over an MC-MOmni network when has a capacity gain of 2π θ

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1 O the Capaity of Multi-Chael ireless Networks Usig Diretioal Ateas Hog-Nig Dai Ka-ig Ng Rayod Chi-ig og ad Mi-You u The Chiese Uiversity of Hog Kog, Hog Kog {hdai,kwg,wwog}@se.uhk.edu.hk Shaghai Jiao Tog Uiversity, Chia wu-y@sjtu.edu. Abstrat The apaity of wireless ad ho etworks is affeted by two key fators: the iterferee aog ourret trasissios ad the uber of siultaeous trasissios o a sigle iterfae. Reet studies foud that usig ultiple haels a separate ourret trasissios ad greatly iprove etwork throughput. However, those studies oly osider that wireless odes are equipped with oly oidiretioal ateas, whih ause high ollisios. O the other had, soe researhers foud that diretioal ateas brig ore beefits suh as redued iterferee ad ireased spatial reuse opared with oidiretioal ateas. But, they oly foused o a sigle-hael etwork whih oly allows fiite ourret trasissios. Thus, obiig the two tehologies of ultiple haels ad diretioal ateas together potetially brigs ore beefits. I this paper, we propose a ulti-hael etwork arhiteture (alled MC-MDA) that equips eah wireless ode with ultiple diretioal ateas. e derive the apaity bouds of MC-MDA etworks uder arbitrary ad rado plaeets. e will show that deployig diretioal ateas to ulti-hael etworks a greatly iprove the etwork apaity due to ireased etwork oetivity ad redued iterferee. e have also foud that eve a ulti-hael etwork with a sigle diretioal atea oly at eah ode a give a sigifiat iproveet o the throughput apaity. Besides, usig ultiple haels itigates iterferee aused by diretioal ateas. MC-MDA etworks itegrate beefits fro ulti-hael ad diretioal ateas ad thus have sigifiat perforae iproveet. I. INTRODUCTION ireless ad ho etworks typially osist of odes that share oe sigle hael for ouiatios. It is foud i [] that i a ad ho etwork with odes uder a rado etwork plaeet, eah ode has a throughput apaity of Θ(/ log ). Eve uder optial arbitrary etworks, the etwork ould oly offer a per-ode throughput of Θ(/ ). The per-ode throughput is dereased whe the uber of odes ireases. Oe ajor reaso is that all the odes withi the etwork share the sae ediu. he a ode trasits, its eighborig odes are prohibited fro trasittig due to iterferee. O the other had, every ode equipped with a sigle iterfae aot trasit ad reeive at the sae tie (i.e., half-duplex ode). e all suh sigle-hael etworks usig oidiretioal ateas as SC-Oi etworks. I a rado etwork, odes are radoly plaed, ad the destiatio of a flow is also radoly hose. I a arbitrary etwork, the loatio of odes, ad traffi patters a be optially otrolled. Oe approah to iprove the etwork perforae is to use ultiple haels istead of usig a sigle hael i a wireless etwork. The experietal results of [] [7] show that usig ultiple haels a sigifiatly iprove the etwork throughput. Oe possible reaso is that ultiple haels a separate ultiple ourret trasissios i frequey doai. Besides, a wireless ode a be equipped with ultiple etwork iterfaes whih allow ultiple siultaeous trasissios/reeptios to proeed at the sae ode. However, suh etworks i those studies [] [8] equip every ode with oidiretioal ateas whih have liited spatial reuse. Siilarly, we ae suh ulti-hael etworks usig ultiple oidiretioal ateas as MC-MOi etworks. Reet works suh as [9] [6] foud that applyig diretioal ateas istead of oidiretioal ateas to wireless etworks a greatly iprove the etwork apaity. For exaple, the aalytial results i [9] show that usig diretioal atea i arbitrary etworks ahieves a apaity gai of π/ αβ whe both trasissio ad reeptio are diretioal, where α ad β are trasitter ad reeiver atea beawidths, respetively. Uder rado etworks, the throughput iproveet fator is 4π /(αβ) for diretioal trasissio ad diretioal reeptio. Sie the etworks typially use oe sigle hael oly, we all suh sigle hael etworks usig diretioal ateas as SC-DA etworks. Usig diretioal ateas istead of oidiretioal ateas i a ulti-hael wireless etwork is ore beefiial. Therefore, we propose a ovel etwork that itegrates the two tehologies. I this etwork, eah ode is equipped with ultiple iterfaes ad eah iterfae is assoiated with oe diretioal atea that a operate o differet haels. Suh ulti-hael etworks usig ultiple diretioal ateas are alled as MC-MDA etworks that have the followig harateristis. Eah ode is equipped with ultiple etwork iterfae ards (NICs). Eah NIC is outed with a diretioal atea. There are ultiple o-overlappig haels available. Eah atea a swith to these haels quikly. All odes a work i a full-duplex ode, i whih a ode a trasit ad reeive with differet eighbors. Eah ode a ouiate ollisio-freely ad siultaeously with ore tha oe ode usig differet dire-

