Distributed Low-Complexity Maximum-Throughput Scheduling for Wireless Backhaul Networks

Transcription

1 Distiut Lw-Cxity Mxiu-Thuhut Shui Wiss Bkhu Ntwks Au Ki EE Dt, St Uivsity P At, CA 9405 Ei: ki@st.u Ths Siis Ths Pis Rsh L Bu Cx, F 9648 Ei: ths.siis@ths.t Ew W. Kihty ECE Dt, Ri Uivsity Hust, TX Ei: kihty@i.u Astt W itu w-xity istiut stt MAC t tht sut si iv ts i wiss khu twk (WBN). F ity wiss twks, suh xiu thuhut t hs tiusy h t iz us v i ty iti is vi, th uti th ti ik tsissi st t h st is NP-t. F th i t stutus iu y WBN ti tis, w ist itu tiz ith tht svs th ti shui i u sts t st i i th u s i th twk. This is hiv y isvi xiti v st h-thti tis WBN tti h. Gui y th tiz ith, w si istiut t wh, t th ii h st, s it ity ut th ti ik tsissi st. W th itu ith t ut th iiu u sts t t this utti, thus iiizi th st vh. Usi th ysis siutis, w shw tht i ti u t yis w vh wh it v xisti wiss this siiity uts xisti suti istiut stt shui hiss. I. INTRODUCTION Wiss Bkhu Ntwks (WBNs) sty wi istutu vi wissy uti-hi t hihs wi Itt ty its. W si WBN hittus tht us shu Miu Ass Ct (MAC) ts suh s IEEE Shu ss is tttiv k systs ik WBNs us it t y vis w-kw stvti/uiss h ttis ss (.., 80.11) ut s vi thuhut y uts. I shu ss MAC t, s tist syhiz uiti us i su s. Eh sists shui hs w y t hs, th ix uti si sv sts. Dui th shui hs, s ust ti st it- iks tht wi tsit ui th t hs. W itu istiut shu ss MAC t tht sut si iv ts i WBN. Suh xiu thuhut t hs tiusy hi t iz i wiss twks ity tis tw ss. Fist, sti th ti ik tsissi st xty hih xity v i tiz iti is vi: i si [14], Tssius Ehis stish tht sti it- st iks hvi xiu su uu sizs t h shui hs is xiu thuhut iy. Ututy, this uis svi ist th xiu wiht it st (MWIS) h, whih is NP-t i. S, th uti th shui hs shu uh s tht s xh th sss i t sv y ist th ti shui, thy uii tiisti u u th u ss xh sts. This u shu s s s s ssi si it tis th shui hs uti, ix - vh ui th t ti. I this, w xit th WBN t stutu sv th MWIS shui i u sts t st i i th u s i th twk. 1 W ist v tht th WBN tti h sssss usiv uti ty, wh t h uti st it is wys ssi t i ik wh its iti ihs it with h th. W utiiz this ty t vis tw-hs tiz shui ith tht uts MWIS y sutiy xuti t h ik xty i h hs, ss th MWIS ist t h. Bs th tiz ith, w si istiut t i whih s v t MWIS usi t st twh ss xhs. Dsit i i i ts th u iks i th twk, th u th u th t sss xh sti i ti. W vi ith tht svs th i si tiizti tii th iiu u t ss-xh us ui t ut y ist th MWIS iv WBN ty. Fiy, w u t t t tht uts xi ik tsissi st t h. F ity wiss twks, xi shui hs shw t ut t st -ihth si iv ts [7], [11], [16] it usi istiut ts with i u th shui hs []. W shw tht xi shui is suti i th s WBNs, u t yis 0.0% t 75% th vh xisti xi ts, i th th th WBN ty stutu. Th st th is iz s ws: I Sti 1 Th t stutu is i ( wi uti) vs. hysi (it).

