A Unified CrossLayer Framework for Resource Allocation in Cooperative Networks


 Shanon Robertson
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1 3000 IEEE TRNSCTIONS ON WIRELESS COMMUNICTIONS, VOL. 7, NO. 8, UGUST 2008 Unfed CrossLayer Framework for Resource llocaton n Cooperatve Networks We Chen, Member, IEEE, Ln Da, Member, IEEE, Khaled Ben Letaef, Fellow, IEEE, and Zhgang Cao, Senor Member, IEEE bstract Node cooperaton s an emergng and powerful soluton that can overcome the lmtaton of wreless systems as well as mprove the capacty of the next generaton wreless networks. By formng a vrtual antenna array, node cooperaton can acheve hgh antenna and dversty gans by usng several partners to relay the transmtted sgnals. There has been a lot of work on mprovng the lnk performance n cooperatve networks by usng advanced sgnal processng or power allocaton methods among a sngle source node and ts relays. However, the resource allocaton among multple nodes has not receved much attenton yet. In ths paper, we present a unfed crosslayer framework for resource allocaton n cooperatve networks, whch consders the physcal and network layers jontly and can be appled for any cooperatve transmsson scheme. It s found that the farness and energy constrant cannot be satsfed smultaneously f each node uses a fxed set of relays. To solve ths problem, a multstate cooperaton methodology s proposed, where the energy s allocated among the nodes statebystate va a geometrc and network decomposton approach. Gven the energy allocaton, the duraton of each state s then optmzed so as to maxmze the nodes utlty. Numercal results wll compare the performance of cooperatve networks wth and wthout resource allocaton for cooperatve beamformng and selecton relayng. It s shown that wthout resource allocaton, cooperaton wll result n a poor lfetme of the heavlyused nodes. In contrast, the proposed framework wll not only guarantee farness, but wll also provde sgnfcant throughput and dversty gan over conventonal cooperaton schemes. Index Terms Cooperatve networks, crosslayer desgn, resource allocaton, farness, lfetme, convex optmzaton. I. INTRODUCTION MIMO (MultpleInput MultpleOutput systems, where multple antennas can be used at both the transmt and receve ends, have recently been recevng sgnfcant attenton because they hold the promse of achevng huge capacty ncreases and dversty gans over the harsh wreless lnk [1], Manuscrpt receved October 16, 2006; revsed March 19, 2007; accepted May 27, The assocate edtor coordnatng the revew of ths paper and approvng t for publcaton was M. Saqub. Ths work was supported n part by NSFC/RGC under grant No , RGC under grant No , and NSFC under grant No W. Chen and Z. Cao are wth the Department of Electroncs Engneerng, Tsnghua Unversty, Bejng, Chna (emal: {wchen, W. Chen was also wth the Hong Kong Unversty of Scence and Technology. L. Da was wth the Hong Kong Unversty of Scence and Technology. She s now wth Cty Unversty of Hong Kong, 83 Tat Chee ve., Hong Kong, Chna (emal: K. B. Letaef s wth the Center for Wreless Informaton Technology, Dept. of Electrcal and Electronc Engneerng, Hong Kong Unversty of Scence and Technology, Room 3116, cademc Buldng, Clear Water Bay, Hong Kong (emal: Dgtal Object Identfer /TWC /08$25.00 c 2008 IEEE [2]. s such, MIMO s currently consdered as one of the man canddates for meetng the strngent requrement and demand of future wreless networks. Unfortunately, the use of MIMO technology may not be practcal n many wreless networks. For nstance, nodes n a sensor network are usually small, nexpensve, and have typcally severe energy constrants. Cooperatve networkng or communcatons s one potental soluton that can overcome ths lmtaton. The fundamental dea behnd cooperatve networks s based on the fact that the sgnals transmtted by a source node to ts destnaton node can be also receved by other nodes n a wreless envronment. These nodes can then act as relays to process and retransmt the sgnals they receve n a dstrbuted fashon, thereby, creatng a vrtual antenna array through the use of the relays antennas wthout complcated sgnal desgn or addng more antennas at the nodes [310]. Sendonars et al frstly proposed the dea of cooperatve dversty for CDM cellular networks [34]. Laneman et al studed varous cooperatve dversty schemes such as fxed relayng, selecton relayng, and ncremental relayng [56]. The work n [7] compared several cooperaton protocols and presented a spacetme code desgn crtera for amplfyandforward relay channels. In order to ncrease the power effcency of adhoc networks such as sensor networks, cooperatve beamformng va a vrtual array was developed n [89]. [10] further studed the capacty of cooperatve networks from an nformatontheoretc perspectve and showed the feasble poston of relays. The above prevous work manly amed at enhancng the performance n the physcal layer. However, cooperatve communcaton s nherently a network problem, as ponted out n [3], [56]. It would be therefore frutful to take nto account addtonal hgher layer network ssues. There have been some efforts towards ths such as combnng node cooperaton wth RQ n the lnk layer [11], routng n the network layer [12], or resource allocaton n the MC layer [1314]. From a crosslayer perspectve, farness s especally mportant n cooperatve networks snce some nodes may have more chances to be relays, or consume more power n cooperatve transmssons so that ther energy may be used up very fast. In ths scenaro, not only the heavlyused nodes wll suffer from a short lfetme, but also the other nodes wll not be able to acheve the expected cooperatve gan due to the lack of avalable relays. More serously, these selfnterested users or heavlyused termnals may refuse to cooperate n order to save ther energy. The farness and lfetme ssues uthorzed lcensed use lmted to: Stanford Unversty. Downloaded on March 08,2010 at 13:26:29 EST from IEEE Xplore. Restrctons apply.
2 CHEN et al.: UNIFIED CROSSLYER FRMEWORK FOR RESOURCE LLOCTION IN COOPERTIVE NETWORKS 3001 have been consdered n wreless networks wth network layer cooperaton usng packet forwardng rather than sgnal forwardng [1518]. It was demonstrated n [15] that an overconsumpton of some nodes may result n a short network lfetme. In order to stmulate node cooperaton and balance energy consumpton, both reputatonbased and marketbased approaches were presented. Usng a gametheoretc framework, [16] proposed a generous ttfortat strategy for energyconstraned adhoc networks. The marketbased approach was studed n [17] and [18], where a mcroeconomc framework was used to maxmze the node utlty n a dstrbuted fashon. However, user cooperaton n the physcal layer was not taken nto account n these exstng works. Most recently, we appled a marketbased approach for ncreasng the farness and effcency of adhoc wreless networks usng cooperatve beamformng [19]. In partcular, a practcal protocol was presented n [19] to sgnfcantly ncrease the lfetme and throughput of energyconstraned cooperatve networks. In ths paper, our man objectve s to develop an effectve way to optmze the overall performance of cooperatve networks across multple layers smultaneously. Specfcally, we consder energyconstraned networks where cooperatve transmsson s adopted. unfed crosslayer framework for resource allocaton among multple nodes s presented, where farness and effcency are taken nto consderaton. Our objectve s to guarantee that the lfetme of each node can be equal to a target lfetme and that the energy used n transmttng and/or relayng each node s sgnal s equal to ts total energy. Moreover, each node can effcently use the avalable energy to optmze ts performance such as throughput or outage probablty. To acheve ths, we propose an energy allocaton method consstng of multple cooperaton states, where the relay set of each node s not fxed. In partcular, each state corresponds to a set of nodes whch run out of energy durng the prevous states and wll not cooperate anymore. In practce, the state of a node corresponds to a partcular relay set of ths node. In partcular, a base staton or a head node wll announce the set of nodes whch do not have energy to serve as relays n each state. node wll then gnore these nodes when t searches ts relays n a gven state. We shall show that at least one node wll run out of ts energy n each state. Thus, the total number of states wll not be greater than the number of nodes. Based on the energy allocaton results, we allocate the node lfetme among the multple states to determne how long a partcular set of relays can serve a node. The proposed multstate