2 tioal ateas that operate o differet haels. Reetly, DMesh [7] also proposed a siilar arhiteture as ours. DMesh fouses o egieerig issues of siulatio ad experietal studies about the throughput iproveet. However, our work fouses o the theoretial aalysis o the etwork apaity. Besides, our etwork is uh ore geeral, whih a apply to wireless etworks, but DMesh is liited to ireless Mesh Networks. To the best of our kowledge, there is o theoretial aalysis o the apaity of suh etworks. This paper oetrates o fidig the apaity bouds for a MC-MDA etwork ad explorig the beefits of this etwork. The reaider of the paper is orgaized as follows. e have suarized our ajor otributios ad outlied the ai fidigs i Setio II. Setio III desribes the atea odel ad our proposed iterferee odel, whih will be used i our aalysis. I Setio IV, we preset the aalytial results of the trasport apaity of arbitrary etworks. Setio V gives the aalytial results of the throughput apaity of rado etworks. e suarize our work i Setio VI. II. CONTRIBUTIONS AND MAIN RESULTS A. Major otributios The priary researh otributios of our paper a be suarized as follows.. e forally idetify MC-MDA etworks that haraterize the features of ulti-hael wireless etworks with ultiple diretioal ateas at eah ode. The apaity of MC-MDA etworks has ot bee studied before.. e derive the upper bouds o the apaity of MC- MDA etworks uder arbitrary etworks ad rado etworks. 3. e also ostrut a arbitrary etwork ad a rado etwork, where both the lower bouds of the two etworks have the sae order of the upper bouds, whih eas that the derived upper bouds a be quite tight. 4. Our theoretial results show that itegratig diretioal ateas with ulti-hael etworks a irease etwork oetivity ad redue iterferee, resultig i iproved etwork apaity. Ipliatios fro the aalytial results are also give. Before presetig our ai results, we eed to give the assuptios ad the otatios first. e adopt the otatios show i Table I throughout this paper. I this paper, all odes are equipped with the sae type of ateas, whih have the sae beawidth (geerally less tha π). Kyasaur ad Vaidya [8] argued that the uber of iterfaes should ot be greater tha the uber of haels (i.e., ) beause surplus iterfaes are wasted if is greater tha. But, this oditio is oly valid whe the etworks adopt oidiretioal ateas. he diretioal ateas are used i the etworks, this oditio a be relaxed to that, i suh etworks, a be greater tha. More speifially, a be π. ith wider rages of the uber of iterfaes, the deployet of ateas to a ode is easier. Due to this additioal property, we a ahieve higher apaity i the ' TABLE I NOTATIONS USED IN THIS PAPER!"#$%&$ ( ) *+ '',- '. #$%#$/0&$%. &$ JHK ABCDEFGB HICFCI L LMNOMRSTU VNORSTU VNOPQ 3 π π π π ) π ( ) π ( π MNOMPQ :;5<<=>?687<6=@95:=? Fig.. The apaity regios uder differet is ot to sale) i arbitrary etworks (figure etworks. Detailed disussio will be give i Setio II-B.3. B. Suary of results Sie the apaity of a MC-MDA etwork depeds o the ratio of, we preset the results aordig to the ratio of.. Results for Arbitrary Networks As show i Fig., the trasport apaity 3 of a (, )- etwork has two regios as follows aordig to ratio of to (fro Theore ad Theore 3). ) he π ) ), the trasport apaity is Θ( ) (whih is π ) bit-eters/se (seget A-B i Fig. ) with a apaity gai of π over a MC-MOi etwork (seget A -B ). ) he π ) ), the trasport apaity is Θ( ) bit-eters/se (seget B-C i Fig. ), whih is idepedet of beawidth.. Results for Rado Networks As show i Fig., the throughput apaity 4 of a (, )- etwork has three regios as follows aordig to ratio of to (fro Theore 4 ad Theore 5). ) he π ) log ), the throughput apaity is Θ( 4π log ) (whih is log ) bits/se (seget 3 the trasport apaity is that the etwork trasport oe bit-eter per seod whe oe bit has bee trasported a distae of oe eter withi oe seod. 4 e just osider the aggregate throughput apaity of the whole etwork, whih is easured i ters of bits/se.

3 # 4 # 4π log # 3+,/0 *+,*/0 3+,-. log ( ) loglog π log '() %& $ log log log loglog log log log log log log log log loglog π ( ) log ( ) π π log!"! Fig.. The apaity regios uder differet i rado etworks (figure is ot to sale) π *+,*-. D-E i Fig. ) with a apaity gai 4π over a MC- MOi etwork (seget D -E ). ) he π ) log ) ad also O(( π ) log log ( log ) ), the throughput apaity is Θ( ) (whih is π ) bits/se (seget E-F i Fig. ), ad the apaity gai over a MC-MOi etwork is π (seget E -F ). 3) he π ) log log ( log ) ), the throughput apaity is Θ( log log log ) bits/se (seget F-G i Fig. ), whih is idepedet of beawidth. 3. Coparisos with Other Networks e osider a arbitrary etwork whe = =, whih has a apaity of π (poit H i Fig. ). Suh etwork is a SC-DA etwork, whih a be regarded as a speial ase of a MC-MDA etwork. Siilarly, a SC-Oi etwork is a speial ase of a MC-MOi etwork whe = = (poit A i Fig. ). he π ) ), the apaity of a MC-MDA etwork is aily affeted by the iterferee. The iterferee a be itigated by assigig ourret trasissios uder differee haels. Thus, if the uber of iterfaes is fixed, ireasig the uber of haels is helpful to redue the iterferee. Speifially, whe the uber of haels is ( π ), (i.e., poit B i Fig. ), all trasissio a be regarded as ollisio-free. However, whe the uber of haels is ireased further ad π ) ), the apaity is affeted by the iterfae ostrait. The apaity drops eve faster whe the ratio of ireases sie the apaity is