2 II, w viw th stt th t i istiut shu ss ts th ity t-stutu wiss twks. I Sti III, w i th WBN it th xiu thuhut shui iy. I Sti IV, w itu th h-thti tis th WBN tti h whih t tiz ith tht svs th MWIS i i ti. W itu th istiut t tht izs th tiz ith i Sti V. W sv th vh iiizti i Sti VI vut th t i Sti VII. Sti VIII us. II. RELATED WORK Th vii thuhut uts thuh istiut shui hs iv siiit ttti ty. Rt wk ssii with st t th ty stutu si u th wi ik it stits: ith uti-h systs wh y iy it xists (u t th h-ux i stit), si-h systs wh th iy it sy it (u t hi tis) xist. Wiss twks ity tis: Giv th iiuty th MWIS, vius wk hs us vuti th suti hs s xi shui. O h, it ws shw i [7] [16] tht xi shui i uti-h twks hiv t st 50% th xiu thuhut i si-h ws uti-h ws stivy. O th th h, [11] shw tht this ut ss t 1/8 i si-h systs. I iti, whi xisti xi shui iths si ui ss xhs kw th iti, thy t uy-sii iu ss ts. Th siitis suh ts ti th vh iu y th shui hs y itt v. Rty, Mi t. itu istiut iz shui ith xiu thuhut shui i utih twks [8], hwv, th iti t t hiv th ui jtiv hs t sii th. T-stutu wiss twks: T-stutu twks hv ss usi thius tht xit thi si stutu. I [10], istiut shu ss t is si t th PHY stt tt th tht hiv y DCF. Th wk i [15] uss kt shui y uts usi thius wii twks. Bth [10] [15] us tw- i wh h th iks tivt i h. This thiu utiizs y tw ik tivti sts, h t iz si iv ts. Th wk i [1], sis syhus istiut xiu thuhut t uti-h systs. I [1] it is s shw tht uti-h systs w istiut huistis s Lst Quu Fist (LQF) t thuhut ti. Hwv, i suh huistis, th u sts ui t ut th ti shu t h is y iv i ts v xity. I tst t [1] [1], u t hivs xiu thuhut st it stits WBNs y ity svi th MWIS i u-u u sts, t st i i th u s i th WBN. III. SYSTEM MODEL W st th WBN y Ntwk Gh (NG) G(N,L) wh h (u, v) L siiis tht s u v th st N withi hv stish i wiss ik. Eh hs si i tsiv t tsit iv siutusy. This iy it stit hiits tw iks shi i shu siutusy. Sy it stits is tw iks tht t sh ut t st thi si tsitt iv s withi. This iiti sy it stit s t uiti wh h sussu ik tsissi uis sussu DATA/ACK xh. Sy it itit y ssii it hs (th uy s s stu s) t s withi tw-h ihh. W wi WBNs with iy it y, uti-h WBNs with th iy sy it, si h. Ns i th WBN uit usi i t stutu vi t th hysi ty i y th s wiss xiitis. This stutu xiity y i. It s iiity iu y y- uti ts. I this s, th WBN ty hs t th ti s th uti uts whih wu s sv s. F siiity w si th WBN ty t stti tht sy it th it hs th WBN t hs suss. This hs uti-h WBNs, wh y iy it xists. I th si h s, this hiv y iti ts [10] y i th WBN t i hx stutu []. A ik tsissi st is st iks tht tsit siutusy sujt t th WBN shui stits. Si uti-h syst hs iy it stits y, y thi i th NG is ik tsissi st (.., th ik st {,, k,,, } i th NG Fi. 1). I si-h syst, ik tsissi st ust s stisy th sy it stits (.., th ik st {, h, k, } i th NG Fi. 1). Th WBN ts with syhiz stutu. At h, th shui hs is ik tsissi st t tsit ui th ti t tsissi hs. Eh ik tsissi st is st y 0-1, L 1 vt i whih h uit ty t ix ss t ik shu tsissi. W t th st ik tsissi sts i th WBN y I(G). Eh ik is st y uu iti t its tsitt. Lt A (k) Q (k) th u ivs t th u kts y i th uu t th ii ti st k, stivy. Lt th iv t t ik λ. A iv t vt λ = {λ, L} is si i th xists ti ik tsissi sts

