The SIR Meta Distribution in Poisson Cellular Networks with Base Station Cooperation

Transcription

1 The SIR Meta Dstrbuton n Posson Cellular Networs wth Base Staton Cooperaton Qme Cu, Senor Member, IEEE, Xnle Yu, Yuanje Wang, and Martn Haengg, Fellow, IEEE Abstract The meta dstrbuton provdes fne-graned nformaton on the sgnal-to-nterference rato SIR compared to the SIR dstrbuton at the typcal user. Ths paper frstly derves the meta dstrbuton of the SIR n heterogeneous cellular networs wth downln coordnated multpont transmsson/recepton CoMP, ncludng jont transmsson JT, dynamc pont blanng DPB, and dynamc pont selecton/dynamc pont blanng DPS/DPB, for the general typcal user and the worst-case user the typcal user located at the Vorono vertex n a sngle-ter networ. A more general scheme called JT-DPB, whch s the combnaton of JT and DPB, s studed. The moments of the condtonal success probablty are derved for the calculaton of the meta dstrbuton and the mean local delay. An exact analytcal resson, the beta approxmaton and smulaton results of the meta dstrbuton are provded. From the theoretcal results, we gan nsghts on the benefts of dfferent cooperaton schemes and the mpact of the number of cooperatng base statons and other networ parameters. Index Terms Base staton cooperaton, CoMP, meta dstrbuton, mean local delay, cellular networ, HetNets, Posson pont process, stochastc geometry. A. Motvaton I. INTRODUCTION Wth the ncreasng demand for data rate over cellular networs, the heterogeneous networs are deployed more and more densely []. In order to reduce the addtonal nter-cell nterference caused by networ densfcaton and heterogenety, coordnated multpont transmsson/recepton CoMP s a ey technology n cellular networs. Stochastc geometry provdes unfed mathematcal tools to model and analyze wreless networs wth randomly placed nodes, ncludng cellular networs [2]. In order to comprehensvely assess the benefts of CoMP, the meta dstrbuton of the sgnal-to-nterference rato SIR, a recently ntroduced performance metrc [3], needs to be studed. The standard mean success probablty defned as the complementary cumulatve dstrbuton functon CCDF of the SIR of the typcal user answers the questons Gven a Manuscrpt date November 3, 27. Qme Cu and Xnle Yu are wth School of Informaton and Communcaton Engneerng, Bejng Unversty of Posts and Telecommuncatons, Bejng, 876, Chna e-mal: cuqme@bupt.edu.cn, xnleyu@hotmal.com. Yuanje Wang s wth the State Key Laboratory of Ralway Traffc Control and Safety, Bejng Jaotong Unversty, Bejng, 44, Chna emal: wang.yuanje@outloo.com. Martn Haengg s wth the Dept. of Electrcal Engneerng, Unversty of Notre Dame, IN, 46556, USA e-mal: mhaengg@nd.edu. The wor was supported n part by the Natonal Nature Scence Foundaton of Chna Project under Grant 64758, n part by the Hong Kong, Macao and Tawan Scence and Technology Cooperaton Projects under Grant 26YFE229, n part by the Project of Chna under Grant B66, and n part by the U.S. Natonal Scence Foundaton under Grant CCF SIR threshold, what fracton of users n the whole networ can acheve successful transmsson on average?, whle the meta dstrbuton provdes more fne-graned nformaton for the ndvdual lns than the standard success probablty and answers more detaled questons such as What fracton of users n a cellular networ acheve 9% ln relablty gven a SIR threshold? [3]. Lettng Φ denote the pont process of base statons BSs, the meta dstrbuton s the CCDF of the condtonal success probablty P s PSIR > Φ, whch s the CCDF of the SIR of the user at the orgn gven the pont process Φ. The meta dstrbuton s formally defned by [3] as F, x F Ps, x PP s > x, x [, ]. 2 Snce t s dffcult, most lely mpossble, to calculate the meta dstrbuton drectly from the defnton n 2, we focus on the moments of P s. The b-th moment of P s s denoted by M b EP s b x b d F Ps x bx b FPs xdx, b C, 3 and the standard success probablty can be ressed as p s M,.e., the frst moment of the condtonal success probablty. The exact meta dstrbuton can then be obtaned by the Gl-Pelaez theorem [4] from the purely magnary moments M jt EP s jt, j, t R +. It s noteworthy that the meta dstrbuton s the dstrbuton of the condtonal success probablty P s, whle the standard success probablty captures only the mean of P s. B. Related Wor Due to the smple form of ts probablty generatng functonal PGFL [5, Theorem 4.9], the Posson pont process PPP s the most tractable model for the analyss of wreless networs based on stochastc geometry. Most of the pror wors focus on the standard success coverage probablty M n our notaton as the performance metrc [6] [9]. The standard success probablty s the spatal average over the channel fadng and the pont process. Ths mportant value In theory, the SIR threshold can be set arbtrarly. In practce, snce dfferent user servces have dfferent requrements of qualty of servces QoS, the operator chooses the threshold accordng to dfferent QoS requrements.

