RELAY deployment in wireless networks has been recognized

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1 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 5, JUNE Outage Probablty of Multuser Relay Networks n Nakagam-m Fadng Channels Nan Yang, Student Member, IEEE, Maged Elkashlan, Member, IEEE, and Jnhong Yuan, Member, IEEE Abstract We evaluate the performance of downlnk multuser relay networks MRNs equpped wth a sngle amplfy-andforward AaF relay. A thorough and exact analyss s conducted to analyze the outage probablty of MRNs under dssmlar Nakagam-m fadng condtons. More specfcally, we derve new closed-form expressons for the outage probablty and the probablty densty functon pdf of the hghest end-to-end sgnalto-nose rato SNR assocated wth the strongest destnaton wth the sngle user and Raylegh fadng as specal cases. In partcular, we provde new results for channel-state nformaton CSI-based-gan relayng and fxed-gan relayng. We then demonstrate that the achevable dversty order s equal to ether the frst-hop fadng parameter or the product of the second-hop fadng parameter and the number of destnatons. Furthermore, we derve compact closed-form expressons for the moments of the hghest end-to-end SNR, from whch other moment-based measures such as the average SNR and the amount of fadng are deduced. Our results hghlght the performance mprovements offered by opportunstc schedulng and reveal the mpact of the relay locaton wth unbalanced hops on the overall performance. Varous numercal examples llustrate the proposed analyss. Index Terms Cooperatve transmsson, Nakagam fadng, opportunstc schedulng. I. INTRODUCTION RELAY deployment n wreless networks has been recognzed as one of the core ngredents n the promoton of robust hgh-speed data transmsson over challengng wreless envronments,. By placng a relay between the source and the destnaton, t s possble to provde hgher transmsson rates wth broader coverage. Snce the basc dea behnd relayng s to assgn a dedcated relay to help forward sgnals from Manuscrpt receved September 3, 9; revsed January, ; accepted January,. Date of publcaton February 7, ; date of current verson June 6,. Ths work was supported n part by Australan Research Councl Dscovery Proect DP8794 and n part by Qatar Natonal Research Fund under the Natonal Prortes Research Program Grant NPRP Ths paper was presented n part at the th IEEE Personal, Indoor, and Moble Rado Communcatons Symposum, Tokyo, Japan, September 9. The revew of ths paper was coordnated by Dr. M. Dohler. N. Yang s wth the School of Informaton and Electroncs, Beng Insttute of Technology, Beng 8, Chna and also wth the School of Electrcal Engneerng and Telecommuncatons, Unversty of New South Wales, Sydney, NSW 5, Australa e-mal: nan.yang@unsw.edu.au. M. Elkashlan s wth the Wreless Technologes Laboratory, CSIRO ICT Centre, Marsfeld, NSW, Australa and also wth the School of Electrcal Engneerng and Telecommuncatons, Unversty of New South Wales, Sydney, NSW 5, Australa e-mal: maged.elkashlan@csro.au. J. Yuan s wth the School of Electrcal Engneerng and Telecommuncatons, Unversty of New South Wales, Sydney, NSW 5, Australa and also wth the CSIRO ICT Centre, Marsfeld, NSW, Australa e-mal:.yuan@unsw.edu.au. Color versons of one or more of the fgures n ths paper are avalable onlne at Dgtal Obect Identfer.9/TVT..488 the source to the destnaton, relay networks are also nterpreted as cooperatve communcatons. Tradtonally, the performance analyss of relayng has been vewed n the context of pont-to-pont lnks havng a sngle source and a sngle destnaton. However, future broadband wreless networks are expected to provde ubqutous hghspeed nformaton transfer to a large number of destnatons.e., moble users at a reasonable cost and qualty of servce. In such pont-to-multpont lnks, the desred data rate may not be allowed f the destnatons are n the far-off dstance from the access pont or n locatons wth severe shadowng. Therefore, good lnk qualty and hgh-data-rate HDR coverage n large areas should be provded by explotng novel cooperatve network archtectures. Hence, the obectve of ths paper s to study the ont nteracton of multuser transmsson and relayng by quantfyng the performance of cooperatve multuser networks n terms of the outage probablty. Among varous cooperaton protocols proposed n the open lterature 3, we are concerned wth the amplfy-and-forward AaF relayng due to ts low complexty and cost-effectveness. AaF can be separated nto two man categores: channel-state nformaton CSI-based-gan relayng, whch adopts channel measurements to regulate a constant nstantaneous transmt power at the relay; and fxed-gan relayng, whch smply apples a constant scalng gan to the receved sgnal to regulate a constant average transmt power at the relay. Although CSIbased-gan relayng offers addtonal performance gans relatve to fxed-gan relayng, fxed-gan relayng has become attractve from a cost-conscous and hardware-mplementaton pont of vew, due to ts smplcty and ease of deployment. In ths paper, we consder both AaF categores. Drven by the potental applcaton of such wreless relays, an upsurge of nterest has been observed n studyng the end-toend performance of relays n varous network confguratons. The outage probablty s an effectve performance measure, and some poneerng studes have attempted to fnd closedform solutons for ths measure n pont-to-pont dual-hop lnks.e., a sngle source communcates wth a sngle destnaton asssted by a sngle relay, as llustrated n Fg. a. Among them, 4 proposed a semanalytcal approach for the evaluaton of the outage probablty of dual-hop systems wth CSI-basedgan relays over Nakagam fadng channels. In 5, expressons were obtaned for the statstcs of the end-to-end sgnal-tonose rato SNR of dual-hop systems wth CSI-based-gan relays over Raylegh fadng channels. The same authors n 6 studed the outage probablty and the error performance of dual-hop systems for fxed-gan relays over Raylegh fadng channels. Closed-form upper bounds for the dstrbutons of the /$6. IEEE