3 I j I(G) us α j [0, 1) suh tht: M M λ α j I j,, α j < 1 (1) j=1 j=1 Th st si ts s th siiity i t y Λ. A shui ith is xiu thuhut ith ( st ith) i vy si st iv ts, th v uu siz h ik s t w t iiity, tht is, i su E[Q (k)] < vy ik i th WBN. I [14], Tssius Ehis si syhiz stt syst wh h sists si st. Thy stish tht sti t h st k ik tsissi st with xiu su uu sizs is xiu thuhut ith [14]: h i j k (1,) 1 (7,4) 6 (4,) 11 (10,5) 14 (1,6) 18 (8,) 16 (,1) h (11,5) 17 (4,) 15 (9,4) 1 (1,1) 5 (6,4) 1 (5,) j 8 (6,) i 7 - Ntwk Gh - Muti-h Ctti Gh (,) 10 (5,) 4 (,) k (,) 9 I (k) = x I I(G) Q T (k)i () Th sut ws shw t h ity tis, ity ik shui stits i.i. iv sss. It hs s xt t sss iv sss, uti-st s [1]. Dsit its uivsity, this ith ssus tht th shui u is y tiz t wh istt iti is ty issit t th t. Futh, t ti th ti ik tsissi st uis svi Mxiu Wiht It St (MWIS) t h, whih is NP-t i. I uti-h systs, th us t Mxiu Wiht Mthi (MWM) utti, whih is yi xity (O(N )). T th st u kw, kw istiut ith xists t xty sv ith MWIS MWM, y ty stutu. W t shw tht th WBN stutu ws t v th v iiutis sv th MWIS istiutivy, i i ti, th uti-h si h systs. Du t s iittis, w th t [6] s. IV. LINEAR COMPLEXITY SCHEDULING I this sti, w itu st h-thti tis tiu t th ik it tishis th WBN stutu. Usi ths tis, w vis tiz ith tht vy sv th MWIS i i ti. A. Gh-thti tis WBNs Th ik it tishis i th WBN tu y Ctti Gh (CG) G(V,E). Eh vtx v i th CG ss t ik i th NG th WBN h i th CG siiis tht th tw iks i NG si t th tw it vtis th iti. Th CG ws us t stt th sii ik it stits tt uti-h si-h systs i uii. F x, th CG i Fi. 1() ss t uti-h WBN. A it st i CG ss t ik tsissi st i its si Fi. 1. () A Ntwk Gh (NG) () si uti-h Ctti Gh (CG). Eh CG vtx ss t ik i NG. Th it w h CG vtx i ts th su ssi y th uti ith (Sti IV-B). Th it is v h CG vtx i t th ti uti i (SEQ1,SEQ) ssi y th vh iiizti ith (Sti VI). NG. Thus, svi th MWIS CG, wh th wiht h vtx is u t th si ik s uu k t th ii, s t xiu thuhut uts. Th h-thti tis th si-h uti-h WBN CG suiz y th wi : WBN Ciu L: Giv WBN CG, th is wys t st vtx v suh tht v its ihs iu. Mv, i v is v th CG, th suti suh sssss th s ty. Th Ciu L vs ut usiv ty WBN CG v y shwi tht it hs CG vtis si t iks t t s i th si NG. F x, wh th NG i Fi. 1() ss t uti-h syst, ik its y with iks th vtis th iks i th si uti-h CG (Fi. 1()) iu. Siiy, wh th NG i Fi. 1() ss t si-h syst, ik its y with iks,,, it shw tht iks iu i th si si-h CG. Th Ciu L is ui us it tis th t y whih ti wiht-xh tis shu xut -CG-vtx sis t vy ut th MWIS. M siiy, u shui ith uis ix uti su, -ut i y uti ith. Th uti ith utiizs th Athuh it is ssi t v tht th Ciu L s hs - iks, y ik stisyi th L suis t sv th MWIS.