iverse-proportioal to the ratio of. Besides, fro Fig., a MC-MDA etwork requires less haels to reah the ollisio-free oditio (at poit B), opared with a MC-MOi etwork (at poit B ). he a rado etwork is osidered, a SC-DA etwork is a speial ase of a MC-MDA etwork whe = = (poit I i Fig. ). Ad a SC-Oi etwork is a speial ase of MC-MOi etworks whe = = (poit D i Fig. ). I a rado plaeet, a MC-MDA etwork has a apaity gai of 4π over a MC-MOi etwork if is O(( π ) log ). The reaso is that diretioal ateas a greatly iprove the etwork oetivity. Sie usig diretioal ateas a redue iterferee, a MC-MDA etwork has a apaity gai of π over a MC-MOi etwork whe π ) log ) ad also O(( π ) log log ( log ) ). he π ) log log ( log ) ), siilar to a MC-MOi etwork, the apaity of a MC-MDA etwork is oly affeted by the flow bottleek i a ode. MC-MDA etworks are proisig to iprove the etwork apaity. Sie diretioal ateas a greatly irease the spatial reuse, the sae haels a be reused i differet diretios without ollisios, but oidiretioal ateas aot. So, the uber of iterfaes a be greater tha the uber of haels. The axiu uber of ateas o a ode i a MC-MDA etwork a be π. However, i a MC-MOi etwork, is always ot greater tha [8]. Thus, i a arbitrary plaeet, a MC-MOi etwork has at ost a apaity bouded by Θ( ). he has the axiu value π, a MC-MDA etwork a have a apaity gai π π over a MC-MOi etwork. For exaple, whe is π 4 ad is 3, the axiu uber of iterfaes is 4. The, we have the apaity gai 6 (early 3) ties over a MC-MOi etwork whih has the sae uber of haels ( = 3) ad 3 iterfaes at eah ode. But, the uber of ateas should ot be set too large. Oe ajor reaso is that a sigle iterfae a oly share the apaity gai of π, whih dereases whe the uber of iterfaes ireases. Let us osider the sae exaple etioed above for illustratio. A sigle iterfae a share a apaity gai oly 4/ 6. The uber of ateas is also liited by the size ad ost of ateas. Choosig the uber of ateas eeds osiderig soe egieerig issues suh as the devie ost, the size of ateas ad the iterferees aog the ateas. However, our work just fouses o theoretial perforae aalysis. How to hoose the proper uber of iterfaes is our future work. III. MODEL I a MC-MDA etwork, eah ode is equipped with diretioal ateas that a be approxiated by the followig atea odel. Besides, sie iterferee aog ourret trasissios is a ajor reaso affetig the etwork apaity, we propose a reeiver-based iterferee odel ad derive the oditio that a trasissio is suessful. A. Atea Model I this paper, we osider a diretioal atea odel that is used i previous works [9], [3] [5]. Sidelobes ad baklobes are igored i this odel. The reasos why we siplify the odel are suarized as follows. First, eve i a ore realisti odel, the sidelobes are too sall to be igored. For exaple, the ai gai is ore tha 00 ties of the gai of sidelobes whe the ai beawidth is less tha 40 i the oe-sphere odel [0]. Seodly, sart ateas ofte have ull apability that a alost eliiate the sidelobes ad baklobes. Ref. [8] derives the ipat of ull apability of sart ateas o the etwork apaity. More oplexed atea odels will be osidered i the future work.

4 Fig. 3. diretioal atea The Atea Model X X X X r (+ ) r Fig. 4. The Reeiver-based Iterferee Model Our proposed odel assues that a diretioal atea gai is withi a speifi agle, where is the beawidth of the atea. The gai outside the beawidth is assued to be zero. At ay tie, the atea bea a oly be poited to a ertai diretio, as show i Fig. 3, i whih the atea is poitig to the right. Thus, the probability that the bea is swithed to over eah diretio is /π. B. Reeiver-based Iterferee Model Based o the protool odel i [], we propose a reeiverbased iterferee odel with extesios of diretioal ateas. Our odel oly osiders diretioal trasissio ad diretioal reeptio, whih a axiize the beefits of diretioal ateas. If ode X i trasits to ode X j over a hael, the trasissio is suessfully opleted by ode X j if o odes withi the regio overed by X j s atea bea will iterfere with X j s reeptio. Therefore, for every other ode X k siultaeously trasittig over the sae hael, ad the guard zoe > 0, the followig oditio holds. { Xk X j ( + ) X i X j or X k s bea does ot over ode X j () where X i ot oly deotes the loatio of a ode but refers to the ode itself. I this odel, eah ode is equipped with oe sigle diretioal atea that a operate over haels. Fig. 4 shows that a trasissio fro ode X k will ot ause iterferee to X i s trasissio sie the atea bea of X k does ot over reeiver X j. Gupta ad Kuar [] established a physial odel i whih the suess probability of a trasissio is related to the Sigal-to-Iterferee-Noise Ratio (SINR). he the fadig fator is greater tha two (it is oo i a real world), the physial odel is equivalet to the iterferee odel. Thus, we will oly osider the iterferee odel i this paper. IV. TRANSPORT CAPACITY FOR ARBITRARY NETORKS Sie the apaity of a MC-MDA etwork is affeted by two fators, i.e., the iterferee aog ourret trasissios ad the uber of siultaeous trasissios o a iterfae, we derive differet upper bouds whe osiderig these two fators, respetively i Setio