cooperaton n whch the heavlyused nodes are not always forced to serve as relays s a natural extenson of the conventonal cooperaton protocols wth fxed sets of relays. The proposed framework wll be appled nto cooperatve beamformng and selecton relayng. Numercal results show that wthout an approprate resource allocaton, unfar cooperaton wll result n a sgnfcant decrease n the lfetme of heavlyused nodes. In contrast, the proposed framework cannot only guarantee farness, but also provde sgnfcant throughput or dversty gan over the conventonal cooperaton schemes. The remander of the paper s organzed as follows. Secton II presents the system model. In Secton III, a multstate cooperaton methodology wth ts energy allocaton scheme s presented. Secton IV nvestgates the effcent tme allocaton over multple states. Numercal results and some mplementaton ssues wll be dscussed n Secton V and VI, respectvely. Fnally, concludng remarks are presented n Sectons VII. Throughout ths paper, the followng notatons wll be used. The superscrpt T shall stand for the transpose of a matrx X or a vector x. The nequalty x y mples that x y for any. The geometrc multplcty of an egenvalue λ(x s denoted by geomult X (λ(x[20]. For a set X, the operator X denotes the number of elements n the set. The operator \ denotes the dfference of two sets. For an event ω, the ndcator functon shall be denoted by I(ω, wherei(ω =1s ω s true. Otherwse, I(ω =0. vector wth all of ts elements equal to 1 s denoted by 1. Fnally, 0 shall denote the all zero matrx/vector wth the approprate sze. II. SYSTEM MODEL Consder a wreless network whch conssts of N source/relay nodes. The source node set s denoted by S = {1,...,N}. Each source node transmts to ts destnaton node d(, whch may not belong to S. Leth j denote the channel power gan between node and j. In our work, both statc channels and tmevaryng channels can be taken nto account. For networks wth tmevaryng channels, whch we shall refer to as tmevaryng networks, smallscale fadng s assumed to be Raylegh so that the nstantaneous channel gan h j s a random varable wth an exponental dstrbuton wth mean value h j. For networks wth statc channels, whch we shall refer to as statc networks, h j = h j. It s assumed that h j = h j and the nose power at the recevers s denoted by σ 2.ll nodes are assumed to be energyconstraned and the total energy of each node s denoted by E total. Wth user cooperaton, each source node may employ some nodes to serve as relays. Each cooperatve transmsson wll be assumed to occur over two tmeslots, where the source transmts to ts relays n the frst tmeslot and relaysretransmt the sgnal to the destnaton n the second tmeslot. Snce for a partcular sourcedestnaton (sd par, some nodes may be far away from both the source and the destnaton, only neghbors are selected n order to ncrease power effcency and avod error propagaton. n average channel gan threshold g s assgned to each node. Then, node can only choose the nodes j satsfyng h j g to serve as ts relays. The set of potental relays for node s denoted by R. Intutvely, g should be an ncreasng functon of node s average sd channel gan. Ths s because the worse an sd channel s, the more relays the source node wll need. In networks where the sd dstances are approxmately the same, we can smply assgn the same threshold to each node. In ths paper, we assume that the source node tself can act as a relay n the second tmeslot. That s, R. ssume that all relay nodes are chosen from the source node set S. Hence, R Sfor any S. In the MC layer, each source node wth ts relays can employ an orthogonal channel to avod multuser nterference. Wthout loss of generalty, FDM s assumed throughout ths paper. Furthermore, each node s assumed to have a saturated queue wth always packet avalablty. In ths paper, two typcal cooperaton schemes are consdered: cooperatve beamformng and selecton relayng. The uthorzed lcensed use lmted to: Stanford Unversty. Downloaded on March 08,2010 at 13:26:29 EST from IEEE Xplore. Restrctons apply.
3 3002 IEEE TRNSCTIONS ON WIRELESS COMMUNICTIONS, VOL. 7, NO. 8, UGUST 2008 Cooperatve Beamformng a j = TBLE I COOPERTION MTRIX Cooperaton Matrx ( g j h d(j + n R j h nd(j 2 I{=j} (gj + n Rj hnd(j n R j h nd(j R j 0 / R j Pr{h j ξ j } R Selecton relayng a j = n R Pr{h j jn ξ j } j 0 / R j. In wreless networks, all the nodes are expected to have the same lfetme T. s a result, each node should consume all of ts energy, E total, smultaneously at the end of the target node lfetme nterval [0, T ]. No nodes are allowed to have resdual energy after T. Otherwse, ts lfetme can be longer than T.On the other hand, the energy utlzed n transmttng and/or relayng each node s nformaton should be equal n order to guarantee farness. In noncooperatve networks where nodes transmt drectly wthout employng relays, ths can be easly satsfed snce each node s energy consumed s utlzed to transmt ts own nformaton. In cooperatve networks, ths s not naturally acheved and hence an approprate resource allocaton s desred. cooperatve beamformng s a rateoptmal cooperaton scheme adopted n statc networks. In ths case, the vrtual antenna array formed by the relays wll retransmt the sgnals wth beamformng n order to acheve the hghest throughput. Selecton relayng, on the other hand, s adopted n tmevaryng networks [5]. s selecton relayng s adopted, only the relays whch can decode the source nformaton correctly wll retransmt. The set of nodes that decode correctly n a tmeslot s often referred to as the decodng set D. We denote the energy that node j consumed n transmttng/relayng sgnals of node to be E ( j III. ENERGY LLOCTION IN COOPERTIVE NETWORKS In ths secton, we address the energy allocaton problem n cooperatve networks. Two basc constrants for energy allocaton as well as a lnear relatonshp between the energy allocated and energy consumed are presented. In order to satsfy the two constrants, a multstate cooperaton methodology s proposed. We then provde a geometrcal approach to allocate the energy statebystate va a fntestep teraton algorthm. In each state, the energy allocaton s obtaned by solvng a lnear equaton.. Energy llocaton Vector and Consumpton Vector: Constrants and Relatonshp Let us denote the energy consumpton vector e C = [e C 1,...,e C N ]T,wheretheth element e C = n j=1 E(j s the total energy consumed by node. Note that the energy allocated to a node s the total energy utlzed n transmttng and relayng ths node s nformaton. We denote the energy allocaton vector e =[e 1,...,e N ]T,wheretheth element = n. Snce the node s energyconstraned and all e j=1 E( j nodes should consume all of ther energy wthn ther target lfetme, the energy constrant s wrtten as e C = E max 1 (1 Due to the farness requrements, e should be equal for each node. Snce the total energy consumed by all nodes s N n=1 Emax = NE max, the far allocaton constrant should guarantee that e = NE max /N. Therefore, such constrant s gven by e = E max 1. (2 In a noncooperatve network, the two constrants (1(2 are naturally satsfed snce E ( j =0for j. s cooperatve transmsson s adopted, e and e C wll be shown to be lnearly related. In order to derve such a relatonshp, a cooperaton matrx s defned as follows. Defnton 1 (Cooperaton Matrx: The cooperaton matrx s defned to be a matrx = [a j ] N N, where the element a j denotes the energy rato that node contrbutes to node j. Thats, a j = E (j N n=1 E(j n. (3 Note that the cooperaton matrx s determned by the cooperaton scheme and the relay sets R, for = 1,...,N.Snce N =1 a j = 1, T s a stochastc matrx [20]. 1 The cooperaton matrces of two cooperaton schemes are gven n Table I, wth the detaled dervaton gven n ppendx I. ccordng to (3, the energy that node consumes for transmttng/relayng the sgnal of node j can be presented as E (j = a j e j. Hence, the total energy consumpton of node s obtaned as e C = N j=1 a je j. s a result, the energy allocaton and consumpton can be related by e = e C. (4 By substtutng (1 and (2 nto (4, we have E max 1 = E max 1. (5 ccordng to Defnton 1 and (5, the cooperaton matrx should be a doublystochastc matrx n order to satsfy both (1 and (2. Unfortunately, cannot satsfy (5 n general because t cannot be doublystochastc for networks wth randomlylocated nodes. 1 s defned n [20], a stochastc matrx s a nonnegatve matrx n whch each row sum s equal to 1. In addton, a doublystochastc matrx s a nonnegatve matrx n whch each row sum as well as each column sum s equal to 1. uthorzed lcensed use lmted to: Stanford Unversty. Downloaded on March 08,2010 at 13:26:29 EST from IEEE Xplore. Restrctons apply.