4 Ciu L t ssi uiu t h CG vtx. This -ut uti su s tis th t i whih th CG vtis shu xh thi wihts. Th su is us y th MWIS ith t ut MWIS t h ist th MWIS. Nxt, w si th uti MWIS iths i ti. B. Euti Aith Giv th WBN CG, th uti ith iks y CG vtx stisyi th Ciu L ssis t it 1. Nxt, this vtx is iit th s vtx stiiiti st ts th u CGs y ssii s i isi, uti vtis th CG hv si. Th sii sts th ith s ws. Euti Aith: A h ICG is iitiiz t h CG. At vy itti i, s s ICG is t ty: (i) st ity vtx v i ICG suh tht v its ihs stitut iu (ut t xist y th Ciu L), (ii) v y th u i, (iii) t v its jt s ICG. Fi. 1 its th utut th uti ith i th uti-h WBN CG. Th su h vtx is shw w its i. C. MWIS ith Th tiz MWIS shui ith tks s iut wiht CG tus it st. It sists th Wiht-Exh Phs th It St Cstuti Phs suiz w st i ti i [6]: Wiht Exh Phs: Iitiiz vtx st TRAV t ty. As, iitiiz th siu wiht RW h vtx t its wiht. Bs th uti su, tvs h vtx th CG i isi. F h vtx v tvs, i its RW is sitiv: (i) it t th TRAV st, (ii) s its RW t z, (iii) s th RW its ihs i CG y v s RW; thwis ski t th xt vtx. It St Cstuti Phs: Iitiiz vtx st I t ty. Tvs vy vtx v i TRAV i si. Vtx v wi iu i st I s s I s t ti y ih v. T I iustts th Wiht Exh Phs T II iustts th wi It-st Cstuti Phs iiti st wihts i th CG Fi. 1(). MWIS Th: Th it st I ut y th th It-st Cstuti Phs th MWIS shui ith is MWIS. P: W stt y ti tht, i vtx-itwiht h, th wiht y it st is u u y th iity y iu v this h I vtx-it-wiht h G, stc ius is iu v i vy vtx G is t t st s y ts C s its wiht is. [9]. Giv this t, w v tht it is ssi t i iu v i th CG whs iity is tiht w u t th wiht st I. Th, th iity th iu v us th wiht I, I is MWIS. Th iu v is stut s ws. By th Ciu L, th st sisti y vtx v its ihs tht hv ut t v iu. W t suh iu y C v. Nw, stut iu st C tii RW v is h iu C v vtx v i th TRAV st tht ws uit ui th Wiht Exh hs - RW v is th siu wiht v wh visit ui th Wiht Exh hs. Ciu st C is iu v us (i) vy vtx v i th TRAV st is v y xty s y ius i C s its iiti wiht W v (y th stuti C) (ii) vy vtx v t it th TRAV stisvytsts y ius s its iiti wiht (thwis, v wu hv t th TRAV st). W w t shw y titi tht th wiht I is tihty w-u y th iity C. Sus I hs stity ss wiht th th iity C. Th ith (1) vtx i I is v y w ius C th its wiht () th ius i C whih t iu y vtx i I. Ci (1) s t h us I is sust TRAV, y stuti, y vtx v i TRAV is v y xty s y ius i C s v s wiht RW v. Ci () s t h us vy iu i C vs xty vtx I. This shw s ws. Si CG vtis tvs i vs wh uii I, vy iu C v i C tis t st I s vtis: i v s jt TRAV ihs with hih su u is i I (i., v hs t iit yt I y th ti it is tvs), th v its wu hv t I. I iti, iu ti th vtx it st. Bs th v, C is iu v w-u t th wiht I. Thus, I is MWIS. Bth hss th MWIS