Location-Aware Cross-Tier Coordinated Multipoint Transmission in Two-Tier Cellular Networks

Transcription

1 Locaton-Awae Coss-Te Coodnated Multpont Tansmsson n Two-Te Cellula Netwoks Ahmed Hamd Sak and Ekam Hossan axv:45.876v cs.ni] 8 Sep 4 Abstact Mult-te cellula netwoks ae consdeed as an effectve soluton to enhance the coveage and data ate offeed by cellula systems. In a mult-te netwok, hgh powe base statons (BSs) such as maco BSs ae ovelad by lowe powe small cells such as femtocells and/o pcocells. Howeve, cochannel deployment of multple tes of BSs gves se to the poblem of coss-te ntefeence that sgnfcantly mpacts the pefomance of weless netwoks. Multcell coopeaton technques, such as coodnated multpont (CoMP) tansmsson, have been poposed as a pomsng soluton to mtgate the mpact of the coss-te ntefeence n mult-te netwoks. In ths pape, we popose a novel scheme fo Locaton-Awae Coss- Te Coopeaton (LA-CTC) between BSs n dffeent tes fo downlnk CoMP tansmsson n two-te cellula netwoks. On one hand, the poposed scheme only uses CoMP tansmsson to enhance the pefomance of the uses who suffe fom hgh cosste ntefeence due to the co-channel deployment of small cells such as pcocells. On the othe hand, uses wth good sgnalto-ntefeence-plus-nose ato (SIN) condtons ae seved dectly by a sngle BS fom any of the two tes. Thus, the data exchange between the coopeatng BSs ove the backhaul netwok can be educed when compaed to the tadtonal CoMP tansmsson scheme. We use tools fom stochastc geomety to quantfy the pefomance gans obtaned by usng the poposed scheme n tems of outage pobablty, achevable data ate, and load pe BS. We compae the pefomance of the poposed scheme wth that of othe schemes n the lteatue such as the schemes whch use coopeaton to seve all uses and schemes that use ange expanson to offload uses to the small cell te. Keywods: Mult-te cellula netwoks, multcell coopeaton, CoMP, ange expanson, outage pobablty, stochastc geomety. I. INTODUCTION Deployment of mult-te netwoks s an attactve soluton to satsfy the eve-nceasng uses demand fo hghe data ates and netwok coveage. Unlke tadtonal snglete netwoks, mult-te netwoks consst of dffeent classes of base statons (BSs) such as femto base statons and pco base statons. These BSs opeate smultaneously n the same geogaphcal aea and dffe n tansmt powe, coveage ange, and spatal densty ]. Howeve, wth co-channel deployment of multple netwok tes, coss-te ntefeence degades netwok pefomance n tems of coveage and thoughput. Fo example, maco uses located n the close vcnty of a small cell may be vctmzed by tansmssons to small cell uses. The concept of coopeaton has been poposed as one soluton to addess the ntefeence poblem ], 3]. Fo A. H. Sak and E. Hossan ae wth the Depatment of Electcal and Compute Engneeng, Unvesty of Mantoba, Wnnpeg, Canada (emals: Ahmed.Sak@umantoba.ca, Ekam.Hossan@umantoba.ca). Ths wok was suppoted by a Stategc Poject Gant (STPGP 4385) fom the Natual Scences and Engneeng eseach Councl of Canada (NSEC). Maco BS DL eceve Powe Use Use 3 Pco BS Use Maco Dstance Pco + Theshold (db) Fg.. A two-te cellula netwok wth a macocell and a pcocell whee the ange of coopeaton s defned by a postve theshold. Whle each of Use and Use s seved by only one BS that esults n the maxmum eceved powe fom any of the two tes, Use 3 s connected to moe than one BS one BS fom each te that esults n the maxmum eceved powe fom that te. example, coodnated multpont (CoMP) tansmsson (also efeed as netwok MIMO) s one fom of coopeaton n whch multple BSs communcate wth each othe to cancel out the ntefeence and mpove the oveall system pefomance by jontly tansmttng the uses data concuently 4] 7]. In CoMP, BSs use backhaul lnks to exchange uses data and/o contol nfomaton whee these lnks ae capacty-lmted n pactce and affect the pefomance of the weless system 8]. Multcell coopeaton solutons such as CoMP could be effectve to mtgate the effect of coss-te ntefeence n mult-te netwoks. Fo example, n the two-te macocellpcocell netwok shown n Fg., although the powe eceved at Use 3 fom the sevng maco BS s hghe than that of the ntefeence esultng fom the closest pco BS, the ntefeence powe fom the closest pco BS can be compaable to the useful sgnal powe whch esults n a low value of sgnalto-ntefeence-plus-nose ato (SIN). Theefoe, the maco BS can coopeate wth the ntefeng pco BS to seve Use 3 jontly. Ths wll not only elmnate the stongest ntefeence sgnal, but also ncease the useful eceved sgnal powe by takng the advantage of the use s poxmty to that ntefee thus mpovng the SIN. Howeve, usng coopeaton mght be unnecessay n some cases. Fo example, n Fg., the useful sgnal powe eceved at Use and Use fom the sevng maco BS and pco BS, espectvely, s suffcently Pco

2 hghe than the powe eceved fom the stongest ntefee,.e., the pco BS fo Use and the maco BS fo Use. Fo these two uses, the gan of coopeaton may not be hgh compaed to the costs of jont pocessng and usng the backhaul netwok to exchange uses data especally when the capacty of the backhaul lnks s lmted. In ths pape, to mpove coveage and thoughput n a twote macocell-pcocell netwok, we popose a novel locatonawae coss-te coopeaton (LA-CTC) scheme n whch maco BSs and small cells can coopeate to seve a use jontly only f the use suffes fom hgh ntefeence due to the deployment of pco cells. Ths use s then efeed to be seved n CoMP mode. Othewse, f the powe eceved fom the ntefeng BS at the use s not hgh enough to cause sevee ntefeence, dect lnk tansmsson s used to seve the use wthout coopeaton and the use s efeed to be seved n non-comp mode. Note that the man focus of ths wok s on mtgatng the effect of coss-te ntefeence. As shown n Fg., we defne a egon aound the pcocell n whch a use s seved by CoMP tansmsson; othewse, the use s connected dectly to one BS,.e., maco BS o pco BS. That s, when the ato of the eceved powe fom the maco BS at any use to the eceved powe fom the pco BS exceeds a pedefned theshold (geate than ), ths mples that the useful sgnal s suffcently hghe than the ntefeence, and thus, coopeaton s unnecessay and the use (e.g. Use ) s seved by the maco BS only. On the othe hand, f the ato s less than the pedefned theshold and stll geate than, coopeaton s benefcal snce the ntefeence powe s compaable to the useful sgnal powe (e.g., fo Use 3). Fnally, f the ato s less than, the use (e.g., Use ) s dectly connected to the small cell snce the small cell s stonge than the maco cell n ths case. The man motvaton of the poposed scheme s to povde bette coveage n mult-te netwoks whle consdeng the lmtaton of the backhaul netwok. Fo example, assume a macocell-pcocell netwok whee each macocell has p andomly-located pco BSs wthn ts coveage aea. Snce coopeaton n the LA-CTC scheme s only possble between BSs belongng to dffeent tes, only p backhaul lnks pe macocell ae equed to enable coopeaton between a maco BS and pco BSs n ts coveage (a sta-connected backhaul netwok). Now consde anothe scheme whee coopeaton s also allowed between BSs belongng to the same te. In ths case both maco and pco BSs ae equed to exchange use s data n ode to pefom jont tansmsson (a fully-connected mesh backhaul netwok). Fo example, when coopeaton s lmted between pco BSs wthn the same macocell, ths needs ( p ) backhaul lnks to connect any two pco BSs. In addton, each maco BS should be able to exchange uses data wth at least ts fst q neghbos as well as the p pco BSs n ts coveage. In total, at least p +p+q backhaul lnks ae equed to enable coopeaton between the BSs. Although the latte scheme offes a bette coveage when compaed to the LA-CTC scheme, only p backhaul lnks ae needed fo the Ths theshold s efeed to as the coopeaton theshold, whch wll be defned late n Secton III.B. LA-CTC scheme. Theefoe, wth the LA-CTC scheme, thee s a sgnfcant savng n the numbe of backhaul lnks when the numbe of pco BSs pe macocell s lage. Note that othe technques, such as ange expanson (also efeed as flexble cell assocaton), have also been poposed to mpove the pefomance of mult-te netwoks and balance the load fo all tes. Fo example, n a two-te netwok wth ange expanson, uses fom the maco-te ae offloaded to the small cell te, whee the assocaton to the small cells s based. That s, a postve bas facto s added to the powe of the plot sgnals of the small cell base statons to convnce the maco uses who ae close to a small cell coveage bounday to connect to that small cell even f the powe eceved fom the maco BS s stonge than that eceved fom the small cell base staton 9], ], ]. We analyze the pefomance of the poposed LA-CTC scheme fo downlnk tansmsson n a two-te cellula netwok. We use tools fom stochastc geomety to model the netwok whee the locatons of the BSs n each te ae dstbuted accodng to a two-dmensonal ndependent homogeneous Posson Pont Pocess (PPP) ]. Each te of BSs s chaactezed by ts avalable tansmt powe, ntensty, and path-loss exponent value. In ode to evaluate the pefomance of the poposed scheme, we deve closed-fom expessons fo the outage pobablty and data ate. Futhemoe, we use ou analytcal model to deve expessons fo outage pobablty fo the ange expanson scheme, the scheme wth full coopeaton whee all uses ae seved by coss-te coodnated CoMP tansmsson, as well as the tadtonal scheme whee nethe coopeaton no ange expanson s used. The pefomances of the dffeent schemes ae compaed n tems of outage pobablty, aveage achevable data ate, and load pe BS. The esults show that the poposed coopeaton scheme outpefoms the tadtonal ange expanson scheme fo multte netwoks n tems of both outage and data ate, whle t has hghe load pe BS. Compaed to the full coopeaton scheme, the poposed scheme educes the amount of uses data exchange ove the backhaul netwok as measued by the load pe BS. In addton, the outage pefomance of the poposed scheme appoaches that wth full coopeaton fo a wde ange of values of coopeaton theshold. The contbutons of the pape can be summazed as follows: We popose a novel use-centc locaton-awae coss-te coopeaton (LA-CTC) scheme that uses CoMP tansmsson fo uses who expeence hgh levels of ntefeence powe compaed to the powe level of the useful sgnal eceved fom the sevng BS. We defne a ange of ntefeence powe based on whch the tansmsson mode (.e., CoMP o non-comp tansmsson) s decded by each use ndvdually. We use stochastc geomety to evaluate the pefomance of the poposed scheme n tems of the outage pobablty, aveage ate, and load pe BS as ou key metcs. We compae the poposed scheme wth othe schemes such as the ange expanson scheme, full coopeaton scheme, as well as a non-coopeatve scheme n whch a use s seved by the stongest BS only. The deved expessons

