THere are increasing interests and use of mobile ad hoc

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1 1 Adaptve Schedulng n MIMO-based Heterogeneous Ad hoc Networks Shan Chu, Xn Wang Member, IEEE, and Yuanyuan Yang Fellow, IEEE. Abstract The demands for data rate and transmsson relablty constantly ncrease wth the explosve use of wreless devces and the advancement of moble computng technques. Multple-nput and multple-output (MIMO) technque s consdered as one of the most promsng wreless technologes that can sgnfcantly mprove transmsson capacty and relablty. Many emergng moble wreless applcatons requre peer-to-peer transmssons over an ad hoc network, where the nodes often have dfferent number of antennas, and the channel condton and network topology vary over tme. It s mportant and challengng to develop effcent schemes to coordnate transmsson resource sharng among a heterogeneous group of nodes over an nfrastructure-free moble ad hoc network. In ths work, we propose a holstc schedulng algorthm that can adaptvely select dfferent transmsson strateges based on the node types and channel condtons to effectvely releve the bottleneck effect caused by nodes wth smaller antenna arrays, and avod the transmsson falure due to the volaton of lower degree of freedom constrant resulted from the channel dependency. The algorthm also takes advantage of channel nformaton to opportunstcally schedule cooperatve spatal multplexed transmssons between nodes and provde specal transmsson support for hgher prorty nodes wth weak channels, so that the data rate of the network can be maxmzed whle user transmsson qualty requrement s supported. The performance of our algorthm s studed through extensve smulatons and the results demonstrate that our algorthm s very effectve n handlng node heterogenety and channel constrant, and can sgnfcantly ncrease the throughput whle reducng the transmsson delay. Index Terms Moble computng, ad hoc networks, schedulng, dstrbuted, cross-layer desgn, MIMO. 1 INTRODUCTION THere are ncreasng nterests and use of moble ad hoc networks wth the prolferaton of moble, networkenabled wreless devces, and the fast progress of computng technques and wreless networkng technques. As the number, CPU power and storage space of wreless devces contnue to grow, there s a sgnfcant ncrease n data transmsson demand to support data ntensve moble computng and applcatons, such as multmeda streamng, gamng, as well as transmsson of a large amount of montorng data. To meet the hgh data rate requrements, more and more wreless devces are equpped wth multple antennas. Wth multple antennas at the transmtter and/or recever, a MIMO (multple-nput-multple-output) system takes advantage of multplexng to smultaneously transmt multple data streams to ncrease the wreless data rate and dversty to optmally combne sgnals from dfferent transmsson streams to ncrease the transmsson relablty and range. The benefts of MIMO lead many to beleve t s the most promsng technque of emergng wreless technologes. MIMO s promnently regarded as a technology of choce for next generaton wreless systems such as IEEE 82.16, IEEE 82.11n, and the thrd and fourth generaton cellular systems. It s also beng consdered for supportng peer to peer moble applcatons over an nfrastructure free ad-hoc network. Although MIMO technques have been wdely studed n a more centralzed and nfrastructure-based cellular system, there are very lmted work and bg challenges n extendng MIMO technque nto a fully dstrbuted system over an nfrastructure-free wreless ad hoc network. Dfferent from an nfrastructure-based system, t s dffcult for nodes to Shan Chu s wth Motorola Solutons, One Motorola Plaza, Holtvlle, NY E-mal: shan chu@hotmal.com. Xn Wang and Yuanyuan Yang are wth the Department of Electrcal and Computer Engneerng, Stony Brook Unversty, Stony Brook, NY E-mal: {xwang,yang}@ece.sunysb.edu. coordnate n channel evaluatons and transmssons n a dstrbuted manner. The fast varaton of channel condton and network topology, the nconsstency n node densty as well as the dfferent traffc demands and servce requrements of nodes lead to more open challenges n coordnatng dstrbuted node transmssons. Moreover, n a moble computng envronment, the network could be heterogeneous, whch ncurs addtonal challenges to MIMO MAC desgn. Frst, network nodes may be equpped wth dfferent number of antennas. The exstence of nodes wth smaller antenna array szes may lead to sgnfcant network performance reducton. Ether the concurrent number of transmssons n a neghborhood needs to be lmted n order to meet the decodng constrant of recevers equpped wth a lower number of antennas, or the lower-antenna nodes wll be sgnfcantly nterfered by neghborng nodes transmttng a larger number of streams at the same tme. Second, the transmsson envronment could be heterogeneous, wth channel condtons dfferent between each node par and varyng over tme, leadng to the varaton of the smultaneous streams allowed between a node par. These two factors jontly determne the number of orthogonal channels (.e., degree of freedom) an envronment allows. It s crtcal for transmtter nodes to be aware of the allowed degree of freedom of a lnk for the correct decodng at recever nodes. Recently, there have been some efforts n developng algorthms and protocols for applyng MIMO technques to ad hoc networks [1] [1], however, t s far from trval to extend the soluton n homogeneous cases to heterogeneous cases especally n a dstrbuted system. To the best of our knowledge, there has not been any effort to specfcally allevate the transmsson lmt thus performance degradaton due to the network heterogenety n a dstrbuted, peer to peer, ad hoc transmsson envronment. To enable more powerful moble computng and applcatons n a practcal system, the objectve of ths work s