2 YANG et al.: OUTAGE PROBABILITY OF MULTIUSER RELAY NETWORKS IN NAKAGAMI-m FADING CHANNELS Fg.. Schematc of a dual-hop lnk, where there s practcally no drect lnk between the source and the destnatons. a Sngle user: pont-to-pont dualhop lnk. b Multuser: pont-to-multpont dual-hop lnk.e., MRN. end-to-end SNR of dual-hop systems wth CSI-based- and fxed-gan relays over Nakagam fadng channels were derved n 7. A useful framework was outlned n 8 for obtanng the exact outage probabltes of dual-hop systems wth CSI-basedand fxed-gan relays over Nakagam fadng channels. In a multuser cellular wreless network, dfferent moble users sharng the rado channel are lable to experence dfferent channel condtons at the same tme. In such pont-to-multpont applcatons, the lmted avalable rado resources must be allocated among all moble users n the most effcent manner. Recently, opportunstc schedulng has attracted much attenton as a schedulng method, n whch the tme-varyng nature of a wreless channel s captured. Although all users share the common downlnk channel, only one user s scheduled to transmt data at a tme n a tme slot. Intutvely, servng the user wth the best channel condtons s expected to maxmze the throughput and mprove the system performance of the sngleantenna broadcast channel 9. Ths effect obtaned wth such schedulng s referred to as multuser dversty,. In many current systems, a pluralty of remote destnatons or equvalently moble users may be ncluded n geographcally dspersed areas, leadng to poor non-lne-of-sght NLOS performance. Hence, drect transmssons between the base staton and the destnatons close to the cell boundary may not guarantee relable communcaton at hgh data rates. In ths regard, the key concept of cooperatve transmsson s appled to multuser cellular wreless networks to assst transmssons between both ends. Introducng a relay n such a stuaton s a promsng approach toward the potental mprovements n channel qualty by overcomng sgnfcant loss of sgnal strength along the propagaton path n NLOS envronments. In addton, ntroducng a relay n exstng networks s consdered to be an attractve means to extend the network coverage and/or capacty dstrbuton. Ths concept s an example of pont-tomultpont dual-hop lnks.e., a sngle source communcates wth multple destnatons asssted by a sngle relay, and we refer to ths applcaton as multuser relay networks MRNs, as llustrated n Fg. b. Deployment of relays n multuser cellular wreless networks s currently a promsng opton for emergng standards such as the IEEE 8.6 moble multhop relay MMR network and the wreless world ntatve new rado WINNER 3. Although the performance of pont-to-pont dual-hop lnks s well understood, knowledge about the mpact of relays n a multuser or network context s lmted. Currently, very few works have analytcally evaluated the performance of MRNs. Among them, 4 derved closed-form upper bound solutons for the outage capactes of a sngle fxed-gan relay. Through an approxmaton of the SNR dstrbuton, 5 derved the average throughput of a sngle CSI-based-gan relay. On the other hand, some efforts n 6 nvestgated the outage probablty of a sngle CSI-based-gan relay. Whle the aforementoned works assumed Raylegh fadng, the performance of MRNs over generalzed Nakagam-m fadng has not been nvestgated. It s well known that Nakagam-m fadng covers a wde range of fadng scenaros that are typcal n realstc wreless relay applcatons va the m parameter, whch ncludes the Raylegh fadng m as a specal case. Moreover, the effect of unbalanced hops on MRNs has not been thoroughly studed yet. In practce, the rado lnk between the source and the relay typcally exhbts good lne-of-sght LOS condtons, whch s a realstc assumpton for a scenaro where both termnals are postoned above the rooftops, whle the lnks between the relay and the destnatons suffer severe multpath n hghly congested bult-up areas. In ths paper, we focus on the realstc potental gans of MRNs wth AaF relayng n unbalanced generalzed Nakagam-m fadng condtons. In dong so, new expressons are derved for the cumulatve dstrbuton functon cdf and the probablty densty functon pdf of the hghest end-toend SNR assocated wth the strongest destnaton. The prmary contrbutons of ths paper are dstnct from the prevous works n the followng aspects: We present a unfed analytcal framework for the exact performance evaluaton of MRNs based upon the drect mplementaton of the statstcs of the per-hop SNR. The novelty of the proposed framework les n the fact that we avod the serous analytcal dffculty arsng from complcated Laplace transforms assocated wth the momentgeneratng functon mgf-based approach. We derve exact closed-form expressons for the outage probablty of MRNs equpped wth a CSI-based-gan relay. Ths scenaro has prevously been studed n 6 for the specal case of Raylegh fadng but not the more generalzed Nakagam-m fadng. 3 We derve exact closed-form expressons for the outage probablty of MRNs equpped wth a fxed-gan relay. Ths scenaro has never been addressed n prevous works for ether Raylegh fadng or Nakagam-m fadng. 4 We derve the achevable dversty order of MRNs. Ths result proves that both CSI-based- and fxed-gan relayng yeld the same dversty order, whch s equal to ether the

