Energy Harvestng Cooperatve Networks: Is the 1 Max-Mn Crteron Stll Dversty-Optmal? Zhguo Dng, Member, IEEE and H. Vncent Poor, Fellow, IEEE arxv:143.354v1 [cs.it] 3 Mar 214 Abstract Ths paper consders a general energy harvestng cooperatve network wth M source-destnaton (SD pars and one relay, where the relay schedules only m user pars for transmssons. For the specal case of m = 1, the addressed schedulng problem s equvalent to relay selecton for the scenaro wth one SD par and M relays. In conventonal cooperatve networks, the max-mn selecton crteron has been recognzed as a dversty-optmal strategy for relay selecton and user schedulng. The man contrbuton of ths paper s to show that the use of the max-mn crteron wll result n loss of dversty gans n energy harvestng cooperatve networks. Partcularly when only a sngle user s scheduled, analytcal results are developed to demonstrate that the dversty gan acheved by the max-mn crteron s only M+1 2, much less than the maxmal dversty gan M. The max-mn crteron suffers ths dversty loss because t does not reflect the fact that the source-relay channels are more mportant than the relay-destnaton channels n energy harvestng networks. Motvated by ths fact, a few user schedulng approaches talored to energy harvestng networks are developed and ther performance s analyzed. Smulaton results are provded to demonstrate the accuracy of the developed analytcal results and facltate the performance comparson. I. INTRODUCTION Smultaneous wreless nformaton and power transfer (SWIPT has recently receved a lot of attenton. Compared to conventonal energy harvestng technques, SWIPT can be used even f wreless nodes do not have access to external energy sources, such as solar and wnder power. The key dea of SWIPT Z. Dng and H. V. Poor are wth the Department of Electrcal Engneerng, Prnceton Unversty, Prnceton, NJ 8544, USA. Z. Dng s also wth the School of Electrcal, Electronc, and Computer Engneerng, Newcastle Unversty, NE1 7RU, UK.
2 s to collect energy from rado frequency (RF sgnals, and ths new concept of energy harvestng was frst proposed n [1] and [2]. Partcularly by assumng that the recever has the capablty to carry out energy harvestng and nformaton decodng at the same tme, the tradeoff between nformaton rate and harvested energy has been characterzed n [1] and [2]. Motvated by the dffculty of desgnng a crcut performng both energy harvestng and sgnal detecton smultaneously, a practcal recever archtecture has been developed n [3], where two recever strateges, power splttng and tme sharng, have been proposed and ther performance have been analyzed. The concept of SWIPT was ntally studed n smple scenaros wth one source-destnaton par, where the use of co-channel nterference for energy harvestng was consdered n [4] and the combnaton of multple-nput multple-output (MIMO technologes wth SWIPT was nvestgated n [5]. SWIPT has been recently appled to varous mportant communcaton scenaros more complcated than the case wth one source-destnaton par. For example, n [6] the applcaton of SWIPT to multple access channels has been consdered, where a few solutons for system throughput maxmzaton have been proposed. Broadcastng scenaros have been consdered n [7] and [8], where one transmtter s to serve two types of users, energy recevers and nformaton recevers, smultaneously. In [9] the jont desgn of uplnk nformaton transfer and downlnk energy transfer has been consdered, where sophstcated algorthms for energy beamformng, power allocaton and throughput maxmzaton have been proposed. The dea of SWIPT has also been appled to wreless cogntve rado systems, where opportunstc energy harvestng from RF sgnals has been studed n [1]. The applcaton of SWIPT to cooperatve networks s mportant snce the lfetme of the relay batteres can be extended by effcently usng the energy harvested from the relay observatons. In [11] a greedy swtchng approach between data decodng and energy harvestng has been proposed for the case wth one source-destnaton par and one relay. In [12] the outage performance acheved by amplfed-and-forward (AF relayng protocols has been developed, and the use of decode-and-forward (DF strateges has been nvestgated n mult-user energy harvestng cooperatve networks [13]. Relay selecton has been studed
3 n a broadcastng scenaro where energy harvestng was carred out at the destnatons, nstead of relays [14]. The mpact of the random locatons of wreless nodes on the path loss and the outage performance has been characterzed by applyng stochastc geometry n [15]. In conventonal cooperatve networks, the max-mn crteron has been recognzed as a dversty-optmal selecton strategy [16] [18]. Take a DF cooperatve network wth one source-destnaton par and M relays as an example. Provded that the -th relay s used, the capacty of a DF relay channel s mn{log(1+ ρ h 2,log(1+ρ g 2 }, where ρ s the transmt sgnal-to-nose rato (SNR, h s the channel gan between the source and the relay, and g s the channel gan between the relay and the destnaton. Obvously the max-mn crteron,.e. max{mn{ h 2, g 2 },1 M}, s capacty optmal and can acheve the maxmal dversty gan, M. But s ths concluson stll vald when energy harvestng relays are used? The man contrbuton of ths paper s to characterze the performance of the max-mn selecton crteron n energy harvestng cooperatve networks. We frst construct a general framework of energy harvestng cooperatve networks, where M pars of sources and destnatons communcate wth each other va a relay. Among the M user pars, the relay wll schedule m of them to transmt. It s mportant to pont out that the problem of relay selecton for the scenaro wth one source-destnaton par s a specal case of the formulated framework by settng m = 1. When only a sngle user s scheduled, the exact expresson for the outage probablty acheved by the max-mn crteron s developed by carefully groupng the possble outage events and then applyng order statstcs. Based on ths obtaned expresson, asymptotc studes of the outage probablty are carred out to show that the dversty gan acheved by the max-mn crteron s only M+1 2, much less than the full dversty gan, M. The reason for ths loss of the dversty gan s that the max-mn crteron treats the source-relay channels and the relay-destnaton channels equally. However, when an energy harvestng relay s used, t s mportant to observe that the source-relay channels become more mportant. For example, the sourcerelay channels mpact not only the recepton relablty at the relay, but also the relay transmsson power. Recognzng ths fact, a few modfed user schedulng approaches are developed, whch s the second
4 contrbuton of ths paper. Partcularly for the case of m = 1, an effcent user schedulng approach s proposed, and analytcal results are developed to demonstrate that ths approach can acheve the maxmal dversty gan. Ths approach can be extended to the case of m > 1, by applyng exhaustve search. A greedy user schedulng approach s also developed by assumng that the relay always has data to be sent to all the destnatons. The use of ths greedy approach yelds closed-form expressons for the outage probablty and dversty order, whch can be used as an upper bound for the other approaches. Smulaton results are also provded to demonstrate the accuracy of the developed analytcal results and facltate the performance comparson among the addressed user schedulng approaches. II. SYSTEM MODEL Consder a cooperatve communcaton scenaro wth M source-destnaton pars and one energy harvestng relay. The M users compete for the wreless medum, and the relay wll schedule m user pars over 2m tme slots, m M. All the channels are assumed to be ndependent and dentcally (..d. quas-statc Raylegh fadng, and ths ndoor slow fadng model s vald for many applcatons of wreless energy transfer, such as wreless body area networks and smart homes [14] and [15]. In Secton V, the mpact of the path loss and the random locatons of the users on the outage performance wll be studed 1. It s assumed that the relay has access to global channel state nformaton (CSI, whch s mportant for the relay to carry out user schedulng. Durng the j-th tme slot, consder that the -th user par s scheduled to transmt ts message s, where the detals for user schedulng wll be provded n the next two sectons. The power splttng strategy wll be used at the DF relay. Partcularly the relay wll frst drect the observaton flow to the detecton crcut, and then to the energy harvestng crcut f there s any energy left after successful detecton [3] and [13]. Therefore the observaton at the relay s gven by y r = P(1 θ h s +n r, (1 1 Note that when the users are randomly deployed, the effectve channel gans,.e. the combnatons of Raylegh fadng and large scale path loss, can be stll approxmated as ndependent and dentcally exponentally dstrbuted varables [15].