2 2 gves some basc nformaton on the SIR performance of the networ, but does not provde nformaton on the dstrbuton of the ndvdual ln relablty. In order to gan more fnegraned nformaton for each ndvdual ln n the networ, the meta dstrbuton s formally defned n [3]. The b-th moment for b C of the condtonal success probablty for both Posson bpolar networs and downln cellular networs s derved for the calculaton of the exact meta dstrbuton. Besdes, a smple approxmaton of the meta dstrbuton s obtaned by matchng the frst and second moment wth a beta dstrbuton, whch provdes an excellent match wth the meta dstrbuton. In [], the meta dstrbuton and mean local delay of the typcal devce-to-devce D2D user and cellular downln user are analyzed for the D2D communcaton underlad Posson downln cellular networs n whch the D2D users form a Posson bpolar networ. In [], the meta dstrbuton of the SIR s analyzed for both upln and downln cellular networs wth fractonal power control. [2] uses the sgnal-to-nterference-plus-nose rato SINR nstead of the SIR to analyze the meta dstrbuton of mllmeter-wave D2D networs. Due to the unque features of mllmeter-wave communcaton, a generalzed beta dstrbuton approxmaton related to the nose level s proposed to obtan a better ft for the meta dstrbuton. In the framewor of 3GPP, downln CoMP transmsson s categorzed as jont transmsson JT, dynamc pont selecton DPS, dynamc pont blanng DPB, and coordnated schedulng/beamformng CS/CB [3], [4]. These cooperaton technques have been studed n pror wor usng stochastc geometry, but not n terms of the meta dstrbuton. Wth JT, the user receves multple desred sgnals that are jontly transmtted on the same tme-frequency resource by a subset of all BSs n the networ. In [5], the authors use stochastc geometry to analyze JT for downln heterogeneous cellular networs modeled as multple ters of ndependent PPPs. The general user and the user located at the cell-corner the so-called worst-case user are consdered. In order to evaluate the performance, the standard success probablty s derved under the assumpton that the BSs have no channel state nformaton CSI. The case of full CSI s evaluated n terms of dfferent performance metrcs dversty gan and power gan. [6], [7] analyze these performance metrcs of both the general and worst-case users n heterogeneous cellular networs wth spatotemporal cooperaton technques ncludng JT, base staton slencng, and the Alamout space-tme code, and the decodng technques ncludng hybrd automatc repeat request HARQ and maxmum rato combnng MRC. [8] uses a sngle-ter Posson cellular networ model to analyze non-coherent JT n term of the SINR dstrbuton and studes the effect of mperfect CSI and ntra-cluster schedulng on non-coherent JT. Wth DPS, the user receves a desred sgnal that s transmtted by the BS wth the best nstantaneous channel condton n a cooperaton set. In [8], under the assumpton that the SINR threshold s greater than db, the standard success probablty s analyzed usng stochastc geometry for K-ter downln heterogeneous Posson cellular networs n whch the servng BS can be selected from all BSs dynamcally. Wth DPB, the BSs wth domnant nterference to the user n the cooperaton set are slenced based on the average receved power, where the average s taen over the fadng. It s noteworthy that DPB s sometmes called nter-cell nterference coordnaton ICIC. [9] models BSs as a homogeneous PPP and provdes lct ntegral ressons for the success probablty for cellular networs wth ICIC. In order to obtan hgher performance gans, DPS s usually combned wth DPB, namely DPS/DPB. Currently, there s no analyss of DPS/DPB based on stochastc geometry n Posson cellular networs. There are very few wors on the jont use of JT and DPB. It s mentoned n a 3GPP techncal report [3], but t s only stated that DPS/DPB may be combned wth JT, n whch case multple base statons can be selected for data transmsson n the tme-frequency resource. [9], [2] ntroduces a novel combned algorthm between JT and DPB ICIC, but only Monte Carlo smulaton results are provded. [2] proposes an adaptve method wth the combnaton between JT and ICIC and focuses on the jont parameter optmzaton of JT and ICIC. A rgorous analyss for ths combned scheme s stll an open ssue. C. Contrbutons In ths paper, we focus on the meta dstrbuton of the SIR n heterogeneous cellular networs wth the downln CoMP schemes ncludng JT, DPB, and DPS/DPB. A more general scheme called JT-DPB, whch s the combnaton of JT and DPB, s studed. We consder two types of typcal users the general user and the worst-case user. The general user corresponds to the general typcal user, whle the worst-case user s the typcal user located at the Vorono vertex n a sngleter networ,.e., the typcal cell-corner user. The contrbutons of the paper are: A more general scheme called JT-DPB s analyzed rgorously for the frst tme, wth JT and DPB as specal cases. We show the comparson of JT, DPB, and the combned JT-DPB scheme. Furthermore, we gve the frst rgorous analyss of DPS/DPB based on stochastc geometry. We derve the b-th moment of the condtonal success probablty for both types of users wth the combned JT- DPB scheme. For DPS/DPB, the frst moment standard success probablty for the general user and the frst and second moment for the worst-case user are derved. The mean local delay, whch s the -st moment of the condtonal success probablty, s derved for both types of users wth JT-DPB. The crtcal values of the phase transton from fnte to nfnte mean local delay are also obtaned. We calculate the exact meta dstrbuton for the worstcase user wth JT-DPB and the beta dstrbuton approxmaton of the exact meta dstrbuton for almost all schemes and user types. We show that the beta approxmaton provdes an accurate match for the exact meta dstrbuton obtaned by analyss and smulaton. By the analyss of the meta dstrbuton, we provde more fne-graned nformaton of the ln relablty for the study of CoMP.