2 2 to desgn a holstc framework for schedulng to adaptvely coordnate sharng of transmsson resources among heterogeneous nodes n a varyng physcal operatonal scenaro. The man contrbutons of ths paper are as follows. Our schedulng algorthm concurrently consders antenna array sze, channel condton, traffc demand and multuser dversty. In each transmsson duraton, the algorthm opportunstcally schedules the nodes to transmt and determnes the set of antennas to use at a selected node n a dstrbuted manner, explotng multuser dversty and antenna selecton dversty to sgnfcantly mprove the transmsson throughput and relablty. Through prorty-aware schedulng, our algorthms also support servce dfferentaton whle reducng the transmsson delay and ensurng the farness among nodes. As nterference algnment and transmtter sde nterference cancelaton technques brng too much computatonal overhead and are dffcult to apply n a mesh network scenaro, we only consder recever-sde nterference cancelaton n ths work. Our algorthms specfcally allevate the constrants caused by node heterogenety and the lower-rank channel, by adaptvely selectng dfferent transmsson strateges based on both the antenna array szes of nodes n a neghborhood and the degree-of-freedom the transmsson channel allows. We mathematcally formulate the problem to maxmze the weghted network throughput, and propose a centralzed schedulng algorthm as a performance benchmark. Dfferent from the lterature work [3] [5], [11], [12] whch are based on the smple antenna model, our formulaton takes nto account the dfferent transmsson rates between dfferent nodes and antenna pars and the constrant on the degree of freedom due to the channel condton between a node par. Our schedulng algorthm thus can select nodes and antennas wth better transmsson rate n a tme slot, takng advantage of mult-user dversty and antenna selecton dversty for a hgher throughput. The algorthm also consders the transmsson prorty and balances the network load, whle avodng transmsson falure by not overloadng a lower-rank channel wth more streams. We perform extensve smulaton to nvestgate the mpact of varous factors on the performance of MIMO schedulng, and to evaluate the effectveness of our proposed algorthm. Specfcally, to examne the mpact of channel heterogenety, we studed the mpact on the spatal channel both due to antenna array sze and lneof-sght component. By consderng more realstc channel condtons and network heterogenety, our results are expected to provde some nsght for the mplementaton of the algorthms n practcal networks. Our smulatons consder both the statc network topology and a dynamc network topology wth moblty modeled by the random waypont model. However the model dynamcs affect only path loss, and do not change our assumpton of quas-statc Rcan fadng durng a transmsson duraton. The rest of the paper s organzed as follows. We present the related work n Secton 2. Secton 3 dscusses the background nformaton ncludng MIMO technologes and ther applcaton n heterogeneous networks. In Secton 4, the system model s defned and the problem s mathematcally formulated, followed by Secton 5, where a centralzed algorthm s proposed to solve t. Secton 6 presents the adaptve dstrbuted schedulng algorthm and the protocol to mplement the algorthm. Smulaton results are provded n Secton 7 and the paper s concluded n Secton 8. 2 RELATED WORK Over the past several years, the applcaton of MIMO technology n networks has undergone a fast development. Earler studes have been performed to develop schedulng schemes to select the best users to transmt based on certan crtera n a multuser MIMO-based cellular network. An overvew of schedulng algorthms n MIMO-based fourthgeneraton wreless systems s gven n [13]. In recent years, many efforts have been made to support MIMO transmsson n local area wreless networks. In [1], spatal dversty (e.g. space tme codng (STC)) s explored to combat fadng and acheve robustness. SPACE-MAC, proposed n [2], enables denser spatal reuse patterns wth the ad of transmtter and recever beamformng. In [14], the desgn and mplementaton of a cross-layer system s presented to enable spatal multplexed transmssons from multple devces to the access pont n the uplnk drecton n a wreless LAN envronment, whle coordnatng node transmssons n a mult-hop meshed network s much more challengng, and the challenge ncreases f the network s heterogeneous. Recently, there have also been some efforts n applyng MIMO wth nterference cancellaton and algnment nto wreless networks through testbed study [15] [17] for MIMO applcaton n wreless LANs. However, wreless LANs are dfferent from wreless mesh and ad hoc networks n the network scale. The algorthm and framework for wreless LANs cannot be smply appled to mult-hop wreless networks, especally n the challengng dstrbuted case. Specfcally, nterference algnment requre sgnfcantly more complcated coordnaton and extra nformaton of channel condtons n a mult-hop ad hoc network scenaro, whch makes t dffcult to apply n a practcal network. Moreover, although nterference algnment s consdered to provde the maxmum degree of freedom theoretcally, t does not guarantee to acheve the maxmum throughput. Therefore, t s out of the scope of ths work. Due to the dffculty of modelng the benefts and constrants of MIMO transmssons n ad hoc networks, only a lmted number of efforts focus on the network performance from the optmzaton perspectve. A centralzed algorthm s presented n [3] to solve the jont routng, schedulng and stream control problem subject to the farness constrant n mesh networks wth MIMO lnks. In [4], the authors characterze the rado and nterference constrants n mult-hop wreless MIMO networks and formulate a mult-hop jont routng and MAC problem to study the maxmum achevable throughput subject to these constrants. The problem of jontly optmzng power and bandwdth allocaton at each node and mult-hop/mult-path routng n a MIMO-based ad hoc network s studed n [5], and a soluton procedure s developed to solve ths cross-layer optmzaton problem. In [11], [12], the authors study the lnk layer model for mult-hop MIMO networks based on accurate accountng of DoF consumpton and a node-level orderng scheme was proposed to dentfy the role of each node n performng nterference cancellaton. In [18], optmal stream schedulng

3 3 for MIMO lnks s studed for a sngle collson doman, and t was shown that optmum throughput s acheved when the task of nterference cancellaton s shared equally between every transmtter and every recever. Although these efforts are mportant, the aforementoned work [3], [4], [11], [12] have assumed smplfed physcal model and overlooked the mpact of channel condton by assumng streams have homogeneous data rate. In fact, these smplfcatons may not only sgnfcantly compromse the network performance, but also make the optmal model formulated far from the practcal network condton. In addton, none of them provdes a feasble soluton to effcently coordnate node transmssons n a practcal dstrbuted scenaro. The features and performance of a few antenna technques are presented n [6], however there s no desgn to enable the selecton of a specfc antenna technque, whch s the major challenge n MAC desgn. A number of dstrbuted schemes have also been proposed for MIMO MAC desgns. In [7], the authors dscuss key consderatons for MIMO MAC desgn, and develop a centralzed algorthm and a dstrbuted algorthm to mprove the transmsson farness. Based on CSMA/CA for control sgnal exchanges, t s hard for the algorthm to support cooperatve transmssons. In [8], spatal multplexng wth antenna subset selecton for data packet transmsson s proposed, based on nodes