5 where θ s the power splttng factor, P s the transmsson power at the source, h denotes the channel gan from the -th source to the relay, and n r denotes the addtve whte Gaussan nose. As dscussed { } n [13], the optmal value of θ for a DF relay s max 1 ǫ h,, the maxmal value of θ 2 constraned by successful detecton at the relay, where ǫ = 22R 1 P and R denotes the targeted data rate. The power obtaned at the relay after carryng out energy harvestng from the -th user par s gven by P r = ηp [ h 2 ǫ ]+, (2 where η denotes the energy harvestng coeffcent, and [x] + denotes max{x,}. At the (m+j-th tme slot, the relay forwards s to the -th destnaton, and the receve SNR at ths destnaton s gven by SNR = P g 2, (3 where P denotes the relay transmsson power allocated to the -th destnaton, and g denotes the channel gan between the relay and the -th destnaton. Note that P r s not necessarly equal to P, dependng on the used relay strategy, as dscussed n the followng sectons. III. THE PERFORMANCE ACHIEVED BY THE MAX-MIN CRITERION A. User schedulng based on the max-mn crteron In ths secton, the performance acheved by the user schedulng strategy based on the max-mn crteron s studed. Partcularly we wll focus on the case that the relay selects only one user par,.e. m = 1, and more dscussons about the case wth m > 1 wll be provded n the next secton. Note that the scenaro addressed n ths secton can be shown mathematcally the same as the problem of relay selecton for the case wth one source-destnaton par and M relays. Therefore the results obtaned for the addressed schedulng problem wll be also applcable to the max-mn relay selecton cases. Snce only one user par s scheduled, the energy harvested from the -th source wll be used to power the relay transmsson to the -th destnaton,.e. P = P r. The max-mn user schedulng strategy can be descrbed as follows:
6 The relay frst fnds out the worst lnk of each user par. Denote z = mn{ h 2, g 2 }. The user par wth the strongest worst lnk s selected,.e. the -th user par s selected because = argmax{z 1,...,z M }. Provded that the relay can decode the -th source s message correctly, the SNR at the correspondng destnaton s gven by SNR = ηp ( h 2 ǫ g 2. (4 B. Performance evaluaton The outage probablty acheved by the max-mn based schedulng scheme can be wrtten as follows: P o P ( h 2 < ǫ +P ( ( h 2 ǫ g 2 < ǫ 1, h 2 > ǫ, (5 whereǫ 1 = ǫ. Although the outage probablty acheved by the max-mn crteron s shown n a smple term η as n (5, t s challengng to evaluate ths probablty. The reason s that the use of the schedulng strategy has changed the statstcal property of the channel gans. For example, h 2 s no longer exponentally dstrbuted. The densty functon of mn{ h 2, g 2 } can be found by usng order statstcs, and the key step s to restructure the expresson of the outage probablty shown n (5 nto a form to whch the densty functon of mn{ h 2, g 2 } can be appled. In the followng theorem, the exact expresson for the outage probablty acheved by the max-mn scheme s provded. Theorem 1: When a sngle user s scheduled, the outage probablty acheved by the max-mn user schedulng strategy s gven by M ( P o = e ǫ M ( 1 ( 1 e (2 1ǫ (6 2 2 1 ( ( M 1 e +M ( 1 ǫ e (2+2ǫ + e (2+2ǫ e (2+2ǫ e ǫ β(ǫ, 2+1 2+2 + (1 e 2ǫ M ( ( M 1 e +M ( 1 2(+1ǫ e 2(+1ǫ e (2+1ǫ β(ǫ ǫ,, 2 2(+1 where β(y, = ǫ e (2+1y ǫ1 y dy, and ǫ = ǫ+ ǫ 2 +4ǫ 1. 2
7 Proof: See the appendx. The expresson shown n (6 can be used to numercally evaluate the outage probablty acheved by the max-mn schedulng approach, as shown n Secton V. In addton, t can also be used for the analyss of the dversty gan acheved by the max-mn approach, as shown n the followng theorem. Theorem 2: When a sngle user par s scheduled, the dversty order acheved by the max-mn user schedulng approach s M+1 2. Proof: See the appendx. For the addressed topology, there are M ndependent pathes gven M user pars, whch means that the maxmal dversty gan s M. And Theorem 2 ndcates that the max-mn schedulng approach cannot acheve ths maxmum dversty. As a benchmark scheme, recall a conventonal cooperatve network that has the same topology as the one descrbed n Secton II. Wthout loss generalty, let P = P,.e. the relay transmsson power s the same as the source power. It can be easly verfed that the max-mn approach can acheve the optmal dversty gan, M, as shown n the followng. The outage probablty acheved by the max-mn approach s P o = P( h 2 < ǫ+p( g 2 < ǫ, h 2 > ǫ = P( h 2 < ǫ, g 2 > ǫ+p( h 2 < ǫ, g 2 < ǫ+p( g 2 < ǫ, h 2 > ǫ = P(mn{ h 2, g 2 } < ǫ ǫ M, (7 where the last step s obtaned by usng the probablty densty functon (pdf shown n (21 and applyng the hgh SNR approxmaton. Comparng (7 to (2, one can observe that the performance of the max-mn schedulng approach n two system setups s sgnfcantly dfferent, and new effcent user schedulng strateges are needed for energy harvestng cooperatve networks. IV. MODIFIED USER SCHEDULING STRATEGICS A. Schedulng a sngle user par A straghtforward approach of user schedulng for the energy harvestng scenaro s descrbed as follows:
8 Construct a subset of user pars contanng all the destnatons whose source nformaton can be decoded correctly at the relay. Denote ths subset as S { S : h 2 ǫ}. Select a destnaton from S to mnmze the outage probablty of the relay transmsson. Denote the ndex of the selected user by,.e. = argmax{( h 2 ǫ g 2, S}. The outage probablty acheved by ths user schedulng strategy can be expressed as follows: P o P( S = +P ( ( h 2 ǫ g 2 < ǫ 1, S > (8 M = P( S = + P ( ( h 2 ǫ g 2 < ǫ 1 S = n P( S = n, }{{} n=1 T 1 where S denotes the cardnalty of the set. Denote x = h 2, and order x as x (1 x (M. The probablty of P( S = n can be calculated as follows: P( S = n = P(x (M n < ǫ,x (M n+1 > ǫ (9 = M! ( 1 e ǫ M n e nǫ, (M n!n! for n M, where the last equaton s obtaned by applyng the jont pdf of x (M n and x (N n+1 [19] and [2]. On the other hand T 1 can be smply expressed as follows: T 1 = [P((x ǫy < ǫ 1, S, S = n] n, (1 where y = g 2. In the followng we frst consder the case of n 1. The condtons of T 1, S and S = n, mply x ǫ, whch means that the condtonal CDF of x s gven by F x S, S 1(x = e ǫ e x, (11 e ǫ for x ǫ. The two condtons, S and S = n, do not affect y whch s stll exponentally dstrbuted. Therefore the factor T 1 can be calculated as follows: ( e T 1 = (E ǫ e ǫ 1 n y ǫ y (12 e ǫ = (1 2 ǫ 1 K 1 (2 ǫ 1 n,