3 3 ae n the closed-ntegal fom. We analyze the pefomance of the dffeent schemes unde dffeent system paametes by vayng the BS ntenstes, path-loss exponents, coopeaton ange, and equed SIN thesholds. Then, we hghlght the nsghts obtaned fom the analyss and show the mpact of the afoementoned paametes on the netwok behavo. We show that the poposed LA-CTC scheme s pomsng fo mpovng the netwok outage and achevable spectal effcency whle consdeng the load of maco BSs. Futhemoe, we show that the pefomance of the LA- CTC scheme les n the mddle between the pefomance of the tadtonal ange expanson-based netwoks and full coopeaton netwoks. The est of ths pape s oganzed as follows. A evew of the elated wok s pesented n Secton II. The system model, dffeent modes of opeaton of the uses, pobablty of a use to opeate n a cetan mode, as well as the dstance analyss fo the uses n dffeent modes ae pesented n Secton III. In Secton IV, the outage pobablty and egodc ate ae obtaned fo the uses n dffeent modes. Fnally, the pefomance evaluaton esults ae pesented n Secton V and the pape s concluded n Secton VI. II. ELATED WOK Pevous woks on mult-te netwoks and multcell coopeaton can be dvded nto two geneal goups. In the fst goup, statstcal modelng technques, such as stochastc geomety, ae used to analyze netwok pefomance and obtan statstcally-optmal decson paametes ] 7]. In the second goup, nstantaneous optmal decsons ae obtaned by usng the nstantaneous nfomaton of the netwok based on some objectve functon 8] ]. Note that the statstcallyoptmal paametes mght not be optmal on a shot tmescale, howeve, obtanng nstantaneous optmal paametes costs moe sgnalng and computatons. In ] the authos povde a geneal famewok to analyze and evaluate the pefomance of a cellula netwok wth K tes of BSs. In ths model, ndependent PPPs ae used to captue the andomness of the locatons of BSs as well as the dffeences n tansmt powe, popagaton envonment, and BS spatal densty. In addton, analytcal expessons fo outage pobablty, achevable data ate, and load pe BS ae obtaned. In ], the model s extended whee the assocaton to dffeent tes s based (ange expanson). It shows that ange expanson degades the oveall netwok pefomance n tems of outage and ate. On the othe hand, n the context of multcell coopeaton, the authos n 3] popose two clusteng schemes fo CoMP tansmsson n mult-te netwoks whee clusteng s pefomed on a pe-use bass and the pefomance s evaluated n tems of outage pobablty. It s assumed that the backhaul netwok s deal and the numbe of coopeatng BSs n each cluste s constant. Whle the fst clusteng scheme foms a goup of N BSs whch esults n the ecepton of the N stongest sgnals at the eceve, the second clusteng scheme selects the N closest BSs to the eceve whee one BS s chosen fom each te. In 4], the authos popose a coopeaton scheme to mtgate the co-te ntefeence fo sngle-te netwoks n whch a use-centc decson cteon s used to decde whethe to be seved wth o wthout coopeaton. The decson s based on the dstance between the use and ts fst two neghbong BSs and some decson paametes. All BSs ae assumed to be able to exchange uses data to pefom jont tansmsson wth powe splttng. The authos use stochastc geomety to nvestgate the effect of lmted channel state nfomaton (CSI) at the tansmtte. The authos n 5] popose anothe clusteng scheme fo sngle-te netwoks whee the clustes ae fomed n a andom manne by goupng the BSs that le n the same Voono cell of an ovelayng PPP wth low ntensty. In ths pape, BSs that belong to the same cluste coopeate to nullfy the ntefeence by exchangng the CSI data. In 6], the authos use stochastc geomety to evaluate the mpact of the ovehead delay on the pefomance of CoMP tansmsson n mult-te netwoks whee wth zeo-focng beamfomng (ZFBF) as a pecodng scheme. In 7], a macocell-femtocell netwok wth sngle maco use and maco BS s consdeed whee all femto BSs ae cogntve. To mtgate the coss-te ntefeence, the maco use s assumed to geneate a busy tone such that femto BSs defe the tansmssons f the eceved powe s geate than a pedefned theshold. The authos use stochastc geomety to obtan the outage pobablty and aveage data ate. The authos n 8] deve closed-fom expessons fo the bas facto of ange expanson n a pcocell-macocell netwok fo downlnk and uplnk. Futhemoe, a coopeatve schedulng scheme between maco and pco BSs s poposed to mtgate the effect of hgh ntefeence n the expanded egons whee smulatons ae used to evaluate the netwok and the poposed scheme. In 9], a game-theoetc appoach s used to study the mpact of the backhaul constants on the pefomance of femtocell netwoks wth CoMP tansmssons. A coopeatve game s fomulated such that each femtocell chooses the coopeaton stategy and exchanges uses data to ts coopeatve patne ove ethe wed o weless backhaul. The objectve of the poposed game s to balance the tadeoff between the achevable spectal effcency and delay. The authos n ] and ] use factonal pogammng to obtan the optmal powe, channel, and pecodng coeffcents allocaton fo CoMP tansmssons n sngle-te and two-te cellula netwoks, espectvely. In both woks, the optmzaton poblem ams at maxmzng the enegy effcency (bt/joule) unde co-te o coss-te ntefeence, powe budget, and backhaul lnk capacty constants. To the best of ou knowledge, the concepts of coss-te BS coopeaton along wth locaton-awae BS coopeaton, whch ae ntoduced n ths pape, have not been exploed pevously n the lteatue. Note that, we have poposed a smla locaton-awae BS coopeaton scheme fo sngle-te cellula netwoks n ] to mtgate the effect of the co-te ntefeence. We use the tem locaton-awae n the sense that the locatons of both uses and BSs ae consdeed to make a decson on whethe coopeatve tansmsson wll be used o not.