IV-A. To illustrate that the upper bouds are quite tight, we ostrut a etwork that a ahieve the lower bouds havig the sae order of the upper bouds i Setio IV-B. A. Upper Boud Siilar to a MC-MOi etwork [8], the trasport apaity of a MC-MDA etwork is also liited by by two ostraits: iterferee ostrait ad iterfae ostrait. () Iterferee Costrait: the iterferee aroud a reeiver is affeted by the uber of iterferig odes i its eighborhood, whih is deteried by the size of the iterferee regio. he we use diretioal ateas at both trasitter ad reeiver eds, the oditio iterferee zoe is (π) portio of that whe oidiretioal ateas are used at both eds [9]. e derive the first boud whe osiderig the iterferee ostrait ad have the followig theore. Theore : The apaity of a ulti-hael etwork equipped with diretioal ateas is O( ) biteters/se. Copared to a ulti-hael etwork usig oidiretioal ateas per ode, the apaity gai is π. Proof: e preset a proof of the boud i Appedix A. It is proved i [8] that the apaity of a MC-MOi (, )- etwork is bouded by π. Copared with this result, a MC-MDA (, )-etwork has a apaity gai of π. () Iterfae Costrait: we osider the iterfae ostrait of a MC-MDA etwork. Sie every ode has iterfaes, there are iterfaes i the whole etwork. Eah iterfae a support at ost bits/se ad the axiu distae that a bit a travel i the etwork is Θ() eters. Thus, the iterfae boud of the etwork is O( ) bit-eters/se. Cobiig the two ostraits, the etwork trasport apaity is O(MIN O (, )) bit-eters/se. The iiu boud of the is a upper boud o the etwork apaity. The, we have the followig theore o the trasport apaity of a arbitrary etwork. Theore : The upper boud o the trasport apaity of a (, )-etwork is show as follows. i) he π ) ), the trasport apaity is O( ) bit-eters/se. ii) he π ) ), the trasport apaity is O( ) bit-eters/se. The etwork apaity of a (, ) etwork is O( ) biteter/se, whih athes the result obtaied by [9]. Thus, a SC-DA a be regarded as a speial ase of a MC- MDA etwork whe = =. he π ) ), a MC-MDA etwork has a apaity gai π over a MC- MOi etwork, ad a apaity gai π over a SC-Oi etwork, where a be greater tha. B. Costrutive Lower Boud I this setio, we ostrut a etwork that a ahieve the apaity of Ω(MIN O (, )) bit-eters/se i order to show that the upper boud derived i Setio IV-A is tight. First, we divide the uit-area ito a uber of equal-sized ells ad eah ell has the sae uber of odes. I eah ell, to esure ollisio-free trasissios, we separate trasitters ad reeivers at differet positios ad their atea beas are aied to proper diretios. For exaple, all trasitters adjust their ateas to 30 east of

5 Fig. 5. A possible plaeet of odes withi a ell due orth ad all reeivers poit their reeivig ateas to 30 west of due south. The, we illustrate that this plaeet a guaratee orret trasissios for all ouiatig pairs. Usig the result i [8], we a exted the result of a (, )-etwork to that of a (, )-etwork. This lea also holds for a etwork usig diretioal ateas istead. Lea : [8] A (, )-etwork a support at least half of the apaity supported by a (, ] ) etwork. e exhibit a seario where the lower boud is ahieved. () Eah ode is equipped with a diretioal atea with beawidth. Let g = i( π ), whih will be used to, 36π alulate the apaity of the etwork. e divide the uit-area plae ito 36πg 36πg equal-sized rhobi. Thus, eah ell has odes. Sie the total area is, eah ell has a size of 36πg. 36πg All sides of every rhobus has a legth l = si. () As show i Fig. 5, we further divide every ell ito ( π + ) equal-sized sub-rhobi ad plae 36πg odes ito 8 positios, whih are divided ito two groups, aely (A) R,R,...,R9 (white dots i Fig. 5) ad (B) T,T,...,T9 (blak dots i Fig. 5). So there are πg odes i eah positio. Nodes that are plaed at group (A) play as reeivers ad those plaed at group (B) at as trasitters. Two eighborig trasitters are separated at least π ties side-legth of a sub-rhobus. After soe derivatios, we obtai the trasissio rage ( r is the diagoal legth (see Fig. 5)) r = π π Here, r = k 7πg(+os ) + si., si ad os a be regarded as ostats. So πg, where k is a ostat. (3) Cosider a pair of ouiatig odes X i ad X j that are loated i T5 ad R5, respetively. The X i s atea is adjusted to fae X j, ad the atea of X j is poited to X i as well. Thus, X j is oly affeted by the odes that are i the sae lie as X i. Fro Fig. 5, the earest iterferig odes withi the ell, other tha those loated i T5, ust be loated i T3, whih is at least a distae of r( + ) away fro X j (where = π > 0). Thus, uder the iterferee odel Eq. (), the trasissio betwee odes X i ad X j is ot affeted by other trasissios i the etwork, ad this result holds for all ouiatig pairs. I a (, )-etwork, there are at ost / pairs of odes that a trasit. Eah pair trasits at a rate of / over a distae r. Hee, the total trasport apaity of the etwork is ot greater tha r = k πg/(). Reall g = i( π, 36π ). Thus, total apaity is bouded by k π π if g =, otherwise it is bouded by k 6 if g = 36π. Hee, the apaity of a (, )-etwork to be Ω(MIN O (, )) bit-eters/se. By Lea, we exted the result fro a (, )-etwork to a (, )-etwork. Thus, the apaity of a (, )-etwork is Ω(MIN O (, )) bit-eters/se. As, the apaity is Ω(MIN O (, )). The lower boud has the sae order of the upper boud. Thus, the upper bouds that we obtaied i Setio IV-A are tight. So we have a theore o the ahievable apaity. Theore 3: There is a plaeet of odes ad a assiget of traffi patters suh that. π ) ), the trasport apaity is i) he Ω( ) bit-eters/se. ii) he Ω( ) bit-eters/se. C. Soe Possible Ipliatios π ) ), the trasport apaity is Usig diretioal ateas to ulti-hael etwork is beefiial to to iprove the etwork apaity. Diretioal ateas a separate ultiple ourret trasissios ad irease spatial reuse. A sall uber of haels a be reused i differet diretios without ollisios. So, the uber of iterfaes a be greater tha the uber of haels, whih is differet fro the results i [8]. Sie has the axiu value π, a MC-MDA etwork a have a apaity gai π π, whih has a sigifiat ireet over a MC-MOi etwork. But, the uber of ateas should ot be set too large. A sigle iterfae a oly share the apaity gai of π, whih is dereasig whe the uber of iterfaes ireases. The uber of ateas is also liited by the size ad ost of ateas. There is a trade-off betwee the uber of ateas ad the ost. ith dereasig the beawidth, the apaity is growig fastly. However, the apaity will ot grow arbitrarily high whe the beawidth dereases further ad eve approahes to zero. Yi et al. [9] have observed that whe the beawidth is too sall, the iterferee has bee fully redued ad there is o ay further iproveet by dereasig the beawidth of the ateas. Atually, whe the beawidth is arrow eough (ore speially, less tha a ertai agle) a trasissio a yield a high suess probability. That is, the trasissio a be regarded as ollisio-tolerat [9]. It is observed that if the beawidth is less tha π (i.e., 5 ) ad odes are ot desely distributed ad both diretioal ateas are used at the trasitter ad the reeiver, the the probability of a suessful trasissio is greater tha 99%. V. THROUGHPUT CAPACITY FOR RANDOM NETORKS Differet fro arbitrary etworks, the apaity of rado etworks is affeted by three ajor fators [8]: etwork oetivity, iterferee, ad destiatio bottleek. So we derive differet upper bouds uder differet fators i Setio V-A. e evaluate rado etworks with throughput apaity istead of trasport apaity beause throughput apaity is

6 ooly used to evaluate rado etworks (e.g., [], [8] ad [9]). I order to prove that the upper bouds are quite tight, i Setio V-B, we ostrut a etwork that a ahieve the lower bouds havig the sae order of the upper bouds. A. Upper Boud As we etioed before, the apaity of ulti-hael rado etworks usig diretioal ateas is liited by the followig three ostraits [8]. ) Coetivity ostrait: he we say a etwork is oeted, we ea that a etwork is oeted whp 5. This ostrait is eessary for a rado etwork to esure that the etwork is oeted. he eah ode is equipped with diretioal ateas i a rado etwork, a high oetivity a be gaied. Previous work [9] foud that the upper boud of a rado etwork usig diretioal ateas at both the trasitter ad the reeiver is O( log ) bits/se. This boud is also appliable to MC-MDA etworks. ) Iterferee ostrait: The apaity of ulti-hael rado etworks usig diretioal ateas is also ostraied by iterferee. Thus, siilar to arbitrary etworks, by Theore, a rado etwork with diretioal ateas have O( ) bit-eters/se. Sie eah pair of souredestiatio i a rado etwork is separated by a distae of Θ() eter o average, the etwork apaity of rado etworks is at ost O( ) bit/se. 3) Destiatio bottleek ostrait: The apaity of a ulti-hael etwork is restrited by the flows 6 toward a destiatio ode. Before alulatig the upper boud uder bottleek ostrait, we eed to boud the axiu uber of flows for a destiatio ode first. I a rado etwork, a ode radoly hooses its destiatio. Thus, it is possible that a ode assebles ultiple flows. Let F() be the axiu uber of flows for a destiatio ode. The proess of hoosig a destiatio ode a be regarded as radoly throwig a ball ito a bi, whih is siilar to [0]. Hee, we use the result of [0] ad have Lea to boud the axiu uber of flows for a destiatio ode. Lea : The axiu uber of flows F() fro other odes to a hose destiatio is Θ( log log log ), whp. I a (, )-etwork, eah hael supports a axiu data rate of bits/se. Suppose that ode X l that is the destiatio of the axiu uber of flows F(). Hee, the total data rate at ode X l with ateas is bits/se. Sie ode X l has F() ioig flows, the data rate of the flow with the iiu rate is at ost F() bits/se. Hee, the iiu per-ode throughput apaity is ot greater tha F(), whih iplies that the etwork apaity is at ost O( F() ) bits/se. Substitutig F() by Lea, log log the etwork apaity is at ost O( log ) bits/se. Cobiig the three bouds uder the three ostraits, we obtai that the etwork apaity is at 5 I this paper, whp eas with probability / 6 The traffi fro a soure ode to a destiatio ode is alled a flow. ost O(MIN O ( log,, log log log )) bits/se. Thus, we have the followig theore o the upper boud o the apaity of rado etworks. Theore 4: The upper boud o the apaity of a rado etwork is as follows. ) he π ) log ), the throughput apaity is O( 4π log ) bits/se with a apaity gai of over a MC-MOi etwork. ) he π ) log ) ad also O(( π ) log log ( log ) ), the throughput apaity is O( ) bits/se with a apaity gai of π over a MC-MOi etwork. 