model holds very accurately at least to -4 db, whch covers even cell edge users. Lemma. Gven postve real numbers {a, a 2... a n }, whch correspond to the receved power from each BS at the typcal moble user and defnng b j a, whch corresponds to aj Fg. 2. Coverage regons n two-ter network. Macro BS locatons red crcles correspond to actual 4G deployment. Femto BSs black squares are dstrbuted as PPP. Also P femto P macro/000, λ femto 5λ macro. B. Coverage Regons Before gong nto the analyss, let us frst buld a lttle ntuton about the coverage regons n heterogeneous networks. Coverage regons are plotted n two steps. Frst, we randomly place K dfferent types of BSs on a 2-D plane accordng to the

3 aforementoned ndependent PPPs. Second, the space s fully tessellated followng the maxmum SIR connectvty model. Due to the dfferences n the transmt powers over the ters, the coverage plots do not correspond to a classcal Vorono tessellaton also called Drchlet tessellaton [8]. In fact, they closely resemble a crcular Drchlet tessellaton, whch s also called a multplcatvely weghted Vorono dagram, where the dstance between the ponts BSs s multpled by postve weghts [9]. The coverage regons for a two-ter network for example comprsng macro and femtocells are depcted n Fgs. and 2 for two cases: the macro-cell BSs are dstrbuted accordng to PPP our model, and 2 the macro-cell BSs correspond to an actual 4G deployment over a relatvely flat urban regon. The femtocells are dstrbuted accordng to an ndependent PPP n both cases. Qualtatvely, the coverage regons are qute smlar n the two cases. The random model used n ths paper could lkely be further mproved by ncorporatng a pont process that models repulson or mnmum separaton dstance, such as determnantal and Matern processes [4], [5], but we leave such extensons whch wll erode tractablty to future work. We wll provde more quanttatve support of our model n the numercal results secton. III. SIR DISTRIBUTION AND COVERAGE PROBABILITY A typcal moble user s sad to be under coverage f t s able to connect to at least one of the BSs. In the case when all the ters have same SIR threshold β >, coverage probablty can be easly vsualzed as the complementary cumulatve dstrbuton functon CCDF of the effectve receved SIR. Wth ths understandng, we now derve the probablty of coverage for a randomly located moble user both for the open and closed access strateges. Fg. 4. Coverage regons n three-ter network. Macro BS locatons red crcles correspond to actual 4G deployment. Pco green trangles and femto black squares BSs are dstrbuted as ndependent PPPs. Also P pco P macro/00, P femto P macro/000, λ pco 2λ macro, λ femto 8λ macro. Fg. 3. Coverage regons n three-ter network. All the ters,.e., macro red crcles, pco green trangles, femto black squares, are modeled as ndependent PPPs. P pco P macro/00, P femto P macro/000, λ pco 2λ macro, λ femto 8λ macro. In Fgs. 3 and 4, the coverage regons are now shown wth an addtonal pco-cell ter. As s the case n the actual networks, we assume that the macro-cells have the hghest and the femtocells have the smallest transmt powers, wth pco-cells n between. Therefore, the coverage regons for the femtocells are n general much smaller than the other two ters, partcularly when they are nearby a hgher power BS. Smlarly, we observe that the coverage footprnt of pcocells ncreases when they are farther from the macro BSs. These observatons hghlght the partcularly mportant role of smaller cells on the cell edges. A. Open Access We frst assume the open access strategy where a typcal moble user s allowed to connect to any ter wthout any restrcton. The man result for the probablty of coverage n ths setup s gven by Theorem. Theorem. The coverage probablty for a typcal randomly located moble user assumng Raylegh fadng for all the lnks s: P c {λ }, {β }, {P } Cα where Cα can be expressed as: λ P 2/α λ P 2/α β 2/α, β >, 2 Cα 22 csc 2 α. 3 α Proof: The coverage probablty n a K-ter network under maxmum SIR connectvty model s P c P SIRx > β K,x Φ