wth two antennas and a smple network topology. In [9], each transmtter or recever greedly accommodates the number of data and nterference streams up to a pre-determned maxmum number. However, t s possble for dfferent recevers to make conflctng decsons on the transmsson requests to accept, wastng the decodng capablty or exceedng the decodng lmt of the recevers. In addton to ssues assocated wth each scheme dscussed above, most MIMO MAC schemes mplctly assume that the channel condton s known. In practce, coordnatng channel measurement tself s a bg challenge n presence of a group of competng nodes, especally n a dstrbuted and a dynamc ad hoc network. In exstng work, the number of antennas or the pre-determned decodng lmt s often used as the constrant of transmsson and recevng wthout consderng the actual physcal channel varaton thus the smultaneous streams allowed by a channel, whch may result n the transmsson falure. In [1], a cooperatve multplexng scheme s proposed, however, t does not consder the heterogenety of antenna arrays n ad hoc networks. We have made an effort to provde an adaptve and dstrbuted soluton consderng the heterogenety of antenna array szes of network nodes n [19]. In ths paper, we further nvestgate the mpact of channel condton on the degree of freedom of MIMO channels. We remodel the problem to more accurately capture the transmsson constrants due to both the number of antennas and channel condtons. We also modfy the dstrbuted algorthms and perform more extensve smulatons to demonstrate the functonalty of the proposed dstrbuted algorthms. In addton, we propose a centralzed soluton wth a proved approxmaton rato to serve as the benchmark of the dstrbuted algorthm. 3 BACKGROUND AND MOTIVATION As mentoned n Secton 1, wth multple antennas at the transmtter and/or recever, multple data streams may be transmtted between a transmsson node par, whch s called spatal multplexng. At the recever, each antenna receves a superposton of all of the transmtted data streams. In a rch scatterng envronment where the transmsson channels for dfferent stream are dfferentable and ndependent,.e. orthogonal, an ntended recever node can separate and decode ts receved data streams based on ther unque spatal sgnatures. Ths multplexng gan can provde a lnear ncrease (n the number of antenna elements) n the asymptotc lnk capacty. Wth multple transmsson paths, the transmsson qualty could be very dfferent. Instead of sendng dfferent data through each transmttng antenna, spatal dversty may be exploted to mprove transmsson relablty. There are dfferent types of dversty technques. Wthout channel nformaton, dependent streams can be transmtted on dfferent antenna elements over multple tme slots and mprove transmsson qualty through space tme codng. When channel nformaton s avalable, a subset of antennas that can transmt sgnals at better qualty can be selected for transmssons through selecton dversty, whch s shown to outperform space-tme codng [2]. As a more powerful yet more sophstcated scheme, data streams can be properly coded accordng to the channel nformaton and sent through dfferent transmt antennas,.e., through precodng, to acheve the maxmum throughput at the recever. In ths secton, we frst present the problems due to the lmtaton of channel degree of freedom and heterogeneous number of network nodes. We then ntroduce the potental strateges to address the ssues, and the tradeoff between dfferent strateges. In MIMO communcatons, the spatal channels between two neghborng nodes n and n k whch have N ant and Nk ant antenna elements respectvely can be represented as a N ant N ant matrx: H k = h 11 h h 1N ant h 21 h h 2N ant h N ant k 1 h N ant k 2... h N ant k N ant k. (1) The (p, q)-th entry of H k, h pq, s the spatal channel coeffcent between the p-th antenna of node n k and q-th antenna of node n. Each h pq can generally be represented as [21]: h pq = κ κ + 1 σ le jθ + 1 κ + 1 CN (, σ2 l ), (2) where the frst term denotes the lne-of-sght (LOS) component wth a unform phase θ, and the second term corresponds to the aggregaton of reflected and scattered paths, usually modeled as a crcular symmetrc random varable wth varance σ l. The parameter κ s called K-factor, whch s the rato of the energy n the LOS path to the energy n the scattered paths. When the LOS component s very weak,.e. the propagaton medum s rch scatterng, the channel can be well modeled by Raylegh fadng. When the LOS component between transmtter and recever s strong and/or there exst fxed scatters/sgnal reflectors n addton to random man scatters, Rcan fadng condtons hold and a hgher correlaton s observed between the elements of H k. The degree-of-freedom of a MIMO channel s an mportant metrc to descrbe the dmenson of space that the transmtted sgnals can be projected onto (so the recever can dfferentate the sgnals), and the number of streams allowed to smultaneously transmt between a par of nodes. The

4 Rate N r =1,OCSM N r =1,OTPC N r =2,OCSM N r =2,OTPC N r =3,OCSM N =3,OTPC r Fg. 1. Illustraton of a heterogeneous MIMO network. degree-of-freedom s defned as the rank of the channel matrx H k, or equvalently the number of non-zero egenvalues of H k. From (1), t s obvous that the degree-of-freedom of the channel between n and n k depends on the number of antennas at nodes n and n k, and the lnear ndependency of the matrx whch depends on the scatterng condtons between n and n k. In Fg. 1, the four nodes, each equpped wth an antenna array, are n the transmsson range of each other. In, f node 1 s a selected recever n a tme slot, n order to ensure ts correct decodng only one stream targeted to 1 s allowed to transmt n ts neghborhood. Moreover, when both nodes 1 and 2 are selected as recevers, even though node 4 would be able to transmt up to 2 streams to node 2 (as shown n dashed lnes), f smply schedulng the transmssons based on the mnmum number of streams allowed n a neghborhood [1], only one stream s allowed to be transmtted around node 2 at a transmsson tme (e.g., ether transmttng from node 3 or from node 4). That s, wthout dfferentatng node types, the maxmum number of streams allowed to transmt at any tme slot s constraned by the canddate recever whch has the smallest array. On the other hand, f every recever smply consders ts own decodng constrant [9], a hgher number of transmssons could lead to serous nterference and potental decodng falure at nodes wth a lower number of antennas. In addton, when the channel between node 4 and 2 can only support one transmsson,.e. the degree-of-freedom s 1, but two streams are transmtted, the streams cannot be decoded at the recever. The examples ndcate that t would lead to ether sgnfcant throughput reducton n order to not nterfere wth a