3 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 5, JUNE frst-hop fadng parameter or the product of the secondhop fadng parameter and the number of destnatons. 5 We derve exact closed-form expressons for the moments of the hghest end-to-end SNR. Based on these new statstcal results, we evaluate other relevant statstcs that characterze the SNR dstrbuton, such as the average SNR and the amount of fadng AoF. Our new analytcal expressons have the addtonal mert of characterzng the sngle-user pont-to-pont dual-hop scenaro as a specal case. Moreover, our generalzed analyss serves as a tool to examne the mpact of unbalanced hops by takng nto account dfferent per-hop average faded SNRs and/or dfferent per-hop fadng severtes on the system performance. Furthermore, a desgn crteron s presented to guarantee performance mprovements offered by opportunstc schedulng. Specfcally, by varyng the per-hop transmsson dstances, our results reveal the role of the relay locaton n determnng the multuser advantage. II. SYSTEM DESCRIPTION A. Cooperatve Transmsson and Channel Model Consder the multuser wreless relay communcaton system depcted n Fg. b. The source communcates wth K destnatons va the cooperatve lnk wth the ad of a sngle AaF relay. The drect lnk between the source and the destnatons s absent. The relay transmsson s constraned to a half-duplex mode, by dvdng the transmsson perod nto two consecutve sgnalng ntervals. In sgnalng nterval, the source transmts the sgnal to the relay; n sgnalng nterval, the relay amplfes the receved sgnal wth a scalng gan factor and then forwards the scaled replca toward the destnaton that has the most favorable end-to-end channel qualty. We denote the average symbol energes avalable at the source and the relay as E S and E R, respectvely. Furthermore, let the modulated sgnal transmtted by the source be denoted as x. In sgnalng nterval, the receved sgnal at the relay from the source can be represented as y r E S d η SR αx + n r where α denotes the channel complex fadng coeffcent between the source and the relay, and n r s the addtve whte Gaussan nose AWGN component wth varance N at the relay. The path loss s ncorporated n the sgnal propagaton, where d SR s the dstance between the source and the relay, and η s the path loss exponent. We defne G as the scalng gan appled at the relay. Hence, n sgnalng nterval, the receved sgnal at the kth destnaton from the relay s expressed as y d E R d η RD β kg E S d η SR αx + n r + n dk where β k denotes the channel complex fadng coeffcent between the relay and the kth destnaton, n dk s the AWGN component wth varance N at the kth destnaton, and d RD s the dstance between the relay and the destnaton. Accordngly, the equvalent nstantaneous end-to-end SNR of the kth destnaton eq,k can be obtaned from as eq,k,k,k + G N 3 where α d η SR E S/N, and,k β k d η RD E R/N are the nstantaneous faded SNRs of the frst and second hop, respectvely. From 3, we note that eq,k s nfluenced by the choce of G. The pdf of the per-hop nstantaneous faded SNR Z {,,k } follows a gamma dstrbuton,.e., f Z mm m m Γm e m where Γ s the gamma functon defned by 7, eq as Γx tx e t dt. The average faded SNR of the frst and second hops s gven by E and E,k, respectvely, where E s the expectaton. The fadng parameter of the frst and second hops s denoted as m and m, respectvely. It s assumed that the destnatons are located wthn the same fadng envronment. In such a homogeneous envronment, all the destnatons have the same per-hop fadng severty m and per-hop average faded SNR. On the other hand, the dualhop transmsson s assumed unbalanced, where the frst and second hops experence dfferent per-hop average faded SNRs.e., and/or dfferent per-hop fadng severtes.e., m m. Ths s typcal n most relay applcatons, where the relay and the destnatons are located n dfferent fadng envronments. The correspondng cdf of Z can be wrtten as m, m Γ m, m F Z 5 Γm Γm where, s the lower ncomplete gamma functon defned by 7, eq , and Γ, s the upper ncomplete gamma functon defned by 7, eq B. Opportunstc Schedulng n Cooperatve Transmsson Tme-dvson multple access TDMA s assumed to facltate the sharng of the downlnk channel among the multple destnatons. Specfcally, we adopt the opportunstc multuser schedulng proposed n 9. In ths polcy, only one destnaton wth the hghest nstantaneous end-to-end SNR out of K destnatons s scheduled for transmsson. Ths polcy holds for a number of tme-slotted downlnk multuser systems ncludng Qualcomm s HDR 8 and hgh-speed downlnk packet access 9. In ths case, f s denotes the hghest nstantaneous end-to-end SNR of the scheduled destnaton.e., strongest destnaton, then the polcy s formulated as where eq,k s defned n 3. 4 s max k K { eq,k} 6

4 YANG et al.: OUTAGE PROBABILITY OF MULTIUSER RELAY NETWORKS IN NAKAGAMI-m FADING CHANNELS 3 The schedulng algorthm s mplemented at the relay, whch s assumed to have knowledge of the relay destnaton lnks of the K destnatons that are scheduled for transmsson. At the begnnng of each schedulng perod, the relay broadcasts a destnaton synchronzaton message DSM, whch contans the synchronzaton nformaton and the plot sgnalng transmsson order. We assume that the destnatons are perfectly synchronzed to the relay n tme. Ths s a common assumpton n TDMA cellular networks,. Each destnaton conveys plot sgnals to the relay accordng to the transmsson order receved n the DSM. At the relay, channel estmaton s conducted based on a plot sgnal sequence sent by the K destnatons. The relay dentfes and selects the strongest destnaton and then feeds back the ndex of the strongest destnaton to the source. The destnaton selecton s assumed to be done at dscrete nstants of tme t nt, where T s the selecton perod. Ths