4 E SIRx > β K,x Φ a E [ SIRx > β ] x Φ b P h x lx λ P > β dx R 2 c I x [ λ E exp R β I x 2 λ R 2 L Ix P lx ] dx β dx, 4 P lx where a follows from Lemma under the assumpton that β >, b follows from Campbell Mecke Theorem and c follows from the fact that the channel gans are assumed to be Raylegh dstrbuted. Here L Ix. s the Laplace transform of the cumulatve nterference from all the ters when the randomly chosen moble user s beng served by the th ter. Snce the pont processes are statonary, the net nterference does not depend on the locaton x. Therefore, we denote L Ix by L I whch s gven by L I s E [exp si x ] E a b c j j j E Φj E Φj x j Φ j/x exp x j Φ j/x E h x j Φ j/x exp 2λ j j spj h xj lx j [ exp spj h xj lx j ] + sp j lx j 0 + sp j r α rdr e 2λsPj2/α 0 r 0 exp t+rα dt dr, where a follows from the Raylegh fadng assumpton.e., h exp, b follows by PGFL and, c results from smple algebrac manpulaton. Usng the propertes of Gamma functon, 5 can be further smplfed to L I s exp s 2/α Cα k where Cα s the Baccell s constant and λ P 2/α 5, 6 Cα 22 csc 2 α. 7 α Usng 6 and 4 the coverage probablty P c s P c {λ }, {β }, {P } λ Cα R 2 L Ix λ P 2/α β P lx λ P 2/α β 2/α dx. 8 Ths completes the proof. Settng K, leads back to the sngle-ter case and leads to the followng corollary. Corollary. In a sngle ter cellular network.e., K, P c λ, β, P can be expressed as: P c λ, β, P. 9 Cαβ2/α Ths corollary states that P c n a sngle-ter network s ndependent of the densty of the BSs and s solely dependent upon the SIR threshold value. Ths s consstent wth the fact that ncreasng the densty of the BSs decreases the dstance of the typcal moble from the BS to whch t s connected and hence ncreases the receved power. At the same tme, t decreases the dstance of the moble from the nterferes wth the same factor and hence ncreases the nterference power wth the same factor. Perhaps surprsngly, both the effects cancel each other and hence coverage probablty does not depend on the densty of BSs. Corollary 2. Assumng β β, P c {λ }, β, {P } can be expressed as: P c {λ }, β, {P }. 0 Cαβ2/α Ths s perhaps an unexpected result snce t states that the coverage probablty s not affected by the number of ters. In fact, t s exactly the same as that of the sngle-ter case. Therefore, more BSs can be added n any ter wthout affectng the coverage and hence the net network capacty can be ncreased lnearly wth the number of BSs. B. Closed Access Under closed access scheme, a moble user s allowed to connect to only a subset of ters and the rest of the ters act purely as nterferers. The man result of coverage probablty under closed access s gven by Proposton. Proposton. Assumng a moble user s allowed to connect to only a subset B of the K ters, the coverage probablty can be expressed as: P c {λ }, {β }, {P } B λ P 2/α //α Cα K λ, P 2/α Proof: The proof follows drectly from the proof of Theorem. Corollary 3. Assumng the threshold of each ter to be same and equal to β and the transmt power of each ter to be same