shui ith ui u ittis u t th u s i CG, whih is i tu u t th u iks L i th WBN. Si th WBN ty is t, th shui ith uis tt (N-1) sts, whih is i i th u s i th twk. V. DISTRIBUTED PROTOCOL Th MWIS ith th vius sti is tiz xut th vtis th WBN CG. W w us this ith t si stt istiut t xut y th s i th WBN tht sti svs th MWIS i i ti. I th wi, w us th t siti th uti-h s. Th si h s ws sii sts is si i ti i [6]. Th stutu th istiut t is t i Fi.. Eh t tsissi hs t sv kts h s is y shui hs. Eh shui hs is u tw su-hss: th Wiht-xh hs th It-st Cstuti

5 TABLE I WEIGHT EXCHANGE PHASE FOR CG OF FIG. 1(B) FOR SOME INITIAL WEIGHTS. CGVERTICES SHOWN AND VISITED IN INCREASING LABEL ORDER. I--st CG vtis thi siu wihts RW TRAV Itti (= #) h k i j Iiti wihts , h 0 1, h, k , h, k,,,, j,,,,,, Shui Phs Wiht Phs MWIS Phs Dt Tsissi Phs Dt Pkt 1. Dt Pkt N Ru 1. Ru MxSEQ1 Ru 1. Ru MxSEQ Fi.. Th istiut t stutu. hs t y Wiht-hs MWIS-hs stivy. Th Wiht-hs MWIS-hs sist sv syhiz us h. Eh ik i th WBN NG is t tiu Wiht-hs u t tiu MWIShs u. Dui h Wiht-hs u, th s th ik t this u st th siu wiht this ik (i sitiv) th ths s th s withi thi ut th siu wihts thi jt iks. Dui h MWIS-hs u, th s th ik t this u st thi it st shi sttus (i th ik hs t s xu th shui hs) th s withi thi ut th it st shi sttus thi jt iks. Th siz h u, th u us hs, th iti ut whih u h shu stt i th t tis stt ti. This iti is ut i s vit th uti u isuss i Sti IV. Thus, th shui hs yis stt vh. W isuss vh iiizti i Sti VI xt st th shui hs his. TABLE II INDEPENDENT-SET CONSTRUCTION PHASE AFTER WEIGHT-EXCHANGE PHASE OF TABLE I. MWIS IS A SUBSET OF THE TRAV SET. VERTICES IN TRAV SET SHOWN AND VISITED IN DECREASING LABEL ORDER. MWIS shi: 1: MWIS, -1: MWIS It j k h A. Oi P Euti Pu Th shui hs is s tw uti sus SEQ1 SEQ t i th Wihths th MWIS-hs stivy. W th u tht ts SEQ1 SEQ th P Euti u. This u ssis s sutiy t CG vtis s th h ik is w t xh its wiht MWIS sttus iti. Th s tw iks tht hv th s SEQ1 (SEQ) uti w t tsit wiht-xh (it st shi) t kt ui th s Wiht-hs (MWIS-hs) u. Th tis hw t ut th CG vtis i ti wy is isuss i Sti VI. Fi. 1() shws th (SEQ1,SEQ) uti i v h ik s i shw i th uti-h CG. Th tis th istiut t shui hs (Wiht-Phs MWIS-Phs) iv xt. B. Oi Shui Phs W stt th t ti uti-h itti s tsitt-it h ssit sh. Eh hs ssi h tsissis hs hv ssi suh tht th is sy it. Wh is shu t iv tiu tsitt ui, it swiths t th h tht tsitt. F this us, s iti shu th wiht-hs th MWIS-hs tht iits wh whih ik thy shu tsit iv iti. 1) Wiht-Phs: Eh itis -ik siu wiht t wh h siu wiht ty is iitiiz t u t th ik s uu siz t th ii th shui hs. At th ii h wiht-hs u i, th s vy i-ut ik st s siu wiht RW thi hs i RW is sitiv. Th, th it s ik s withi thi t th siu wiht thi jt iks y RW. As sut, th s vy ik whs CG vtx is jt t ik s CG vtx hs hih SEQ1 wi t th siu wiht tht ik. Mv, ik whih th siu wiht s z tiv is iit y its s th st th wiht-hs th MWIS-hs tht wi w. Csuty, th s suh ik wi t st its wiht tht ik