node wth lower number of antennas or transmsson falure f the node heterogenety and channel rank constrant are not consdered n the MAC desgn. Addtonal ssues wll arse f some of the channels are weak, and cannot support good qualty transmsson. These practcal problems ndcate that effectve schedulng algorthms need to be desgned to allevate the bottleneck effect and to provde good system performance under any transmsson envronment. A few strateges may help. Frst, when the recever has multple antennas, the constrants to transmssons due to the lower degree-of-freedom between node pars may be mtgated wth the formulaton of cooperatve vrtual MIMO array. In Fgure 1, nodes 1, 2 and 3 can transmt concurrently to node 4 and explot multplexng gan to mprove the throughput. Second, addtonal capacty gan can be acheved wth the exploraton of mult-user dversty and antenna selecton dversty, n whch case, the transmtter nodes and the antenna to use from a node are opportunstcally selected based on the channel condtons between dfferent nodes and antennas. Thrd, when the recever has very few antennas (Node 1 n Fg. 1 ), ts transmtter could employ precodng to optmally weght the transmssons from multple antennas to mprove the data rate. As shown n the example, opportunstc multplexng N t Fg. 2. Comparson of multplexng and transmtter precodng wth vared transmtter/recever antenna array sze. transmsson generally allow multple nodes to smultaneously transmt to a recever that has multple antennas, and a sender wth multple antennas can also transmt multple streams to a set of nodes. The many-to-many nature of the transmsson makes precodng dffcult to be appled across multple transmtters and recevers, as the calculaton of the precodng weghts nvolves multple channel matrces and s much more complcated than the one-to-many case n cellular network. Therefore, precodng t s not used smultaneously wth the cooperatve multplexng. In Fg. 2, a smple experment s performed to compare the data rates acheved by opportunstc transmtter precodng (OTPC) and opportunstc and cooperatve spatal multplexng (OCSM) [1] under a topology where two transmtter nodes are around one recever node wth..d faded channels. The performances of the two are compared wth the varaton of the number of antennas at each node. When the recever antenna array sze s small, transmtter precodng s seen to outperform mult-user multplexng as power gan s more sgnfcant. However, wth more recevng antennas, cooperatve multplexng starts to outperform precodng. From ths smple example, we can see that t s mportant to select an approprate transmsson strategy accordng to specfc constrants, n order to acheve optmum possble performance. Instead of transmttng the same sgnal from multple antennas wth approprate weghtng to ncrease the rate of one stream as done n conventonal beam-formng scheme, n ths work each data packet s transmtted only through one selected stream and selected streams from all canddate antenna pars form many-to-many cooperatve MIMO transmssons to mprove the total network capacty. Therefore, precodng s only assumed to weght the transmssons when multple streams are selected to transmt between a node par. 4 SYSTEM MODEL We consder channel resource allocaton among an ad hoc network of nodes whch have dfferent number of antenna elements and experence dfferent channel condtons. For a group of nodes that share the transmsson resource, one node par s often scheduled to transmt at a tme n the tradtonal MIMO schemes. However, the chance of havng multple strong spatal paths between a node par s small, whch lmts the transmsson rate. Instead, our schedulng schemes support many-to-many transmssons between nodes usng vrtual MIMO arrays, and take advantage of mult-user dversty and antenna selecton dversty to sgnfcantly mprove the transmsson relablty and throughput. Specfcally, to address the challenges due to the network heterogenety, our algorthms adaptvely and flexbly schedule node transmssons usng dfferent MIMO technques, n-

5 5 cludng spatal multplexng, selecton dversty, and precodng, based on the node constrants and channel condtons. For the convenence of presentaton, n ths secton, we frst ntroduce some notatons used n ths paper, and then formulate the problem mathematcally and prove ts NPhardness. 4.1 Stream and Stream Characterstcs A stream s defned to be an ndependent flow of sgnals transmtted from a transmt antenna to a target node and dentfed by a trplet (I tx, I rc, I ant ), where I tx /I rc s the ndex of the transmtter/recever node, and I ant s the ndex of the antenna that nvolves n the transmssons of the stream. Wth the explotaton of selecton dversty, the antennas wth the strongest channel condtons among the canddate ones are selected to transmt the data streams. For a transmtter node wth several streams selected, f the streams target for the same recever, precodng s performed among the selected transmttng antennas wth the power optmally allocated to acheve the maxmum data rate between a node par; otherwse, the power s evenly dstrbuted over the selected antennas for streams targetng for dfferent recevers for the processng smplcty. In order for recever nodes to decode data streams and suppress nterference streams concurrently, the number of streams transmtted or receved at a node s subject to certan constrant. Due to the broadcast nature of wreless channels, streams are categorzed as data streams and nterference streams. A data stream from node n to node n k s receved by n s neghborng node n j as an nterference stream. Denote the degree-of-freedom of the channel between n and n k as DoF (, k), t s clear that n k can dfferentate streams from n only f the number of streams s no more than DoF (, k), whch depend on both the antenna numbers of n and n k and the correlaton level of the channel between the two nodes. Denote the set of all actve recevng nodes (.e., the target recevers of some transmtter nodes) around node n s transmsson range as R actve, as the transmttng constrant, the number of transmttng streams from n should be no larger than N tx = mn k R actve DoF (, k). Note that transmtter sde nterference cancellaton s out of our scope here. Smlarly, to avod erroneous decodng at a recever node n k, the number of smultaneous receved streams Nk rc (ncludng both data streams and nterference streams) should be lmted. Wth use of vrtual MIMO array, the sze of antenna array Nk ant generally provdes the metrc of spatal resoluton at a recever n k, and hence the total receved streams should not exceed the recevng constrant Nk rc = N k ant. The characterstcs of a stream are captured by two parameters, stream prorty P(s) and stream capacty C (s). The stream prorty depends on the servce type and queung delay of the data packet to be sent wth the stream. The value of P(s) s ntally set to the servce prorty of the assocated packet, and ncreases as the queung tme of the packet ncreases. The stream capacty descrbes the maxmum achevable rate of a stream transmsson, whch depends on recevng sgnal to nterference and nose rato of the stream, and can generally be represented as follows: C (s) = log 1 + P s h s N I N ant + 1 P d(s) q h q h q h s, q I(s) (3) where P s s the transmsson power of the stream s, h s s the channel vector from the selected transmtter antenna and the recever of stream s, N s the nose varance, Nd(s) ant s the number of antennas at the recever of stream s and I(s) s the set of streams that nterfere the recevng of stream s at the recever. The value of C (s) can be estmated at a transmtter based on the estmated channel condton and nterference level durng the schedulng. 