5 and equal to P, the coverage probablty can be expressed as: B P c {λ }, β, {P } λ Cα/α λ, 2 Above corollary states that f the thresholds and transmt powers of all the ters are same, closed access has a lower B coverage than open access by a factor of λ K. λ Coverage Probablty P c IV. NUMERICAL RESULTS Smulaton: PPP Smulaton: Actual Data Smulaton Grd Analyss SIR Threshold β expected, theoretcal and smulated results match reasonably well for β > but nterestngly, the theoretcal results also provde a tght upper bound to the exact soluton even untl β 4dB.4. Ths suggests that under the gven smulaton parameters, the lkelhood of connectng to multple BSs even through β 4dB s very low. Coverage Probablty P c db db 5 db Densty of Femto BSs λ 2 Fg. 6. Effect of changng the densty of femto-cell BSs on the coverage probablty n a two-ter HCN K 2, α 3, P 0, P 2., λ, β db. Fg. 5. Coverage probablty n a two-ter HCN K 2, α 3, P 000, P 2, λ 2 2λ, db. Whle a random PPP model s the best that can be hoped for n modelng unplanned ters, such as femtocells, ts accuracy n modelng planned BS locatons, such as those of macrocells, s open to queston. Therefore, we begn ths secton by verfyng the PPP assumpton for macro-cells from a coverage probablty perspectve. To acheve ths, we consder a two-ter network n three dfferent scenaros: the macro-cell BSs are dstrbuted accordng to PPP our model, 2 the macro-cell BSs correspond to an actual 4G deployment, and 3 macrocell BSs are dstrbuted accordng to hexagonal grd model. The second ter s modeled as an ndependent PPP n all the three cases. We compare the coverage probabltes for these scenaros n Fg. 5. We observe that the actual coverage probablty les n between the coverage probabltes acheved by the PPP and the grd model. Ths s due to the fact that the lkelhood of havng a domnant nterferer s hghest n PPP and least n grd model, wth the real scenaro beng nbetween. Ths comparson shows that the PPP assumpton s nearly as accurate as the grd model n the case of macrocells, wth PPP provdng a lower bound and grd model provdng an upper bound to the actual coverage probablty. We now valdate the man coverage probablty result, gven by Theorem, by comparng t wth the smulaton result n Fg. 5. It should be noted that P c gven by Theorem s an exact soluton to coverage probablty when β > but s an upper bound on the coverage probablty when β <. As Coverage Probablty P c db db 5 db Transmt Power P 2 Fg. 7. Effect of changng the transmt power of femto-cell BSs on the coverage probablty n a two-ter HCN K 2, α 3, P 00, λ, λ 2 50, β db. After valdatng the man result, we agan look at the coverage probablty n the context of a two-ter HCN where a femtocell network s overlad on a conventonal macro-cell network. In partcular, we study the effect of BS densty and transmt power of the femtocell ter on the overall coverage probablty n Fgs. 6 and 7, respectvely. We note that the coverage probablty has a strong dependence on the SIR threshold of the femtocell ter. In fact, coverage probablty can be theoretcally mproved f the femtocell ter threshold s lower than that of the macro-cell ter threshold β.

6 Ths s due to the fact that more BSs can now connect to the femtocell ter due to a lower threshold, hence leadng to an mproved coverage. On the other hand, the addton of femtocell ter reduces the coverage probablty f the SIR threshold of femtocell ter s hgher than that of the macro-cell ter. V. CONCLUSION In ths paper, we have developed a tractable model for K- ter downlnk HCNs. The BS locatons of each ter are modeled by an ndependent PPP. The BSs across ters may dffer n terms of the transmt power, the supported data rate and the BS densty. Whle a random model s the best that can be hoped for n modelng unplanned ters, such as femtocells, we have shown that ths model s also as good as the popular grd model for modelng the planned ters, such as macro-cells, by comparng t wth an actual 4G macro-cell deployment. Usng ths model, we have obtaned smple closed form expressons for the coverage probablty of a randomly located moble user n both the open and the closed access scenaros. Future extensons to our approach from physcal layer perspectve could be n the study of HCNs that employ multple antennas, spread spectrum, power control, nterference