can be accommodated by ntroducng a short guard nterval every selecton perod, durng whch the relay performs channel estmatons for the relay destnaton lnks and necessary comparsons for destnaton selecton. To avod the frequent selecton operaton between destnatons, we assume that the relay updates the destnaton selecton only durng ths guard nterval before each data burst and that the length of the data burst s on the order of the channel coherence tme. III. OUTAGE PROBABILITY ANALYSIS We characterze the performance of MRNs n terms of the outage probablty. In partcular, we provde a unfed approach to derve the cdf of s, under the assumpton of two dfferent gan constrants,.e., CSI-based- and fxed-gan relayng, n Nakagam-m fadng channel condtons. By relyng on these statstcal results, the outage probablty s evaluated. The outage probablty P out s defned as the probablty that s drops below a predetermned specfed SNR threshold th. Therefore, P out s gven by P out Pr s < th F s th 7 where F s th s the cdf of s, evaluated at th. In the followng, we evaluate exact closed-form expressons for F s th under the CSI-based-gan constrant F s,csi th and the fxed-gan constrant F s,fx th. A. CSI-Based-Gan MRN: General Results In ths subsecton, we consder MRNs equpped wth a CSIbased-gan relay. CSI-based-gan relayng adopts channel measurements to counteract the fadng effects of the source relay lnk. When the nose statstcal nformaton s not avalable at the relay, the scalng gan s set to G / α d η SR E S 5. In ths case, the nstantaneous end-to-end SNR of the kth destnaton can be rewrtten as eq,k,csi,k +,k. 8 Although systems wth CSI-based-gan relayng are expected to outperform those wth fxed-gan relayng, ths mprovement comes at the nevtable cost of ncreased power consumpton, channel-nformaton overhead, and crcutry footprnt. Theorem CDF of s,csi n CSI-Based-Gan MRNs: Under the assumpton of unbalanced hops.e., m m and and the fact that the K relay destnaton lnks undergo ndependent dentcally dstrbuted..d. fadng channels, the cdf of the hghest nstantaneous end-to-end SNR s,csi of the strongest destnaton s derved as F s,csi + mm m K Γm K m K K m n n n n t t n t!t! n t n t+ φ φ m K s m s m+φ e m + m K m m K K m s n m n m φ m m s K! n m! where K v denotes the vth-order modfed Bessel functon of the second knd, φ m q n q n + n + + n m, n K, and n m. Proof: See Appendx A. Consequently, the outage probablty of MRNs wth CSIbased-gan relayng s evaluated by substtutng 9 nto 7 as 9 P out,csi F s,csi th. Ths s a new exact outage probablty expresson that s vald for arbtrary nteger values of the fadng parameter m. It s worth notng that can be calculated n closed form, nvolvng only fnte summatons of exponentals, powers, and Bessel functons. Corollary PDF of s,csi n CSI-Based-Gan MRNs: The pdf of s,csi s gven by f s,csi mm m Γm K t φ s K φ s m m K K n n n n t n t!t! n t n t+ m K m n m n m φ m m s K! n m! m+φ e m + m K Θ K m sθ Θ K m s Θ

5 4 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 5, JUNE where the followng notatons are ntroduced for compactness: Θ φ + + s/ m / +m / K, and Θ m m K /. Proof: The proof follows by dfferentatng 9 wth respect to and usng 7, eq for the dervatve of K v. B. CSI-Based-Gan MRN: Results for Specal Cases We now show two specal cases for the P out,csi of CSIbased-gan MRNs, as dscussed below. Specal Case : For the sngle-user pont-to-pont dual-hop lnk n Nakagam-m fadng, P out,csi s found by settng K n as P out,csi Γm where Ξ, k, s m m k k s e m + m th K m s m k s k! m m ++s Ξ, k, s m +k th th m m m m s+k Note that a smlar expresson to has been derved n 8, eq.. Moreover, when balanced hops are assumed e.g., m m and, the expresson n serves as an alternatve representaton to, eq. 8, whch was expressed n terms of the Meer s G-functon defned n 7, eq In addton, the pdf of s,csi for the sngle-user scenaro n Nakagam-m fadng s extracted by settng K n as f s,csi Γm m m k m +k e m + m k Ξ, k, s s Θ K m s Θ Θ K m s Θ 3 where Θ + k + s/ m / +m /, Θ m m /, and Ξ, k, s s defned n. Specal Case : For the multuser pont-to-multpont dualhop lnk n Raylegh fadng, P out,csi s found by settng m m n as P out,csi + K th K K K e + K th K K th. 4 It s noted that 4 concdes wth the prevously publshed result n 6, eq. 5. Ths demonstrates the effectveness and the valdty of our result n.. Subsequently, the pdf of s,csi for the multuser scenaro n Raylegh fadng s obtaned by settng m m n as f s,csi K K K K e + K Θ K Θ Θ K Θ 5 where Θ / +K /, and Θ K /. Note that the result n 5 can also be obtaned by takng the dervatve of 4 wth respect to th and settng th. C. Fxed-Gan MRN: General Results We now turn our attenton to fxed-gan relayng, where the fadng coeffcents of the source relay lnk are not known to the relay. As such, the relay ntroduces a fxed scalng gan of G /CN to ts receved sgnal 6, where C +. Therefore, the end-to-end nstantaneous SNR of the kth destnaton s gven by eq,k,fx,k,k + C. 6 From an mplementaton standpont, the need for low complexty and/or processng power savngs have motvated the study of fxed-gan relayng, although from a performance standpont, fxed-gan relayng s not expected to perform as well as CSIbased-gan relayng. Theorem CDF of s,fx n Fxed-Gan MRNs: Assumng unbalanced hops and the fact that the K relay destnaton lnks experence..d. fadng condtons, the cdf of the hghest nstantaneous end-to-end SNR s,fx of the strongest destnaton s derved as F s,fx + mm m Γm K t m K K K m C m n n n n t n t!t! n t n t+ m +φ K m m++φ e m m m K C K m φ n m n m m φ K! n m! 7 where K v denotes the vth-order modfed Bessel functon of the second knd, φ m q n q n + n + + n m, n K, and n m. Proof: See Appendx B.