9 where K n ( denotes the modfed Bessel functon of the second knd. Recall that xk 1 (x 1+ x2 2 ln x 2 for x, [13], whch means T 1 ǫln 1. The overall outage probablty can be approxmated as follows: ǫ P o = ( M 1 e ǫm + (1 2 ǫ 1 K 1 (2 ǫ 1 n M! ( 1 e ǫ M n e nǫ (13 (M n!n! n=1 M ( ǫ M + ǫ n ln 1 n M! ǫ (M n!n! ǫm n. n=1 When ǫ, t s straghtforward to show logpo logǫ M, whch results n the followng lemma. Lemma 1: The proposed user schedulng strategy can acheve the full dversty gan M. Compared to the max-mn based approach, the proposed schedulng strategy can acheve a larger dversty gan. The reason for ths performance mprovement s that the source-relay channels have been gven a more mportant role for use schedulng, compared to the relay-destnaton channels. Partcularly the sourcerelay channels have been consdered when formng S and also selectng the best user from the set, whereas the relay-destnaton channels affect only the second step. B. Schedulng m user pars The approach proposed n the prevous subsecton can be extended to the case of schedulng m user pars, as descrbed n the followng. Construct a subset of user pars, S, as defned n Secton IV-A. Fnd all possble combnatons of the users n S, denoted by {π 1,,π ( }, where each set mn{m, S } contans mn{m, S } users,.e. π = {π (1,...,π (mn{m, S }}. For each possble combnaton, π, 1 ( S mn{m, S } Calculate the accumulated power obtaned from energy harvestng, mn{m, S } j=1 P rπ (j. Dstrbute the overall power among m destnatons equally,.e. P = mn{m, S } j=1 P rπ (j mn{m, S }. Fnd the worst outage performance among the mn{m, S } users n π, denoted by P o,π. Select the combnaton whch mnmze the worst user outage performance,.e. = argmn{p o,π1,,p o,π ( mn{m, S } }.
1 Ths schedulng approach s to exhaustvely search all possble combnatons of the S user pars, and one combnaton wll be selected f t can mnmze the outage probablty for the worst user case. Provded that there s a large number of users to be scheduled, the complexty of ths exhaustve search scheme can be nfeasble due to the large number of the possble combnatons. Note that n ths paper, we consder only the equal power allocaton strategy, whereas other power allocaton strateges, such as the sequental water fllng scheme proposed n [13], can also be appled. It s dffcult to analyze the performance acheved by the exhaustve search approach, snce the channel gans from dfferent combnatons mght be correlated. Instead, we wll propose a greedy approach whch s applcable to delay tolerant networks, and also serves as an upper bound for the system performance. C. Greedy user schedulng approach Frst order all the source-relay channels and the relay-destnaton channels,.e. h (1 2... h (M 2 and g (1 2... g (M 2. The greedy user schedulng approach can be descrbed as follows: Construct a subset of user pars, S, as defned n Secton IV-A. Schedule mn{m, S } sources wth the best source-relay channel condtons durng the frst mn{m, S } tme slots,.e. the mn{m, S } sources wth the followng channels, h (M mn{m, S }+1 2... h (M 2. Calculate the accumulated power obtaned from energy harvestng, mn{m, S } j=1 P r(m j+1. Schedule mn{m, S } destnatons wth the best relay-destnaton channel condtons durng the second mn{m, S } tme slots,.e. the mn{m, S } destnatons wth the followng channels, g (M mn{m, S }+1 2... g (M 2, wth equally allocated transmsson power, denoted by P mn{m, S } = mn{m, S } j=1 P r(m j+1 mn{m, S }. Note that the scheduled destnatons are not necessarly the partners of the scheduled sources, so ths greedy approach assumes that the relay always has data to be transmtted to all the destnatons.
11 Based on the above strategy descrpton, the outage probablty at the -th best destnaton, 1 mn{m, S }, can be wrtten as follows: P o P( S = + M P ( P mn{m, S } g (M +1 2 < (2 2R 1 S = n P( S = n. (14 n=1 And the followng lemma provdes the exact expresson of the above outage probablty. Lemma 2: The outage probablty acheved by the greedy user schedulng approach s gven by: P o P( S = + m T 2 P( S = n+ n=1 where P( S = n s defned n (9, T 2 = ( M M k= T 3 = M! M (M!( 1! l= +2b mǫ1 (l+ k K 1+ k+1 1 (2 ǫ 1 (l+(m+k +1 m and a j,k = ( 1m j m m j+1 (k+1 m j+1. Proof: See the appendx. ( M ( 1 k k k+ M n=m+1 ( M ( 1 l (1 T l l+ 4, T 4 = n m 1 k= d m,k, d m,k = ( n! (n m 1!m!m T 3 P( S = n, (15 ( K (n 1! n 2 (k +nǫ 1, 1 2((k+nǫ 1 n 2 ( m 2a j,k (mǫ 1 (l+ 2K j j (2 mǫ 1 (l+ j=1 (j 1! ( n m 1 k ( 1 k, b k = ( 1 m mm, (k+1 m Although the outage probablty expresson n Lemma 2 can be used for numercal studes, ths form s qute complcated and cannot be used for analyzng dversty gans. For the specal case of m = 1, asymptotc studes can be carred out and the achevable dversty gan can be obtaned, as shown n the followng lemma. Lemma 3: When schedulng only a sngle user par,.e. m = 1, the dversty gan acheved by the greedy user schedulng approach s M. Proof: See the appendx. The fact that the greedy user schedulng approach can acheve the full dversty gan s not surprsng, snce the greedy approach outperforms the dversty-optmal one descrbed n Secton IV-A. V. NUMERICAL RESULTS In ths secton, computer smulatons wll be carred out to evaluate the performance of the user schedulng approaches addressed n ths paper. To smplfy clarfcatons, we term the user schedulng