4.2 Types of Nodes and Slots Our algorthm s TDMA-based, n whch the tme doman s dvded nto transmsson duratons (TD). A TD conssts of several tme slots and covers one round of control sgnal exchange and fxed-sze data frame transmsson. The data transmsson rate wthn a frame can vary based on the channel condton. For a channel wth hgher qualty, more effcent codng can be used to encode the symbols at a hgher rate. A lnk between a transmtter-recever par s halfduplex, so that a node can ether transmt or receve but not at the same tme. Denote the set of nodes n the transmsson range of node n as V A, the recevng constrant of node n k as Nk rc. Snce a node wth a hgher value of Nk rc can generally decode more streams, we use Nk rc as a metrc for measurng the recevng capablty of n k. The average recevng capablty of nodes n V A rc s then represented as N = 1 V A k V N rc A k. rc Compared wth N, f Nk rc rc N, node n consders n k as a rch node as t has relatvely hgher recevng capablty among the neghborng nodes of n ; otherwse, n k s consdered as a poor node and could potentally become a recevng bottleneck n the neghborhood. Note that when all nodes have the same number of antennas, the network contans only rch nodes, and t s degenerated to the homogeneous network case. As dscussed n Secton 3, the lmted decodng capablty of a poor recever constrans the maxmum number of streams (ncludng both data streams and nterference streams) allowable n ts neghborhood. To reduce the constrant, we dvde the transmsson slots nto P-slots and R- slots and assume dfferent transmsson strateges towards poor nodes and rch nodes respectvely. In a P-slot, the number of concurrent transmsson streams s lmted by the recevng constrant of the targeted poor node, and transmtter precodng may be utlzed to optmze the lnk rate. In an R-slot, as only rch nodes serve as the recevers, multuser spatal multplexed transmssons are opportunstcally scheduled for a hgher throughput. 4.3 Problem Formulaton In a TDMA-based MIMO ad hoc network, packets are generated constantly. It s thus practcal to schedule the transmsson of packets n each transmsson duraton (TD) wth the purpose of optmzng temporary network performance. Suppose there s a set of nodes N = {n 1, n 2,..., n Nn } n the network. Based on ther queung packets, node n has a set of canddate streams S, where the destnaton node of the q-th stream s q S s denoted as d(s q ). Let the parameter set {y q } (y q {, 1}, = 1,..., N n, q = 1,..., S ) denote whether the q-th canddate stream of node s transmtted n the current TD. If a stream s q s transmtted, y q = 1; otherwse, y q =. Smlarly, {t } and {h } (t, h {, 1}, = 1,..., N n ) are used to denote the transmtter and recever node assgnment n the current TD respectvely. If

6 6 node n s selected as a transmtter/recever node, we have t = 1/h = 1, otherwse t = /h =. If t = h =, node n s recognzed as an dle node. The assgnment of a stream to a specfc antenna of a transmtter s represented by the parameter a qk (a qk {, 1}, = 1,..., N n, q = 1,..., S and k = 1,..., N ant ), where a qk = 1 f stream s q s assgned to transmt from antenna k of node n. The transmsson rate of stream s q s mpacted by both the strength of the stream (, d(s q ), k) (denoted as S(s q )), and the nterference level at recever node d(s q ) (denoted as I(d(s q ))). The prorty of stream s q depends on the prorty of ts assocated packet and s denoted as P(s q ). The schedulng process selects a set of streams to transmt among all the canddate ones n the current TD. The objectve of the schedulng s to maxmze the sum of prortyweghted capacty of the scheduled streams, so that both data rate and prorty can be jontly optmzed. The problem s formulated as follows: max U = y q C (S(s q ), I(d(s q )))P(s q ); (4) n N s q S a qk 1, = 1, 2,..., N n, k = 1,..., N ant ; (5) s q S y q N tx, = 1, 2,..., N n ; (6) s q S h m V smq Sm d(smq )= y mq + h m V smq Sm d(smq ) y mq N rc, = 1, 2,..., N n ; (7) t + h 1, = 1, 2,..., N n ; (8) a qk y q t, a qk y q h d(sq ), = 1,..., N n, q = 1,..., S, k = 1,..., N ant ; (9) t, h, y q, a qk {, 1}. Constrant (5) ensures that an antenna can only transmt one scheduled stream at most n each slot. Equaton (6) constrans the total number of transmtted streams from n should be no more than ts transmttng constrant value N tx, whch depends on the antenna numbers of n and all ts neghborng recevers as well as the channel ndependency level between n and every recever. Equaton (7) provdes the constrant at recever n where the total number of recevng streams ncludng data streams (the frst term on the left sde) and nterference streams (the second term on the left sde) s restrcted to be no more than ts recevng n order to decode the recevng packet. Equaton (8) represents that nodes n the network are halfduplex; and equaton (9) ensures the parameters to have the correct relatonshp. So far, we formulate the problem of heterogeneous stream schedulng as an nteger programmng problem wth the objectve functon n (4) subject to constrants (5)-(9). Note that the the strength of a stream S(s q ) wll reduce f more streams are scheduled to transmt from n, whch wll be ncorporated durng the stream schedulng process. As the nterference I(d(s q )) wll not be known untl the schedulng s completed, we wll use an average nterference level estmated from the past transmssons. In addton, a recever cannot cancel the nterference when the total number of streams t receves s beyond ts decodng capablty or t does not have channel knowledge, or the nterference s due to decodng errors as a result of naccurate channel knowledge constrant value N rc or a-synchronzaton [9]. The last two types of nterference s ncluded n the measured nterference. As our MAC desgn ensures that the number of concurrent transmssons from one-hop transmtters s below the decodng capablty of each recever