6 YANG et al.: OUTAGE PROBABILITY OF MULTIUSER RELAY NETWORKS IN NAKAGAMI-m FADING CHANNELS 5 Usng 7, the outage probablty of MRNs wth fxed-gan relayng s calculated as P out,fx F s,fx th. 8 To the best of our knowledge, ths result s new. It s observed from 8 that ths explct outage probablty expresson contans only standard functons and can therefore be easly and effcently evaluated numercally wth software packages such as MATLAB. Corollary PDF of s,fx n Fxed-Gan MRNs: The pdf of s,fx s shown as f s,fx mm m Γm K t m m K K K m C n n n n t n t!t! n t n t+ m ++φ e m m +φ K m n m n m m φ K! n m! Δ K m φδ Δ K m φ Δ 9 where we ntroduce Δ + φ/ m / and Δ m m K C/ for compactness. Proof: The proof follows by dfferentatng 7 wth respect to and usng 7, eq for the dervatve of K v. D. Fxed-Gan MRN: Results for Specal Cases Next, we present two specal cases for P out,fx of fxed-gan MRNs as follows. Specal Case : For the sngle-user pont-to-pont dual-hop lnk n Nakagam-m fadng, P out,fx s found by settng K n 8 as P out,fx Γm where Ξ, k m m k e m th K m k m k! m m ++k Ξ, k m ++k th m C m m C th m +k. It s noted that s equvalent to 8, eq. 8. Hence, our result n 8 stands for a generalzaton of the sngle-user dualhop relay lnk. Furthermore, the pdf of s,fx for the sngle-user pont-topont dual-hop lnk n Nakagam-m fadng s found by settng K n 9 as f s,fx Γm m m k Ξ, k m ++k e m Δ K m φ Δ Δ K m φ Δ where Δ Δ + k/ m /, Δ m m /, and Ξ, k s defned n. Specal Case : For the multuser pont-to-multpont dualhop lnk n Raylegh fadng, we provde a new result for P out,fx by settng m m n 8 as P out,fx + C th K e th K K K K K C th. 3 Ths specal case has not prevously been addressed. Moreover, the pdf of s,fx for the multuser pont-tomultpont dual-hop lnk n Raylegh fadng s found by settng m m n 9 as K C K f s,fx K K e Δ K Δ Δ K Δ 4 where Δ /, and Δ K C/.Note that the result n 4 can be also obtaned by takng the dervatve of 3 wth respect to th and settng th. E. Achevable Dversty Order We now focus on the achevable dversty order of MRNs to gan further nsght nto the mpact of unbalanced hops on the system performance. Let d log P out lm 5, log / ρ be the achevable dversty order, where ρ s a nonzero fnte constant. Therefore, the achevable dversty order of MRNs s quantfed by d MRN mn{m,m K}. 6 Our result reveals that the dversty order acheved s ontly determned by the per-hop fadng parameters and the number of destnatons. From 6, we characterze the qualtes of the frst and second hops by m and m K, respectvely, and refer to the lower qualty hop as the weaker hop. It can be concluded from 6 that the system performance s lmted by the weaker hop.