12 approaches descrbed n Secton IV-A, IV-B and IV-C as Approach I, Approach II, and Approach III, respectvely. We frst focus on the scenaro where only a sngle user s scheduled. In Fg. 1 the accuracy of the developed analytcal results about the outage probablty shown n Theorem 1, (13, and Lemma 2, s verfed by usng smulaton results, where the targeted data rate s R = 4 bts per channel use (BPCU, and the energy harvestng effcency coeffcent s η = 1. As can be seen from the fgure, the developed analytcal results match the smulaton results exactly. In Fg. 2 the outage probabltes acheved by dfferent user schedulng approaches are examned wth more detals, where analytcal results are used to generate the fgure. As a benchmark, the scheme wth a random selected user s also shown n the fgure, and ts outage performance s the worst among all the schedulng approaches. On the other hand, Approach III, the greedy user schedulng approach, can acheve the best outage performance. The maxmn schedulng approach can outperform random relayng, snce ts dversty gan can be mproved when more users jon n the competton, as shown n Theorem 2. However, t wll result n some performance loss compared to Approach I and Approach III, snce t cannot acheve the full dversty gan, as ndcated n Theorem 2. 1 1 1 1 2 Outage probablty 1 3 1 4 Max mn, M=3 Appoarch I, M=3 Approach III, M=3 Max mn, M=6 Appoarch I, M=6 Approach III, M=6 1 5 Sold lnes: Smulatons Dashed lnes: Analytcal resutls 1 6 1 15 2 25 3 35 4 SNR Fg. 1. Analytcal results vs computer smulatons. Only one user par s scheduled, η = 1. The targeted data rate s R = 4 BPCU. Snce the man focus of ths paper s to study the performance of the max-mn user schedulng approach,
13 1 1 1 Outage probablty 1 2 1 3 1 4 M=6 M=3 1 5 Random relayng Max mn approach Approach I 1 6 Approach III 1 15 2 25 3 35 4 SNR Fg. 2. Comparson of varous user schedulng approaches. Only one user par s scheduled. η = 1. The targeted data rate s R = 2 BPCU. Fg. 3 s provded n order to closely examne the dversty order acheved by ths approach. Partcularly the analytcal results developed n Theorem 1 are used to generate the curves of outage probabltes. To clearly demonstrate achevable dversty gans, auxlary lnes wth the dversty order of M+1 2 are also shown as a benchmark. As can be seen from the fgure, the outage probablty curves for the max-mn approach are always parallel to the benchmarkng curves. Recall that the dversty order s ndcated by the slope of an outage probablty curve. Therefore Fg. 3 confrms that the dversty order acheved by the max-mn approach s M+1 2, as ndcated by Theorem 2. The reason for ths loss of dversty gans s that the max-mn approach treats the source-relay channels and the relay-destnaton channels equally mportant when user schedulng s carred out. However, when an energy harvestng relay s used, the source-relay channels become more mportant, snce they affect not only the transmsson relablty durng the frst phase, but also the transmsson power for the second phase. In Fgs. 4 and 5 we wll focus on the scenaro when multple user pars are scheduled. Partcularly, n Fg. 4 we compare the outage performance acheved by the three schemes, the max-mn approach and the two approaches proposed n Secton IV. The total number of the user pars s M = 1 and two user pars wll be scheduled. Snce the scheduled users experence dfferent outage performance, n the fgure we show the outage performance for the user wth the strongest SNR and also the user wth the
14 1 1 2 Outage probablty 1 4 1 6 1 8 1 1 Max mn Approach, M=3 3 SNR 2 Max mn Approach, M=4 5 SNR 5/2 Max mn Approach, M=5 1 SNR 3 Max mn Approach, M=6 2 SNR 7/2 1 12 1 15 2 25 3 35 4 SNR Fg. 3. Verfcaton of the dversty order for the max-mn schedulng approach. Only one user par s scheduled. η = 1. The targeted data rate s R = 2 BPCU. 1 1 1 Max mn approach, the worst case Approach II, the worst case Approach III, the worst case Max mn approach, the best case Approach II, the best case Approach III, the best case Outage probablty 1 2 1 3 1 4 1 15 2 25 SNR Fg. 4. Comparson of varous user schedulng approaches. The total number of user pars s M = 1, η = 1 and two user pars are scheduled, m = 2. weakest SNR. As can be observed from the fgure, Approach III, the greedy user schedulng approach, can acheve the best outage performance, and the max-mn approach acheves the worst performance. But t s worthy to pont out that Approach II outperforms the max-mn approach at a prce of hgh computatonal complexty, snce Approach II needs to enumerate all possble combnatons of the user pars. In Fg. 5, we evaluate the accuracy of the analytcal results developed n Lemma 2, by comparng the outage probablty calculated usng (15 to computer smulatons. The total number of the user pars
15 1 1 1 Outage probablty 1 2 1 3 1 4 1 5 R=3 BPCU R= 5 BPCU Best user case, smulaton Worst user case, smulaton Best user case, analytcal Worst user case, analytcal 1 6 1 15 2 25 3 35 4 SNR Fg. 5. Analytcal results vs computer smulatons. The total number of user pars s M = 6, η = 1 and three user pars are scheduled, m = 3. s M = 6 and three user pars wll be scheduled. As can be observed from the fgure, the developed analytcal results match the computer smulatons exactly. Fnally we present some smulaton results when η < 1 and the large scale path loss s consdered. Partcularly consder a dsk wth the relay at ts center and ts dameter as 4 meters. The M pars of sources and destnatons are unformly deployed n ths dsc, and the used path loss exponent s 2. In Fg. 6 and Fg. 7, the performance of the user schedulng approaches for the cases of m = 1 and m = 2 are shown, respectvely. As can be seen from Fg. 6, the use of the user schedulng approaches can mprove the system performance compared to the random relayng scheme. Another observaton from both fgures s that, among all the opportunstc schedulng approaches, the max-mn approach acheves the worst performance, and the greedy approach outperforms the other user schedulng approaches, whch s consstent to the prevous fgures. VI. CONCLUSIONS In ths paper, we consdered an energy harvestng cooperatve network wth M source-destnaton pars and one relay, where the relay schedules only m user pars for transmssons. It s mportant to pont out that for the specal case of m = 1, the addressed schedulng problem s the same as relay selecton for the