4 4 III. SYSTEM MODEL AND ASSUMPTIONS A. Two-te Cellula Netwok Model We consde downlnk tansmsson n a two-te macocellpcocell netwok whee both tes ae ndependent wth dffeent spatal denstes, path-loss exponents, and tansmt powes. BSs belongng to the same te {, } have the same tansmt powe P. Locatons of BSs n the th te ae modeled accodng to a two-dmensonal homogeneous PPP Φ wth spatal ntensty λ. The uses ae spatally dstbuted accodng to some ndependent statonay pont pocess Φ u (e.g., a homogenous PPP) wth ntensty whch s assumed hgh enough (compaed to λ ) such that each BS has at least one use to seve. Fo statstcal analyss, wthout any loss of genealty, we consde a typcal use at the ogn ]. Dung a tansmsson nteval, a use seved by a maco BS and/o a pco BS n a patcula channel wll expeence ntefeence fom the othe maco BSs and pco BSs. Howeve, thee wll be no nta-cell ntefeence assumng that dffeent uses n a cell ae seved usng othogonal tme-fequency esouces (e.g. OFDMA). Dffeent macocells and pcocells can use the same channels (.e., a co-channel deployment scenao s consdeed). All tansmttes and eceves ae equpped wth a sngle antenna. Wthout loss of genealty, Fg. shows a ealzaton of a two-te cellula netwok whee a macocell netwok te (o maco-te) s deployed as te and ovelad wth a dense and lowe powe pcocell netwok te (o pco-te) as te. Fo a genec pont y, we defne x as the BS belongng to the th te that esults n the stongest long-tem aveage eceved powe at ths pont. That s, x = ag max x Φ {P x y α } () whee dffeent path-loss exponents {α } =, ae used fo downlnk modelng at each netwok te and denotes the Eucldean dstance. B. Mode of Opeaton and Use Assocaton: Locaton-Awae Coss-te Coopeaton Based on the eceved powe fom each te, each use ndependently chooses ts mode of opeaton though o wthout coopeaton. In ths context, we defne two modes of opeaton: non-comp and CoMP tansmsson modes. In the non-comp mode of opeaton, the use s connected to the BS that esults n the maxmum long-tem aveage eceved powe egadless of the coespondng netwok te,.e., maco-te o pco-te. In the CoMP mode of opeaton, the use s seved by two BSs that coopeate wth each othe to jontly tansmt data to ths use. In ths mode, one BS s selected fom each te based on the maxmum eceved powe at the use. That s, the uses ae splt nto thee dsjont goups: non-comp maco uses, non-comp pco uses, and CoMP uses. To elaboate, f the eceved sgnal powe fom the stongest BS at the use s suffcently hghe than that eceved fom the hghest ntefee, the use opeates n the non-comp mode snce the coopeaton between the sevng BS and ths ntefeng BS s not necessay n ths case. On the othe Y coodnates X coodnates Fg.. A two-te cellula netwok wth a maco-te (squaes) ovelad wth lowe powe and dense pcocells (ed ccles). Sold black lnes show the coveage aea of each cell fo a tadtonal two-te netwok, whle the dashed lnes show the coopeaton egons that suound each pcocell n whch coopeaton s pefomed between the maco and pco tes. hand, f the eceved sgnal powe at the use fom the stongest ntefeng BS s compaable to the useful sgnal powe eceved fom the stongest BS, the use opeates n the CoMP mode. In ths case, the netwok takes advantage of the poxmty of the ntefeng BS to the use and makes t to coopeate wth the use s sevng BS to jontly tansmt data to the use. Ths not only mtgates the effect of the hghest ntefee, but also nceases the powe level of the useful sgnal. We defne B as the set of BSs that seve a typcal use, whch can be wtten as follows: B = {x }, f P α P α {x }, f P α P α β {x, x }, f < P α P α non-comp maco non-comp pco < β CoMP whee x ( {, }) s defned n () and ( {, }) s the dstance fom the typcal use to the stongest BS n the th te. β s coopeaton theshold whch epesents the ato between the powes eceved fom the sevng maco BS and the stongest pco BS, espectvely. Ths theshold defnes the level of coss-te ntefeence beyond whch the use swtches to the CoMP mode. That s, f the stongest ntefeence powe eceved fom the pco-te P nt at some maco use s n the ange β P α < P nt < P α, ths use swtches ts mode to be seved va coopeaton. As shown n Fg., Use s eceved powe fom the maco BS s stonge than that fom the pco BS plus the theshold (db) and Use s eceved powe fom the pco BS s stonge than that fom the maco BS. Theefoe, Use and opeate n the non-comp mode whee they ae assocated ()

5 5 wth the maco BS and pco BS, espectvely. On the othe hand, although Use 3 s eceved powe fom the maco BS s hghe than that fom the pco BS, the eceved powe fom the pco BS plus the theshold (db) s hghe. Theefoe, Use 3 opeates n the CoMP tansmsson mode whee t s connected to both the BSs. Fo the poposed LA-CTC scheme, coopeaton theshold β s an mpotant desgn paamete and plays a key ole n contollng the gans obtaned by usng ths scheme. That s, the hghe the coopeaton theshold, the lage s the coopeaton egon whch mpoves the oveall system pefomance whle nceasng the amount of data exchange ove the backhaul netwok as well as the load pe BS. On the othe hand, the lowe the coopeaton theshold, the smalle s the coopeaton egon whch educes the backhaul sgnalng between BSs and the load pe BS whle sacfcng some oveall pefomance gan n tems of outage and data ate. C. Dstance Analyss Let q M, q P, and q C denote the pobablty that a typcal use s n non-comp mode and seved by the maco BS (.e., non-comp maco use), n the non-comp mode and seved by the pco BS (.e., non-comp pco use), and n the CoMP mode, espectvely. Condtoned on each event, n the followng lemma, we deve the pobablty densty functons (PDFs) of the dstance between a typcal use at the ogn and ts sevng BS(s) n the dffeent modes of opeaton. Fo a typcal use n the CoMP mode, we denote by f C () the jont PDF of the dstances between the typcal use and ts two sevng BSs x and x,.e., maco BS and pco BS, espectvely. Fo a non-comp maco use, we denote by f () the PDF of the dstance between a maco use and ts sevng maco BS x. Fnally, f () s the PDF of the dstance between a non-comp pco use and ts sevng pco BS x. Lemma. The PDFs of the dstances between a typcal use and ts sevng BS(s) ae f () = πλ ( ( ) exp π λ + λ βp α α P q α (3) M f () = πλ exp q P π ( ) (λ α α P + λ P α, (4) f C () = 4π λ λ q C exp π ( λ + λ whee and q M = πλ exp q P = πλ exp ( ) α α P P α < < π (5) ( ) α βp α P α, and ( ( ) λ + λ βp α α P α d (6) ( ) π (λ α α P + λ P α d, (7) q C = q M q P. (8) Poof: See Appendx A. Fo the specal case when α = α = α, q M and q P can be expessed n a closed-fom as q M = λ P α λ P α + λ (βp ) α, q P = λ P α. (9) λ P α + λ P α Futhemoe, t can be seen that when the coopeaton theshold β s set to (no coopeaton), the pobablty that a typcal use opeates n the CoMP mode educes to zeo,.e., q M + q P =. That s, a use assocate only wth the stongest BS n tems of eceved powe. IV. ANALYSIS OF OUTAGE POBABILITY AND AVEAGE ATE In ths secton, we chaacteze the SIN fo downlnk tansmsson to a typcal use n dffeent modes of opeaton. Then, we deve closed ntegal-foms fo the outage pobablty and egodc ate of downlnk tansmsson fo the poposed LA-CTC scheme. A. SIN Analyss Based on the mode selecton ctea n (), the eceved sgnal powe at a typcal use can be wtten as P h, Pj g j, X + Y x B x α j= x }{{} Φ j\b x α j j, +Z () }{{} useful sgnal nte-cell ntefeence whee h, and g j, ae the small-scale fadng coeffcents fo the lnks between the typcal use and the sevng and ntefeng BSs, espectvely. {h,, g j, } CN (, ) ae..d. ccula complex Gaussan andom vaables. That s, h, and g j, ae aylegh-dstbuted andom vaables, the channel powe envelope s exponentally-dstbuted as Exp(), and the phase shft n unfomly dstbuted n, π]. X and Y j, ae two zeo-mean and unty-vaance andom vaables that epesent the jontly tansmtted data by set B of the sevng BSs and the data sent by the ntefeng BSs, espectvely. Z CN (, σz) s the addtve whte nose at the eceve. No CSI s assumed at BSs and that the channel coheence tme s geate than o equal to the fame duaton. Note that, deally, the ntefeence sgnals eceved at a use ae dependent snce ntefees could be coopeatng as well. Howeve, as can be seen n (), Y,j s ae ndependent. The atonale behnd ths assumpton s as follows. Gven that two BSs (at dstance z and z fom a locaton y) ae coopeatng and ntefeng to a cetan use located at y, we know that: (a) the eceved ntefeence powe fom two coopeatng BSs s P g z α + P g z α whch has a Laplace tansfom (LT) 3 of L actual (s) = ( + (θ + θ)s) whee θ = P z α, (b) the eceved ntefeence powe s assumed to be P g z α + P g z α that has a LT 4 of L assump (s) = (+θs) (+θs), and (c) the outage 3 The eceved powe fom any two coopeatng BSs s exponentallydstbuted. Moe detals about the dstbuton ae gven n Appendx B. 4 Afte the assumpton, the eceved powe fom any two coopeatng BSs becomes a sum of two ndependent exponentally-dstbuted andom vaables wth dffeent means whch s equvalent to a hypeexponental andom vaable wth mean θ + θ and vaance θ + θ.