3) he π ) log log ( log ) ), the throughput apaity is O( ) bits/se. log log log he π ) log ), a MC-MDA etwork has a apaity gai 4π / over a MC-MOi etwork. The reaso is that diretioal ateas greatly iprove the etwork oetivity. Siilar to MC-MOi etworks, the ratio of to has o ipat o the etwork apaity. he π ) log ) ad also O(( π ) log log ( log ) ), the apaity of a MC-MDA etwork is O( ), whih has a apaity gai of π over a MC-MOi etwork. he π ) log log ( log ) ), log log the apaity of a MC-MDA is O( log ), whih is the sae as a MC-MOi etwork. A SC-DA etwork a be regarded as a speial ase of a MC-MDA etwork whe = =. he = =, the apaity of a SC-DA etwork a fall ito O( log ) or O( ), whih is related to log (log or ). Siilarly, a SC-Oi etwork a be regarded as a speifi ase of a MC-MOi etwork, whe = =. B. Costrutive Lower Boud To prove that the upper boud i Setio V-A a be quite tight, we begi to ostrut a etwork ad the desig a routig shee ad a trasissio shedulig ehais as follows. Step (Torus Divisio): we divide the uit-area plae ito eve-sized squares. The size of eah square suffies three ostraits etioed previously. Step (Routig Costrutio): we desig a routig shee that assigs a flow to a ode with balaed flows at eah ode. I the followig, we will fid that the total flows assiged to ay ode is oly deteried by the square size. Step 3 (Trasissio Shedulig): we osider a (, )-etwork. To esure the etwork satisfies two additioal ostraits (whih was used i [8] ad will be desribed i details later), we propose a trasissio shedulig ehais to esure a ollisio-free trasissio withi that hael. Fially, we obtai the apaity of a (, )- etwork. Usig Lea etioed i Setio IV-B (whih also holds for a rado etwork usig diretioal ateas), we exted the result to a (, )-etwork ad obtai the ostrutive lower boud. Step (Torus Divisio): e divide the uit-area plae ito equal-sized squares. The size of eah square deoted by a() ust satisfy the three ostraits etioed i Setio V-A.

7 It is foud i [8] that whe the size of eah square is greater tha a ertai value, eah square ust otai a ertai uber of odes. So, it a guaratee suessful trasissios fro soure odes to destiatio odes. e state their lea here. 50 log Lea 3: [8] If a() is greater tha, eah ell has Θ(a()) odes per ell, whp. 00 log To siplify the alulatio, we take for a large. It is foud i [9] that i a rado etwork, usig diretioal ateas at both the trasitter ad the reeiver a redue the iterferig area by ( π ). Sie the uber of odes is proportioal to the size of the area, the uber of iterferig odes is redued by ( π ). I other words, the iterfereetolerat apability of a ode is ireased by ( π ). Thus, for 00 log a (, )-etwork, a() is equal to ax(, ( π ) ). To esure the flow bottleek ostrait, we take ( F() ) as aother possible value for a(), where F() = Θ( log log log ) (by Lea ). The, we have 00 log a() = i(ax(, ( π ) ), ( F() ) ) () If a ode i ell B iterferes with aother trasissio i ell A, this ell is alled a iterferig ell. e prove that the uber of iterferig ells aroud a ell is a ostat, whih is idepedet of a() ad. Thus, we have Lea 4. Lea 4: The uber of ells that iterfere with ay give ell is bouded by a ostat k (where k = 8(+ ) 4π ), whih is idepedet of a() ad. Proof: The detailed proof is stated i Appedix B. Step (Routig Costrutio): e ostrut a siple routig shee that hooses a route with the shortest distae to forward pakets. A straight lie deoted by S-D lie is passig through the ells that soure ode S ad destiatio ode D are loated. Pakets are delivered alog the ells lyig o the soure-destiatio lie. The, we hoose a ode withi eah ell lyig o the straight lie to arry that flow. The ode assiget is based o load balaig. The flow assiget proedure is divided ito two sub-steps. Step (a): soure ad destiatio odes are assiged. For ay flow that origiates fro a ell, soure ode S is assiged to the flow. Siilarly, for ay flow that teriates i a ell, destiatio ode D is assiged to the flow. After this step, oly those flows passig through a ell (ot origiatig or teriatig) are left. Step (b): we assig the reaiig flows. To balae the load, we assig eah reaiig flow to a ode that has the least uber of flows assiged to it. Thus, eah ode has early the sae uber of flows. It is foud i [] that the uber of S-D lies passig through ay ell is O( a()), whp. Sie a() is hose based o Eq. () ad is greater tha 00 log/, eah ell has Θ(a()) odes (by Lea 3). Besides, eah ell has O( a()) flows ad Step (a) assigs early the sae uber of flows. So, Step (b) assigs to ay ode i the etwork at ost O(/ a()) flows. Cobiig with Step (a), the total flows assiged to every ode is O(+F()+/ a()), whih is also doiated by O(/ a()) (ote that a() is at ost (/F()), hee F() is at ost / a()). Step 3 (Trasissio Shedulig): e osider a shedulig shee for a (, )-etwork. Ay trasissios