3 3 From the theoretcal results, we gan nsghts on the benefts of dfferent cooperaton schemes and the mpact of the number of cooperatng base statons and other networ parameters. A. Cellular Networ Model II. SYSTEM MODEL We consder a K-ter PPP heterogeneous cellular networ model where the BSs belongng to -th ter are dstrbuted n R 2 accordng to a homogeneous PPP Φ wth densty λ and transmt power P,,..., K. We focus on the typcal user at the orgn, R 2. In order to study the downln CoMP, we denote the cooperaton set by C Φ, where Φ K Φ,.e., BSs from all networ ters can cooperate. The typcal user receves the same message that s transmtted synchronously by a subset of C, whch s denoted by O C. Let n denote the sze of C and m denote the sze of O, hence m n. For JT, O C,.e., m n. The receved channel output at the typcal user can be wrtten as x O P /2 νx x α/2 h x w x X+ x C c P /2 νx x α/2 h x w x X x +N, 4 where the frst sum s the desred sgnal from the cooperatng BSs n subset O, the second sum s the nterference from the non-cooperatng BSs, and N s addtve whte Gaussan nose,.e., N N C, σ 2 ; νx denotes the ndex of the networ ter of the BS located at x,.e., νx ff x Φ ; h x denotes the Raylegh fadng between the typcal user at the orgn and the BS at x, h x N C, and h x s..d.; w x s the precoder at BS x; α > 2 s the path loss onent; C c Φ\C denotes the BSs that are not n the cooperaton set; X denotes the channel nput symbol transmtted by the BSs n O, and X x denotes the channel nput symbol transmtted by the BS whch s at x C c. We assume that X and X x are..d. unform random varables wth zero mean. Snce the typcal heterogeneous cellular networs are nterference-lmted and the nose has very lmted effect on the success probablty [8], [5], [22], we focus on the nterference-lmted regme,.e., N s gnored. B. General and Worst-case Users Fg. shows an llustraton of two types of typcal users the general user and the worst-case user. For the general user, we focus on the typcal user located at the orgn n a K-ter heterogeneous cellular networ, the cooperaton set C conssts of the n BSs wth the strongest average receved power for arbtrary n Z +,.e., C arg max {x,...,x n} Φ P νx x α, n Z +. 5 Pror research has demonstrated that CoMP can mprove system spectral effcency and, n partcular, sgnfcantly enhance the cell-edge spectral effcency [3], [23]. Hence, n order to study the cell-edge performance, n addton to the general user, we consder another type of users named the worst-case user as n [5], whch s located at the Vorono Mcro BS General User a General User Pco BS Macro BS Worst-case User Macro BS b Worst-case User Fg.. Two types of typcal users general user and worst-case user. vertex n a sngle-ter networ n R 2 modeled by a homogeneous PPP Φ wth densty λ and transmt power P. A Vorono vertex s a locaton at equal dstance from three BSs. In ths case, t s natural that we restrct the sze of the cooperaton set to n {, 2, 3} snce the user has three equdstant BSs. Wthout loss of generalty, we assume a Vorono vertex to be located at,,.e., we condton on Φ havng a Vorono vertex at,, and we place a user at ths locaton. Hence the cooperaton set C of ths user s the subset of three BSs that are all closest to the orgn. Denotng the locaton of the -th closest BS to the orgn by x, the cooperaton set s wth x x 2 x 3 D. A. SIR Model C {x, x 2, x 3 }, 6 III. META DISTRIBUTION FOR JT-DPB Wth JT, all the BSs n a cooperaton set jontly transmt the same message to a target user on the same tme-frequency resource,.e., all the BSs n a cooperaton set are servng BSs for the target user,.e., m n. Wth DPB, n a cooperatng set, the user receves a message that s transmtted by the BS wth strongest average receved power and the other BSs n the cooperaton set are slenced,.e., m. Gven a user, for both CoMP schemes, the servng BSs are unquely determned by the transmt power, dstance, and m. The only dfference between JT and DPB s m. Hence, we consder a more general scheme, whch s the combnaton of JT and DPB, called JT- DPB. In ths scheme, JT s the specal case where m n, and DPB s the specal case where m. The cases where m 2, 3,..., n are combned JT-DPB schemes.

4 4 In the framewor of 3GPP LTE, JT s categorzed nto coherent and non-coherent JT [3]. The coherence 2 of JT refers to the ablty to form precoders that lot the phase and potental ampltude relatons between channels assocated wth dfferent servng BSs [28]. In the case of coherent JT t s assumed that the networ has detaled CSI of the servng lns from the BSs n the cooperaton set to the same sngle user [29]. Based on the CSI shared among all cooperatng BSs, the transmtted sgnals from dfferent BSs are jontly precoded wth pror phase algnment and tght synchronzaton across BSs to acheve coherent combnng at the served user. There s currently no support n the 3GPP LTE specfcatons for the user to report ths nd of detaled CSI for multple BSs n the cooperaton set, and thus there s currently no lct support for coherent JT. In contrast, for non-coherent JT, a BS n the cooperaton set only nows ts own CSI and cannot get the CSI from other cooperatng BSs. The transmtted sgnals from each servng BS are ndvdually precoded based on the CSI of that specfc servng ln, therefore a phase msmatch exsts between the multple useful sgnals at the served user. Non-coherent JT s supported n 3GPP LTE specfcaton Release. In ths secton, we consder a specal case of non-coherent JT, n whch the BSs have no CSI and the transmtted sgnals from each servng BS are wthout precoder,.e., w x, and blnd demodulaton s used at the user [5] [7], [3], [3]. Throughout ths secton, JT refers to ths scheme. The SIR of the general user wth JT-DPB s gven by SIR JT-DPB x O P /2 2 νx x α/2 h x g x C P νx x α h c x 2. 7 For every,..., K, let Ξ { x α /P, x Φ }. By the mappng theorem [5, Theorem 2.34] and the superposton property [5, Secton 2.5] of the PPP, Ξ K Ξ s a nonhomogeneous PPP on R + wth ntensty functon λx K λ πδp δ x δ, x R +, 8 where δ 2/α. We sort the elements of Ξ n ascendng order, such that x α / P νx x 2 α / P νx2 x 3 α / P νx3, defne x α / P νx as the -th element n the ordered set, and name the normalzed path loss. The SIR n 7 can be re-wrtten as m h x /2 2 SIR JT-DPB g, 9 g where g h x 2 and m n. n+ 2 In tradtonal communcaton theory, coherence refers to the ablty to trac the phase of the sgnal, whle n 3GPP LTE, t has a more general meanng, ncludng jont precodng [23] [27]. Smlarly, for n, 2, 3, the SIR of the worst-case user wth JT-DPB s P x α/2 2 m h x x α/2 2 h x SIR JT-DPB x O w P x α h x 2 x α g a x Φ\C 3 n+ m 2 D α/2 h x n+, D α g + x α g 4 where m n, and a follows snce the dstances between the target user and ts three nearest BSs are equal,.e., x x 2 x 3 D. B. Moments Lemma Condtonal success probablty for the general user wth JT-DPB The condtonal success probablty for the general user wth JT-DPB s gven by P s n+ + G m, where G m / m s the parallel connecton n the sense of parallel resstors of the normalzed path loss values,..., m. Proof: Accordng to the assumpton of Raylegh fadng and the ndependence of the fadng coeffcents h x, h x2,..., h xm, m /2 h x 2 s onentally dstrbuted wth mean m. Lettng I n+ g, the condtonal success probablty for the general user wth JT-DPB can be derved as P s PSIR JT-DPB g > Ξ m 2 P h x /2 > I Ξ I E Ξ E E a n+ n+ m m n+ + m m, g Ξ g Ξ where a follows snce g h x 2 are..d. onental wth unt mean. Theorem Moments of P s for the general user wth JT-DPB For every n, the b-th moment M b of the