16 1 1 1 Outage probablty 1 2 1 3 1 4 Random relayng, R=2BPCU Max mn approach, R=2BPCU Approach I, R=2BPCU Approach III, R=2BPCU Random relayng, R=3BPCU Max mn approach, R=3BPCU Approach I, R=3BPCU Approach III, R=3BPCU 1 15 2 25 3 35 SNR Fg. 6. Comparson of varous user schedulng approaches. η =.5. The total number of user pars s M = 6, and one user par s scheduled, m = 1. 1 1 1 Outage probablty 1 2 1 3 1 4 Max mn approach, the best case Approach II, the best case Approach III, the best case Max mn approach, the worst case Approach II, the worst case Approach III, the worst case 1 15 2 25 3 35 SNR Fg. 7. Comparson of varous user schedulng approaches. η =.5. The total number of user pars s M = 6, and two user pars are scheduled, m = 3. The targeted data rate s R = 2 BPCU. scenaro wth one source-destnaton par and M relays. The man contrbuton of ths paper s to show that the use of the max-mn crteron wll result n loss of dversty gans, when an energy harvestng relay s employed. Partcularly when only one user s scheduled, analytcal results have been developed to demonstrate that the dversty gan acheved by the max-mn crteron s only M+1 2, much less than the maxmal dversty gan M. Motvated by ths performance loss, a few user schedulng approaches talored to energy harvestng networks have been proposed and ther performance s analyzed. Smulaton
17 results have been provded to demonstrate the accuracy of the developed analytcal results and facltate the performance comparson. When developng user schedulng approaches, only recepton relablty s consdered, and t s assumed that the network s delay tolerant. It s a promsng future drecton to study how to acheve a balanced tradeoff between recepton relablty and user delay. REFERENCES [1] L. R. Varshney, Transportng nformaton and energy smultaneously, n Proc. IEEE Int. Symp. Inf. Theory (ISIT, Toronto, Canada, Jul. 28. [2] P. Grover and A. Saha, Shannon meets Tesla: wreless nformaton and power transfer, n Proc. IEEE Int. Symp. Inf. Theory (ISIT, Austn, TX, Jun. 21. [3] X. Zhou, R. Zhang, and C. K. Ho, Wreless nformaton and power transfer: Archtecture desgn and rate-energy tradeoff, IEEE Trans. Wreless Commun., vol. 61, no. 11, pp. 4754 4767, Nov. 213. [4] L. Lu, R. Zhang, and K.-C. Chua, Wreless nformaton transfer wth opportunstc energy harvestng, IEEE Trans. Wreless Commun., vol. 12, no. 1, pp. 288 3, Jan. 213. [5] Z. Xang and M. Tao, Robust beamformng for wreless nformaton and power transmsson, IEEE Wreless Commun. Letters, vol. 1, no. 4, pp. 372 375, Jan. 212. [6] H. Ju and R. Zhang, Throughput maxmzaton for wreless powered communcaton networks, IEEE J. Sel. Areas Commun., to appear n 214, (avalable at http://arxv:134.7886. [7] R. Zhang and C. K. Ho, MIMO broadcastng for smultaneous wreless nformaton and power transfer, IEEE Trans. Wreless Commun., vol. 12, no. 5, pp. 1989 21, May 213. [8] K. Huang and E. Larsson, Smultaneous nformaton and power transfer for broadband wreless systems, IEEE Trans. Sgnal Process., vol. 61, no. 23, pp. 5972 5986, Dec. 213. [9] L. Lu, R. Zhang, and K.-C. Chua, Mult-antenna wreless powered communcaton wth energy beamformng, IEEE Trans. Commun., (submtted, Avalable on-lne at arxv:1312.145. [1] S. Lee, R. Zhang, and K. Huang, Opportunstc wreless energy harvestng n cogntve rado networks, IEEE Trans. Wreless Commun., vol. 12, no. 9, pp. 4788 4799, Jul. 213. [11] I. Krkds, S. Tmotheou, and S. Sasak, RF energy transfer for cooperatve networks: Data relayng or energy harvestng? IEEE Commun. Letters, no. 11, pp. 1772 1775, 212. [12] A. A. Nasr, X. Zhou, S. Durran, and R. A. Kennedy, Relayng protocols for wreless energy harvestng and nformaton processng, IEEE Trans. Wreless Commun., vol. 12, no. 7, pp. 3622 3636, Jul. 213. [13] Z. Dng, S. M. Perlaza, I. Esnaola, and H. V. Poor, Power allocaton strateges n energy harvestng wreless cooperatve networks, IEEE Trans. Wreless Commun., to appear n 214 (avalable at http://arxv.org/abs/137.163.
18 [14] D. S. Mchalopoulos, H. A. Suraweera, and R. Schober, Relay selecton for smultaneous nformaton transmsson and wreless energy transfer: A tradeoff perspectve, Avalable on-lne at arxv:133.1647. [15] Z. Dng and H. V. Poor, Cooperatve energy harvestng networks wth spatally random users, IEEE Sgnal Process. Lett., vol. 2, no. 12, pp. 1211 1215, Dec. 213. [16] I. Krkds, J. Thompson, S. McLaughln, and N. Goertz, Max-mn relay selecton for legacy amplfy-and-forward systems wth nterference, IEEE Trans. Wreless Commun., vol. 51, pp. 316 327, Jun. 29. [17] S. Talwar, Y. Jng, and S. Shahbazpanah, Jont relay selecton and power allocaton for two-way relay networks, IEEE Sgnal Process. Lett., vol. 18, no. 2, pp. 91 94, Feb. 211. [18] L. Song, Relay selecton for two-way relayng wth amplfy-and-forward protocols, IEEE Trans. Vehcular Technology, vol. 6, no. 4, pp. 1954 1959, May 211. [19] H. A. Davd and H. N. Nagaraja, Order Statstcs. John Wley, New York, 3rd ed., 23. [2] Z. Dng, Y. Gong, T. Ratnarajah, and C. Cowan, On the performance of opportunstc cooperatve wreless networks, IEEE Trans. Commun., vol. 56, pp. 1236 124, Aug. 28. [21] I. S. Gradshteyn and I. M. Ryzhk, Table of Integrals, Seres and Products, 6th ed. New York: Academc Press, 2. [22] K. Alam and K. T. Wallenus, Dstrbuton of a sum of order statstcs, Scandnavan Journal of Statstcs, vol. 6, no. 3, pp. 123 126, 1979. [23] Z. Dng and H. V. Poor, The use of spatally random base statons n cloud rado access networks, IEEE Sgnal Process. Lett., vol. 2, no. 11, pp. 1138 1141, Nov 213. APPENDIX Proof of Theorem 1 : To smplfy notaton, defne x = h 2 and y = g 2, and the outage probablty n (5 can be expressed as follows: P o P(x < ǫ+p((x ǫy < ǫ 1,x > ǫ. (16 The schedulng strategy has changed the statstcal property of x and y, but the densty functon of mn{x,y} can be found smply by applyng order statstcs. To use such a densty functon, we need to frst rewrte the outage probablty as follows: P o = P(x < ǫ,x > y+p((x ǫy < ǫ 1,x > ǫ,x > y (17 +P(x < ǫ,x < y+p((x ǫy < ǫ 1,x > ǫ,x < y.