6 wi t shu t tsit ui th t tsissi hs. Fi. () shws x th Wiht-hs ui th ist Wiht-hs u s th uti su shw i Fi. 1(). Bsti s tt i k, thi wiht tsissis st y ws. Th k it s iks (th hvi SEQ1 u t 1) th sti s th k whit s ut thi siu wiht iti thi jt iks,,,,. Nt tht th t th t th s t t st th wiht thi iks, h thy iu t t s. h i j k Fi.. Wiht-hs MWIS-hs i uti-h WBN s th (SEQ1,SEQ) uti sus shw i Fi. 1() ) MWIS-Phs: This hs sists suti sts th iks MWIS-shi sttus i th it- t tsissi st. Th jtiv this hs is t ity ti th it- tsissi st usi th TRAV st ut i th Wiht-hs. At th ii h u i, th s vy i- ut ik st MWIS-shi kt y i ik hs iu i th MWIS tsissi st ui th wiht hs: st its sitiv RW vu ui th Wiht-hs, is t ih y ik whih hs y shu t tsissi. By th h MWIS-hs u, th s h sti ik h tht u ss th MWIS-shi tsissi th s, iit h ik whs CG vtx is jt t th CG vtx ik. Fi. () shws x th MWIS-hs ui th ist MWIS-hs u s th uti shw i Fi. 1(): th s ik tt i k th sti s, th st s tt i k whit iit iks,,, th MWIS tsissi st. VI. OVERHEAD MINIMIZATION Th vh th istiut t is itt y th uti ui t t th MWIS utti th shui hs. This sists t st utis, τ 1 h i j k τ, th u t sts, SEQ1 SEQ, i th Wiht-hs MWIS-hs stivy: Ovh = SEQ1 τ 1 + SEQ τ () Th utis τ 1 τ th hysi y. W st iiu st utis PHY i [6]. Th u t sts SEQ1 SEQ is t st i i th u s i th twk. This u is tyit hs v i twks with sti us (.., st ty), hwv it sti yi hih vh s th twk siz iss. I this sti, w si iths tht xit sti us t iiiz th vh iv y E. () y iiizi th SEQ1 SEQ iv WBN ty. I tst t th istiut t, ths iths tiz xut t th uh sw ti s ty hs i th WBN. Th iths ti th iiu stt - vh s w s th ti tsissi shu ui th t hs th t. Th titi th ti shu t ss y xisti h i hs us, i iti t sti stits, th xist ti stits itt y th t ti. Ou suti tw ii th iiu vh xits th stutu is s st ti usiv us. Gui y u ti iths, w s iv s xssis th s u D-y ts th L t i isihts th ut vh s uti th th th th WBN. A. P iiti: Euti Cstits Th iiu vh utti hs th ti sti uti stits. Ti stits itt whih CG vtis ut ist (h, hv w s), whs sti stits su tht t sss wi i ui th shui hs. SEQ1 Ti Cstits: As isuss i th tiz ith, y CG vtx is sti t stisy th Ciu L wh it is SEQ1-ut. SEQ1 Sti Cstits: I t vi isis tw wiht-xh kts, tw iks tht withi hs i uti-h twks 4 hs i si-h twks t ut y th s SEQ1 u. U th ss stits, this is uivt t th stiti tht CG vtis tht withi shtst ist t ut y th s u. F x, i th twk shw i Fi. 1() u utih ss stits, ik t ut y th u y th ik i th twk. SEQ Ti Cstits: Ri SEQ uti stits, CG vtis sus t i thi ihs thi it tsissi st shi sttus i vs with st t th SEQ1 su. H, h i (v, v ) jt CG vtis: i SEQ1 v <SEQ1 v SEQ v <SEQ v (4)