5 5 condtonal success probablty P s for the general user n downln cellular networs wth JT-DPB s M b u n 2F b, δ; δ; m un du, δ u <u < <u n< 2 where δ 2/α, and 2 F s the Gaussan hypergeometrc functon. Proof: The jont probablty densty functon of, 2,..., n s gven by [5, Eqn. 37],.e., for < < < n, f r πδ K n λ P δ π K n λ P δ rn δ r δ. 3 Usng the PGFL of the non-homogeneous PPP Ξ, M b follows as M b E P s b E <r < <r n< r n n+ + G m b b λxdx f rdr, + x m where λx s gven n 8. Lettng L b s 4 can be wrtten as M b r n <r < <r n< L b r + sx b m r λxdx, 4 f rdr. 5 Usng the Gaussan hypergeometrc functon 2 F, L b s can be re-wrtten as L b s λxdx πδ πrn δ r n K λ P δ K λ P δ + sx b r n + sx b 2F b, δ; δ; s r n x δ dx. 6 Substtutng 6 and 3 nto 5, the b-th moment can be wrtten as n M b δq n <r < <r n< r δ qr δn2f b, δ; δ; m r n r dr, 7 where q π K λ P δ. By changng the varable of ths ntegraton,.e., lettng u qr δ, 2 follows. Lemma 2 Condtonal success probablty for the worst case user wth JT-DPB For n, 2, 3, the condtonal success probablty for the worst-case user wth JT-DPB s gven by 3n P s + G m D α α, 8 + G m x where G m md α. 4 Proof: Accordng to the assumpton of ndependent Raylegh fadng, m Dα/2 h x 2 s onentally dstrbuted wth mean md α. The condtonal success probablty of the worst-case user s derved as P s PSIR JT-DPB w > Φ m 2 P D α/2 h x > I Φ E E 3 E a n+ + m I md α 3 n+ Φ g m 3n 4 D α g + x α g 4 md α 4 + md x α α, Φ x α g md α Φ where a follows snce g h x 2 s onentally dstrbuted wth unt mean. Theorem 2 Moments of P s for the worst-case user wth JT-DPB For n, 2, 3, the b-th moment M b of the condtonal success probablty P s for the worst-case user n downln cellular networs wth JT-DPB s gven by M b u + m n3b u 2 F b, δ; δ; m where δ 2/α. When b N, 9 can be smplfed to du, 9 + n3b m M b 2F b, δ; δ; m 2. 2 Proof: The probablty densty functon of the common dstance D s [5, Eqn. 4] f D r 2π 2 λ 2 r 3 e λπr2, r. 2

6 6 The b-th moments of P s are derved as M b E P s b E + Gm D α n3b a E D + n3b m r D n3b + m + b 4 + G m x α + md x α b α mr x α b α n3b 2πλxdx 2π 2 λ 2 r 3 e πλr2 + m πλr F b, δ; δ; m n3b 2π 2 λ 2 r 3 + m πλr 2 2F b, δ; δ; m b 2πλxdx f D rdr dr dr, 22 where a follows from the PGFL of the homogeneous PPP Φ and b s derved by usng the Gaussan hypergeometrc functon 2 F. By the substtuton u πλr 2, the result n 9 s obtaned. If b N, then >, 2 F b, δ; δ; /m >. By usng ntegraton by parts, 9 can be wrtten as + n3b m M b 2F b, δ; δ; m 2. M Varance n n2 n3 n, worst n2, worst n3, worst a M db n n2 n3 n, worst n2, worst n3, worst db b Varance M 2 M 2 Fg. 2. M and the varance.e. M 2 M 2 of Ps for non-coherent JT where n m, 2, 3, and α 4. Remar : Notce that 2 s the jont probablty of b successful transmssons. Remar 2: It s noteworthy n 2 and 9 that M b for both the general user and the worst-case user wth JT-DPB are ndependent of the transmt power P and the densty λ of ter n nterference-lmted heterogeneous networs. For M, ths observaton was made n [5, Remar ] for JT and [8, Eqn. 3] for non-comp. Fg. 2 and Fg. 3 show the standard success probablty p s M and the varance of the condtonal success probablty for both types of users for non-coherent JT and DPB respectvely. M shown n Fg. 2a has been derved n [5], and M for the general user wth DPB shown n Fg. 3a corresponds to the result n [9]. Moreover, Fg. 4 compares these results of the combned scheme JT-DPB < m < n, non-coherent JT m n, and DPB m. Remar 3: It s remarable that the maxmum varance ncreases when n ncreases, as shown n Fg. 2b, Fg. 3b, and Fg. 4b. C. Mean Local Delay M For a certan wreless ln, the local delay, denoted by L, s the number of transmsson attempts untl the frst success f the transmtter s allowed to eep transmttng pacets [32]. Thus, the mean local delay can be ressed as EL EEL Φ, where the nner ectaton s over the fadng and the outer ectaton s over the pont process Φ. Under the assumpton that every transmsson success event s condtonally ndependent gven Φ 3, L s geometrcally dstrbuted wth condtonal success probablty P s condtoned on Φ,.e., PL Φ P s P s,, 2, 3, Hence, we have EL EEL Φ E M, 24 P s.e., the mean local delay s the -st moment of the condtonal success probablty, whch s denoted by M. In 3 The condtonal ndependence follows from the ndependence of the fadng random varables from one transmsson to the next.