4 CHEN et al.: UNIFIED CROSSLYER FRMEWORK FOR RESOURCE LLOCTION IN COOPERTIVE NETWORKS 3003 B. Multstate Cooperaton: To Cooperate or Not to Cooperate Snce the energy constrant (1 and the farness constrant (2 cannot be satsfed smultaneously as relay sets are fxed, each node should be allowed to use a dfferent relay set n a dfferent cooperaton state n order to satsfy (1(2. Ths motvates us to develop a resource allocaton framework consstng of multple cooperaton states, whch we shall refer to as multstate cooperaton. In each state k, therelayset R (k for any node s fxed. Therefore, the cooperaton state s gven by the set of all relay sets. Let e (k denote the energy allocaton vector n state k, wheretheth element s the total energy utlzed n transmttng and relayng node s nformaton n state k. lsolete C (k denote the energy consumpton vector n state k, wheretheth element s the total energy consumed by node n state k. For a multstate cooperaton, the energy allocaton e (k, k =1,...,K, should satsfy the energy and farness constrants gven by { K k=1 ec (k = K k=1 (ke (k =E max 1 K k=1 e (k =E max 1, where K s the total number of states and (k s the cooperaton matrx assocated wth state k. In order to obtan e (k that satsfes (6, a statebystate energy allocaton methodology s presented. In each state, the energy allocaton should satsfy the constrant (6 e (k =e C (k, (7 whch can always be satsfed by the method descrbed next n Secton IIIC. However, e C (k may not be equal to ec j (k for any j n general. Therefore, some nodes wll consume all of ther energy n state k whle others may stll have resdual energy to be allocated and consume n the followng states. In each state, only the nodes that have resdual energy can transmt and serve as relay for others. Iteratvely, we can allocate the resdual or remanng energy statebystate untl all of the energy s allocated n the fnal state. Let e max (k denote the resdual energy vector, where the th element e max (k s the node s resdual energy n state k. In the ntal state 1, e max (1 = E max 1. fter energy allocaton n state k, the resdual energy e max (k +1of the next state s gven by e max (k +1=e max (k e (k. (8 Constraned by e max (k, the relay set of node n state k, R (k, sgvenby R { (k { = : hj g,e max j (k > 0,j S } { } e max (k > 0 e max (k =0 (9 In Secton IIIC, we wll present an energy allocaton method, where at least one node wll run out of energy n each state. Therefore, all of the energy can be allocated wthn K<Nstates so that the resdual energy vector n the fnal state K satsfes e max (k +1=0. (10 ccordng to (7, (8, and (10, (6 s satsfed n a Kstate cooperaton. C. Energy llocaton n One State In ths part, the analytcal result for energy allocaton n one state s presented. For a partcular state k, the cooperaton matrx (k s determned by the relay sets gven by (9. Thus, by substtutng (4 nto (7, t follows that e (k should be a nonnegatve soluton to the followng equaton (ke (k =e (k. (11 and satsfy the resdual energy constrant n state k, whch s gven by e (k e max (k. (12 Snce T (k s a stochastc matrx, there must be an egenvalue λ((k = λ( T (k = 1 [20]. Thus, Eqn. (11 must have nontrval solutons. The soluton space s the egenspace of (k wth respect to λ((k = 1. Inorderto obtan e (k, anetwork decomposton methodology s frst ntroduced n order to fnd an orthogonal bass set of the soluton space. We shall show that each bass characterzes the far energy allocaton of a subnetwork. For convenence, we can gnore the ndex k n ths part snce we are consderng energy allocaton of a partcular state. In each state, the cooperatve network can be decomposed nto dsjont subnetworks, where the nodes n dfferent subnetworks do not cooperate wth each other. 2 Intutvely, the energy allocatons n each subnetwork are ndependent. Mathematcally, for a gven cooperaton matrx, the whole network S can be decomposed nto M dsjont subnetworks S (m = {n (m 1,...,n (m },form =1,...,M, whch satsfy S (m the followng (1 S = M m=1 S(m, wth S (m S (n =, m n. (2 a j = a j =0, S (m, j S (n, m n. (3 Each subnetwork S (m cannot be decomposed nto multple dsjont subsets satsfyng propertes (1 and (2. For a gven, the whole network S can be decomposed nto S(m satsfyng the above three propertes by usng a graphtheoretc algorthm as shown n ppendx II. It s noted that each subnetwork S(m wll have ts own cooperaton matrx (m. Let P denote the permutaton matrx wth elements p uv = I {( u = n (m,v = + m 1 r=1 S(r }. Then, (m s the mth dagonal block n the dagonal block matrx P T P = (1... (m. (13 Havng establshed the network decomposton methodology, we shall present the optmal energy allocaton n one state n the followng theorem. 2 In the frst state, a network may consst of only one subnetwork f ts topology s not clustered. Due to the reshapng of the relay sets, some nodes wll not cooperate wth others and be solated n the followng states. In ths case, however, a network must be decomposed nto more than one subnetwork. uthorzed lcensed use lmted to: Stanford Unversty. Downloaded on March 08,2010 at 13:26:29 EST from IEEE Xplore. Restrctons apply.
5 3004 IEEE TRNSCTIONS ON WIRELESS COMMUNICTIONS, VOL. 7, NO. 8, UGUST 2008 e 2 max E e (1 e (2 e (3 max E e 1 e 3 max E Fg. 2. MultState Cooperaton n the Tme Doman Fg. 1. Geometrcal representaton of multstate energy allocaton Theorem 1: The optmum energy allocaton vector that satsfes (11(12 s obtaned by [ ( ] M e e max = mn b (m, (14 S m=1 (m b (m where the orthogonal bass b(m = [b (m 1,...,b (m N ]T, for m =1,...,M, are nonnegatve. They are gven by 0 ( m 1 r=1 S(r 1 ] 1 [ ] b (m = P [ (m 0 ( M r=m+1 S(r ( S (m 1 1. (15 where (m s a S (m 1 by S (m matrx obtaned by deletng an arbtrary one row of (m I. D. Energy llocaton n Multple Cooperaton States Havng obtaned the energy allocaton n one state, our proposed multstate energy allocaton algorthm can be presented as follows. Multstate Energy llocaton lgorthm Step 0: Intalze k =1,andsete max (1 = E max 1; Step 1: Generate R (k for =1,...,N,by(9and (k by (3; Step 2: Decompose the network nto S (m, m =1,..., M, usng the network decomposton algorthm; Step 3: Obtan the energy allocaton vector e (k by (13(15; Step 4: Calculate the resdual energy vector e max (k +1by (8; Step 5: k = k +1; Step 6: If e max (k > 0, go to Step 1; End It must be noted here that Theorem 1 shows that the energy allocaton s always feasble snce the soluton s nonnegatve. One can easly see that Eqn. (11 s soluton space characterzed by b (m, m =1,...,M, contans nfnte number of energy allocaton vectors satsfyng (12. The optmum energy allocaton presented n Theorem 1 chooses the soluton vector wth the maxmum 1norm. The optmalty of choosng such an energy allocaton vector can be explaned as follows. From (9, t can be seen that the number of each node s relays decreases as the state ndex k ncreases. Notce that the energy effcency s hgher as more relays are used. Clearly wth an ncreasng state ndex k the energy effcency wll go down. Therefore, as much as possble energy should be allocated n each state durng the teraton. Ths s why we choose the energy allocaton vector wth the maxmum 1norm. From (14 t can be also seen that wth the proposed energy allocaton, at least one node n each S (m satsfes e (k = e max (k. Ths mples that at least one node runs out of energy n each state. Therefore, (10 can be satsfed by a Kstate (K N energy allocaton. From a geometrcal perspectve, the multstate energy allocaton can be characterzed by a K part curve n R N startng from the orgn pont 0 to E max 1. The kth part of the curve, e (k, belongs to the egenspace of (k wth respect to λ((k = 1, and the cumulatve energy allocaton vector s bounded by the supercube characterzed by E max 1. For nstance, Fg. 1 shows an energy allocaton n a three node network. In ths case, node 2 s allocated all of ts energy n state 1, whle nodes 1 and 3 stll have resdual energy. Next, node 1 s allocated all of ts resdual energy n state 2. Fnally, node 3 runs out of ts energy n the last state. Gven the above energy allocaton outcome, node 2 wll always transmt wth ts state 1 s relay set R 2 (1={1, 2, 3} throughout ts entre lfetme. Node 1, however, wll use ts state 1 s relay set, R 1 (1={1, 2, 3} and state 2 s relay set, R 1 (2={1, 3}, n a tme sharng manner. Node 3 wll have three dfferent relay sets, namely, {1, 2, 3}, {1, 3}, and{3}, whch are used n dsjont tme duratons. Fg. 2 shows how multstate cooperaton evolves n the tme doman, where one can see that all nodes wll have the same lfetme. How to determne the optmal state duraton wll be addressed n the next Secton. IV. OPTIML STTE DURTION FOR MULTISTTE COOPERTION In ths secton, we nvestgate how long the set of nodes R (k can serve as the relay set of node. Gven a target lfetme, we shall allocate the whole lfetme over the multple uthorzed lcensed use lmted to: Stanford Unversty. Downloaded on March 08,2010 at 13:26:29 EST from IEEE Xplore. Restrctons apply.