6 6 pobablty s a deceasng functon of the Laplace tansfom of the ntefeence (see Appendx B). Hence, L actual (s) L assump (s) and the ndependence assumpton gves a lowe bound on the outage pobablty. Thus, the eceved SIN at a typcal eceve s gven by SIN(B) = P h, x α x B j= P j x g Φ j\b j, x αj + σz. () Note that n (), the effect of the use s mode of opeaton s eflected n B. That s, dffeent modes of opeaton lead to dffeent levels of the useful sgnal powe (hghe/lowe) and aggegate ntefeence powe (lowe/hghe). To show the mpotant ole that the poposed mode of opeaton plays n mpovng the level of the eceved SIN at the typcal use, we consde the followng scenao. Consdeng the no-coopeaton case (.e., when β = ) and the cell assocaton based on the stongest sgnal powe egadless of the BS te, the sevng BS x s selected as follows: x = ag max {P x α }. () x Φ Φ In ths case, maco uses, who ae close to the boundaes of the deployed pco cells coveage, expeence hgh ntefeence, and consequently low SIN. In the poposed scheme, these uses ae lkely to change the mode of opeaton to use CoMP tansmsson nstead of sngle cell tansmsson. The poposed scheme nceases the powe level of the useful sgnal and educes the total ntefeence powe by focng the stongest ntefee to coopeate wth the ognal tansmtte. The educton n the ntefeence powe level along wth the ncease of the useful sgnal powe enhances the eceved SIN at the CoMP use. B. Outage Pobablty Usng the nstantaneous SIN gven n (), we can obtan the outage pobablty O of the oveall system. Hee, outage pobablty s defned as the pobablty that the eceved SIN s less than a pedefned theshold τ. Note that τ s a desgn paamete and t s chosen to satsfy cetan qualty-of-sevce equements of uses. We denote by O M, O P, and O C the outage pobablty of a andomly located use condtoned on ts mode of opeatons,.e., non-comp maco use, non- CoMP pco use, and CoMP use, espectvely. Fo example, the outage pobablty of a andomly located use gven that t opeates n the non-comp maco mode s obtaned by O M = E x P SIN(B = {x }) τ]]. (3) Snce the thee modes,.e., non-comp maco mode, non- CoMP pco mode, and CoMP mode, ae mutually exclusve, the oveall outage pobablty n the netwok can be obtaned by usng the law of total pobablty as follows: O = q M O M + q P O P + q C O C (4) whee q M, q P, and q C ae gven n Lemma. The followng theoem gves the outage pobabltes fo a typcal use unde dffeent modes of opeaton. Theoem. The outage pobabltes fo a typcal use gven that ths use opeates as a non-comp maco use, o as a non- CoMP pco use, o as a CoMP use ae O M = exp + O P = exp O C = exp A τσ z P α ] j= L Ij ( τ α P ) f ()d (5) ] τσ ( ) z τ P L α α Ij P f ()d (6) j= τσ z P α = L I j j= = τ P α f C ()d (7) whee A s defned n (34), f (), f () and f C () ae gven n Lemma, and ( ) ( ) τ L α Ij P = exp πλ j τ Pj α α ) j P α j F (( ] aj τ ) α j, αj L I j (s) = exp πλ j (sp j ) α j F (( sp j ) α j j, α j { β, = and j = a j =, othewse u F(y, α) = du. (8) + uα y Poof: See Appendx B. Theoem povdes geneal closed ntegal-fom expessons fo the outage pobabltes fo a andomly located use. Note that the functon F(y, α) can be evaluated numecally. Futhemoe, n some specal cases F(y, α) educes to smple closed-fom expessons (see Appendx C). The expessons n Theoem can be used to obtan the pefomances fo some specal cases by vayng β, α, and coss-ntefeence mtgaton scheme. In the followng, we ntoduce thee man schemes, namely, ange expanson (E), full coopeaton (FC), and two-te netwok wth stongest BS assocaton and no coopeaton (T) schemes, whch wll be compaed to ou poposed scheme. ) Non-coopeatve two-te cellula netwok wth ange expanson (E): E s a non-coopeatve scheme n whch the assocaton to the pco-te s based such that some maco uses ae offloaded to the stongest pco BS even though the eceved powe fom ths pco BS s less than that fom the maco BS, hence, the ange of the pcocell s expanded. To elaboate, n Fg., the coopeaton egons of ou poposed scheme become a pat of the pco BSs coveage aeas and uses n these egons become pco uses. In othe wods, t can be seen that E offloads each CoMP use to ts stongest pco BS whee these uses swtch to the non-comp mode. That s, the postve bas to the pco-te assocaton becomes β. Note that, β efes to both coopeaton theshold of LA-CTC scheme and bas facto of E scheme dependng on the context. Fo the descbed scheme, accodng to Theoem, the outage of the maco-te (as gven n (5)) emans unchanged, whee the outage of the pco-te can be obtaned as n the followng coollay.

7 7 Coollay. (ange expanson) In the specal case of a non-coopeatve two-te netwok wth a based assocaton to the pco-te, the outage pobablty of a andomly located pco use s gven by O E P whee f E = exp (x) = πλ qp E ] τσ ( ) z τ P L α α Ij P f E ()d (9) j= ( ) exp π (λ α α P + λ βp α n whch L Ij ( ) s gven n Theoem wth a j = β j and qp E = q P + q C. Poof: We follow the same poofs as n Appendx B and Appendx A, espectvely, whle eplacng P by βp. Hence, the oveall outage pobablty of E s gven by O E = q M O M + qp E OP E () whee O M s gven by (5), and qp E = q P + q C whee q P and q C ae gven n Lemma. In ths case, the closest ntefee fom the maco-te to ( ) α α P a typcal pco use s at least at a dstance of βp α nstead of. That s, the maco BS coespondng to whch the eceved powe at the pco use s the hghest s consdeed as the closest ntefee. Futhemoe, fo the pco use n the expanded pcocell coveage aea, cf. Fg., the hghest ntefeence sgnal fom the maco-te s even hghe than ts useful sgnal eceved fom the sevng pco BS. Ths means that the SIN of ths use s less than. Ths mples that the E scheme degades system pefomance compaed to ou poposed scheme. ) Fully-coopeatve two-te cellula netwok (FC): In ths scheme, any typcal use, egadless of ts locaton, connects to the stongest BS fom each te,.e., all uses opeate n the CoMP mode. The outage pobablty n ths case s povded n the followng coollay. Coollay. (Full coopeaton) In the specal case of a fully-coopeatve two-te cellula netwok, the oveall outage pobablty of the netwok s gven by O F C = exp + τσ z P α = L I j j= = τ P α f j ( j )d () whee f j ( j ) s gven n (3) and L I j ( ) s gven n Theoem. Poof: We use the fact that the BS wth the stongest eceved sgnal at the typcal use fom the th te s the neaest BS to ths typcal use among all BSs n ths te. That s, the dstance to the stongest BS s aylegh dstbuted,.e., f j ( j ) = πλ j j exp πλ j j ] and the jont PDF of the dstance s the multplcaton of the two dstbutons because of the ndependence between the two andom vaables. By pluggng the PDF of the dstance n (43) and followng the j= poof of O C n Appendx B, we obtan the esults n () whee q C = and q M = q P =. In ths case, the closet ntefees fom the maco-te and the pco-te to any use s at least at a dstance of and, espectvely. Hence, the pefomance of all uses s mpoved and the oveall outage pefomance s bette compaed to the LA-CTC scheme, howeve, ths enhancement comes at the expense of the ovehead due to data exchange between the two coopeatng BSs. 3) Intefeence-lmted tadtonal two-te cellula netwok (T): In ths case, each use assocates wth the stongest BS fom any te as defned n (). The outage pobablty can be obtaned fom Theoem as n the followng coollay. Coollay 3. (No coopeaton wth stongest BS assocaton) In the specal case of a two-te cellula netwok when each use assocates wth the BS that esults n the hghest aveage eceved powe, the total outage pobablty smplfes to O T = ( ) () + τ α F τ α, α whee the netwok opeates n the ntefeence-lmted egme and α = α = α. Poof: By usng the esults n Theoem and substtutng α = α = α, β = and σ z =, we obtan O T M = OT P (5), q C =, and q M and q P ae as gven n (9). Then, the oveall outage pobablty s obtaned as n (). In ths scheme, the closest ntefee fom the pco-te to a typcal maco use s at least at a dstance compaed to ( ) α βp α P α n the case of LA-CTC scheme. That s, the stongest ntefee s close to the use whch degades the oveall pefomance compaed to ou poposed scheme. It can be seen that, n ths case, the outage pobablty s ndependent of the BS ntensty and tansmt powe. That s because, the assocaton s based on the hghest sgnal eceved fom any BS whch means that the outage pobablty does not change when moe BSs ae deployed o the tansmt powe s nceased whle assumng the same path-loss exponent. Note that the esults pesented n Coollaes,, and 3 ae consstent wth the pevous esults n ] 3] on mult-te cellula netwoks. Futhemoe, the same esult n Coollay 3 can be obtaned fo the non-coopeatve snglete cellula case by substtutng λ = λ + λ and assumng that both tes ae dentcal n powes (P = P ) and path-loss exponents (α = α), o smply substtutng λ =. Ths esult s consstent wth the pevous esults on sngle-te netwoks n 3]. C. Aveage Egodc ate Based on the condtonal outage pobabltes defned n Theoem, we deve expessons of the egodc ates fo a typcal use when t opeates n dffeent modes. The egodc ate s measued n nats/sec/hz whee t epesents the spectal effcency of tansmsson to a use. Usng the ndependence