19 Convertng the jont probabltes to condtonal probabltes, the outage probablty s gven by P o = P(x < ǫ x > yp(x > y+p((x ǫy < ǫ 1,x > ǫ x > yp(x > y (18 +P(x < ǫ x < yp(x < y+p((x ǫy < ǫ 1,x > ǫ x < yp(x < y. Snce the ncomng and outgong channels at the relay are ndependent and dentcally dstrbuted, we have P(x > y = P(x < y = 1. Consequently the outage probablty can be expressed as n the followng 2 form: P o = 1 2 E y x>y{p(x < ǫ x > y} + }{{} Q 1 + 1 2 E x x<y{p(x < ǫ x < y} + }{{} Q 3 1 2 E y x>y{p((x ǫy < ǫ 1,x > ǫ x > y} (19 }{{} Q 2 1 2 E y x>yp((x ǫy < ǫ 1,x > ǫ x < y }{{} Q 4, where E{ } denotes the expectaton operaton. The ratonale to have the above expresson s followng. Take Q 1 as an example. Q 1 can be calculated n two steps. The frst step s to calculate Q 1 by treatng y as a constant and usng the condton x > y. The second step s to calculate the expectaton of the probablty by usng the densty functon of y. Snce x > y, y = mn{x,y}, and the densty functon of y can be found easly. In the followng the four terms Q wll be evaluated ndvdually. 1 Calculatng Q 1 : We start from the calculaton of Q 1, the frst terms n (19. In partcular, Q 1 can be expressed as follows: Q 1 = ǫ ǫ y f x x>y,y (xdxf y x>y (ydy, (2 where f x x>y,y (x s the pdf of x condtoned on a fxed y and x > y, and f y x>y (y s the pdf of y also condtoned on x > y. To fnd the two condtonal pdfs, we frst defne x = h 2 and y = g 2, z = mn{x,y }, and z = mn{z,1 M}. From order statstcs [19], the pdf of z s f z (z = 2e 2z, and the pdf of z can be found as follows: f z (z = 2Me 2z( 1 e 2z. (21
2 Condtoned on x > y, the pdf of y s the same as z,.e. f y x>y (y = f z (y. On the other hand, condtoned on a fxed y and x > y, the cumulatve dstrbuton functon (CDF of x can be found as follows: F x x>y,y (x = e y e x e y, (22 where the factor e y at the denomnator s to ensure F x x>y (x 1 when x. By usng the obtaned condtonal pdfs, Q 1 can be calculated as follows: Q 1 = ǫ ǫ z f x x>y (xdxf z (zdy (23 ǫ ( = e ǫ 1 e 2z M e z dz. By applyng bnomal expansons, we obtan the followng: M ( M ( 1 Q 1 = e ǫ ( 1 e (2 1ǫ. (24 2 1 2 Calculatng Q 2 : Recall that Q 2 = E y x>y {P((x ǫy < ǫ 1,x > ǫ x > y}. The condtonal densty functons, f x x>y,y (x and f y x>y (y, obtaned n (21 and (22 can be used agan. An mportant step to calculate Q 2 s to determne the doman of ntegraton. The constrans, x > y and x < ǫ 1 y +ǫ, mply that y < ǫ 1 y +ǫ. Together wth the addtonal constrant, x > ǫ, the ntegraton doman for Q 2 s gven by y < x < ǫ 1 y +ǫ, f ǫ < y < ǫ, (25 ǫ < x < ǫ 1 y +ǫ, f y < ǫ where ǫ ǫ+ ǫ 2 +4ǫ 1 s the postve root of y 2 ǫy ǫ =, due to the constrant y < ǫ 1 2 y +ǫ. Wth the obtaned ntegraton doman, Q 2 can be rewrtten as follows: Q 2 = = ǫ ǫ ǫ+ 1y + ǫ ǫ ǫ ǫ+ 1y ǫ ǫ y f x x>y (xdxf y x>y (ydy (26 ( e ǫ e ǫ ǫ 1 y e y f x x>y (xdxf y x>y (ydy f y x>y (ydy }{{} Q 21 ( ǫ e y e ǫ ǫ 1 y + f y x>y (ydy. ǫ e y } {{ } Q 22
21 Now applyng the condtonal pdf of y, the frst factor, Q 21, n the above equaton can be expressed as follows: Q 21 = 2Me ǫ ǫ (1 e ǫ 1 y e y( 1 e 2y dy (27 ( ( M 1 1 e = 2Me ( 1 ǫ (2+1ǫ 2+1 Smlarly the factor Q 22 can be calculated as follows: Q 22 = 2M = 2M ǫ ǫ ǫ e (2+1y ǫ 1 y dy. (e y e ǫ ǫ 1 y e y( 1 e 2y dy (28 ( M 1 ( 1 ( e (2+2ǫ e (2+2ǫ 2+2 e ǫ ǫ By combnng (27 and (28, the factor Q 2 can be expressed as follows: ǫ e (2+1y ǫ 1 y dy. Q 2 = 2M e ǫ ǫ ( ( M 1 e ( 1 ǫ e (2+2ǫ e (2+1y ǫ 1 y dy. 2+1 + e (2+2ǫ e (2+2ǫ 2+2 (29 3 Calculatng Q 4 : Recall that Q 4 = E y x>y P((x ǫy < ǫ 1,x > ǫ x < y. Agan t s mportant to determne the ntegraton doman of Q 4. Partcularly, the ntegral constrants, y < ǫ 1, x > ǫ and x < y, x ǫ mply the negraton doman of x < y < ǫ 1 x ǫ and ǫ < x < ǫ, where the nequalty of x < ǫ s due to x < ǫ 1 x ǫ,.e. x2 ǫx ǫ 1 <. By applyng the obtaned ntegraton doman, Q 4 s calculated as follows: Q 4 = = ǫ ǫ ǫ ǫ ǫ 1 x ǫ f y x<y,x (ydyf x x<y (xdx (3 x (e x e ǫ 1 x ǫ e x f x x<y (xdx, where the last equaton follows from the symmetry of ncomng and outgong channels,.e. f y x<y,x (y = f x y>x,y (x. Smlarly we have f x x<y (x = f y x>y (y, whch yelds the followng expresson of Q 4 : Q 4 = 2M ǫ ǫ ( M 1 = 2M ( 1 e x ǫ 1 x ǫ e 2x( 1 e 2x dx (31 ( 1 ( e 2(+1ǫ e 2(+1ǫ 2(+1 ǫ ǫ e (2+1x ǫ 1 x ǫ dx.
22 On the other hand, Q 3 can be easly calculated as Q 3 = F z (ǫ = (1 e 2ǫ M, where F z (z s the CDF correspondng to the pdf n (21. Therefore, the overall outage probablty can be expressed as M ( P o = e ǫ M ( 1 ( 1 e (2 1ǫ 2 2 1 ( ( M 1 e +M ( 1 ǫ e (2+2ǫ 2+1 + (1 e 2ǫ M ( M 1 +M 2 + e (2+2ǫ e (2+2ǫ 2+2 ( 1 ( e 2(+1ǫ e 2(+1ǫ 2(+1 ǫ e ǫ e (2+1y ǫ 1 y dy e (2+1ǫ ǫ ǫ e (2+1x ǫ 1 x dx, and the proof of the theorem s completed. Proof of Theorem 2 : To smplfy the analytcal development, we let η = 1, whch means ǫ 1 = ǫ. Note that ths smplfcaton has no mpact on the developed analytcal results, snce the dversty order s obtaned at hgh SNR. As shown n (19, the outage probablty can be expressed as P o = 1 4 2 l=1 Q l. In the followng the asymptotc study for the four terms wll be carred out ndvdually. 1 Asymptotc study of Q 1 : The am of the asymptotc study s to convert Q 1 n a form of tǫ d, where t should be a constant, not a functon of ǫ, and d wll be used to determne the dversty order. By applyng seres expanson of exponental functons, Q 1, the frst term n (19, can be expressed as follows: M ( ( M ( 1 Q 1 = e ǫ ( 1 k (2 1 k ǫ k 1 2 1 k! k= = e ǫ ( 1 k ǫ k M ( M ( 1 (2 1 k 1. k! k=1 (32 Compared to the expresson of Q 1 n (6, the above form s more complcated, but facltates the asymptotc studes as shown n the followng. Recall the followng two propertes about the sums of bnomal coeffcents: [21] M ( M ( 1 j =, (33 for j (M 1, and M ( M ( 1 M = ( 1 M M!. (34