7 SEQ Sti Cstits: Th MWIS hs ws sti us th th Wiht-hs us it y uis s t ss MWIS-shi tsissis ist i wiht-xh kts. Ssi th t MWIS-shi tsissis uis iks sujt t iy it i uti-h NG (sujt t th iy sy it i si-h NG) t t ut y th s SEQ u. U th si-h uti-h ss stits, this is uivt t th stiti tht jt CG vtis t ut y th s. 0 1 h i j k T {,} B. Suti: Miiu-Euti Aiths W w st iths t ut th iiu u Wiht-hs us (SEQ1) MWIS-hs us (SEQ) y uti-h WBN. Th sih s is tt i sii shi u i [6]. W ist itu th Hihst- Vtx Th, ui i svi th ivi th iths tiity s: Hihst- Vtx Th: I th iiu-vh suti, th hihst SEQ1 SEQ uti s ui y si th s CG vtx. Th Hihst- Vtx Th vis stti it tw th suti. Si th hihst th SEQ1 SEQ wi ui y si vtx i th CG, it suis t si ist tht iiizs SEQ1 SEQ stti vtx. Th iiu-vh suti is th iiu suti v ists. W ssit th ist h WBN R with it t T R t t R. W th si iths tht t h it t T R t ut th iiu SEQ1 R iiu SEQ R, thus iiizi E. () th R ist. Giv T R, u iths utiiz th Hihst- Vtx Th t sti th st iks i T R ssui th t R hs th hihst. Fusi WBN R, w t si th thy tht svs its si ist usi th iths: th Cit T Cstuti Aith, Mi-SEQ1 utti ith, Mi-SEQ utti ith. Th tiity s u i [6]. 1) Cit T Cstuti Aith: F y R i th twk, it t T R is stut s ws: Th it t is iitiiz t th s s th NG xt tht R is sit t th t th it t. Evy i th it t is ssi wiht u t th u hi is hs, vy ik i th twk h is st y th si - t vtx i th it t. Fi. 4 iustts th stuti th it t T R th WBN i Fi. 1(), wh th t R is th jt t iks. Fi. 4. Th T, it t stuti th twk shw i Fi. 1() wh th t is th jt t iks. ) Mi-SEQ1 utti ith: This ith is th iiu SEQ1 siz ssui th hihst ix ik is jt t R, th t th it t. Th ituiti hi th ith is t i th iiu SEQ1 siz th sst suts th it t th t usivy i th iiu SEQ1 siz h iti suts s thy ji t ui suts t th hih vs, uiti t th t th it t. Th ith s s ws. Dt th vtis th it t, t th wst v u th ii it t T R y L, ssit with h vtx v i th t tw istitt sts: S v Chi v. S v tis th SEQ1 us th iks jt t v i th NG whi Chi v tis th SEQ1 us th iks si t vy hi vtx v v i th ii it t. Th tts ths tw sts vy ui th iiu-siz utti u s thy just t stisy th SEQ1 sti ti stits. At th th utti u, th xiu-vu t i th S st R is u t th si iiu vu SEQ1 R. This hs ssui tht th t vtx R i th it t sts th ik with th hihst SEQ1, i t th Hihst- Vtx Th. W w t si th sts t iv t th S st th t R T R. F h vtx v th it t, t its hi vtis wiht y Ch v W v stivy. F vy vtx v i th ii sut, iitiiz th S v st t {1,,...W v } th Chi v sttty. Stti v L 1 th t usivy u t th t v 0, tvs h vtx th s v th t. F h suh vtx v, tvs h hi vtx v v is S v vus suh tht th SEQ1 Sti Cstits stisi. M siiy, isi th vus S v st, th SEQ1 uti s th iks st y th sut t t v w tiiz xi sti us ui vius ittis; hwv, this ws ssui th s i th S v it th s i th S sts th th hi v (whih is y t th s i t th SEQ1 Sti Cstits isuss i). Th, t