8 8 popety used n (4), the aveage egodc ate fo a use s gven by = q M M + q P P + q C C (3) whee M, P, and C ae the egodc ate of a typcal use gven that t opeates n the non-comp mode and seved by a maco BS, n the non-comp mode and seved by a pco BS, and n the CoMP mode, espectvely, and the assocaton pobabltes ae gven n Lemma. In the followng theoem, we deve an expesson fo the egodc ate of a andomly located CoMP use. Note that the ate of non-comp uses,.e., maco o pco uses, can be obtaned followng the same pocedue and the oveall aveage egodc ate fo a use n the netwok can be obtaned fom (3). The expessons fo the egodc ate of downlnk tansmsson fo the E, FC, and T schemes follow the same pocedue. Theoem. The egodc ate fo a typcal CoMP use s C = O C ] τ=et dt (4) whee O C s gven n (7) and ] τ=f(t) means eplacng each τ by f(t) fo some functon f : t τ. Poof: The egodc ate fo a andomly located CoMP use s defned as C = E E SIN ln( + SIN(B = {x, x })] whee the expectaton s taken wth espect to the dstance between the use and ts sevng BSs. That s, the egodc ate can be ewtten as C = E SIN ln( + SIN(B) f C ()dx A = A = A ] P ln( + SIN(B)) > t] f C ()d dt P SIN(B) > e t ] ] f C ()d dt and by usng the the defnton of O C gven n (43), we obtan the esult n (4). V. NUMEICAL ESULTS AND DISCUSSION A. Pefomance Metcs and Values of System Paametes In ths secton, we compae the poposed LA-CTC scheme wth thee schemes n the lteatue dscussed n Sectons IV-B, IV-B, and IV-B3. The fst scheme s the flexble cell assocaton, whch s efeed to as ange Expanson (E), whee ts oveall outage pobablty s defned n (). In the second scheme, efeed to as Full Coopeaton (FC), each use n the netwok s seved by two BSs. That s, each use s connected to one BS fom each te that esults n the stongest aveage eceved powe. The oveall outage pobablty fo ths scheme O F C s gven n Coollay. Fnally, the thd scheme s the tadtonal scheme (T) fo a two-te cellula netwok n whch a typcal use s seved only by the stongest BS and no basng s used (.e., β = db). The oveall outage pobablty fo ths scheme O T s gven n Coollay 3. The compason s pefomed n tems of outage pobablty, spectal effcency, and load pe BS. Whle the fst two metcs have been defned befoe, the load pe BS s defned as the aveage numbe of uses connected to a BS n any te. Usng the ndependence assumpton between pont pocesses of BSs and uses, the load pe BS fo the fou schemes can be obtaned as gven n Table I. TABLE I LOAD PE BS FO THE CONSIDEED SCHEMES: LA-CTC, ANGE EXPANSION, FULL COOPEATION, AND TADITIONAL Scheme Maco BS Pco BS LA-CTC E FC T λ (q M + q C) λ q M λ λ (q M + q C) λ (q P + q C) λ (q P + q C) λ λ q P Fo the numecal evaluaton, the tansmt powes of a maco BS and a pco BS ae assumed to be 37 dbm and dbm, espectvely, whle the themal nose powe σz s 4 dbm. Independent and dentcally dstbuted (..d.) ccula complex andom vaables wth zeo mean and unt vaance ae consdeed to smulate the channels. The maco-te has an ntensty of λ = (5 π). Unless othewse stated, the ntensty of BSs n the pco-te s 5 tmes that of the maco-te,.e., λ = 5(5 π) and the ntensty of uses = (5 π). Fo the evaluaton of outage pobablty, the theshold τ s set to db. B. Valdaton of Analyss In Fg. 3, we valdate ou analyss by compang the oveall outage pobablty (.e., CCDF of SIN at τ) fo the LA-CTC scheme obtaned fom both the analyss (4) and smulaton. Monte Calo smulatons va MATLAB ae used whee the smulaton aea s km km and the esults ae aveaged ove 6 teatons. In each ealzaton, the pefomance s evaluated fo a typcal use at the ogn whee the BSs ae deployed accodng to two ndependent PPPs. It can be seen that the analytcal esults (see the expessons gven n (4) and Theoem ) match exactly wth the smulaton esults fo all SIN thesholds whch eflects the accuacy of ou analyss. Theefoe, fom now and on, we use the analytcal expessons to evaluate the system pefomance. C. Outage Pobablty Fg. 4 shows the effect of vayng both the path-loss exponents and the BS ntensty on the oveall outage pobabltes fo the LA-CTC and E schemes. Fom ths fgue, t can be seen that the poposed LA-CTC scheme has two advantages ove the E scheme. Fstly, the oveall outage pobablty fo the LA-CTC scheme s bette than that fo E scheme fo all the dffeent values of path-loss exponents.

9 9 Oveall Outage Pobablty Analyss (Theoem ) Smulaton SIN Theshold, (db) Fg. 3. Analyss vs. smulaton: Oveall outage pobablty fo the LA- CTC scheme whee λ = (5 π), λ = 5(5 π), P = 37 dbm, P = dbm, β = 4 db, α = α = 4, and σz = 4 dbm. Outage Pobablty non-comp maco uses, LA-CTC. non-comp pco + CoMP uses, LA-CTC maco uses, E pco + offloaded uses, E 5 5 Bas Facto, (db) Fg. 5. LA-CTC vs. ange Expanson: Outage pobablty vs. the coopeaton theshold (bas facto) β whee λ = (5 π), λ = 5(5 π), P = 37 dbm, P = dbm, α = α = 4, and σz = 4 dbm. Oveall Outage Pobablty LA-CTC, = 3.5, = 3 LA-CTC, = 3.5, = 3.5 LA-CTC, = 3.5, = 4 E, = 3.5, = 3 E, = 3.5, = 3.5 E, = 3.5, = / Fg. 4. LA-CTC vs. ange Expanson: Oveall outage pobablty fo dffeent path-loss exponents vs. the ato of BS ntensty whee λ = (5 π), P = 37 dbm, P = dbm, β = 4 db, and σz = 4 dbm. Futhemoe, n some cases, e.g., when α = α, wth the E scheme, the outage pobablty deteoates wth nceasng pco BS ntensty, whle wth the LA-CTC scheme the outage pobablty mpoves unde the same condtons. The poposed scheme outpefoms the E scheme snce fo a maco use t elmnates the hghest ntefee fom the pco-te when the hghest eceved ntefeence powe s wthn a pedefned ange,.e., β P α < P nt < P α. Moeove, t uses ths ntefeng BS as a coopeaton patne along wth the ognal sevng maco BS to seve ths use. Fo the case when α = α, whle usng the E scheme, nceasng the pco BS ntensty lmts the effect of the themal nose and the netwok opeates n the ntefeencelmted egme n whch the nte-bs ntefeence domnates the pefomance. Consequently, the outage pobablty emans constant when the pco BS ntensty s hgh enough to cancel the effect of both the basng and the themal nose. On the hand, the outage pobablty fo the LA-CTC scheme s mpoved fo the same case (.e., when α = α ), because the poposed scheme mtgates the hghest ntefee fom the pco BS and also uses t as a sevng tansmtte. Fo the case when α s hghe than α, the pco BSs become moe solated fom the the maco BSs whch, n tun, educes the effect of ntefeence and mpoves the oveall outage pobablty fo the LA-CTC and E schemes. Howeve, the mpovement n outage due to the LA-CTC scheme s much hghe than that due to the E scheme because of the same eason mentoned n the pevous case. Fnally, n the case when α s less than α, the outage pefomance deteoates fo the two schemes. Howeve, the poposed LA-CTC scheme lmts the pefomance loss by usng coopeaton between the ognal sevng BS and ts hghest ntefee fom the othe te to seve the use n CoMP mode. Fg. 5 depcts the effect of nceasng the coopeaton theshold (bas facto) β on the outage pefomance of each opeaton mode fo the poposed scheme and the E scheme. Snce the outage pobablty of the offloaded uses s added to the outage of the pco uses n the E scheme, fo a fa compason, n Fg. 5 we add the outage of CoMP uses n the LA-CTC scheme to the pco uses outage as well (.e., q P O P + q C O C ). In Fg. 5, fom the pespectve of maco uses, as the coopeaton theshold (bas facto) nceases, both schemes mpove the outage pefomance compaed to the T scheme (.e., when β = db). Ths mpovement s due to offloadng maco uses wth poo SIN condtons to the pcote (n the E scheme) o to the CoMP tansmsson mode (n the LA-CTC scheme). Although offloadng uses mpoves the outage of the maco-te n the E scheme, t degades the outage of the pco-te and the oveall netwok as shown n Fg. 5. Ths degadaton n outage occus because each offloaded use connects to a pco BS that does not esult n the stongest eceved powe; hence, the use s SIN deteoates. On the othe hand, n the LA-CTC scheme, CoMP uses ae seved by both the BSs whch boosts the SIN of these uses and compensates fo the loss ncued n the E scheme. That s, the LA-CTC scheme povdes a bette outage fo the CoMP uses compaed to the offloaded uses n the E scheme whle mantanng the same maco-te pefomance. In Fg. 6, t can be seen that the oveall outage pobablty of