6 CHEN et al.: UNIFIED CROSSLYER FRMEWORK FOR RESOURCE LLOCTION IN COOPERTIVE NETWORKS 3005 TBLE II UTILITY FUNCTIONS Utlty Functon ( ( g e Cooperatve Beamformng U j R (k h jd( (k,k = 12 t (k log 1+c (k e (k (g,c t (k (k = + R (k 2 j R (k h jd( σ2 2h d( /σ 2 R (k =1 [ ( ( e Selecton Relayng U (k,k = j D exp (22r 1σ 2 j R (k exp( ξ /h j 2h jd( e (k/t n j (k t (k D R (k ( [ ( ] j D exp ξ h j j/ D 1 exp ξ h j ] h jd( h jd( h nd( states of a node. The optmal state duraton, whch determnes the number of tme each state would exst or how long R (k can serve as the relay set of node, wll maxmze the node s utlty such as throughput or outage probablty. unfed utlty maxmzaton problem s then formulated for each node. ssume that R (k serves as a relay set of node for t (k seconds. Here, t (k denotes the duraton of state k. Subject to the target lfetme T, the lfetme constrant for each node s presented by K t (k =T. (16 k=1 The node s utlty n state k s denoted by u (k, whch can be represented as a functon of the average power n state k. Note that the average power n state k s proportonal to the energy allocated to ths state and the nverse of the state duraton. Thus, we have, u (k =U,k ( e (k t (k, (17 where the utlty functon U,k (x s an ncreasng functon of x satsfyng U,k (0 = 0 and U,k (x 0. In statc networks, cooperatve beamformng s adopted to maxmze the sd throughput. Hence, we shall use the sd throughput to measure the utlty of each node n ths case. In tmevaryng networks, selecton relayng s adopted to reduce the outage probablty. Therefore, the probablty that outage does not occur, 1 P out, s adopted to measure the utlty of each node. The utlty functons are gven n Table II wth the detaled dervaton shown n ppendx IV. Gven the above, the state duraton optmzaton problem for node can then be formulated as ( K t Maxmze (k e k=1 T U (k,k t { (k K Subject to k=1 t (k = T (18 t (k 0, where the objectve functon s the average utlty, whch represents the energy effcency of node. It s also guaranteed, mathematcally, that each node can acheve a target lfetme by the constrants. From Secton III, node s not allocated any energy n state k f e (k =0. Thus, t (k =0as e (k =0. In ppendx III, we show that U,k (x n Table II s a concave functon satsfyng U,k (x 0. Then, by dfferentatng the objectve functon of (18, we get [ ( ] ( 2 t (k t (k 2 T U e (k,k = [e (k]2 e t (k T t 3 U (k,k. (k t (k Clearly (18 s a convex optmzaton problem and as a result, the KurashKuhnTucker (KKT condton s a necessary and suffcent condton for optmalty. By usng the KKT condton, the analytcal optmal soluton for cooperatve beamformng s gven by t (k = T c (ke (k K l=1 c (19 (le (l. In the case where the KKT condton does not have an analytcal soluton, wth the decomposton prncple [pp , 21], the optmal soluton of (18 wll be gven by t (k =max { 0,f 1,k (y}. (20 where [ f 1,k ( (y denotes ] the nverse functon of f,k(x = d x e dx T U (k,k x, and y s the optmal soluton to the followng unconstraned problem { K max 0, f 1,k (y T }. e Maxmze y + U (k,k f 1 yf 1,k k=1,k (y (y (21 By (20(21, (18 can be reduced nto a onedmensonal optmzaton problem so that the complexty wll be greatly decreased. V. NUMERICL RESULTS In ths secton, numercal results are presented to compare the performance of cooperatve networks wth and wthout resource allocaton. Both statc networks wth cooperatve beamformng and tmevaryng networks wth selecton relayng are consdered. For the sake of far comparson, drect transmsson where nodes do not use any relay s also consdered. Ths provdes a baselne reference to compare cooperaton gans. s cooperaton wthout resource allocaton s adopted, the power consumed for transmttng/relayng each node s sgnal s equal to the transmsson power of drect transmsson. Ths knd of cooperaton shall be referred to as full cooperaton. The pathloss factor s assumed to be 4 throughout Secton V. Let D j denotes the dstance between node and node j, thenh j = D 4 j. 3 Fnally, the nose power at the recever s assumed to be σ 2 = Here, we do not normalze the channel gan snce t can be scaled by the transmsson power. 4 Here, we drop the unt n the followng text. uthorzed lcensed use lmted to: Stanford Unversty. Downloaded on March 08,2010 at 13:26:29 EST from IEEE Xplore. Restrctons apply.
7 3006 IEEE TRNSCTIONS ON WIRELESS COMMUNICTIONS, VOL. 7, NO. 8, UGUST 2008 TBLE III SYSTEM PRMETERS FOR NUMERICL RESULTS Statc Network wth Cooperatve Beamformng sd Node coordnates (Topology Relay Selecton par source destnaton Threshold 1 (0,1 (5,0 2 (1,0 (5, (0.8,0 (5,0 4 (0.5, 3/2 (5,0 Tmevaryng Network wth Selecton Relayng sd Node coordnates (Topology Relay Selecton par source destnaton Threshold 1 (0.5, 3/2 (0.5, 3/2 2 (0,0.1 (1, (0,0.1 (1, (0.5, 3/2 (0.5, 3/2 Fg. 3. Throughput curves of drect transmsson, cooperatve beamformng wth full cooperaton and multsate cooperaton. The state duratons of node 4, whch are also gven n Table IV, are presented.. Statc Network wth Cooperatve Beamformng Consder a statc network wth four source nodes 14, randomly located n a unt crcle. ll the source nodes transmt to a common destnaton node. The coordnates of all source and destnaton nodes as well as the relay selecton threshold are gven n Table III. In statc networks, the channel gan s gven by h j = h j. Note that g =0.25 for =1,...,4. Thus, nodes 1 and 4 wll not serve as relay for each other whle nodes 2 and 3 serve all source nodes 14. Besdes, nodes 2 and 3 wll consume more power than others n each cooperatve transmsson snce ther lnks to the destnaton node are better. ssume that E total = s drect transmsson s adopted, the transmsson power of each node s 1000 and therefore the lfetme of each node s 1. s full cooperaton s adopted, t can be obtaned that the maxmum node lfetme s For the sake of far comparson, we set the target lfetme of multstate cooperaton to be By usng the multstate energy allocaton algorthm and (19, we obtan the energy and tme allocaton results shown n Table IV. Fg. 3 presents the aggregate throughput curves wth drect transmsson, full cooperaton, and multstate cooperaton. It can be seen that full cooperaton acheves the hghest total rate TBLE IV ENERGY ND TIME LLOCTION FOR COOPERTIVE BEMFORMING State 1 State 2 State 3 Node 1 e t Node 2 e t Node 3 e t Node 4 e t of all nodes before node 2 runs out of ts energy n t = However, nodes 2 and 3 run out of ther energy very quckly due to hgher power consumpton compared to that of drect transmsson. fter that, nodes 1 and 4 cannot fnd a relay and have to transmt drectly for over 63% of ther lfetme. In ths scenaro, ther power effcency s very poor. Wth the proposed multstate cooperaton, however, each node can use all nodes j satsfyng h j = g as relays for at least 67% of ther lfetme, as shown n Table IV. lthough the transmsson power of each node s lower than that of full cooperaton at the begnnng, all nodes can beneft from the beamformng gan much longer. s a result, the energy effcency ncreases. For nstance, the aggregate throughput of drect transmsson, full cooperaton and multstate cooperaton s 7.38 bt/hz, 9.46 bt/hz, and bt/hz, respectvely. bout 69.3% and 32.1% gans are obtaned by multstate cooperaton over drect transmsson and full cooperaton, respectvely. Next, we compare the farness of the full cooperaton and multstate cooperaton. Here, the farness of cooperaton s characterzed by two parameters: Increase n lfetme and ncrease n throughput compared to drect transmsson. s shown n Table V, the heavlyused nodes 2 and 3 suffer from a shorter lfetme snce ther power consumpton s ncreased compared to drect transmsson. More serously, snce they run out of energy very soon, ther throughput s also reduced compared to drect transmsson. Hence, nodes 2 and 3 do not beneft from cooperaton ndeed. The proposed multstate cooperaton, however, can guarantee that all nodes lfetme s equal to the target lfetme. Moreover, the ncrease n throughput of all nodes s approxmately equal. In partcular, note that node 1 acheves the hghest ncrease n throughput. Ths s smply due to the fact that node 1 s throughput s the smallest wthout cooperaton. B. Tmevaryng Networks wth Selecton Relayng Consder a tmevaryng network wth four sd pars satsfyng h j =1for =1,...,4, where the coordnates of the source and destnaton nodes are gven n Table III. The target rate of each sd par s 1 bt/s/hz. Here assume that the thresholds h =0.5, = 1,...,4. It can be seen that nodes 1 and 4 are far away from each other so that they wll not use each other as a relay. In addton, such a network topology s symmetrc wth respect to the xaxs. Therefore, the performance of nodes 1 and 4 s the same, as well as that of nodes 2 and 3. We shall use the average SgnaltoNose uthorzed lcensed use lmted to: Stanford Unversty. Downloaded on March 08,2010 at 13:26:29 EST from IEEE Xplore. Restrctons apply.