7 6 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 5, JUNE Furthermore, t s remarked that both CSI-based- and fxed-gan relayng acheve the same dversty order. IV. MOMENT ANALYSIS OF s We now derve the moments of s that can effcently be appled to uncover other statstcal measures. For nstance, the average SNR and the AoF can be evaluated wth the frst-order and second-order moments. In addton, the moments of s can be used to study hgher order metrcs such as the skewness and the kurtoss that characterze the dstrbuton of s. By defnton, the nth moment of s s gven by E n s n f s d. 7 Armed wth the results for the pdf provded n Secton III, we present exact closed-form expressons for the moments of s as follows. A. CSI-Based-Gan MRN Theorem 3 Moments of s,csi n CSI-Based-Gan MRNs: The nth moment of s,csi s gven by E s,csi n n m π K m K K Γm K n n n t φ s φ s n m n m K! n m! n t n t!t! n t n t+ m +s φ m n+m +φ s+ m + m n φ s K Γn + + φ + s Γn +m + φ s Γ n + m + φ + F n + + φ + s, n + + φ + s + ; n + m + φ + ; z CSI 8 where F a, b; c; z s Gauss s hypergeometrc functon, and z CSI 4m m K m + m K. Proof: Ths result s obtaned by substtutng nto 7 and solvng the resultant ntegral wth the ad of 3, eq We now present two specal cases for whch the moments of s,csi n 8 reduce to a relatvely smple form. In partcular, we nvestgate the average SNR and the AoF of CSI-based-gan relayng. Average SNR: By settng n n 8, the average SNR can be deduced n closed form as π φ φ s,csi Γm Ωm,m,K s s m m m +φ s + m K Γχ + Γm + φ s + Γ m + φ + 3 +s m χ χ + F, χ +;m + φ + 3 ; z CSI where χ + φ + s, and Ωm,m,K m K m n n n t K n m n m K K K! n m! n t n t!t! n t n t+ φ 9. 3 AoF: The AoF characterzes the severty of the fadng. Wth 8, the AoF can be expressed n a smple closed form as A F,CSI E s,csi. 3 E s,csi B. Fxed-Gan MRN Theorem 4 Moments of s,fx n Fxed-Gan MRNs: The nth moment of s,fx s gven by E s,fx n n Γm K n n t m n m m n m n m K K! n m! n t n t!t! n t n t+ K K n n m C K n φ Γn + m Γn + + φ F n + m,n+ + φ; ; z Fx 3

8 YANG et al.: OUTAGE PROBABILITY OF MULTIUSER RELAY NETWORKS IN NAKAGAMI-m FADING CHANNELS 7 where F a, b;;z s the hypergeometrc functon, and z Fx m K C. Proof: Ths result s obtaned by substtutng 9 nto 7 and solvng the resultant ntegral wth the ad of 7, eq Smlar to the prevous subsecton, two specal cases for the moments of s,fx n 3 are presented as follows. Average SNR: By settng n n 3, the average SNR can be calculated n closed form as s,fx Γm Ωm,m,K m m C K φ Γm + Γ + φ + F m +,+ φ +; ; z Fx 33 where Ωm,m,K s gven by 3. AoF: Usng 3, a smple closed-form expresson for the AoF s gven by A F,Fx E s,fx. 34 E s,fx Fg.. CDF of s,csi and s,fx for E S /N db, d SR.3, and d RD.7wth m m. V. N UMERICAL RESULTS AND DISCUSSION In ths secton, we provde numercal examples to valdate the theoretcal analyss developed n the prevous secton. The performance of a network of K destnatons s examned under varous channel parameters. More specfcally, we study the effect of a relay wth unbalanced hops on the end-to-end outage probablty by varyng the per-hop average faded SNR.e., and and the per-hop fadng parameter.e., m and m. Throughout ths secton, we assume that the source relay destnaton dstance s normalzed to unty such that d SR + d RD. In addton, the varance of the fadng coeffcents s also normalzed to unty wth E α and E β k. Moreover, equal average energes are assumed at the source and the relay wth E S E R, and the nformaton symbols are modulated usng bnary phase-shft keyng. As such, the frst-hop and the second-hop average faded SNR s attenuated by d η SR and d SR η, respectvely, and are ontly nfluenced by the path loss and the shadowng of obstacles. The path loss exponent η s an emprcal constant that s often measured to descrbe the nfluence of the transmsson medum. For outdoor propagaton, η ranges between and 5, dependng upon the rado envronment 4. For nstance, n free space, η s equal to, and when obstructons are present, η has a larger value. In ths secton, our results concentrate on the practcal example of a hghly shadowed urban area wth η 4. We use the symbol to mark ponts generated va Monte Carlo smulatons. In each fgure, the ponts precsely match the analytc curves, hghlghtng the accuracy of the analyss. Fg. depcts the cdf of the hghest end-to-end SNR assocated wth the strongest destnaton n varous Nakagam-m fadng envronments wth m m and m m. The curves are plotted across an SNR threshold range from to Fg. 3. PDF of s,csi and s,fx for E S /N db, d SR.3, and d RD.7wth m m. db for K and 4, usng 9 for CSI-based-gan relayng and 7 for fxed-gan relayng. Compared wth a sngle-user scenaro.e., K, selectng the hghest end-to-end SNR lnk n a multuser scenaro.e., K 4 mproves the system performance as expected. Ths performance mprovement s further llustrated n Fg. 3, whch presents the correspondng pdf shown n and 9 for m m. A. Impact of the Relay Locaton We nvestgate the mpact of the relay locaton on the outage probablty of MRNs by settng d SR d RD thereby, whle keepng m m. Ths descrbes a wreless network where the relay s not placed halfway between the source and the destnatons. Moreover, the relay and the destnatons are both located wthn the same fadng envronment. In other words, the sgnals receved at the relay and the destnatons experence the same fadng condtons. Here, we consder the fadng condtons descrbed as follows. When m m, t corresponds to the case where both the relay and the destnatons are wthn a densely populated area wth more severe