10 Oveall Outage Pobablty LA-CTC FC E T Oveall Egodc ate (nats/sec/hz) LA-CTC FC E T Bas Facto, (db) Fg. 6. LA-CTC vs. Tadtonal, ange Expanson, and Full Coopeaton: Oveall outage pobablty vs. the coopeaton theshold (bas facto) β whee λ = (5 π), λ = 5(5 π), P = 37 dbm, P = dbm, α = α = 4, and σz = 4 dbm Bas Facto, (db) Fg. 7. LA-CTC vs. Tadtonal, ange Expanson, and Full Coopeaton: Oveall aveage egodc ate vs. the coopeaton theshold (bas facto) β whee λ = (5 π), λ = 5(5 π), P = 37 dbm, P = dbm, α = α = 4, and σz = 4 dbm. the poposed scheme les between those of the tadtonal and the full coopeaton schemes. Futhemoe, compaed to the E scheme, the poposed scheme sgnfcantly mpoves the oveall outage pobablty of the system. As the coopeaton theshold (bas facto) nceases, moe uses ae seved va coopeaton and the pefomance of the LA-CTC scheme appoaches that of the FC scheme. When β, the gap between the two cuves esults fom the outage of the non- CoMP pco uses whch s not affected by nceasng the bas facto. On the othe hand, the gap between the pefomance of the E scheme and the T scheme nceases when the bas facto nceases. Ths s because, a hghe β causes moe uses to be offloaded to the pco-te and seved wth SIN less than db, hence, the oveall outage pobablty deteoates. That s, the LA-CTC scheme outpefoms both the T scheme and the E scheme n tems of oveall outage pobablty whle appoachng the pefomance of the full coopeaton scheme. D. Spectal Effcency In tems of the oveall aveage achevable ate, t can be seen n Fg. 7 that the LA-CTC scheme mpoves the pefomance of the netwok compaed to the T scheme as the coopeaton theshold nceases. Ths esult s consstent wth that n Fg. 5. By usng CoMP tansmsson, the poposed scheme nceases the SIN of uses who eceve hgh ntefeence fom the pco-te, by nceasng the useful sgnal powe along wth deceasng the ntefeence powe. On the othe hand, as the bas facto nceases, the oveall aveage egodc ate deteoates wth the E scheme compaed to both the T and LA-CTC schemes. Ths s also consstent wth the esults n Fg. 5 snce the offloaded uses have lowe SIN compaed to that they had befoe the offloadng. As expected, the full coopeaton scheme offes the hghest achevable data ate, howeve, the data ate offeed by the LA-CTC scheme appoaches that of the full coopeaton scheme when the value of the coopeaton theshold s hgh enough. In ode to show the mpact of usng the dffeent schemes on the ate of the legacy uses, Fg. 8 compaes the pe- fomance of the LA-CTC scheme to that of E scheme n tems of the mnmum aveage egodc ate the netwok can povde to a use by any of ts tes. The mnmum aveage use ate offeed by a cetan BS can be defned as the ato of the aveage egodc ate defned n Secton IV-C to the numbe of uses pe ths BS defned n Table I. Fo example, the mnmum aveage ate offeed by a maco BS to ts uses when adoptng the LA-CTC scheme s obtaned as q M M +q C C λ q M +q C (q M +q C ) whee the mnmum ate offeed by a pco BS s q P P +q C C λ q P +q C (q P +q C ). Hence, the mnmum aveage ate offeed by the netwok fo the LA-CTC scheme can be obtaned as { qm M + q C C λ mn (q M + q C ), q } P P + q C C λ (q P + q C ). (5) Smlaly, the mnmum aveage use ate offeed by the netwok fo the E scheme can be obtaned by as { M λ E P λ mn, q M q P + q C }. (6) Fo the T scheme, the mnmum ate s equal to that of E scheme when β goes to, whle fo the FC scheme, t s equal to that of LA-CTC when β appoaches nfnty. It can be seen n Fg. 8 that, fo the E scheme, as the bas facto nceases, the aveage use ate offeed by the netwok mpoves up to a maxmum pont. Afte ths pont, the ate offeed by pco BSs stats to lmt the netwok pefomance due to the ncease n the numbe of uses pe pco BS, hence, the mnmum ate stats to degade. Ths effect s less sevee n the LA-CTC scheme as the ncease n the numbe of uses pe pco BS due to the ncease n the coopeaton theshold s compensated by the mpovement n the oveall ate of the CoMP uses offeed by the netwok. That s, the mnmum aveage use ate emans almost constant fo hgh bas facto values. It can also be seen that the pefomance of the poposed scheme appoaches the pefomance due to full coopeaton when β s hgh enough. In addton, the mnmum ate offeed by each of the LA-CTC and E schemes s bette than that of the T scheme fo all β >.

11 .8 Mnmum Aveage Use ate (nats/sec/hz) LA-CTC FC E T Aveage BS Load (use pe BS) Macocell te, LA-CTC Pcocell te, LA-CTC Macocell te, FC Pcocell te, FC Macocell te, E Pcocell te, E Bas Facto, (db) Fg. 8. LA-CTC vs. ange Expanson: Mnmum aveage use ate fo dffeent BS ntenstes vs. the coopeaton theshold (bas facto) β whee λ = (5 π), λ = 5(5 π), = (5 π), P = 37 dbm, P = dbm, α = α = 4, and σz = 4 dbm. 5 5 Bas Facto, (db) Fg. 9. LA-CTC vs. ange Expanson and Full Coopeaton: Aveage load pe BS vs. the coopeaton theshold (bas facto) β whee λ = (5 π), λ = 5(5 π), = (5 π), P = 37 dbm, P = dbm, α = α = 3.5, and σz = 4 dbm. E. Aveage Load pe BS Fg. 9 shows the mpact of nceasng the coopeaton theshold (bas facto) on the aveage load pe BS fo the E scheme, FC scheme, as well as the LA-CTC scheme. It can be seen that, as the bas facto nceases, the E scheme educes the numbe of uses pe maco BS compaed to the T scheme wthout basng (.e, when β = db), by offloadng some of the maco uses to the pco-te based on the eceved powes at these uses. On the othe hand, the FC scheme nceases the numbe of uses pe both maco BS and pco BS compaed to the T scheme n a two-te cellula netwok snce t seves all uses by usng coopeaton between BSs n the two tes. Fnally, t can be seen that the LA-CTC scheme keeps the same numbe of uses pe maco BS whle nceasng the numbe of uses pe pco BS when compaed to the T scheme. Ths s due to the fact that the poposed scheme does not actually offload any uses to a dffeent te. Instead, t changes the mode of opeaton of uses wth bad SIN condtons whch ae now seved by the ognal maco BS along wth the stongest ntefeng pco BS. The load pe BS can eflect the amount of backhaul data exchange equed by each scheme. Fo example, none of the T and E schemes eques any uses data exchange between any two BSs ove the backhaul lnks snce all uses ae seved by a sngle BS all the tme. On the othe hand, among the fou schemes, the FC scheme eques the maxmum amount of backhaul data exchange snce t uses coopeaton to seve all uses. In ou poposed scheme, the amount of backhaul data exchange les between those of the T and E, and FC schemes. In ode to compae the FC scheme wth the poposed scheme, Fg. shows the jont PDF of the dstance of a CoMP use to sevng BSs fo both the schemes. It can be seen n Fg. a that the FC scheme seves all uses by CoMP tansmsson egadless of the locatons n the netwok. Fo the LA-CTC scheme, Fgs. b and c show the effect of nceasng the coopeaton theshold on the aea of coopeaton egon. Wth a hghe coopeaton theshold β, moe uses ae ncluded n the coopeaton egons whch, n tun, nceases the amount of uses data exchange ove the backhaul netwok. Compaed to Fg. a, t can be seen that uses wth good SIN condtons, who ae close to the sevng BS and fa fom the stongest ntefee, do not use CoMP tansmsson to save the esouces of the backhaul netwok. Fg. d shows that the effect of coss-te ntefeence deceases when the path-loss exponent of the pco-te s hghe than that of the maco-te, whch solates the pco cells. That s, CoMP tansmsson s lmted to uses who ae vey close to the pco BSs and thus the amount of equed data exchanges s educed. VI. CONCLUSION We have nvestgated the concept of coss-te coopeaton n two-te cellula netwoks. We have poposed a novel locatonawae coss-te coopeaton scheme that uses downlnk CoMP tansmsson dependng on the locatons of the uses and the neaest maco and pco BSs. Tools fom stochastc geomety have been used to analyze the outage pobablty and aveage ate fo the poposed scheme. The poposed scheme has been compaed wth thee othe schemes, namely, Tadtonal (T), ange Expanson (E), and Full Coopeaton (FC) schemes. The compason has been pefomed n tems of outage pobablty, aveage egodc ate, as well as load pe BS. The esults have shown that the poposed LA-CTC scheme outpefoms both the ange expanson and tadtonal schemes n tems of outage pobablty and aveage egodc ate. Howeve, ths pefomance gan wth the poposed scheme comes at the expense of the ovehead due to the exchange of uses data between the two dffeent BSs. In addton, the pefomance of the poposed scheme appoaches that of the FC scheme fo suffcently hgh coopeaton theshold. In ths way, the LA-CTC scheme povdes a tadeoff between the mpoved outage pobablty and the cost of coopeaton between BSs n tems of load pe BS whch eflects the amount of uses data exchange ove the backhaul netwok. As a futue extenson to ths wok, coopeaton between co-te BSs could be exploted to mtgate the effect of co-te ntefeence as