8 CHEN et al.: UNIFIED CROSSLYER FRMEWORK FOR RESOURCE LLOCTION IN COOPERTIVE NETWORKS 3007 TBLE V INCRESE IN LIFETIME ND INCRESE IN THROUGHPUT OF COOPERTIVE BEMFORMING WITH FULL COOPERTION ND MULTISTTE COOPERTION Node 1 Node 2 Node 3 Node 4 Increase n lfetme Full cooperaton 104% 36% 26% 74% thanks to cooperaton Multstate cooperaton 104% 104% 104% 104% Increase n throughput Full cooperaton 144% 38% 22% 90% thanks to cooperaton Multstate cooperaton 87% 55% 67% 76% Increase n lfetme (% Drect Transmsson, Nodes 1 4 Full Cooperaton, Nodes 1, 4 Full Cooperaton, Nodes 2, 3 Mult state Cooperaton, Nodes 1 4 Outage Probablty Drect Transmsson, Nodes 1 4 Full Cooperaton, Nodes 1,4 Full Cooperaton, Nodes 2,3 Mult state Cooperaton, Nodes 1,4 Mult state Cooperaton, Nodes 2, SNR (db Fg. 4. Increase n node lfetme wth drect transmsson, selecton relayng wth full cooperaton and multstate cooperaton SNR (db Fg. 5. Outage probablty averaged over the lfetme of each node wth drect transmsson, selecton relayng wth full cooperaton and multstate cooperaton Rato (SNR of drect transmsson gven by E total /(T σ 2 to characterze the SNR of the network. The performance wll be compared for varous SNR values from 0 db to 40 db. Wthout loss of generalty, we can normalze lfetme so that T =1and therefore, E total can be determned by the gven SNR. s selecton relayng s adopted, we assume that the transmsson power of the source nodes n the frst tmeslot s E total /(2T. Then, wth full cooperaton, the average total power of the relays of each node should be 3E total /(2T. Due to space lmtaton, we omt the results for energy and tme allocaton of multstate cooperaton. Fg. 4 presents the ncrease n lfetme compared to T wth drect transmsson, full cooperaton and multstate cooperaton. Wth the drect transmsson and multstate cooperaton, the lfetme of each node s equal to T. Wth full cooperaton, however, the lfetme of nodes 1 and 4 s ncreased by about 11% T, whle the lfetme of nodes 2 and 3 s decreased by about 11% T n the hgh SNR regon. It can be seen that the gap between the maxmum and mnmum lfetme ncreases wth SNR. Ths s because the probablty that a relay can decode correctly ncreases wth SNR. By servng more nodes, nodes 2 and 3 wll consume a larger number of power n the hgh SNR regon. When SNR s suffcently hgh, (e.g. SNR 10 db, the probablty that a relay can decode correctly s approxmately equal to 1. Hence, the ncrease n lfetme wll reman constant n the hgh SNR regon. Fg. 5 presents the outage probablty averaged over the lfetme of each node wth drect transmsson, full cooperaton and multstate cooperaton. Snce drect transmsson cannot acheve dversty gan, the outage probablty of all nodes approxmately decays as 1/SNR [2]. Wth full cooperaton, the verage Outage Probablty Drect Transmsson Full Cooperaton Mult state Cooperaton SNR (db Fg. 6. Outage probablty averaged over all nodes of drect transmsson, selecton relayng wth full cooperaton and multstate cooperaton outage probablty of nodes 1 and 4 decays as 0.2/SNR. Ths s because, as shown n Fg. 4, they transmt drectly wthout relay for 20% of ther lfetme, when the outage probablty s approxmately equal to 1/SNR. Snce our proposed framework can effcently allocate tme among the states, nodes 1 and 4 can transmt wth relays much longer whle transmttng drectly for only a very short tme wth much hgher transmsson power. lthough the power of relays s lower compared to full cooperaton, multstate cooperaton can beneft from the longer node lfetme when the spatal dversty gan of 3 s acheved. closer observaton shows that node 2 or 3 s outage probablty wth full cooperaton s equal to that uthorzed lcensed use lmted to: Stanford Unversty. Downloaded on March 08,2010 at 13:26:29 EST from IEEE Xplore. Restrctons apply.
9 3008 IEEE TRNSCTIONS ON WIRELESS COMMUNICTIONS, VOL. 7, NO. 8, UGUST 2008 wth multstate cooperaton. Ths s smply due to the fact that wth multstate cooperaton, the average total power of node 2 or 3 relays s also equal to 3E total /(2T.Fg.6 presents the outage probablty averaged over all nodes. The average outage probablty of full cooperaton s approxmately equal to 0.1/SNR, whle that of our proposed framework s 1/SNR 2.5. VI. IMPLEMENTTION ISSUES So far, we have presented, mathematcally, a unfed framework for far and effcent resource allocaton n cooperatve networks. In ths Secton, we wll turn our attenton to some mplementaton ssues of the proposed framework.. Halfdecentralzed Implementaton The whole resource allocaton scheme proposed n Sectons III and IV can be mplemented by a halfdecentralzed protocol, whch conssts of two steps, namely, centralzed multstate energy allocaton and dstrbuted state duraton optmzaton. To perform multstate energy allocaton, a control node, whch can be a base staton, an access pont or smply an elected head node, needs to collect each node s local nformaton of the average sourcerelay (sr and/or relaydestnaton (rd channel gans, whch can be obtaned locally va 1 the average power gan estmaton based on plot energy; or 2 Global Postonng System (GPS. By usng the multstate energy allocaton algorthm, the control node wll determne the energy allocaton vectors, and then broadcast them over the whole network. Based on the energy allocaton results, each node can locally formulate and solve ts own state duraton optmzaton problem (18, whch only requres local parameters nformaton such as the sr and rd channel nformaton. No nformaton exchange s needed among nodes n ths stage. Note that many wreless systems, such as cellular networks or WLNs, have centralzed controllers. The control node only needs to collect the average channel gan nformaton and broadcast the energy allocaton results durng the ntal network confguraton. s such, the proposed framework wll not nduce much overhead n practce. B. Dstrbuted Implementaton In some networks, havng a centralzed controller mght be mpractcal. Thus, a fully dstrbuted mplementaton wll be hghly desred n ths case. marketbased dstrbuted protocol s presented n ths part to acheve an approxmately far cooperaton. By jontly consderng (1 and (2, we have N N E (j = E ( j, =1,...,N. (22 j=1, j j=1, j The above equalty mples a market rule, namely, that the energy a node contrbutes to others should be equal to what others contrbute to ths node. Based on ths market rule, a local parameter referred to as the energy reward s ntroduced for each node. In the dstrbuted protocol, the energy reward of a node ncreases when the node helps others and decreases when the node uses others as relays. When a gven node requres cooperaton, the relays energy that t can use wll depend on ts current energy reward. In ths way, farness s guaranteed and overusng the heavlyloaded nodes s avoded, both n dstrbuted manners. smlar method has been proposed n [19] to engneer a dstrbuted far cooperaton protocol n practce. Fnally, t should be ponted out that dstrbuted far cooperaton s stll an open problem. The proposed framework, however, provdes a performance upperbound for the dstrbuted solutons. VII. CONCLUSION In ths paper, we presented a unfed crosslayer framework for far and effcent resource allocaton n cooperatve networks. Usng the proposed framework, all nodes can run out of energy smultaneously and each node s allocated an equal number of energy so that farness s guaranteed. The proposed approach s based on the use of a multstate cooperaton methodology where the total energy s allocated among the nodes statebystate va a geometrc and network decomposton approach. Gven the energy allocaton results, the optmal state duraton of each node s found so as to maxmze each node s utlty such as throughput or outage probablty. By dervng the cooperaton matrces and utlty functons, we appled the proposed framework nto cooperatve beamformng and selecton relayng. The performance of cooperaton networks wth and wthout resource allocaton was also compared. It was demonstrated that the proposed framework guarantees an equal lfetme of all nodes. Ths s n contrast to the unfar cooperaton whch wll result n a sgnfcant decrease n the lfetme of heavlyused nodes. For nstance, the decrease n lfetme of heavlyused nodes s 36% for cooperatve beamformng and 11% for selecton relayng wth full cooperaton. Furthermore, the proposed framework can acheve 32% throughput gan over full cooperatve beamformng. For selecton relayng, the proposed framework can guarantee a dversty gan greater than 2 on average compared to full cooperaton, whch acheves only a dversty gan of 1 on average. In partcular, due to the lack of farness, full cooperaton wll result n a short lfetme of those heavlyloaded nodes. s a result, other nodes have to transmt drectly wthout relay assstance. Thus, the dversty gan of full cooperaton s lower than that of multstate cooperaton. PPENDIX ENERGY CONSTRINTS ND COOPERTION MTRICES DERIVTION. Cooperatve beamformng Note that h jn g j for any n R j. s cooperatve beamformng s adopted, a source node can transmt to ts relays n the frst tmeslot, wth relable transmsson rate gven by C s r j ( =log 1+g j Pj s /σ 2, (23 where Pj s denotes the power of node j n the frst tmeslot. In the second tmeslot, the source as well as ts relays transmt uthorzed lcensed use lmted to: Stanford Unversty. Downloaded on March 08,2010 at 13:26:29 EST from IEEE Xplore. Restrctons apply.