7 7.8.6 n n2 n3 n, worst n2, worst n3, worst.8.6 n3, m3 n3, m2 n3, m n2, m2 n3, m3, worst n3, m2, worst n3, m, worst n2, m2, worst M M db a M db a M Varance n n2 n3 n, worst n2, worst n3, worst Varance n3, m3 n3, m2 n3, m n2, m2 n3, m3, worst n3, m2, worst n3, m, worst n2, m2, worst db b Varance M 2 M 2 Fg. 3. M and the varance.e. M 2 M 2 of Ps for DPB where m, n, 2, 3, and α db b Varance M 2 M 2 Fg. 4. Comparson of M and the varance.e. M 2 M 2 of Ps for non-coherent JT m n, DPB m, and the combned scheme JT-DPB < m < n, where α 4. downln Posson cellular networs, the mean local delay shows a phase transton at the crtcal value crtcal, whch means that as the SIR threshold reaches crtcal, the mean local delay wll jump from beng fnte to nfnte [3], [], [33]. An nfnte mean local delay means that the fracton of lns sufferng from hgh delays s non-neglgble. Put dfferently, the dstrbuton of the local delay has a heavy tal. Corollary Mean local delay for the general user wth JT-DPB The mean local delay M for the general user wth JT-DPB s gven by M <u < <u n< u n + where crtcal /δ m. u n /δ m u du, n/u /δ < crtcal, 25 M Proof: By substtutng b n 2, M s gven by <u < <u n< a <u < <u n< u n 2 F u n +, δ; δ; u n /δ m un u /δ m du un u δ du, where a follows snce 2 F, a; c; z az/c. In fact, < crtcal s the convergence condton of the mproper ntegral resson M. For the general user, f the mproper ntegral resson M s convergent, then for every u, u 2,..., u n {u, u 2,..., u n R n : < u < u 2 < < u n },

8 8 we have and thus u n + u n /δ m <, un /δ u m un /δ > u /δ. Snce < u < u 2 < < u n and m n, we have m un /δ nf m. u Hence m > /δ. Consequently, crtcal /δ m. Remar 4: For n m,.e., the message s transmtted by only one BS wthout cooperaton, 25 can be smplfed to the closed-form resson M u + u /δ du δ, < /δ, 26 δ + whch s n lne wth the result n [3, Eqn. 24]. Corollary 2 Mean local delay for the worst-case user wth JT-DPB For the worst-case user, the mean local delay M for n, 2, 3 wth JT-DPB s gven by + 3n m M 2, < crtcal, 27 /δm where crtcal /δ m. Proof: By substtutng b n 9 and usng 2F, a; c; z az/c, M s gven by M u + 3n u m + 3n m 2, < /δ m /δm, /δ m, du /δ m and the crtcal value of s crtcal /δ m. Fg. 5 shows the mean local delay for non-coherent JT and DPB for the general user and the worst-case user wth n, 2, 3 respectvely, and the crtcal value of phase transton can be observed. For both general and worst-case users wth non-coherent JT, the phase transtons occur at, 2, 3 not n db for n, 2, 3 respectvely. In contrast, for DPB, the phase transton occurs at not n db for all users and all n. Remar 5: The mean local delay of the worst-case user s larger than that of the general user. As the sze of cooperaton set n ncreases, the mean local delay of the worst-case user and the general user get closer, as shown n Fg. 5. Remar 6: Interestngly, the crtcal value does not depend on the number of slenced BSs,.e., only the number of jontly transmttng BSs m matters, and crtcal ncreases lnearly wth m. M - M n n2 n3 n, worst n2, worst n3, worst n a non-coherent JT n n2 n3 n, worst n2, worst n3, worst n2 db b DPB n,2,3 n,2,3 db n3 Fg. 5. The mean local delay M for the general and worst-case users for non-coherent JT and DPB, where n, 2, 3, and α 4. The phase transtons can be observed n the fgures. D. Meta Dstrbuton and ts Beta Approxmaton As mentoned earler, the exact meta dstrbuton can be obtaned by the Gl-Pelaez theorem [4] from the purely magnary moments M jt EP s jt, j, t R +, as [3] F Ps x 2 + Im [ ] e jt log x M jt dt. 28 π t Ths formula together wth the moments n Theorem and Theorem 2 show that the entre meta dstrbuton for JT-DPB does not depend on the transmt powers and denstes of each ter. The numercal calculaton of the exact meta dstrbuton accordng to 28 s tedous. Alternatvely, t s often suffcent to approxmate the meta dstrbuton by matchng ts frst and second moment to the beta dstrbuton, resultng n an excellent match. For cellular networs wthout CoMP, the excellent match between the meta dstrbuton and the beta dstrbuton has been confrmed n [3]. The beta dstrbuton s a two-parameter contnuous dstrbuton supported on [, ] and thus a natural canddate to approxmate the dstrbuton of