12 x -6 x (a) Case I: Full Coopeaton scheme, α = α = 4 (b) Case II: LA-CTC scheme, α = α = 4 and β = 4 db x -6 x (c) Case III: LA-CTC scheme, α = α = 4 and β = db (d) Case IV: LA-CTC scheme, α = 3.5, α = 4 and β = 4 db Fg.. LA-CTC vs. Full Coopeaton: Jont PDF of the dstance of a CoMP use to sevng BSs whee λ = (5 π), λ = 5(5 π), and P = 37 dbm, P = dbm. well. In addton, futhe wok s needed to take nto account the effect of non-deal backhaul lnks on the pefomance gan of the poposed scheme. APPENDIX A POOF OF LEMMA Fstly, we deve the pobablty fo a typcal use to opeate n a cetan mode. By defnton, ]] P q M = E P B = x ]] = E P α β, (7) P α and q P = E P B = x ]] = E P P α P α ]] <. (8) Usng (7) and (8) and followng the poof of Lemma n ] wth the pope changes, q M, q P, and q C can be obtaned. Fo the jont PDF f C () of a typcal CoMP use s dstance to the coopeatng maco BS and pco BS, we know fo sue that f the dstance to the maco BS s, the dstance to the pco BS s bounded as follows: ( P P ) α α α < < ( βp P ) α α α, (9) as can be obtaned fom () when B = {x, x }. Theefoe, the condtonal pobablty of > and > gven that the use opeates n the CoMP mode can be wtten as P >, > B = {x, x }] ( (a) = ( P P >, > max, q C P ( ( βp P >, > max, q C P (b) = ( ( ( ) P P > max, q C > ( ( βp P > max, P P ) ) α α α ) α α α α α α α α α ) f ()d (3) whee (a) follows the bound on gven n (9) and f () n (b) s the dstbuton of the dstance to the neaest pont n a homogeneous PPP Φ whch can be deved as follows: P > ] = P Thee ae no BSs n a dsc of adus ] Theefoe, = exp πλ ]. (3) f () = d d ( P > ]) = πλ exp πλ ]. (3) Afte pluggng (3) nto (3), we use the esultng cumulatve CDF (CCDF),.e., P >, > B = {x, x }],

13 3 to obtan the jont PDF f C () of and of a use who opeates n the CoMP mode as follows: f C () = { = ( P >, > B = {x, x }]) 4π λ λ q C exp π ( λ + λ, (, ) A, othewse (33) whee { A = (, ) : and ( ) α ( ) α P P α < < βp P α α α }. (34) Fo f (), we use the event of > gven that the maco use opeates n the non-comp mode,.e., B = {x }, whee the pobablty of ths event s gven by P > B = {x }] = P >, P ] α > β q M P α = ( ) ] βp α α P > α f ()d. q M > P (35) Then, we follow the same pocedue by pluggng (3) nto (35) and takng the the devatve of the CDF,.e., P > B = {x }], wth espect to. Hence, (35) educes to (3). Smlaly, we can obtan the PDF of as n (4). APPENDIX B POOF OF THEOEM Fstly, we deve the outage pobablty of a andomly located non-comp maco use. Usng the defnton of the outage pobablty n (3) fo a non-comp maco use, O M = P SIN(B = {x }) > τ] f ( )d (36) whee the SIN n () can be ewtten as n whch SIN(B = {x }) = P h, α I + σz (37) I = I + I, I = P g, x α x Φ \x I = P g, x α. (38) x Φ I ( {, }) s the total ntefeence powe eceved fom the th te and f ( ) s the PDF of dstance gven n Lemma. Afte ewtng the SIN of the non-comp maco use, we can calculate the CCDF as follows: P SIN > τ] = P h, > τ I + ] σ z P α ] (c) = exp f I ()d = E I exp (d) τσ = exp z P α τ + σ z P α τ + σ z P α ] j= ]] ( τ α L Ij P ) (39) whee (c) follows because the channel fadng powe h, Exp(), and (d) follows fom the defnton of Laplace tansfom. Wthout loss of genealty, we calculate the Laplace tansfom of I and the calculaton of the Laplace tansfom of I follows the same pocedue. L I (s) = E I exp si ]] = E Φ,{g,} exp sp (e) = E Φ x Φ \x E {g,} g, α x Φ \x exp sp g, α (f) = E Φ + sp x α Φ \x ( ) (g) = exp πλ d. + sp α > ]] (4) In the above, (e) follows because of the ndependence assumpton between g, s, (f) follows because the moment geneatng functon of an exponental andom vaable wth paamete µ s µ/(µ t), whle (g) follows the pobablty geneatng functonal of PPP. Now, let u α = (sp ) α and eplacng s wth τα P, we obtan ( α τ ) L I P = exp πλ τ α F (( τ ) α, α (4) whee F (y, α) s defned n (8). Smlaly, we can obtan the Laplace tansfom of I as ( α τ ) ( ) α ) L I P = exp πλ τ P α P α F (( ] βτ ) α, α (4) whee ( ) the closest ntefee n ths case s at least at a dstance α βp α P α nstead of whch was used to obtan the Laplace tansfom of I. By combnng (4) and (4) wth (39) and then substtutng n (36), we obtan (5). The outage pobablty of a non-comp pco use can be easly obtaned as n (6) by followng the same pocedue.