and wth channel estmaton, so the un-cancelable nterference s from transmtters two hops or more away and s thus weaker. Also, the number of antennas of nodes n the ad-hoc network s generally small, so the number of steps needed for nterference cancelaton and the error propagaton s also lmted. Based on the actual decodng qualty, the estmated nterference level can be adjusted, and set hgher to select stronger streams for more relable decodng at the cost of possble reducton of the number of concurrent streams. Further, as our algorthms schedule stronger streams, t helps to sgnfcantly ncrease the sgnal to nterference plus nose rato and mtgate the nterference mpact. Proposton I: The heterogeneous stream schedulng (HSS) problem descrbed above s NP-hard. Proof: Frst we ntroduce a smplfed verson of HSS problem represented by a graph G = (V, E). A vertex v V represents a node n, and an edge e = (v, v k ) denotes that n and n k are neghbors n the network. Assume each node has a canddate stream s for each of ts neghbors, and the gan of schedulng C (s)p(s) s 1 for all s. The transmttng and recevng constrants for all n are N tx = N rc = 1. The optmum schedulng soluton of the smplfed HSS problem s a maxmum set of vertces that can transmt smultaneously whle N tx and N rc are satsfed for transmtter and recever nodes respectvely. The smplfed HSS problem can be proved to be NP-hard by reducng the NPcomplete maxmum ndependent set (MIS) problem to t. For any nstance of MIS represented by a graph G = (V, E ), form a new graph G = (V, E) n the followng way. Keep the vertex set V and replace each edge n E wth a dummy vertex, denoted as a set V d, so that V = V V d. Connect each dummy vertex n V d to the two orgnal end vertces n V. The dummy vertces that represent edges connected to the same vertex n G are also connected n G. It s then straghtforward to see that the optmum schedulng soluton of the smplfed HSS problem n G gves an equvalent soluton of MIS problem n G. 5 CENTRALIZED ALGORITHM Due to the NP-hardness of the problem, an effcent heurstc algorthm s requred to solve the schedulng problem. In algorthm 1, we propose a centralzed algorthm. In lnes 1-7, a set W s constructed to nclude all the canddate streams from every node n the network. In lnes 8-17, the centralzed algorthm greedly schedules the stream wth the hghest weght for transmsson n a TD, whle meetng the constrants n equatons (5)-(9). In lne 11-12, the selected stream s assgned to be transmtted from the correspondng transmtter to the recever. As a node cannot be a transmtter or recever at the same tme, n lne 13, all the canddate streams that have transmsson conflct wth the scheduled stream s = (, d(s q ), k ) are removed from the set W, ncludng the canddate streams that have the node n as the recever, have n d(s q ) as the transmtter, or have node n as the transmtter and are assocated wth the antenna k. It can be proved that the centralzed algorthm s wthn a fxed approxmaton rato to the optmal soluton [22].

7 7 Algorthm 1 Schedulng 1: Intalze: W 2: for = 1 to N n do 3: f s q, q {1,..., S } then 4: w(qk) C (qk)p(q), k {1,..., N ant } 5: W W {w(qk)} 6: end f 7: end for 8: whle W do 9: (, q, k ) = arg max {,q,k} W, the correspondng destnaton node s d(s q ) 1: f Selectng (, d(s q ), k ) satsfes constrants (6) and (7) then 11: Assgn n /n d(s q ) as the transmtter/recever node 12: Schedule the stream (, d(s q ), k ) 13: W W \ {w(qk), q s.t. d(s q ) =, k = 1,..., N ant } {w(qk) = d(s q ), q = 1,..., S, k = 1,..., N ant } {w(qk) =, k = k, q = 1,..., S } 14: else 15: W W \ w( q k ) 16: end f 17: end whle 18: for nodes n T do 19: f streams are towards the same recever then 2: precodng 21: end f 22: end for Proposton 2: The centralzed schedulng algorthm can acheve an approxmaton rato of 1/ ((2 + D) max {N ant } + 2), where D s the maxmum node degree n the network. The centralzed algorthm wth the proved approxmaton rato serves as a benchmark for performance comparson. From the formulaton (4)-(9), t s clear that the schedulng problem has to determne the values of the parameter sets: {t }, {h }, {y q } and {a qk } to assgn a packet to an approprate transmtter antenna n order to maxmze the total weghted rate of the network. In a practcal dstrbuted halfduplex network, t s reasonable to dvde the problem nto two parts: transmtter selecton and stream allocaton, where the frst phase determnes the values of {t } and {h }, and the second phase determnes the value of {y q } and {a qk } to assgn a packet to a specfc transmsson stream. In the next secton, the two subproblems are solved separately. 6 DISTRIBUTED ALGORITHM AND PROTOCOL In order to address the network heterogenety, our algorthm groups transmssons nto two types, transmssons to poor nodes usng P-slots and to rch nodes usng R-slots. The current slot type s determned n a dstrbuted manner by each node and the nodes n a neghborhood reach a consensus through sgnalng exchange. In both types of slots, spatal multplexng, selecton dversty and transmtter precodng are adaptvely utlzed to deal wth varyng traffc demands and channel condtons to mprove the overall network performance. The dstrbuted schedulng algorthm conssts of two phases, namely transmtter node selecton / slot request and stream allocaton. In the frst phase, a set of nodes are frst selected to be transmtter nodes, and each node dfferentates ts packets for poor nodes and rch nodes to determne ts current preference of transmsson slot type. In the second phase, stream allocaton s performed to allocate the data packets of the transmtter nodes to a selected set of antennas wth an approprate MIMO strategy. In the rest of ths secton, we frst present our schedulng algorthm n sequence of the two phases mentoned above. The complete protocol s then ntroduced, where we explan the detaled procedures taken to mplement the algorthm and calculate the requred parameters n a dstrbuted envronment. 