9 9 a condtonal probablty. Its CDF s gven by the regularzed ncomplete beta functon wth shape parameters a, b >,.e., x I x a, b ta t b dt, 29 Ba, b where Ba, b s the beta functon. By matchng frst and second moments wth M and M 2, we obtan the approxmated meta dstrbuton F Ps x I x M β M, β, 3 where β M M 2 M M 2 M 2. 3 Furthermore, system-level smulatons are carred out for comparson. Throughout the smulatons n ths secton, snce the entre meta dstrbuton for JT-DPB s ndependent of the number of networ ters and ther respectve transmt powers and denstes accordng to Theorem and Theorem 2, wthout loss of generalty, we focus on the case of a sngleter networ, and P and λ can be set arbtrarly. Specfcally, we produce PPP realzatons, and then produce realzatons of the Raylegh fadng random varables for each PPP realzaton. Next, we calculate the SIR of the target users and obtan the smulaton results of the success probablty, varance, and meta dstrbuton over the data ponts. The smulaton parameters are: sze of cooperaton set n, 2, 3, path loss onent α 4, transmt power P, densty λ, and smulaton regon [3, 3] 2. In terms of the average number of BSs, there are about 36 BSs n a smulaton regon, whch s certanly suffcent. Fg. 6a shows the meta dstrbuton wth non-coherent JT. For the general user, we confrm the accuracy of the beta dstrbuton approxmatons by comparng the approxmatons wth smulaton results. For the worst-case user, the exact meta dstrbuton can be calculated by usng the Gl-Pelaez theorem. We calculate the exact meta dstrbuton compared wth ts beta dstrbuton approxmaton. Fg. 6b also shows these results for DPB. Fg. 7 compares the beta dstrbuton approxmaton among the combned scheme JT-DPB < m < n, noncoherent JT m n, and DPB m. Remar 7: Non-coherent JT wth n 2 and DPB wth n 3 have smlar performance n terms of M and the meta dstrbuton, as shown n Fg. 4a and Fg. 7. Remar 8: The combned JT-DPB scheme provdes a tradeoff between JT and DPB. For a gven n, ts performance and cost are n between JT and DPB. A. SIR Model IV. META DISTRIBUTION FOR DPS/DPB In ths secton, we study the downln DPS/DPB scheme, whch s a combnaton of DPS and DPB. In ths scheme, there s only one servng BS n the cooperaton set,.e., the sze of set O s m. We agan consder two types of users the general user and the worst-case user. For DPS/DPB, the user receves a message that s transmtted by the BS wth the best nstantaneous channel condton -F p x -F p x x a non-coherent JT n n2 n3 n, worst n2, worst n3, worst x b DPB n n2 n3 n, worst n2, worst n3, worst Fg. 6. Exact meta dstrbuton, ts beta dstrbuton approxmaton, and smulated results where n, 2, 3, α 4, and db. The curves are the beta dstrbuton approxmaton, the round marers are the smulated results, and the trangle marers are the exact meta dstrbuton from 28. -F p x n3, m3 n3, m2 n3, m n2, m2 n3, m3, worst n3, m2, worst n3, m, worst n2, m2, worst Fg. 7. Comparson of the beta dstrbuton approxmaton of the meta dstrbuton for non-coherent JT m n, DPB m, and the combned scheme JT-DPB < m < n, where α 4, and db. x

10 n the cooperaton set, and the other BSs n the cooperaton set are slenced. The servng BS for a certan user s determned by not only the normalzed path loss transmt power and dstance but also the fadng. Hence the servng BS s chosen dynamcally, and the desred sgnal s nstantaneously the best, whch s denoted by S g max h x 2 and,...,n S w max h x 2 D α for the general user and the worstcase user, respectvely.,...,n Smlar to JT-DPB, the SIR of the general user wth DPS/DPB s gven by SIR DPS/DPB g max h x 2,...,n n+ g, 32 and the SIR of the worst-case user wth DPS/DPB for n, 2, 3 s gven by B. Moments max h SIR DPS/DPB x 2 D α,...,n w x α g n+ 3 n+ max h x 2 D α,...,n. 33 D α g + x α g 4 Lemma 3 Condtonal success probablty for the general user wth DPS/DPB The condtonal success probablty for the general user wth DPS/DPB s gven by P s +,..., p n + +, 34 where {,..., n }, and the notaton above the sum means j, j, and p n x n+ + x. Proof: The condtonal success probablty can be ressed as P s P SIR DPS/DPB g > Ξ P max h x 2,...,n > I Ξ, 35 whch s the CCDF of the extreme value of random varables h x 2 gven the pont process Ξ, where I n+ g. The CDF of Y max h x 2 can be derved as,...,n P Y y Ξ P h x 2 y, h x2 2 2 y,..., h xn 2 n y Ξ a n P h x 2 y Ξ, 36 where a follows snce h x 2 are..d. Hence 35 can be re-wrtten as P s P Y > I Ξ n P h x 2 I E n n E n+ Ξ I Ξ g } {{ } A g Lettng a n+ 37 can be anded as A n a + Ξ. 37, the product term A n a,...,a a a a 2 a, where a {a, a 2,..., a n }, then 37 can be ressed as n P s E a Ξ E + + a a 2 a Ξ a,...,a a a,...,a a E a a 2 a Ξ }{{} B The term B n 38 can be ressed as B Ea a 2 a Ξ a,...,a a,..., E n+ a n+,..., E g 2 n+,..., n+ g n+ g g , Ξ Ξ where {,..., n }, and a follows snce g h x 2 s onentally dstrbuted wth unt mean. The fnal result n 34 s obtaned after some smplfcaton. Theorem 3 Frst Moment of P s for the general user wth DPS/DPB The frst moment M of the condtonal success probablty P s,.e., the standard success proba-