14 4 Fo a andomly located CoMP use, gven that B = {x, x }, the outage pobablty s gven by O C = P SIN(B = {x, x }) > τ] f c ()d (43) A whee f c () s the jont PDF of the dstance to the neaest two BSs (one fom each te) to the typcal use,.e., x and x, gven n Lemma. Then, we can ewte the SIN n () as whee SIN(B) = P h, α + P h, α I + σz (44) I = I + I, I = P g, x α x Φ \x I = P g, x α. (45) x Φ \x Befoe calculatng the CCDF of the SIN, we defne a new vaable θ such that θ = P x α. Then, the CCDF of the SIN can be ewtten as P SIN > τ] = P θ h, + θ h, > τ(i + σ z) ]. (46) Snce h, s ae..d. and CN (, ), we obtan ( ) θ h, + θ h, Exp = θ whch means that the CCDF of the SIN can be wtten as ]] P SIN > τ] = E I exp (g) = exp τ + σ z = θ ] τσ z = θ j= L Ij ( τ = θ ) (47) whee (g) follows the defnton of the Laplace tansfom of I j. By followng the same steps n devng (4), we have L Ij (s) = exp πλ j (sp j ) α j F (( ) α j j, α j. (48) sp j By combnng (5), (47), and (48), and then substtutng n (43), we obtan the outage pobablty of a andomly located CoMP use as gven n (7). APPENDIX C F(y, α) SPECIAL CASES The functon F, gven n (8), has a sem-open ntegal and does not gve a closed-fom soluton n geneal. Howeve, ths functon yelds a closed-fom expesson fo some values of α. Fo example, f α s a atonal numbe and can be expessed as α = n n m, n > m whee n and m ae any two postve ntege numbes, the functon F educes to F(y, α) = ( ) n α ln n y α exp ] ] πk n ] (49) α exp k= πk(α ) α whee = s the magnay unt numbe. Ths expesson educes to even moe smple expessons fo specfc values of α. Fo example, f α = 4,.e., m = and n =, F(y 4, 4) educes to F(y α, α) = actan( y). EFEENCES ] A. Damnjanovc, J. Montojo, Y. We, T. J, T. Luo, M. Vajapeyam, T. Yoo, O. Song, and D. Mallad, A suvey on 3GPP heteogeneous netwoks, IEEE Weless Commun., vol. 8, no. 3, pp.,. ] D. Gesbet, S. Hanly, H. Huang, S. Shama Shtz, O. Smeone, and W. Yu, Mult-cell MIMO coopeatve netwoks: A new look at ntefeence, IEEE J. Select. Aeas n Commun., vol. 8, no. 9, pp ,. 3] O. Smeone, O. Somekh, H. V. Poo, and S. Shama, Local base staton coopeaton va fnte-capacty lnks fo the uplnk of lnea cellula netwoks, IEEE Tans. Inf. Theoy, vol. 55, no., pp. 9 4, 9. 4] D. Lee, H. Seo, B. Cleckx, E. Hadoun, D. Mazzaese, S. Nagata, and K. Sayana, Coodnated multpont tansmsson and ecepton n LTE-Advanced: deployment scenaos and opeatonal challenges, IEEE Commun. Mag., vol. 5, no., pp ,. 5] S. Venkatesan, A. Lozano, and. Valenzuela, Netwok MIMO: Ovecomng ntecell ntefeence n ndoo weless systems, n Conf. ec. of the 4st Asloma Confeence on Sgnals, Systems and Computes, 7, pp ] G. Foschn, K. Kaakayal, and. Valenzuela, Coodnatng multple antenna cellula netwoks to acheve enomous spectal effcency, IEE Poc. Commun., vol. 53, no. 4, pp , Aug. 6. 7]. Ime, H. Doste, P. Masch, M. Gege, G. Fettwes, S. Bueck, H.-P. Maye, L. Thele, and V. Jungnckel, Coodnated multpont: Concepts, pefomance, and feld tal esults, IEEE Commun. Mag., vol. 49, no., pp.,. 8] P. Masch and G. Fettwes, On base staton coopeaton schemes fo downlnk netwok MIMO unde a constaned backhaul, n Poc. of IEEE Global Telecommuncatons Confeence (GLOBECOM), 8, pp. 6. 9] K. Okno, T. Nakayama, C. Yamazak, H. Sato, and Y. Kusano, Pco cell ange expanson wth ntefeence mtgaton towad LTE-Advanced heteogeneous netwoks, n Poc. of IEEE Int. Conf. on Communcatons Wokshops (ICC),, pp. 5. ] D. López-Péez, I. Guvenc, G. De La oche, M. Kountous, T. Q. Quek, and J. Zhang, Enhanced ntecell ntefeence coodnaton challenges n heteogeneous netwoks, IEEE Weless Commun., vol. 8, no. 3, pp. 3,. ] H. S. Dhllon,. K. Gant, F. Baccell, and J. G. Andews, Modelng and analyss of k-te downlnk heteogeneous cellula netwoks, IEEE J. Select. Aeas Commun., vol. 3, no. 3, pp ,. ] H.-S. Jo, Y. J. Sang, P. Xa, and J. Andews, Heteogeneous cellula netwoks wth flexble cell assocaton: A compehensve downlnk SIN analyss, IEEE Tans. Weless Commun., vol., no., pp ,. 3] G. Ngam, P. Mneo, and M. Haengg, Coodnated multpont n heteogeneous netwoks: A stochastc geomety appoach, IEEE GLOBE- COM Wokshop on Emegng Technologes fo LTE-Advanced and Beyond 4G (GLOBECOM-B4G 3), 3, pp ] A. Govands and F. Baccell, A stochastc geomety famewok fo analyzng pawse-coopeatve cellula netwoks, axv pepnt axv:35.654, 3. 5] S. Akoum and. Heath, Intefeence coodnaton: andom clusteng and adaptve lmted feedback, IEEE Tans. Sgnal Pocess., vol. 6, no. 7, pp , 3. 6] P. Xa, C.-H. Lu, and J. G. Andews, Downlnk coodnated multpont wth ovehead modelng n heteogeneous cellula netwoks, IEEE Tans. Weless Commun, vol., no. 8, pp , Aug. 3.

15 5 7] C. Lma, M. Benns, and M. Latva-aho, Coodnaton mechansms fo self-oganzng femtocells n two-te coexstence scenaos, IEEE Tans. Weless Commun, vol., no. 6, pp. 3, June. 8] D. Lo pez-pe ez, X. Chu, and I. Guvenc, On the expanded egon of pcocells n heteogeneous netwoks, IEEE J. Sel. Top. Sgnal Pocess., vol. 6, no. 3, pp. 8 94, Jun. 9] F. Pantsano, M. Benns, W. Saad, M. Debbah, M. Latva-aho, On the mpact of heteogeneous backhauls on coodnated multpont tansmsson n femtocell netwoks, n Poc. of IEEE Int. Conf. on Communcatons (ICC),, pp ] A. H. Sak, H. ElSawy, and E. Hossan, Locaton-awae coodnated multpont tansmsson n OFDMA netwoks, n Poc. of IEEE Int. Conf. on Communcatons (ICC), 4, pp ] A. H. Sak and E. Hossan, Enegy-effcent downlnk tansmsson n two-te netwok MIMO OFDMA netwoks, n Poc. of IEEE Int. Conf. on Communcatons (ICC), 4, pp ] F. Baccell and B. Blaszczyszyn, Stochastc Geomety and Weless Netwoks: Volume I Theoy. Now Publshes Inc,, vol.. 3] J. G. Andews, F. Baccell,. K. Gant, A tactable appoach to coveage and ate n cellula netwoks, IEEE Tans. Commun., vol. 59, no., pp , Nov.. Ahmed H. Sak (S ) s a Ph.D. canddate n the Depatment of Electcal and Compute Engneeng, Unvesty of Mantoba, Canada. He eceved the B.Sc. (-7) and M.Sc. (-) degees both n Electoncs and Communcatons Engneeng fom Tanta Unvesty, Tanta, Egypt, and EgyptJapan Unvesty of Scence and Technology (EJUST), Alexanda, Egypt, espectvely. Fo hs academc excellence, he has eceved seveal academc awads ncludng the Unvesty of Mantoba Gaduate Fellowshp (UMGF) n 4-6, the Gaduate Enhancement of T-Councl Stpends (GETS) n 3, and Egyptan Mnsty of Hghe Educaton Excellence Scholashp n -. Ahmed has been a membe n the techncal pogam commttee and a evewe n seveal IEEE jounals and confeences. Hs cuent eseach nteests nclude statstcal modelng of weless netwoks, esouce allocaton n mult-te cellula netwoks, and geen communcatons. Ekam Hossan (S 98-M -SM 6) s a Pofesso (snce Mach ) n the Depatment of Electcal and Compute Engneeng at Unvesty of Mantoba, Wnnpeg, Canada. He eceved hs Ph.D. n Electcal Engneeng fom Unvesty of Vctoa, Canada, n. D. Hossan s cuent eseach nteests nclude desgn, analyss, and optmzaton of weless/moble communcatons netwoks, cogntve ado systems, and netwok economcs. He has authoed/edted seveal books n these aeas ( hossana). D. Hossan seves as the Edto-n-Chef fo the IEEE Communcatons Suveys and Tutoals and an Edto fo IEEE Jounal on Selected Aeas n Communcatons - Cogntve ado Sees and IEEE Weless Communcatons. Also, he s a membe of the IEEE Pess Edtoal Boad. Pevously, he seved as the Aea Edto fo the IEEE Tansactons on Weless Communcatons n the aea of esouce Management and Multple Access fom 9- and an Edto fo the IEEE Tansactons on Moble Computng fom 7-. D. Hossan has won seveal eseach awads ncludng the Unvesty of Mantoba Met Awad n and 4 (fo eseach and Scholaly Actvtes), the IEEE Communcatons Socety Fed Ellesck Pze Pape Awad, and the IEEE Weless Communcatons and Netwokng Confeence (WCNC ) Best Pape Awad. He s a Dstngushed Lectue of the IEEE Communcatons Socety (-5). D. Hossan s a egsteed Pofessonal Engnee n the povnce of Mantoba, Canada.