6.1 Transmtter Node Selecton and Slot Request In ths phase, nodes are dstrbutvely selected as transmtter nodes and ther preference of slot type s decded. Instead of randomly selectng the transmtter nodes, the transmtter selecton phase supports servce dfferentaton and reduces transmsson delay by gvng a hgher transmsson prorty to the streams that are wth packets n hgher servce class and/or have larger queung delay. Addtonally, the type of transmsson slots s dfferentated to support transmssons to heterogeneous nodes. We frst gve the man dea and defne parameters used for the selecton, then we dscuss the detals of the selecton process Basc Plot In MIMO transmssons, n order to not exceed the decodng capacty of nodes, the number of streams that can be smultaneously transmtted n a neghborhood s constraned. Therefore, the number of transmtter nodes selected n our algorthm also has a lmt, whch wll avod unnecessary channel measurement. In addton, the decodng capabltes of recevers, represented by ther recevng constrants n Secton 4.1, are dfferent n a heterogeneous MIMO network. In our algorthm, each node dstrbutvely determnes f t can serve as a transmtter node n a transmsson duraton, and selects the type of slot used for transmsson based on the decodng capacty of ts neghborng recevers. Based on the recevng constrant, an actve node n whch has data to send groups ts neghborng nodes nto poor node set V p and rch node set V r based on the recevng constrant Nk rc of a neghbor n k, whch s broadcast wth the Hello messages sent perodcally at the network layer. We ntroduce a threshold value T T X, whch s calculated separately for each of the two sets. Denote the set of neghborng nodes n concern as V, where V can correspond to V p or V r dependng on whch set s concerned at the calculaton tme. The parameter T T X of n s estmated based on the number of actve nodes around a neghborng node n j V (denoted as Nj actve ) and the recevng constrant of node n j (denoted as Nj rc) as T ( ) T X = mn{1, mn j V N rc actve j /Nj }. To support some transmsson farness, the neghborng transmtters of n j can be evenly allocated the transmsson opportuntes based on the decodng constrant of n j. Therefore, T T X represents the probablty of a node n beng a transmtter n order to ensure all neghbors n V to perform the correct decodng. A node n can be selected as a transmtter f the value of an approprately calculated random varable s below T T X. Recall that we use stream prorty to represent how urgent a stream transmsson s. It s therefore natural to use the average stream prorty to reflect the level of prorty for a

8 8 node to be a transmtter. Denote all canddate streams (.e. the head-of-queue packets wth the number constraned by the number of antennas of n ) of n as a set S and the prorty of a stream s q as P(s q ), the prorty of a node n can be represented by the average prorty of ts canddate streams as P = s q S P(s q )/ S. A node n can calculate the average prorty P of all the N actve actve nodes n ts neghborhood as P = N actve j=1 P j /N actve. The prorty of a node can be attached wth perodc Hello messages sent at the network layer, and updated wth the data packets sent. The prorty of nodes not havng packets sent n a TD can be predcted as tme moves forward. To avod extra sgnalng and control overhead, an actve node n self-decdes f t should be selected as a transmtter node by calculatng an ndex number X T X = ( P P ˆ )/ P + γ. Here the parameter γ s a random number unformly dstrbuted n the range [,1] and generated by a node n at each transmsson duraton (TD) to provde some farness among nodes. Pˆ s the average prorty of canddate streams at node n that are targeted for nodes n V. The factor ( P P ˆ )/ P s used to gve the hgher prorty node a larger probablty for transmsson. In a TD, f X T X < T T X, node n s selected as a transmtter node for recever nodes n V ; otherwse, t has no rght of transmsson. Our transmtter selecton algorthm prefers a node wth a hgher servce level and/or a larger load and hence longer delay, and thus supports QoS and load balancng whle ensurng certan farness. Our selecton s conservatve as t consders the decodng capablty of all the neghborng nodes nstead of only that of the actually selected recever nodes known only after the schedulng Selecton Process To gve prorty to transmssons towards poor nodes, at the begnnng of a transmsson duraton, an actve node n frst determnes whether t needs to ntate a transmsson usng P-slot based on the prorty of ts streams targeted for poor nodes n V p. For the subset of canddate streams n S destned to poor nodes n V p, ther average prorty can be calculated as P p = ( k V p m S,k P(m))/ k V p S,k, where S,k s the set of canddate streams from node n to the poor node n k. Let V = V p and substtute Pˆ by P p for calculatng the ndex X T X, whch s compared wth T T X < T T X, node n can be a transmtter node and ntate a P-slot transmsson. The P-slot streams are selected so that the recevng constrants are satsfed at a targeted poor recever. Otherwse, node calculated based on nodes n V p. If X T X n checks f t can be a transmtter usng R-slot. Smlar to the prevous step, T T X s calculated concernng nodes n V r and X T X s obtaned by lettng P ˆ equal to P r = ( k V r m S,k P(m))/ k V S r,k, where S,k s the set of canddate streams whch are from node n to a rch node n k. Node n s selected as a transmtter node for recever nodes n V r f the updated parameters satsfy X T X < T T X. If a node determnes to be a transmtter node, t broadcasts an RTS message ndcatng the slot type as dscussed n 6.3. After the transmtters and the slot types are confrmed by the recever nodes through CTS transmsson, the transmtter nodes proceed to the second phase of the schedulng descrbed next. 6.2 Stream Allocaton Stream allocaton s performed dstrbutvely at each of the selected transmtter nodes. The selecton gves preference to streams wth hgher prorty. For streams of the same prorty, to acheve a hgher data rate, the allocaton process s solely based on the stream capacty by opportunstcally assgnng a channel wth good condton to a selected stream. For a hgh-prorty stream that does not have hgh-qualty channel, the selecton process reserves more of the total transmttng power for the stream to ensure a hgher transmsson relablty. For a selected transmtter, there s a lmt on the number of streams t s allowed to transmt, n order to meet the recevng constrants at all neghborng recevers. For a selected transmtter n, let N be the number of preselected streams to be transmtted and N allo be the number of streams node n s allowed to transmt, whch s calculated based on feedbacks from neghborng recevers as descrbed n Secton 6.3. Suppose the N canddate streams have L dstnct prorty levels. The recever nodes that the canddate streams are targeted for are then parttoned nto subsets {D 1}, {D2 },..., {DL } accordng to the descendng prortes of the streams, where the set {D j } contans the target recever nodes of the streams wth the j-th hghest prorty level, and the q-th element n {D j } s denoted as D j (q). Recall that a stream s s dentfed by ts transmtter node, recever node and transmtter antenna. Denote the set of antennas that node n has as {A }, and the p-th element s A (p). For a stream of n whch has the recever D j (q) and transmttng antenna A (p), the stream capacty C (, D j (q), A (p)) depends on the stream strength and the estmated nterference level at the recever node D j (q), as dscussed n Secton 4.3. For transmtter node n, there s a set W consstng of all