10 CHEN et al.: UNIFIED CROSSLYER FRMEWORK FOR RESOURCE LLOCTION IN COOPERTIVE NETWORKS 3009 to the destnaton node n a beamformng manner. From the beamformng capacty formula for MISO channel [1], the capacty of the rd channel can be obtaned by C r d j =log ( 1+P Σ j h nd(j /σ 2, (24 n R j where Pj Σ denotes the total power of relays n the second tmeslot. ccordng to the maxflowmncut theorem, the relable transmsson rate should be the mnmum of the sr and r d channel capactes. By takng two tmeslots per transmsson, the capacty of cooperatve beamformng s gven by { Cj CB = 1 2 mn log (1+g j Pj s/σ2, } ( (25 log 1+Pj Σ n R j h nd(j /σ 2, Hence, we have Pj sg j = Pj Σ n R j h nd(j n order to maxmze the power effcency. Snce the energy consumpton n both tmeslots are constraned, the resource allocaton takes both tmeslots nto account so that the total energy constrant s E max = E total. Consder node j usng the relay set R j for tme duraton t j. Wth beamformng, the power that node R j consumes n relayng node j s sgnal s P r d(j = Pj Σh d(j/ n R j h nd(j. Note that the energy consumed by a node s the result of multplyng ths node s transmsson power and ts transmsson tme. Snce the relays only transmt n the second tmeslot, we have E (j =P r d(j t /2=t j Pj Σh d(j/ ( 2 n R j h nd(j.the source node j tself transmts n both tmeslots, wth power Pj s n the frst tmeslot and power P r d(j ( n the second tmeslot. Therefore, we have E (j j = t j Pj Σ h jd(j / n R j h nd( + n R j h nd( /g j /2. One can easly see that E (j =0,f / R j for any cooperaton schemes. By substtutng nto (3, the results n Table I can be obtaned accordngly. B. Selecton Relayng s selecton relayng s adopted, the power that a source node transmts to relays s fxedtobep s so that the relays may know the decodng threshold of the sr channel gan [5], ξ j = (2 2rj 1σ 2 /P s, where r j s the target rate of node j. s a result, only the even tmeslot s consdered by the resource allocaton and the energy constrant s E max = E total P s T /2. Smlar to ppendx I, the energy consumed by a node s determned by multplyng ths node s transmsson power and ts transmsson tme. Wth selecton relayng, the total tme that node R j serves as relay for node j s t j Pr{ D}= t j Pr{h j ξ j }/2. 5 Snce the nodes n D transmt wth equal power P r d(j, t follows that E (j = P r d(j Pr{h j ξ j }/2. By substtutng nto (3, the results n Table I can be obtaned accordngly. 5 Note that Pr{h jj ξ } =1. Wth selecton relayng, a node can always retransmt ts own message n the second tmeslot. PPENDIX B NETWORK DECOMPOSITION Network Decomposton lgorthm Step 0: Intalze M =1,andS (1 = ; Step 1: If M m=1 S(m = S, go to End; Step 2: Choose one node s such that s/ M Let T = {s} and S (M+1 = {s}; Step 3: If T =, go to Step 6; Step 4: T = {n : a np + a pn > 0, p S (M+1, n/ S (M+1 }; Step 5: S (M+1 = S (M+1 T, go to Step 3; Step 6: M = M +1, go to Step 1. End PPENDIX C PROOF OF THEOREM 1 m=1 S(m. We begn by provdng the followng lemma. Lemma 1: The dmenson of the soluton space to (me(m =e(m s 1 wth the soluton partcularly beng a nonnegatve vector. Proof: Snce a j = 0 for j S (n and / S (m, t follows that S (m a (m j = N =1 a j = 1. Therefore, (mt s a stochastc matrx. Thus, there exsts an egenvalue λ( (m =1. By the Gersgorn dsk theorem [20], the spectral radus of (m s ρ( (m =1.LetU denote the set satsfyng U S (m and a j =0for U, j S (m /U. IfU =, the matrx (m s rreducble. Snce a (m > 0 for any, (m s a prmtve matrx. By the PerronFrobenus theorem [20], geomult (m(1 = 1 and there must exst a postve vector e (m satsfyng (m e (m = e (m.ifu, there must be a permutaton matrx Q satsfyng Q T (m Q = [ B( S (m U ( S (m U C ( S (m U U 0 U ( S (m U D U U ] (26 where B, C, and D are matrx blocks wth the approprate sze. Snce Q 1 = Q T, t follows that (m Qx = QQ T (m Qx = Qx. Therefore, the soluton to (m e (m = e (m can be obtaned by, where x s the soluton to Q T (m Qx = x. In the parttoned form, ths equaton s presented by [ ][ ] [ ] B C x1 x1 =. (27 0 D x 2 ccordng to condton (3 of the network decomposton propertes, the matrx C cannot contan the allzero column. Otherwse, the subnetwork S (m can be decomposed. By the Gersgorn dsk theorem, the spectral radus of D s bounded by ρ(d max U =1 D j =max ( 1 S (m U =1 C j < 1. s a result, the soluton to Dx 2 = x 2 much be x 2 = 0. By substtutng x 2 =0nto (22, we have Bx 1 = x 1.SnceB s prmtve and ρ(b =1, t follows that geomult B (1 = 1 and x 1 can be a postve vector. Snce Q s a permutaton matrx satsfyng (26 amd e (m = Qx, t follows that e (m =0for U and e (m > 0, for S (m /U. Havng establshed Lemma 1, we now prove Theorem 1. Proof: From Lemma 1, we know that geomult (m(1 = 1 and rank( (m I = S (m 1. Snce0 a (m j 1, x 2 uthorzed lcensed use lmted to: Stanford Unversty. Downloaded on March 08,2010 at 13:26:29 EST from IEEE Xplore. Restrctons apply.