23 These prospertes are useful to remove the terms at the order of ǫ d, d < M+1 2, from Q 1, as descrbed n the followng. To make the above propertes applcable, we rewrte Q 1 as follows: Q 1 = e ǫ ( 1 k ǫ k k! k=1 M ( M k 1 ( 1 j= ( k 1 j ( 1 k 1 j 2 j j. (35 All the terms wth j for j (M 1 can be removed because of (33. At hgh SNR,.e. ǫ, all the factors wth ǫ k for k (M +2 can be also gnored. So the domnant factor of Q 1 wll be the one at the order of ǫ M+1. By applyng (34, Q 1 can be approxmated as follows: Q 1 ( 1M+1 ǫ M+1 (M +1! M = ( 1M+1 ǫ M+1 2 M ( 1 M M! = 2M ǫ M+1 (M +1! M +1. ( M ( 1 2 M M (36 Therefore the frst factor of the outage probablty expresson n (19 s at the order of ǫ M+1. 2 Asymptotc study of Q 2 : The approxmaton of Q 2 s more dffcult than that of Q 1, snce Q 2 contans an ntegral whch cannot be expressed analytcally. As shown n (29, Q 2 can be re-wrtten as follows: ( M 1 Q 2 = 2M ( 1 e ǫ e (2+2ǫ ( M 1 +2M 2+1 }{{} Q 21 ( M 1 2Me ǫ ( 1 ǫ } {{ } Q 23 ( 1 e (2+2ǫ e (2+2ǫ 2+2 } {{ } Q 22 e (2+1y ǫ y dy. (37 Agan by applyng the propertes n (33 and (34, Q21 can be approxmated as follows: Q 21 = e ǫ ( M 1 e ǫ2 ǫ M M. ( 1 +1 k=1 ( 1 k k! ǫ k k 1 j= ( k 2 j j (38 j Smlarly the factor Q 22 can be approxmated as follows: Q 22 = ( 1M M! ( M 1 ( 1 ( 2 ( ǫ M ǫ M k=1 ( 1 k 2 k 1( ǫ k ǫ k k! ( k 1 j j k 1 j= ( 1 (M 1! = 2( ǫ M ǫm. M (39
24 Dfferent from Q 21 and Q 22, t s dffcult to drectly fnd the the closed form of the asymptotc expresson for the term Q 23. Instead, we wll frst develop the upper and lower bounds on Q 23 and then show that they converge at hgh SNR. Observe that for the ntegral of Q 23, ǫ e (2+1y ǫ y dy, the range of y s from to ǫ, so y at hgh SNR. Therefore the term n the ntegral, e (2+1y, can be approxmated at hgh SNR. Ths observaton motvates us to rewrte Q 23 as follows: ( ( M 1 ǫ Q 23 = ( 1 ( 1 k (2+1 k y k e ǫ y dy (4 k! k= ( ǫ ( 1 k y k ( M 1 = ( 1 (2+1 k e ǫ y dy. k! k= By usng the propertes n (33 and (34, Q23 can be approxmated as follows: Q 23 = ǫ ǫ ( ( 1 y (M 1! 2 y e ǫ y dy Q23, 2 ( 1 (M 1! e ǫ y dy (41 where the approxmaton follows from the fact that y ǫ o and ǫ at hgh SNR. To obtan the upper and lower bounds on Q 23, the use of the nequaltes for exponental functons yelds the followng: 1 ǫ y ǫ 1 e y 1+ ǫ, y for y ǫ. Now the upper bound of Q 23 can be computed as follows: ǫ Q 23 2 y 1 = 2 M =1 ( M 1+ ǫ y dy (42 ( 1 M ǫ M (ǫ +ǫ ǫ +2 ( 1 M ǫ M ln ǫ +ǫ. ǫ At hgh SNR, ǫ, and ǫ ǫ 1 2. Therefore the domnant factors n the upper bound of Q 23 are the terms wth = M and = M 1, whch means Q 23 2 ( (ǫ +ǫ M ǫ M M ǫ M(ǫ +ǫ Mǫ. (43 M 1
25 Combnng (38, (39 and (43, Q 2 can be lower bounded as follows: Q 2 = 2M Q 21 +2M Q 22 2Me ǫ Q23 (44 2 M ǫ M +2 M ( ǫ M ǫ M 2 M (ǫ +ǫ M +2 M ǫ M (a = +2 M ǫ M2 (ǫ +ǫ M 1 2 M ( ǫ M ǫ M Mǫǫ 2 M M 1 ǫǫ ǫ M+1 2, 2 M ǫ M2 ǫ M 1 +ǫ M2 ǫ M 1 where (a s obtaned by keepng only the terms at the order of ǫ M and ǫǫ. The lower bound of Q 23 can be obtaned as follows: ǫ ( Q 23 2 y 1 ǫ dy (45 y ( ǫ = 2 M ǫ ǫ. M M 1 Combnng (45 wth (38 and (39, the upper bound of Q 2 can be asymptotcally shown n the followng: Q 2 = 2M Q 21 +2M Q 22 2Me ǫ Q23 (46 2 M ǫ M +2 ( M ǫ M ǫ M ( 2 M ǫ M ǫ Mǫ M 1 = M2M M 1 ǫǫ ǫ M+1 2. As can be observed from (46 and (44, the upper and lower bounds converge at hgh SNR, whch mples Q 2 ǫ M+1 2. (47 3 Asymptotc study of Q 4 : Frst rewrte Q 4 n the followng expresson: ( M 1 Q 4 = 2M ( 1 e 2(+1ǫ e 2(+1ǫ 2(+1 } {{ } Q 41 ( M 1 ǫ ǫ ( 1 e (2+1ǫ 2M e (2+1x ǫ x dx. }{{} Q 42 (48
26 Comparng (48 to (37, we observe that Q41 s the same as Q 22, and therefore can be approxmated smlarly as follows: Q 41 = Q 22 2( ǫ M ǫm. (49 M Smlar to Q 23, the term Q 42 also contans an ntegral whose analytcal closed-form expresson cannot be found. Followng the prevous steps, we can frst use the seres expanson of e (2+1(ǫ+x to get the followng: Q 42 = = ǫ ǫ ǫ ǫ ( M 1 ( 1 k k= k! ( 1 k= ( M 1 ( 1 k (2+1 k (ǫ+x k e ǫ x dx (5 k! ( k ( k ( 1 2 j j (ǫ+x k e ǫ x dx. j j= And by usng the propertes n (33 and (34, we obtan Q 42 = ǫ ǫ ǫ ǫ ( 1 ( 2 ( 1 (M 1! (ǫ+x e ǫ x dx (51 (M 1! 2 (ǫ+x e ǫ x dx Q42. Agan applyng the upper bound of exponental functons, we have Q 42 ǫ ǫ 2 (ǫ+x 1 1+ ǫ x M 2 ( M 2 = 2 ǫ M 2 (ǫ ǫ +2 +2 2 (ǫ ǫ M M ǫ M+1 2. dx (52 By subsstng ths upper bound to the expresson of Q 4, the lower bound of Q 4 s gven by Q 4 2 M ( ǫ M ǫm 2 M (ǫ ǫ M 2 M Mǫǫ. (53 On the other hand, the lower bound of Q 42 can be expressed as follows: Q 42 ǫ ǫ ( 2 x 1 ǫ x = 2 ( (ǫ ǫ M M ǫ(ǫ ǫ M 1 dx (54.