the capacty parameters of the canddate streams W = L j=1 {C (, Dj (q), A (p)) A (p) {A }, D j (q) {D j }, p = 1,..., {A }, q = 1,..., {D j } }. The procedure of stream allocaton s descrbed n the algorthm 2, where j s the ndex of the prorty level, {A } res s the set of remanng avalable antennas of node n and N res s the resdual number of streams to allocate. The ntal value of N res s set to be the total number of streams for allocaton N allo. As n lnes 2-11, the algorthm starts from the set of canddate streams whch have the hghest prorty (j = 1), and calls the subroutne OPPORTUNISTIC ALLOCATION as n algorthm 3 for each prorty level, untl all the allowed streams have been allocated or the antennas of node n have all been assgned or reserved for streams. In lnes 12-16, power s allocated to the selected antennas based on the transmsson pattern. As descrbed n secton 4.1, precodng s used to maxmze the data rate between a node par f all streams are scheduled to transmt towards the same recever where optmal power allocaton s performed through waterfllng; when streams are towards dfferent recevers, power s smply dstrbuted evenly among the antennas. The subroutne OPPORTUNISTIC ALLOCATION s descrbed n algorthm 2 to allocate k antennas to transmt the streams of the j-th hghest prorty level that are targeted for the recever set {D j res }. The parameter N s the resdual number of antennas avalable for allocaton, the set {A } res contans the canddate antennas of node n for stream allocaton, W j contans the capacty parameters of the streams formulated between the antennas n {A } res

9 9 Algorthm 2 Schedulng 1: Intalze: j = 1, {A } res = {A }, N res = N allo 2: whle N res > do 3: f {D j res } N then 4: OPPORTUNISTIC ALLOCATION({A } res, {D j }, {Dj res }, N ) 5: N res = N res {D j } 6: else 7: OPPORTUNISTIC ALLOCATION({A } res, {D j res }, N, ) 8: N res = 9: end f 1: j j : end whle 12: f All streams are towards one recever then 13: Use precodng and optmal power allocaton 14: else 15: Power s evenly dstrbuted 16: end f Algorthm 3 OPPORTUNISTIC ALLOCATION ({A } res, {D j res }, k, N ) 1: Intalze: l =, W j = {C (, D j (q), Ares (p)) A res (p) {A } res, D j (q) {D j }, p = 1,..., {A } res, q = 1,..., {D j } } 2: whle l < k do 3: W max max W j, {A max, D max } arg max W j 4: Allocate the stream for the recever D max to the antenna A max ; 5: W j W j \ {W (A max, D j (q)) Dj (q) {Dj }, q = 1,..., {D j } }; f there s no other stream target for the recever node D max, also remove {W (A res (p), D max ) A res (p) {A } res, p = 1,..., {A } res }; 6: f D max has sent ndcator of weak channel then 7: f N res > then 8: k k 1, l l + 1, N res 9: else {N res = } 1: k k 1 11: end f 12: end f 13: {A } res {A } res \ A max 14: l l : end whle N res 1; and the recevers n {D j } and l represents the number of streams currently allocated. The allocaton s based on spatal multplexng and selecton dversty, and n sequence of descendng stream qualty. As the allocaton scheme favors stream prorty than stream qualty, n some cases, although the channel condton s severe, a transmsson wth a hgh prorty s stll permtted. To reduce erroneous decodng thus packet loss under the severe channel condton, when a selected stream does not have good enough qualty as ndcated by a weak channel ndcator nclude n the CTS (Secton 6.3), the total number of antennas avalable for allocaton of ths stream s decreased by one to reserve extra transmttng power for the weak stream to mprove ts qualty, as n lnes Implementaton of Schedulng To enable the proposed many-to-many transmsson and better explot varous dversty technques for hgher capacty and relablty, the mplementaton of the dstrbuted schedulng algorthm s TDMA-based, where the tme s dvded nto a serals of transmsson duraton consstng of four phases wth dfferent lengths. The duraton of each phase s fxed and enough for the correspondng message transmsson. Followng the conventon of IEEE DCF, sgnalng messages are named RTS, CTS, DATA and ACK, whch are transmtted durng phase I, II, III and IV respectvely. Note that slot synchronzaton s currently achevable n the IEEE82.11 famly of protocols. By takng advantage of the selecton dversty and mult-user dversty, our scheme could effectvely ncrease the SINR of a receved sgnal, whch would help mprove the accuracy of synchronzaton as well as mtgate the mpact of a-synchroncty n a dstrbuted scenaro. The procedure of sgnal exchange and nformaton acquston for heterogeneous MIMO schedulng s as follows. Phase I: Transmsson Request and Slot Conservaton. At the begnnng of phase I, a node n whch selects tself as a transmtter node as n Secton 6.1 broadcasts an RTS. Before sendng out the RTS, node n selects a set of hghestprorty data packets from ts queue to form N N max canddate streams, where N max s the maxmum number of streams that can be transmtted n a transmsson duraton dependng on the number of antennas of n, and the amount of data queued. The IDs of the target recever nodes of the selected packets, the value N, as well as the ID of node n are then ncluded n the RTS. If n wants to request a P-slot towards node n k, an RTS should further carry an ndcator of P-slot and the calculated average prorty P p. The preamble of a packet s used as the tranng sequence for the channel estmaton purpose. After the RTS s transmtted from the frst antenna of the transmtter node, for both types of slot, the preamble s rotatonally broadcasted through the remanng antennas of the transmtter node wth a short notce sgnal separatng two antennas transmssons, so that the spatal channels between each antenna of the transmtter nodes and the recever nodes can be dfferentated and estmated. An RTS s masked by another random code, called ID code, whch are almost orthogonal for dfferent nodes and assgned smlarly to that n [23], so a recever node can get the channel nformaton of dfferent transmtter nodes from concurrently receved RTSs. Our transmtter node selecton algorthm n Secton 6.1 adaptvely selects a subset of nodes n a neghborhood to partcpate n channel estmatons based on the decodng capabltes of nodes n the neghborhood, whch not only reduces the channel estmaton complexty and avods unnecessary channel estmatons but also constrans the total nterference n a neghborhood for better decodng. Phase II: Transmsson Confrmaton. Upon recevng multple RTSs, a recever correlates ts receved sgnals wth each element n ts set of random codes to dfferentate the tranng sequences from dfferent transmtter nodes, estmates spatal channels and extracts other nformaton ncluded n RTSs. If a node n k receves a request for P-slot transmsson to tself, t sorts all P-slot requests t receves (for tself or for other recever nodes) based on the request prortes. When