i this etwork ust satisfy these two additioal ostraits siultaeously: ) eah iterfae oly allows oe trasissio/reeptio at the sae tie, ad ) ay two trasissios o ay hael should ot iterfere with eah other. e propose a tie-divisio ulti-aess (TDMA) shee to shedule trasissios, whih satisfy the above two ostraits. I this shee, a seod is divided ito a uber of edge-olor slots ad at ost oe trasissio/reeptio is sheduled at every ode durig eah edge-olor slot. Hee, the first ostrait is satisfied. Eah edge-olor slot a be further split ito saller ii-slots. I eah ii-slot, eah trasissio satisfies the above two ostraits. Suppose that a oidiretioal atea eeds haels to separate t ourret trasissios. Ituitively, diretioal ateas a redue the uber of haels to ( π ) beause diretioal ateas a separate the urret trasissios if both the trasitter ad the reeiver use diretioal ateas. Thus, the uber of ii-slots is redued by a fator of ( π ). The, we desribe the two tie slots as follows. Fig. 6 depits a shedule of trasissio o the etwork. (i) Edge-olor slot: First, we ostrut a routig graph i whih verties are the odes i the etwork ad a edge deotes trasissio/reeptio of a ode. I this ostrutio, oe hop alog a flow is assoiated with oe edge i the routig graph. I [8] ad [], it is show that this routig graph a be edge-olored with at ost O(/ a()) olors. The, we divide oe seod ito O(/ a()) edge-olor slots ad eah slot has a legth of Ω( a()) seods. Eah slot is staied with a uique edge-olor. Sie all edges oetig to a vertex use differet olors, eah ode has at ost oe trasissio/reeptio sheduled i ay edge-olor tie slot. (ii) Mii-slot: e further divide eah edge-olor slot ito ii-slots. The, we build a shedule that assigs a trasissio to a ode i a ii-slot withi a edge-olor slot over a hael. e ostrut a iterferee graph i whih verties are the odes i the etwork ad edges deote iterferee betwee two odes. By Lea 4, every ell has at ost a ostat uber of iterferig ells with a fator ( π ), ad eah ell has Θ(a()) odes (by Lea 3). Thus, eah ode has at ost O(( π ) a()) edges i the iterferee graph. It is show that a graph of degree at ost k a be vertex-olored with at ost k + olors []. Hee, the iterferee graph a be vertex-olored with at ost O(( π ) a()) olors. The, we use k 3 ( π ) a() to deote the uber of vertexolors (where k 3 is a ostat). Two odes assiged the sae vertex-olor do ot iterfere with eah other, while two odes staied with differet olors ay iterfere with eah other. So, we eed to shedule the iterferig odes either o differet haels, or at differet ii-slots o the sae hael. e divide eah edge-olor slot ito ( π ) k 3a() ii-slots o every hael, ad assig the ii-slots o eah hael fro to ( π ) k 3a(). A ode assiged with a olor s, s ( π ) k 3 a() is allowed to trasit i ii-slot s

8 Fig. 6. The TDMA trasissio shedule ( π ) k 3 a() o hael (s od ) +. Let us aalyze the apaity of the (, )-etwork. Eah edge-olor slot has a legth of Ω( a()) seods. Eah edge-olor slot is divided ito ( π ) k 3a() ii-slots over every hael. Therefore, eah ii-slots has a a() legth of Ω( ) seods. Sie eah hael a trasit at the rate of ii-slot, λ() = Ω( ) bits a be trasported. Sie have, λ() = Ω( ( ( π ) k 3a() a() π ) k 3 a() a() ( π ) k 3a()+ a() bits/seod, i eah ( π ) k 3a() +, we ) bits/se. Hee, λ() = Ω(MIN O (, )) bits/se. Sie eah flow ( π ) a() is sheduled i oe ii-slot o eah hop durig oe seod iterval ad every soure-destiatio flow a support a per-ode throughput of λ() bits/se, withi oe a() seod, there are Ω(MIN O ( ( ), π a() )) bits trasitted. Thus, the etwork apaity is λ() = Ω(MIN O ( ( ), a() π a() )) bits/se. The, we exted the result to a (, )- etwork, ad the apaity of a (, )-etwork is ( ), a() π a() Ω(MIN O ( )). Fro Eq. (), the size of eah ell is i(ax( 00log, ( π ) ), ( F() ) ), where F() = Θ( log log log ). Substitutig the three values, we have the followig theore. Theore 5: The ostrutive lower boud o the apaity of a (, )-etwork is as follows. ) he π ) log ), a() = Θ( log ), the etwork apaity is Ω( log ) bits/se. ) he π ) log ) ad also O(( π ) log log ( log ) ) ad a() = Θ( ), the etwork apaity is Ω( ). 3) he π ) log log ( log ) ) ad a() = log log Θ(( log ) log log ), the etwork apaity is Ω( log ) bits/se. C. Soe Possible Ipliatios Usig diretioal ateas i ulti-hael etworks a iprove the etwork apaity by ehaig the oetivity ad reduig iterferee. he π ) log ), the apaity is Θ( log ), whih is obtaied uder the oetivity ostrait. This apaity has a apaity gai 4π over a MC-MOi etwork. This result iplies that diretioal ateas a greatly iprove the etwork oetivity. he is ireased to Ω(( π ) log ), the throughput apaity degrades to Θ( ). At that tie, diretioal ateas a sigifiatly itigate the iterferee ad the etwork has a apaity gai of π. O the other had, usig ultiple haels a help to solve the probles of hidde terials ad deafess aused by diretioal ateas. The diretioal hidde terial proble happes whe a trasitter fails to hear a prior RTS/CTS exhage betwee aother pair of odes ad ause ollisios by iitiatig a trasissio to the reeiver of the ogoig trasissio. The deafess proble