11 blty, for the general user n downln cellular networs wth DPS/DPB s gven by M + u,...,u u m n where u {u,..., u n }, and m n x <u < <u n< u n 2 F, δ; δ; u /δ + + u /δ, u /δ n /x Proof: In accordance wth Lemma 3, M follows as M EP s n E + + <r < <r n<,..., p n,..., E du. pn The term E p n x s smlar wth 4 n the proof of Theorem, and can be ressed as Ep n x E + n+ x a λtdt f rdr b <r < <r n< δq n r n n r δ + t rx qr δn 2F, δ; δ; r n rx dr, 4 where a follows from the PGFL of the non-homogeneous PPP Ξ; f r s the PDF of gven n 3; λt s the ntensty of the non-homogeneous PPP Ξ gven n 8; b s derved by usng the Gaussan hypergeometrc functon 2 F and q π K λ P δ. By substtutng 4 nto 4 and changng the ntegral varable u qr δ, the result n 39 s obtaned after some smplfcaton. Lemma 4 Condtonal success probablty for the worst case user wth DPS/DPB For n, 2, 3, the condtonal success probablty for the worst-case user wth DPS/DPB s gven by P s n + + n x α D α 42 Proof: Smlar to 36, the CDF of Y max,...,n Dα h x 2 can be ressed as PY y Φ n P h x 2 y D α Φ. Then the condtonal success probablty can be derved as P s PY > I Φ a E I E 3 n+ D α g x α 4 D α g n P h x 2 I D α Φ n Φ n Φ b n n 3 E g n+ x α Φ c + 4 D α g 3n n + + 4, + x α D α where a follows snce h x 2 s ndependently onentally dstrbuted wth unt mean; b follows from the bnomal theorem a + b n n n a n b ; and c follows snce g h x 2 s ndependently onentally dstrbuted wth unt mean. After some smplfcaton the result n 42 s obtaned. Theorem 4 Frst and second moments of P s for the worst-case user wth DPS/DPB For n, 2, 3, the frst moment M of the condtonal success probablty P s,.e., the standard success probablty, for the worst-case user n downln cellular networs wth DPS/DPB s gven by M n3 n + + 2, 43 2F, δ; δ; and the second moment M 2 of P s s gven by F + 4, n 4F M F Q3, , n 2 9F + 9F 2 + F 3 8Q3, Q4, 3 2 6Q5, 6 2, n 3, 44 where Qu, v F x 2 F 2, δ; δ; x 2, r 2πλx + ur α x α + vr 2α x 2α dx f D rdr.

12 2 Proof: We frst ress the frst moment of P s as M EP s n + + n3 E. 4 + x α D }{{ α } 45 We notce that the term A n 45 has almost the same form as the resson n the proof of Theorem 2. Smlarly, we can get the results n 43. Then for M 2, we tae n 2 as an example, and the dervaton of M 2 wth n, 3 s smlar. When n 2, the condtonal success probablty P s n 42 can be ressed as P s x α D α and we obtan M 2 n the form M 2 EP s E + x α E E D α 2 A x α 4 D α x α D α, + 2 x α D α x 2α D 2α F 2, δ; δ; F 2, δ; δ; πλx r + 3r α x α r 2α x 2α dx f D rdr, where f D r s gven n 2. After some smplfcaton 44 s obtaned. The standard success probablty p s M and the varance of the condtonal success probablty are shown n Fg. 8. C. Meta Dstrbuton and ts Beta Approxmaton We follow the same analyss methods as for JT-DPB. Snce the SIR dstrbuton only depends on the ntensty functon λx n 8, we agan conclude that the meta dstrbuton for DPS/DPB s ndependent of the number of networ ters and ther respectve denstes and transmt powers. Hence, for the convenence of comparson, the smulaton setups n ths secton are the same as n Sec. III. For DPS/DPB, Fg. 9 shows the beta approxmaton for the worst-case user and the smulaton results for both general and worst-case users. Due to the complexty of the hgherorder moments n ths scheme, the beta approxmaton for the general user s too unweldy to gan drect nsght, let alone the exact meta dstrbuton for both types of users. As shown n Fg. 9, for the worst-case user wth DPS/DPB, the accuracy M Varance n Sm. n2 Sm. n3 Sm. n, worst n2, worst n3, worst a M db n n2 n3 n, worst n2, worst n3, worst db b Varance M 2 M 2 Fg. 8. M and the varance.e. M 2 M 2 of Ps for DPS/DPB where n, 2, 3, and α 4. The varance for the general user s obtaned by smulaton. -F p x x n Sm. n2 Sm. n3 Sm. n, worst n2, worst n3, worst Fg. 9. Beta dstrbuton approxmaton and smulaton result for DPS/DPB where n, 2, 3, α 4, and db. The dashed curves are the beta dstrbuton approxmaton, and the sold curves and the round marers are the smulated results.