27 Therefore the upper bound of Q 4 can be shown as follows: Q 4 2 ( ( M ǫ M ǫ M 2 M (ǫ ǫ M ǫ M(ǫ ǫ 2 M ǫ M 2M ( ǫ M Mǫǫ = M2 2 M M 1 ǫǫ ǫ M+1 2, Mǫǫ M 1 M 1 (55 where the approxmaton s carred out by keepng only the terms at ǫ M and ǫ ǫ. Combnng (52 and (55, one can observe that the upper and lower bounds converge at hgh SNR, and the followng concluson can be obtaned Q 4 ǫ M+1 2. (56 Applyng the seres expanson of exponental functons, Q 3 can be smply approxmated as Q 3 2 M ǫ M. Therefore the asymptotc expresson for the overall outage probablty can be obtaned as follows: P o = 1 2 4 Q 4 (57 =1 ǫ M+1 +2ǫ M+1 2 +ǫ M ǫ M+1 2, and the proof of the theorem s completed. Proof of Lemma 2 : Based on the equal power allocaton strategy, the power allocated to each destnaton s gven by 1 m m =1 Pη( x (M +1 ǫ f M n m 1 n n =1 Pη( x (M +1 ǫ f 1 n m 1 Therefore the overall outage probablty wll be ( m n ( P o P( S = + P g (M +1 2 x(m j+1 ǫ < nǫ 1 S = n P( S = n (58 + M n=m+1 n=1 j=1 }{{} ( m ( P g (M +1 2 x(m j+1 ǫ < mǫ 1 S = n P( S = n. j=1 } {{ } T 3 T 2.
28 T 3 can be frst rewrtten as follows: T 3 = P ( y (M +1 α m < mǫ 1 S = n, (59 where α m = m j=1( x(m j+1 ǫ and m n M. The condton of T 3 mples that there are n, n > m, sources whose nformaton can be decoded by the relay and m of the n users wll be scheduled. Therefore the condtonal pdf of α m wll be the same as that of m j=1( x(n +1 ǫ, where x ( are from the parents x, and x, 1 n, are..d. exponentally varables wth the constrant x > ǫ. It s straghtforward to verfy that the CDF of x condtoned on x > ǫ s F x (x = e ǫ e x. Consequently w e ǫ ( x ǫ s smply another exponental varable. Therefore the pdf of α m s the same as the pdf of w m j=1 w (n j+1, the sum of m largest order statstcs chosen from n..d exponental varables. Followng the steps n [22], [23], the pdf of w s gven by f w (w = n m 1 k= d m,k ( m j=1 a j,k e w w j 1 (j 1! +b k e (1+k+1 m w. (6 From [19], the pdf of y (M +1 s f y(m +1 (y = M! (M!( 1! e y (1 e y M. So T 3 can be calculated as follows: T 3 = = = mǫ 1 f w (w w f y(m +1 (ydydw (61 M ( M ( 1 l M! (M!( 1! M! (M!( 1! l= M The ntegral n the above equaton can be calculated as follows: T 4 = n m 1 k= l= ( m a j,k e w w j 1 e d m,k (j 1! j=1 (1 e mǫ 1 (l+ w dw f w (w l l + ( M ( 1 l l l + 1 f w (we mǫ 1 (l+ w dw. }{{} mǫ 1(l+ w dw +b k T4 e (1+k+1 m w e mǫ 1 (l+ w Wth some straghtforward manpulatons, T 4 can be further smplfed as shown n the lemma. T 2 can be frst recalculated as follows: dw. (62 T 2 = P ( g (M +1 2 α n < nǫ 1 S = n, (63
29 where α n = n j=1( x(m j+1 ǫ. Dfferent to α m n (59, the pdf of α n can be found smply as n the followng. The condton of T 2 mples that there are n sources whose nformaton can be decoded by the relay and all these users wll be scheduled. Therefore the condtonal pdf of α n wll be the same as that of n j=1 ( x ǫ. Followng the same arguments as prevously, ( x ǫ s smply an exponental varable, whch means α n s Ch-square dstrbuted,.e. f αn (z = e x x n 1 (n 1!. Therefore T 2 can be calculated as follows: T 2 = = e z z n 1 nǫ 1 z (n 1! M! (M!( 1!(n 1! M! (M!( 1! e x( 1 e x M dydz (64 M ( M ( 1 k ( e z z n 1 z n 1 e z (k+nǫ 1 z dz. k= k k + Combnng (58, (61 and (64, and also wth some algebrac manpulaton, the outage probablty shown n the lemma can be obtaned. The proof of the lemma s completed. Proof of Lemma 3 : When m = 1, the overall outage probablty can be smplfed as follows: P o P( S = + M T 3 P( S = n. (65 The condton that only one user par wll be scheduled can also help to smplfy the expresson of T 3 as follows: T 3 = n = n M k= e y( 1 e y ǫ 1y n 1 ( n 1 M ( 1 k k n=1 d ( 1 e z M dy (66 ( n 1 ( 1 kǫ1 2 +1 K 1 (2 (+1kǫ 1, where the frst equaton follows from the densty functon of the largest order statstcs. Recall the seres representaton of the Bessel functon as follows: xk 1 (x = 1+xI 1 (x (ln x 2 +C 1 2 1+ κ q x 2q lnx, q=1 l= ( x 2 2l+1 ( l x l!(l +2! k=1 l+2 1 k + k=1 1 k (67
3 for x, where κ q s the constant coeffcent assocated to x 2q lnx. Note that the terms of x 2q have been gnored snce they are domnated by the terms of x 2q lnx when x. It s also worthy to pont out that the exact value of κ q has no effect to dversty gans. By applyng the above approxmaton, we can rewrte T 3 as follows: T 3 n M ( M n 1 ( ( n 1 ( 1 ( 1 k 1+ k +1 k= where φ,k = 4(+1kǫ 1. We frst focus on the case of n = M. Snce k= M ( M 1 ( 1 ( 1 k M k= ( M k +1 q=1 1 =. κ q 2 φq,k lnφ,k, ( k ( 1 k =, we have To show that the terms at the order of ǫ q lnǫ, 1 q (M 1, are zero, we frst observe the followng: φ q,k lnφ,k = 4 q (+1 q k q ǫ q 1ln[4(+1kǫ 1 ] (68 = 4 q (+1 q k q ǫ q 1ln[4(+1ǫ 1 ] +4 q (+1 q k q ǫ q }{{} 1lnk. }{{} T 5 T 6 By usng the above separated expresson, we can show that M M k= ( M k ( 1 k snce M k=( M k ( 1 k k q =, 1 q (M 1, and snce ( M M k= ( M k ( 1 k ( M 1 ( 1 +1 T 5 =, (69 ( M 1 ( 1 +1 T 6 =, (7 ( 1 q 1 =, 1 q (M 1. Therefore the term at the order of ǫ lnǫ wll be removed from T 3, and the overall outage probablty can be expressed as M ( M ( ( M 1 ( 1 T 3 M ( 1 k 1+ k +1 k= q=m κ q 2 φq,k lnφ,k Therefore the domnant factor s at the order of ǫ M lnǫ. Smlarly the domnant factors for T 3, 1 n < M, s ǫ n lnǫ. Substtutng ths result nto (65 and also usng the fact that P( S = n ǫ M n, the dversty gan of the overall outage probablty wll be M. And the proof of the lemma s completed..