ours whe a trasitter fails to ouiate to its iteded reeiver, beause the reeiver s atea is adjusted i a differet diretio. Elbatt et al. [3] solved the deafess proble by usig two iterfaes whih are tued to two differet haels. Both the hidde terial ad deafess probles were itigated by sedig busy toes over aother hael fro a oidiretioal atea [4]. Thus, itegratig ultiple haels with diretioal ateas a iprove the etwork perforae further. VI. CONCLUSION Previous studies [] [7] foused o usig ultiple haels i wireless etworks to iprove the etwork perforae. However, sie oly oidiretioal ateas are equipped at every ode i suh etworks, the iproveet o the etwork apaity is liited by high iterferee. Other studies [9] [6] foud that usig diretioal ateas istead of oidiretioal ateas i etworks a greatly iprove the etwork apaity. But, suh sigle-hael etworks usig diretioal ateas oly allow liited ourret trasissios. I this paper, we propose a ovel wireless etwork that itegrates ulti-hael ad diretioal ateas. e derive the upper bouds ad lower bouds o the apaity uder arbitrary etworks ad rado etworks. e have foud that usig diretioal ateas i ulti-hael etworks ot oly a ehae etwork oetivity but also a itigate iterferees. Meawhile, usig ultiple haels also helps to solve the hidde terial ad deaf probles [] aused by diretioal ateas. Therefore, obiig ultiple haels with diretioal ateas a ahieve sigifiat iproveet o the etwork perforae. ACKNOLEDGMENT This researh was partially supported by Natural Siee Foudatio of Chia grat No ad , ad the Natioal Grad Fudaetal Researh 973 Progra of Chia uder Grat No.006CB REFERENCES [] P. Gupta ad P. R. Kuar, The apaity of wireless etworks, IEEE Trasatios o Iforatio Theory, vol. 46, o., pp , 000.

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APPENDIX A Proof of Theore : First, we osider a uit-area plae whih has odes arbitrarily plaed. There are haels available i the etwork. The whole etwork trasports λt bits over T seods. Let the average distae betwee the soure ode ad the destiatio ode of a bit be L. Thus, a trasport apaity of λl bit-eters per seod is ahieved. Let us osider bit b, where b λt. Suppose that bit b oves fro its soure to its destiatio i a sequee of h(b) hops, where the h-th hop traverses a distae of r h b. The, we have λtl λt b= h= rh b (3) e defie H to be the total uber of hops traversed by all bits i T seods, i.e., H = λt h(b). Therefore, the uber of bits b= trasitted by all odes withi T seods is equal to H. Sie eah ode has iterfaes, ad eah iterfae trasits over a hael with rate /, the total uber of bits that a be trasitted by all odes over all iterfaes is at ost T. Therefore, we have H = λt T h(b) (4) b= It is show i [8] that eah hop osues a disk of radius ties the legth of the hop aroud eah reeiver, i.e., rh b. Meawhile, fro the seod oditio of Eq. (), oly whe a ode adjusts its bea toward a reeiver ad the reeiver is oly affeted by the odes withi its atea bea, as show i Fig. 4. O average, proportio of the odes iside the reeptio bea will π iterfere with the reeiver. Thus, the oditioal iterferee zoe area is π [π( rh b ) ] = π 6π (rh b ). λt The, we have the ostrait b= h= 6π (rh b ) T, whih a be rewritte as λt b= h= H (rh b ) 6πT H (5) Sie the quadrati futio is ovex, we have ( λt b= h= H rh b ) λt b= Therefore, obiig (5) ad (6) yields λt b= The, substitutig (4) i (7) gives λt b= h= H (rh b ) (6) h= rh b 6πTH (7) h= rh b T Fially, we substitute (3) i (8), ad obtai 8π (8) λl 8π (9) This proves that the etwork apaity of a arbitrary etwork is O( ) bit-eters/se. Copared with the result i [8], i.e., the apaity of a MC-MOi etwork is, a MC-MDA π etwork has a apaity gai of π. APPENDIX B Proof of Lea 4: Suppose that there is a ell D that a trasit with its 8 eighborig ells. The trasissio rage of eah ode i ell D, r(), is defied as the distae betwee the trasitter ad the reeiver. Sie eah ell has the size a(), r() is o ore tha 3 a() (if iludig the ell itself, there are 9 ells). Fro the iterferee odel Eq. (), the trasissio is suessful oly whe the iterferig odes are ( + )r() away fro the reeiver or the iterferig odes will ot ause iterferee at the reeiver (the beas of the iterferig odes do ot over the reeiver). Let us osider that a trasitter X i withi ell B is trasittig a data paket to a reeiver X j withi ell A. Sie the trasissio rage betwee X i ad X j is r(), the distae betwee two trasitter X k ad X i ust be less tha ( + )r(), if X k auses the iterferee with X j. Thus, a iterferig area is loosely bouded withi a square with a edge legth of 3( + )r(). Meawhile, to esure a suessful trasissio, the beas of the two odes are poited at eah other. Therefore, oly the odes withi the reeivig bea of X j a iterfere with the reeptio at X j. Furtherore, oly whe a trasitter adjusts its bea to the reeiver, it a iterfere with the reeiver. Therefore, the iterferig probability is ( π ). Cobiig the two observatios, there are at ost k = (3(+ )r()) ( a() π ) = 8( + ) iterferig ells. Hee, 4π the uber of iterferig ells is bouded by 8(+ ), whih 4π is a ostat k idepedet of a() ad.