11 3010 IEEE TRNSCTIONS ON WIRELESS COMMUNICTIONS, VOL. 7, NO. 8, UGUST T does not belong to the space spanned by all the rows of ( (m I. s a result, { 1 T e (m = 1 (m e (m = e (m has a unque soluton e (m, 6 whch s gven by [ ] 1 [ e (m = (m 0 ( S (m 1 1 ]. (28 Snce 1 T e (m =1, e (m s nonnegatve. Next, by notng that a permutaton matrx satsfes P 1 = P T,wehave 0 ( m 1 b (m = PP T r=1 P 1 e (m 0 ( M r=m+1 S(r 1 (1 = P... (M (29 0 = P (m e (m 0 0 = P e (m 0 = b (m. Hence, b (m belongs to the soluton space of (11. From (15, we have b (m b (n =0,form n. Therefore, the b (m are orthogonal. That s, b (m, b (n =0. Fnally, we show that the bass vectors are complete. ccordng to (13 and snce P 1 = P T, we have = Pdag( (1,..., (M P T.Snce det ( λi =det(pdet ( dag( (1,..., (M det(p T = M m=1 det ( (m λi (30, t follows that geomult (1 = M m=1 geomult (m(1 = M. Therefore, the dmenson of the soluton space of (11 s M. Thus, the soluton space can be spanned by the bass b (m, m =1,...,M. The energy allocaton vector can be gven by M e = μ (m b (m. (31 m=1 In order to ncrease the energy effcency, as much as possble energy should be allocated n each state durng teraton. Note that e (k s always a nonnegatve vector. The 1norm of e (k s gven by N N e (k 1 = e (k = e (k =1 =1 s a result, maxmzng the 1norm of e (k s equvalent to maxmzng the energy allocated n state k. Constraned 6 s U =, e (m s the socalled Perron vector of (m [22]. by (12, μ (m, m =1,...,M, are chosen to maxmze the 1norm of (31, whch can be obtaned as μ (m = mn e max S (m b (m. (32 Here, (32 can be proved as follows. Snce b (m b (n =0for m n, b (m are orthogonal bass for the soluton space of (11 and are nonnegatve. If there exsts a soluton ẽ to (11, satsfyng ẽ 1 > e 1, then there must be m, satsfyng μ m >μ m. From (32, we have μ (m ( > mn e max S (m /b (m. (33 ( Let j =argmn S (m e max. Clearly, we have e j /b (m > μ mb (m j >e max. (34 Ths contradcts constrant (12. s a result, t follows that μ m maxmze the 1norm of e. PPENDIX D DERIVTION OF UTILITY FUNCTIONS. Cooperatve Beamformng s R (k 2, the capacty s determned by the total power of the vrtual array. Snce Pj sg j = P j Σ n R j h nd(j, the energy used by the vrtual array s e (kg / ( g + j R h (k jd(. Snce the vrtual antenna array only transmts n even tmeslots, the capacty of node n state k s gven by ( e U (k,k t (k = 1 ( 2 log 1+ 2g j R (k h jd( ( e (k (35 σ 2 t (k. g + j R (k h jd( s R (k = 1, we assume that the source repeats ts transmsson n the second tmeslot. 7 Hence, the capacty s obtaned by U,k ( e (k t (k = 12 log ( 1+ 2h d( σ 2 e (k. (36 t (k By dfferentatng the utlty functon twce, we have U c 2,k (x = (k 0. (37 2[1 + c (kx] 2 Hence, U(x s a concave functon. B. Selectve Relayng Usng the methods n ppendx IB and the Raylegh fadng assumpton, we know that the each relay s transmsson power, P r d(, s gven by P r d( 2e = (k t (k exp( ξ/hj. j R (k From [5], the outage threshold s η = (22r 1σ 2. By P r d( substtutng P r d( nto the formula of η,weget 1σ 2 η (k = (22r j R exp( ξ (k /h j 2e (k/t. (38 (k 7 In fact, repetton codng s not rateoptmal compared wth drect wth drect transmsson n each tmeslot. In low SNR regme, however, t s near optmal. uthorzed lcensed use lmted to: Stanford Unversty. Downloaded on March 08,2010 at 13:26:29 EST from IEEE Xplore. Restrctons apply.
12 CHEN et al.: UNIFIED CROSSLYER FRMEWORK FOR RESOURCE LLOCTION IN COOPERTIVE NETWORKS 3011 Usng the method of condtonal probablty, the outage probablty can be gven by P out, (k = Pr h jd( <η (k D Pr {D}, D R (k j D (39 where the probablty of a partcular D can be obtaned by Pr{D} = ( exp ξ [ ( ] ξ 1 exp. (40 h j D j h / D j By usng the generatng functon [13], [22], the condtoned outage probablty s calculated as 8 Pr h jd( <η (k D = [ ( ] 1 exp η (k h j D j D jd( D R (k h jd( h jd( h nd(. (41 n j By substtutng (39(41 nto u (k =1 P out, (k, the results n Table II can be obtaned. Next, we shall show that the utlty functon of selecton relayng s convex n the hgh SNR regme, whch denotes the regme of nterest for outage probablty. From [5], the condtonal outage probablty can be approxmated by a power functon of x = e (k t (k n the hgh SNR regme. That s Pr h jd( <η (k D 1 ( s D 1, D! x h j D j D jd( (42 where s (k = (22r 1σ 2 2 j R exp (k ( ξ. It follows h j that the utlty functon can be approxmately gven by U,k (x 1 Pr{D} ( s D 1 (43 D! x h jd( By dfferentatng t twce, we have U,k(x D R (k Pr{D}( D +1 ( D 1! j D s D x D +2 1 h j D jd( 0. (44 Thus, t follows that the utlty functon of selecton relayng s concave n the mportant hgh SNR regme. [3]. Sendonars, E. Erkp, and B. azhang, User cooperaton dversty part I: system descrpton, IEEE Trans. Commun., vol. 51, no. 11, pp , Nov [4]. Sendonars, E. Erkp, and B. azhang, User cooperaton dversty part II: mplementaton aspects and performance analyss, IEEE Trans. Commun., vol. 51, no. 11, pp , Nov [5] J. N. Laneman and G. W. Wornell, Dstrbuted spacetmecoded protocols for explotng cooperatve dversty n wreless networks, IEEE Trans. Inform. Theory, vol. 49, no. 10, pp , Oct [6] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, Cooperatve dversty n wreless networks: effcent protocols and outage behavor, IEEE Trans. Inform. Theory, vol. 50, no. 12, pp , Dec [7] R. U. Nabar, H. Bolcske and F. W. Kneubuhler, Fadng relay channels: performance lmts and spacetme sgnal desgn, IEEE J. Select. reas Commun., vol. 22, no. 6, pp , ug [8] H. Ocha, P. Mtran, H. V. Poor, and V. Tarokh, Collaboratve beamformng for dstrbuted wreless ad hoc sensor networks, IEEE Trans. Sgnal Processng, vol. 53, no. 11, pp , Nov [9] G. Barrac, R. Mudumba, and U. Madhow, Dstrbuted beamformng for nformaton transfer n sensor networks, n Proc. IEEE IPSN, Calforna, US, pr. 2004, pp [10] G. Kramer, M. Gastpar, and P. Gupta, Cooperatve strateges and capacty theorems for relay networks, IEEE Trans. Inform. Theory, vol. 51, no. 9, pp , Sept [11] L. Da and K. B. Letaef, Crosslayer desgn for combnng cooperatve dversty wth truncated RQ n adhoc networks, n Proc. IEEE Globecom, St. Lous, US, Nov. 2005, pp [12] J. Luo, R. S. Blum, L. J. Greensten, L. J. Cmn, and. M. Hamovch, New approaches for cooperatve use of multple antennas n adhoc wreless networks, n Proc. IEEE VTCFall, Los ngeles, C, Sept. 2004, pp [13] J. Luo, R. Blum, L. Cmn, L. Greensten, and. M. Hamovch, Power allocaton n a transmt dversty system wth mean channel gan nformaton, IEEE Commun. Letter, vol. 9, no. 7, pp , July [14]. K. Sadek, Z. Han, and K. J. R. Lu, Multnode cooperatve resource allocaton to mprove coverage area n wreless networks, n Proc. IEEE Globecom, St. Lous, US, Nov. 2005, pp [15] J. Chang and L. Tassulas, Maxmum lfetme routng n wreless sensor networks, IEEE/CM Trans. Networkng, vol. 12, no. 4, pp , ug [16] V. Srnvasan, P. Nuggehall, C. Chassern, and R. R. Rao, n analytcal approach to the study of cooperaton n wreless adhoc networks, IEEE Trans. Wreless Commun., vol. 4, no. 2, pp , Mar [17] O. Iler, S. Mau, and N. B. Mandayam, Prcng for enable forwardng n selfconfgurng ad hoc networks, IEEE J. Select. reas Commun., vol. 23, no. 1, pp , Jan [18] P. Marbach and Y. Qu, Cooperaton n wreless ad hoc networks: a marketbased approach, IEEE/CM Trans. Networkng, vol. 13, no. 6, pp , Dec [19] L. Da, W. Chen, K. B. Letaef, and Z. Cao, far multuser cooperaton protocol for ncreasng the throughput n energyconstraned adhoc networks, n Proc. IEEE ICC, Istanbul, Turkey, June, [20] R.. Horn and C. R. Johnson, Matrx nalyss. New York: Cambrdge Unversty Press, [21] R. T. Rockafellar, Convex nalyss, 2nd ed. Prnceton Unversty Press, [22] Y. Zhao, R. dve, and T. J. Lm, Outage probablty at arbtrary SNR wth cooperaton dversty, IEEE Commun. Lett., vol. 9, no. 8, pp , ug CKNOWLEDGMENT The authors would lke to thank anonymous revewers and the assocate edtor for ther constructve comments. REFERENCES [1] I. Telatar, Capacty of multantenna Gaussan channels, Eur. Trans. Telecommun., vol. 10, no. 6, pp , Nov./Dec [2] L. Zheng and D. N. C. Tse, Dversty and multplexng: a fundamental tradeoff n multpleantenna channels, IEEE Trans. Inform. Theory, vol. 49, pp , May Due to the random dstrbuton of nodes, we assume that h jd( h nd(, for j n here. If h jd( = h nd(, the results can be found n [22]. We Chen (S 03M 07 receved hs BS and PhD degrees n Electronc Engneerng (both wth the hghest honors from Tsnghua Unversty, Bejng, Chna, n 2002, and 2007, respectvely. From May 2005 to January 2007, He was a vstng research staff n Department of Electronc and Computer Engneerng, the Hong Kong Unversty of Scence and Technology. Snce July 2007, he has been wth Department of Electronc Engneerng, Tsnghua Unversty, where he s currently an ssstant Professor. He vsted the Chnese Unversty of Hong Kong on October, Hs research nterests are n broads areas of nformaton theory, wreless networks and optmzaton theory. He has served as TPC members for a number of major nternatonal conferences, ncludng IEEE ICC, Globecom, and WCNC. He served as student travel grant char of ICC 2008 and cochar of cogntve and cooperatve network workshop In 2004, he receved the uthorzed lcensed use lmted to: Stanford Unversty. Downloaded on March 08,2010 at 13:26:29 EST from IEEE Xplore. Restrctons apply.