9 8 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 5, JUNE Fg. 4. Outage probablty of CSI-based-gan MRNs for th db. Case : d SR.3, andd RD.7. Fg. 6. Outage probablty of CSI-based-gan MRNs for th db. Case : d SR.7, andd RD.3. Fg. 5. Outage probablty of fxed-gan MRNs for th db. Case : d SR.3, andd RD.7. multpath fadng; when m m, t corresponds to the case where both the relay and the destnatons are wthn a moderate multpath envronment wth less scatterng; and when m m 3, t corresponds to the case where the LOS component s domnant between the source and the relay and between the relay and the destnatons. We now show the outage probablty of MRNs for K and K, takng nto account two cases based on d SR and d RD, as follows. Case : The relay s located close to the source such that d SR.3 and d RD.7. Fgs. 4 and 5 plot the outage probablty aganst E S /N for CSI-based-gan relayng usng and fxed-gan relayng usng 8, respectvely. It s obvous that opportunstc schedulng offers a performance boost. For example, when m m 3at the outage probablty of 3 for CSI-based-gan relayng, t can be observed that K s superor by about 5.6 db to the sngle-user scenaro. On the other hand, for fxed-gan relayng, K outperforms the sngle-user scenaro by about 3.5 db. Fg. 7. Outage probablty of fxed-gan MRNs for th db. Case : d SR.7, andd RD.3. Case : The relay s located close to the destnatons such that d SR.7 and d RD.3. Fgs. 6 and 7 plot the outage probablty aganst E S /N for CSI-based-gan relayng usng and fxed-gan relayng usng 8, respectvely. For both CSI-based- and fxed-gan relayng, t can be observed that neglgble gans are attaned by opportunstc schedulng wth K, regardless of the fadng parameter m. Moreover, t s shown that CSI-based-gan relayng yelds a comparable outage probablty wth that of fxed-gan relayng. Ths s what we expect, snce the presence of channel knowledge at the relay s less proftable as the channel qualty of the frst hop decreases, relatve to that of the second hop. These observatons are not surprsng snce the performance of cooperatve transmsson s, n general, restrcted by the weaker lnk. Specfcally, when the relay sts closer to the source, as shown n Case, opportunstc schedulng wll mprove the channel qualty of the second hop.e., weaker lnk, resultng n a consderable performance mprovement. By contrast, when the relay sts closer to the destnatons, as shown n Case, the frst hop.e., weaker lnk becomes the

10 YANG et al.: OUTAGE PROBABILITY OF MULTIUSER RELAY NETWORKS IN NAKAGAMI-m FADING CHANNELS 9 Fg. 8. Outage probablty of MRNs as a functon of K for CSI-basedand fxed-gan relayng wth E S /N db, th db, d SR.3, and d RD.7. lmtng factor. In ths case, opportunstc schedulng wll not mprove the channel qualty of the frst hop, resultng n a lessnotceable performance mprovement. The aforementoned observatons address the key desgn queston on where to place the relay to ensure the performance gans offered by opportunstc schedulng. B. Impact of the Number of Destnatons We now examne the mpact of the number of destnatons on the outage probablty of MRNs. Here, we model the fadng condtons n downlnk relay-asssted cellular applcatons. In ths scenaro, the source and the relay antennas are generally mounted on hgh towers to avod obstructons, resultng n a domnant LOS transmsson n the frst hop wth m. On the other hand, an NLOS transmsson wth m s experenced n the second hop, where the sgnal transmtted from the relay to the destnatons n urban areas s usually reflected from surroundng buldngs, trees, and other obstructons. Based on our prevous observatons, we choose to place the relay closer to the source such that d SR.3 and d RD.7. The outage probablty of CSI-based- and fxed-gan MRNs s plotted versus K n Fg. 8. We observe that the outage probablty decreases as K ncreases. Ths performance mprovement s expected due to the fact that a hgher K provdes a hgher varablty n the nstantaneous end-to-end SNRs among the dfferent destnatons. We also observe the mpact of m on the outage probablty. Intutvely, the outage probablty decreases as the fadng severty decreases.e., m ncreases. Furthermore, t s worth notcng that the multuser advantage s more notable when the two hops are hghly unbalanced e.g., m 4and m. Ths can be explaned by the fact that the performance s profoundly nfluenced by the second hop, whch s the weaker lnk. As a result, ncreasng K mproves the lnk qualty of the second hop, leadng to a consderable reducton n outage probablty. Fnally, we fnd that the outage probablty of CSI-based-gan relayng s lower than that of fxed-gan relayng. Ths s due to the fact that the CSI-basedgan relayng takes nto account the channel condtons on the source relay lnk. VI. CONCLUSION Ths paper has presented new closed-form expressons for the outage probablty of