13 3 M DPS/DPB DPS/DPB, worst Non-coh. JT Non-coh. JT, worst more fne-graned nformaton. As shown n Fg. b, the meta dstrbuton F Ps for non-coherent JT and DPS/DPB s dfferent. There are 6% users n a non-coherent JT networ achevng 9% ln relablty for db, but 5% users n a DPS/DPB networ. Hence, although M s approxmately equal, DPS/DPB has more hghly relable lns than noncoherent JT. The dfference s even more promnent for the worst-case user when 3 db. -F p x a M Non-coh. JT, worst, -3 db DPS/DPB, worst, -3 db Non-coherent JT, - db DPS/DPB, - db Sm. db x b Meta dstrbuton Fg.. M and meta dstrbuton of the general user and the worst-case user wth DPS/DPB or non-coherent JT, where n 3, and α 4. of the beta dstrbuton approxmaton s also confrmed by comparng the approxmatons wth smulaton results. V. INSIGHTS AND IMPORTANCE OF THE META DISTRIBUTION FOR COMP NETWORKS In Fg. 6 and Fg. 9, the accuracy of the beta approxmaton for meta dstrbuton s confrmed. Hence, n ths secton, we use the beta approxmaton to compare and dscuss our results. A. Condtonal Ln Relablty The meta dstrbuton provdes more fne-graned nformaton for the ndvdual lns than the standard success probablty. For example, Fg. shows the standard success probablty M and the meta dstrbuton F Ps of the general user and the worst-case user wth DPS/DPB or non-coherent JT where n 3, and α 4. In Fg. a, t s observed that, for the general user wth non-coherent JT or DPS/DPB, M, DPS/DPB M, JT for db, and the standard success probablty does not provde more nformaton to dstngush these two schemes. However, the meta dstrbuton shows B. Comparson of the Dfferent CoMP Schemes In the pror sectons, we studed non-coherent JT, DPB, and DPS/DPB by analyss, where the non-coherent JT s wthout precoder. Among these dfferent CoMP schemes, the nterference s the same n all schemes for a gven n, and the only dfference s the combnng modes of the desred sgnals, whch are gven by hx 2, DPB only or Non-CoMP max hx 2, DPS/DPB,...,n 2 S h x /2, Non-coherent JT n 2 h x /2, Coherent JT. 46 Here we also smulate the coherent JT and the enhanced non-coherent JT whch uses ndvdual precoders,.e., S n hx 2, and compare these results as shown n Fg. and Fg. 2. Fg. 2 shows that all these CoMP schemes beneft the worst-case user more than the general user, whch verfes the ntuton that CoMP can sgnfcantly mprove the system performance and, especally, enhance the cell-edge coverage. Fg. 2 also shows that the worst-case user wth DPS/DPB n 3 can acheve the smlar performance compared wth the general user wthout cooperaton. Qualtatvely, t s observed from Fg. and Fg. 2 that most of the results are consstent wth our ntuton of CoMP, wth the excepton that the non-coherent JT scheme n Secton III does not provde more gan than the DPS/DPB n Secton IV. We analyze ths phenomenon usng the combnng modes of the desred sgnals gven n 46. The performance gan of JT s closely related to the combnng mode of the desred sgnals. It s easly seen that h x 2 max h x 2,...,n n h x 2 n h x /2 2, and h x 2 n h x /2 2 n hx 2. For non-coherent JT wthout precoder, blnd demodulaton s used at the recever sde. Snce t can not lot the dversty gan of JT, noncoherent JT does not provde a sgnfcant gan compared wth DPS/DPB. However, for enhanced non-coherent JT, because of the ndvdual precodng, the performance of ths scheme s better. For coherent JT, the dversty gan of JT can be fully harvested because of the jont precodng, hence the performance of coherent JT s sgnfcantly hgher than DPS/DPB.

14 4.8 The performance of non-coherent JT wthout precoder can be regarded as a lower bound for the performance of other JT schemes. M Varance No CoMP DPB DPS/DPB Non-coh. JT enon-coh. JT Sm. Coh. JT Sm No CoMP DPB DPS/DPB Sm. Non-coh. JT enon-coh. JT Sm. Coh. JT Sm. a M db db b Varance M 2 M 2 Fg.. Comparson of M and the varance.e. M 2 M 2 of Ps for the general user wth DPB, DPS/DPB, and three JT schemes, where n 3, and α 4. The varance for DPS/DPB, and the curves for non-coherent JT and coherent JT are obtaned by smulaton. -F p x No CoMP beta DPB beta DPS/DPB Sm. Non-coh. JT beta enon-coh. JT Sm. Coh. JT Sm Fg. 2. Comparson of the beta approxmaton or the smulated meta dstrbuton for the general user and the worst-case user wth DPB, DPS/DPB, and three JT schemes, where n 3, α 4, and db. The dashed lnes correspond to the worst-case user. The curve for the worst-case user wth DPS/DPB s obtaned by beta approxmaton. x C. The Sze of the Cooperaton Set As we can see from the fgures n ths paper, for the general user wth all these CoMP schemes, the gan from n to n 2 s larger than the gan from n 2 to n 3. For example, as shown n Fg. 6a, when the relablty value x.6, for the general user wth non-coherent JT, the gan of meta dstrbuton from n to n 2 s about 45%, whle the gan from n 2 to n 3 s about 6%. In contrast, for the worst-case user, ths s not the case, snce the dstances from the servng user to the nearest three BSs are equal. In short, the performance gan assocated wth CoMP decreases as the sze of the cooperaton set ncreases. Hence, gven the overhead assocated wth CoMP, there exsts an optmum sze of the cooperaton set. VI. CONCLUSION Ths paper studes the meta dstrbuton n downln Posson cellular networs wth multple types of CoMP schemes ncludng JT, DPB, and DPS/DPB. We gve a general scheme for JT-DPB wth JT and DPB as specal cases. For each CoMP scheme, we consder two types of users the general user and the worst-case user. The b-th moment of condtonal success probablty s derved for both types of users wth JT-DPB. For DPS/DPB, the frst moment standard success probablty for the general user and the frst and second moments for the worst-case user are derved. We calculate the exact meta dstrbuton for the worst-case user wth JT-DPB and calculate the beta dstrbuton approxmaton of the exact meta dstrbuton for almost all schemes and user types. We show that the beta approxmaton provdes a great match for the exact meta dstrbuton. The mean local delay s also derved for both types of users wth JT-DPB, and the analyss shows that for both the general and worst-case users, the crtcal value of the SIR threshold for fnte mean local delay wth non-coherent JT s /δ n, whle wth DPB t s /δ. By the comparson and analyss of the meta dstrbuton of DPB, DPS/DPB, and JT, we provde more fne-graned nformaton on the ln relablty for dfferent CoMP schemes. The analyss shows: CoMP can sgnfcantly mprove the system performance and, especally, enhance the cell-edge coverage, 2 the performance gan of JT s closely related to the combnng mode, and 3 the performance gan assocated wth CoMP decreases as the sze of the cooperaton set ncreases. Based on the study n ths paper, the future wor ncludes: consderng the spatal correlaton by modelng the networs by more general pont processes, e.g., determnantal pont processes such as the Gnbre process [34], [35]; 2 analyzng the effect of ncomplete/mperfect CSI; 3 extendng to more scenaros, such as MIMO or relayng.

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