13 3012 IEEE TRNSCTIONS ON WIRELESS COMMUNICTIONS, VOL. 7, NO. 8, UGUST 2008 nnovaton fundng for outstandng PhD canddates n Tsnghua Unversty. In 2008, he won the key faculty fundng support plan. He receved the Best Paper ward at IEEE ICC 2006 and the Best Paper ward at IEEE IWCLD Ln Da (S 00M 03 receved the B.S. n electronc engneerng from Huazhong Unversty of Scence and Technology, Wuhan, Chna, n 1998 and the M.S. and Ph. D. degrees n electrcal and electronc engneerng from Tsnghua Unversty, Bejng, Chna, n 2000 and 2002, respectvely. She was a postdoctoral fellow at the Hong Kong Unversty of Scence and Technology and Unversty of Delaware. Snce 2007, she has been wth Cty Unversty of Hong Kong, where she s an assstant professor. Her research nterests nclude wreless communcatons, nformaton and communcaton theory. Dr. Da receved the Best Paper ward at IEEE WCNC Khaled B. Letaef (S 85M 86SM 97F 03 receved the BS degree wth dstncton n Electrcal Engneerng from Purdue Unversty at West Lafayette, Indana, US, n December He receved the MS and Ph.D. Degrees n Electrcal Engneerng from Purdue Unversty, n ugust 1986, and May 1990, respectvely. From January 1985 and as a Graduate Instructor n the School of Electrcal Engneerng at Purdue Unversty, he has taught courses n communcatons and electroncs. From 1990 to 1993, he was a faculty member at the Unversty of Melbourne, ustrala. Snce 1993, he has been wth the Hong Kong Unversty of Scence & Technology where he s currently a Char Professor of Electronc and Computer Engneerng (ECE and Head of the ECE Department. He s also the Drector of the Hong Kong Telecom Insttute of Informaton Technology as well as the Drector of the Center for Wreless Informaton Technology. Hs current research nterests nclude wreless and moble networks, Broadband wreless access, OFDM, CDM, and Beyond 3G systems. In these areas, he has publshed over 280 journal and conference papers and gven nvted keynote talks as well as courses all over the world. Dr. Letaef served as consultants for dfferent organzatons and s currently the foundng Edtorn Chef of the IEEE Transactons on Wreless Communcatons. He has served on the edtoral board of other prestgous journals ncludng the IEEE Journal on Selected reas n Communcatons  Wreless Seres (as EdtornChef and the IEEE Transactons on Communcatons. He has been nvolved n organzng a number of major nternatonal conferences and events. These nclude servng as the Techncal Program Char of the 1998 IEEE Globecom MnConference on Communcatons Theory, held n Sydney, ustrala as well as the CoChar of the 2001 IEEE ICC Communcatons Theory Symposum, held n Helsnk, Fnland. In 2004, he served as the CoChar of the IEEE Wreless Communcatons, Networks and Systems Symposum, held n Dallas, US as well as the CoTechncal Program Char of the 2004 IEEE Internatonal Conference on Communcatons, Crcuts and Systems, held n Chengdu, Chna. He s also the General CoChar of the 2007 IEEE Wreless Communcatons and Networkng Conference, WCNC 07, held n Hong Kong as well as the Techncal Program CoChar of the 2008 IEEE Internatonal Conference on Communcaton, ICC 08 held n Bejng. He served as the Char of the IEEE Communcatons Socety Techncal Commttee on Personal Communcatons as well as a member of the IEEE ComSoc Techncal ctvty Councl. In addton to hs actve research actvtes, Professor Letaef has been a dedcated teacher commtted to excellence n teachng and scholarshp. He receved the Mangoon Teachng ward from Purdue Unversty n 1990; the Teachng Excellence pprecaton ward by the School of Engneerng at HKUST (four tmes; and the Mchael G. Gale Medal for Dstngushed Teachng (Hghest unverstywde teachng award and only one recpent/year s honored for hs/her contrbutons. He s a Fellow of IEEE and an IEEE Dstngushed lecturer of the IEEE Communcatons Socety, and an elected member of the IEEE Communcatons Socety Board of Governors. Zhgang Cao (M 84SM 85 Graduated wth golden medal from the Department. of Rado Electroncs at Tsnghua Unversty, Bejng n 1962.He was a vstng scholar at Stanford Unversty from 1984 to Snce 1962, He has been wth Tsnghua Unversty, where he s currently a Professor of Electronc Engneerng Department, Vce Drector of the Natonal Key Laboratory on Mcrowave and Dgtal Communcatons He has publshed sx books and more than 360 papers on Communcaton and Sgnal Processng felds, and held 20 patents. He has won the frst prze of the Best Engneerng Textbook ward n Chna. For hs research achevements he won nne awards from Natonal Scence Congress, Mnstry of Electronc Industry, State Educaton Commsson, Mnstry of Educaton, Bejng Muncpal Government, etc. He has also won specal grant ssued by the Chnese Government for hs outstandng contrbutons to educaton and research. Hs current research nterests nclude moble communcatons, satellte communcatons, modulaton/demodulaton, codng/decodng, and speech sgnal processng. Prof. Cao s a fellow of Chnese Insttute of Communcatons (CIC and serves as a vce charman of cademc Commttee of CIC. He s also a senor member of IEEE, senor member of Chnese Insttute of Electroncs (CIE, member of the board of drectors of Bejng Communcaton nsttute, and member of board of drectors of Chna Satellte Communcaton Broadcastng & Televson ssocaton. One of PhD students under hs supervsng obtaned 2003 best PhD dssertaton award among all unverstes n Chna. nd hs other PhD student obtaned 2005 best PhD dssertaton award nomnator. He also serves as an assocate edtornchef of CT Electroncs Snca, and the edtor of Fronters of Electronc and Engneerng n Chna, Chnese Journal of Electroncs, Chna Communcatons, Journal of Chna Insttute of Communcatons and Journal of stronautcs. uthorzed lcensed use lmted to: Stanford Unversty. Downloaded on March 08,2010 at 13:26:29 EST from IEEE Xplore. Restrctons apply.
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