MRNs employng a sngle AaF relay over generalzed Nakagam-m fadng channels. In partcular, we have taken nto consderaton the realstc assumpton of unbalanced hops and ponted out the effect of dssmlar perhop fadng parameters and/or dssmlar per-hop average faded SNRs. In dong so, we have derved exact expressons for the cdf and the pdf of the hghest end-to-end SNR assocated wth the strongest destnaton n the cases of CSI-based- and fxed-gan relayng. For each case, we have also obtaned closed-form expressons for the Raylegh fadng scenaro and the sngle-user scenaro as specal cases. Moreover, closedform expressons for the moments of the strongest end-to-end SNR have been presented, whch were substantally appled to nvestgate mportant statstcal parameters such as the average SNR and the AoF. Furthermore, we have studed the mpact of the relay locaton on the system performance and developed a desgn crteron for the relay placement. Our proposed results provde a gudelne to assst cellular nfrastructure provders wth the fundamental problem of relay placement to guarantee the multuser advantage n practce. APPENDIX A PROOF OF THEOREM We now calculate the cdf of the hghest nstantaneous endto-end SNR of the strongest destnaton for CSI-based-gan relayng s,csi. The cdf of the per-hop nstantaneous faded SNRs and,k, k,...,k s gven by 5. Under the assumpton that the K relay destnaton lnks undergo..d. Nakagam fadng, the cdf of s,csi s calculated as where F s,csi I I + Pr s,csi < f d K,k Pr < f + d,k Pr,k > Pr,k < K f d K f d I + I 35 f d F K F,k f d Γ m, m Γm K Γ m, m Γm 36 m m m m m Γm e d. 37

11 3 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 5, JUNE By usng the bnomal expanson n 7, eq.., I can be evaluated as I K K Γ m, K mm m m Γ Γm e m, m Γm m d K m Γm K + mm K K m K Γm Γm K K m Γ m, m e m d. 38 Settng x + n 38, I can be reexpressed as Γ m, m m I + mm e Γm m Γm m K m K K Γm K I 3 39 where I 3 Γ m, m + K x m e m x dx. x 4 Applyng 7, eq , the ncomplete gamma functon n the ntegrand of I 3 can be expanded as Γ m, m + K m! K x e m m m + x K p + x p! p K. 4 To tackle the power of multnomal n 4, we nvoke a useful formula gven by 5, eq. 9, yeldng m m p where θm,k + x p! K p n n n K t φ m φ φ s θm,k s n m n m K! n m! s n t n t!t! n t n t+ x s 4 43 wth n K, n m, and φ m q n q n + n + + n m. Therefore, I 3 can be solved by substtutng 4 nto 4 and applyng 7, eq as φ I 3 m! K e m K m θm,k φ s φ s s x m s e m K x m! K e m K m θm,k φ s K m s φ s m K s m m x dx φ m s m m K. 44 Thus, I s evaluated by nsertng 44 nto 39. Fnally, we obtan F s,csi by usng 36 together wth 39. APPENDIX B PROOF OF THEOREM For fxed-gan relayng, followng a smlar procedure descrbed n Appendx A, the cdf of s,fx shown n 7 s gven by F s,fx + Pr s,fx < f d K,k Pr,k + C < f d Pr,k > Pr,k < C K f d C K f d I + I. 45 The ntegral I n 45 s dentcal to that shown n 36. Smlarly, the ntegral I can be derved by usng bnomal expanson as follows: I Γ Γ m, m, m Γm Γ m, m C Γm + mm m Γm m C K K m m m m m Γm e d K K Γm K K m e m d. 46

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Nan Yang S 9 receved the B.S. degree n electroncs n 5 from Chna Agrcultural Unversty, Beng, Chna, and the M.S. degree n electrcal engneerng n 7 from the Beng Insttute of Technology, where he s currently workng toward the Ph.D. degree n electrcal engneerng. From 8 to, he was a vstng Ph.D. student wth the School of Electrcal Engneerng and Telecommuncatons, Unversty of New South Wales, Sydney, NSW, Australa. Hs research nterests nclude cooperatve communcaton systems, multple-nput multple-output systems, and fadng channels.

13 3 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 5, JUNE Maged Elkashlan M 6 receved the Ph.D. degree n electrcal engneerng from the Unversty of Brtsh Columba, Vancouver, BC, Canada, n 6. From 6 to 7, he was a Research Fellow wth the Laboratory for Advanced Networkng, Unversty of Brtsh Columba. In 7, he oned the Wreless Technologes Laboratory, Commonwealth Scentfc and Industral Research Organzaton CSIRO, Sydney, Australa. Hs research nterests nclude cooperatve wreless networks, fadng channels, and resource allocaton. Dr. Elkashlan s a recpent of the GREAT Award from the Scence Councl of Brtsh Columba and the Unversty Graduate Fellowshp from the Unversty of Brtsh Columba. He also served as a member of the Techncal Program Commttee of the IEEE Wreless Communcatons and Networkng Conference and the IEEE Internatonal Conference on Communcatons. Jnhong Yuan M receved the B.E. and Ph.D. degrees n electroncs engneerng from the Beng Insttute of Technology, Beng, Chna, n 99 and 997, respectvely. From 997 to 999, he was a Research Fellow wth the School of Electrcal Engneerng, Unversty of Sydney, Sydney, NSW, Australa. Snce, he has been wth the School of Electrcal Engneerng and Telecommuncatons, Unversty of New South Wales, Sydney, where he s currently a Professor. He has publshed two books, two book chapters, and more than 5 papers n telecommuncatons ournals and conference proceedngs. Hs publcatons are avalable at Hs current research nterests nclude wreless communcaton, communcaton theory, error-control codng, and dgtal modulaton.