cancelaton or nterference algnment. From the MAC layer perspectve, t would be nterestng to nclude schedulng, resource allocaton and frequency reuse n our setup. REFERENCES [] Chh-Ln I, L.J. Greensten and R.D. Gtln, A Mcrocell/Macrocell Cellular Archtecture for Low- and Hgh-Moblty Wreless Users, IEEE J. Sel. Areas n Communcatons, vol., no. 6, pp , Aug [2] V. Chandrasekhar, J. G. Andrews, and A. Gatherer, Femtocell Networks: A Survey, IEEE Comm. Magazne, vol. 46, no. 9, pp , Sept [3] Pcochp, The Case For Home Base Statons, Whte Paper, Aprl [4] V. Chandrasekhar and J. G. Andrews, Uplnk Capacty and Interference Avodance for Two-Ter Femtocell Networks, IEEE Trans. Wreless Comm., vol. 8, no. 7, pp , July [5] J. Zhang and J. G. Andrews, Dstrbuted Antenna Systems wth Randomness, IEEE Trans. Wreless Comm., vol. 7, no. 9, pp , Sept [6] S. Kshore, Capacty and Coverage n two-ter Cellular CDMA Networks, Ph.D. Dssertaton, Prnceton Unversty, [7] Qualcomm R&D, LTE Advanced: Heterogeneous Networks, whte paper, Qualcomm Inc., [8] A. D. Wyner, Shannon-Theoretc Approach to a Gaussan Cellular Multple-Access Channel, IEEE Trans. Inform. Th., vol. 40, no., pp , Nov [9] J. Xu, J. Zhang, and J. G. Andrews, On the Accuracy of the Wyner Model n Cellular Networks, submtted. [Onlne] Avalable: [0] S. Catreux, L. J. Greensten, and P. F. Dressen, Smulaton Results for an Interference-Lmted Multple-Input Multple-Output Cellular System, IEEE Comm. Lett., vol. 4, pp , Nov [] A. Ganz, C. M. Krshna, D. Tang, and Z. J. Haas, On Optmal Desgn of Multter Wreless Cellular Systems, IEEE Comm. Magazne, vol. 35, no. 2, pp , Feb [2] E. Ekc and C. Ersoy, Mult-Ter Cellular Network Dmensonng, ACM Wreless Networks, vol. 7, no. 4, pp. 40-4, July 200. [3] F. Baccell, M. Klen, M. Lebourges and S. Zuyev, Stochastc Geometry and Archtecture of Communcaton Networks, Telecommuncaton Systems, vol. 7, no., pp , 997. [4] D. Stoyan, W. S. Kendall, and J. Mecke, Stochastc Geometry and ts Applcatons. Wley seres n probablty and mathematcal statstcs. Wley, New York, 2 nd ed., 995. [5] F. Baccell and B. Blaszczyszyn, Stochastc Geometry and Wreless Networks. NOW: Foundatons and Trends n Networkng, 200. [6] M. Haengg, J. G. Andrews, F. Baccell, O. Dousse, and M. Franceschett, Stochastc Geometry and Random Graphs for the Analyss and Desgn of Wreless Networks, IEEE J. Sel. Areas n Communcatons, vol. 27, no. 7, pp , Sept [7] J. G. Andrews, F. Baccell and R. K. Gant, A Tractable Approach to Coverage and Rate n Cellular Networks, submtted to IEEE Trans. Comm. [Onlne] Avalable: [8] F. Aurenhammer, Vorono Dagrams - A Survey of a Fundamental Geometrc Data Structure, ACM Computng Surveys, vol. 23, no. 3, pp , Sept. 99. [9] P. F. Ash and E. D. Bolker, Generalzed Drchlet Tessellatons, Geometrae Dedcata, vol. 20, no. 2, pp , 986. b APPENDIX PROOF OF LEMMA a j a j b a j a j a a + b j a j n b + b n. 3 /b + We frst prove the result for m by contradcton and then show that t can be trvally extended to the case of a general m. We frst observe that 3 s satsfed f only one of the b s s greater than. Now assume that two b s are greater than one and wthout loss of generalty, assume that they are b and b 2. Ths mples /b and /b 2 0,. Therefore, /b and + /2,. Thus, /b + n 2 /b + n /b n > + 3 /b +, /b +, 4 whch s n contradcton wth 3. Snce 3 does not even hold for two b s greater than one, t proves that the only one of the b s can be greater than one. Smlarly for the case of general m, t s easy to observe that 3 s trvally satsfed f at most m of the b s are greater than /m. Now assume that m+ b s are greater than /m and wthout loss of generalty, assume that they are b, b 2,..., b m+. Proceedng as n 4, n n /b + > + m+2 /b +, 5 whch s n contradcton to 3. Therefore, at most m b s can be greater than /m.