8 Fi. 5. Fi. 4 T {,} S St Chi St Lv 0 {11,1} {5,6,8,9,10} Lv 1 {8,9,10} {5,6} {1,,... 7} {1,,,4} Lv {1,,} {4,5} {6,7} {1,} {,4} {} {} {} {} {} Fi S Chi Sts th Cit T T, shw i iti SEQ1-siz tiity, isi th vus i S v st shu whi svi s uh sti us s ssi. T hiv this jtiv, w is th vu vy t i S v y th iiu vu suh tht th sti stit is t vit: S v (Chi v S v ) =φ (v,v ) Ch v Ch v (5) Th Chi st v is w u t th ui th S sts its hi. Th S st v is th u t th iiu xihi st W v ts suh tht th sti ti stits tu s ws: S v (Chi v S v Chi v ) =φ, (6) vtx v Ch v : x S v suh tht SEQ 1x >SEQ 1x x S v (7) Fi. 5 shws th i S Chi sts h vtx i th it t Fi. 4. ) Mi-SEQ Cutti Aith: This ith usivy is th iiu SEQ siz usi sii ituiti s tht ii th iiu SEQ1 siz: stt y ii th iiu SEQ siz th sst suts it t th usivy i th iiu SEQ siz h iti suts s thy ji t ui suts t th hih v th it t. Th ith is s ws: Dt th wst v u th it t T R y L th st th hi h vtx v y Ch v. St th hi h vtx v y thi wiht i si t th siti h hi v v i th st ist y v. Stti v L 1 th t usivy u t v 0, tvs th t vtis t th s v. F vy vtx v tvs, ut its wiht w v s ws: w v = x(w v + v ) v Ch v Th i wiht th t vtx is th iiu SEQ P Euti ix R. Fi. 5 shws th i wiht h vtx i th it t Fi. 4 t xuti th Mi-SEQ Cutti Aith. C. Miiu vh D-y Ts th L Fwi th sts th iiu vh utti iths i th vius sti, it is ssi t iv si s xssis th si s D- y ts th L wh h - hs th s u hi D, s L hs wy th t s t. Th iiu siz utti us th si-h uti-h WBNs t i t III. W sv tht, ix th L, th TABLE III MINIMUM (SEQ1,SEQ) OVERHEAD FOR D-ARY TREES WITH DEPTH L Mi SEQ1 Mi SEQ Muti-h D + LD D 1 LD 1 Si-h D + D +D (L ) 1 LD D D 1 vh is s uti th D is uti i uti-h WBNs - yi i si-h WBNs. O th th h, th vh is s uti th L ix D is i i th uti-h si-h twks. H, ts tht w y th yi ss vh with u t th ts tht w y th. VII. PERFORMANCE EVALUATION W u t with xi shui ts si thy it i istiut. F uti-h systs, Li Sh uti xi t i [7] s H s ith istiut wiht this [4]. Th ti xity this ith is is O(N) wi twks its xity iss y i th iy it stits tk it ut [8]. M itty, xity is y iv i th v ss with tiisti u u th vh. T th st u kw, u us y vi y th wk i [7], whih siis t t ut xi shus i N us si-h systs. W th t th t i [] s Mxi it with th si-h vsi u t. Ovh. Th Mxi t is ti stt uss tsissi hittu sii t us: shui hs w y t tsissi hs h. Th shui hs Mxi sists stt u us wh s shu t st ID ss ui u shu iusi / xusi ss ui th u. M siiy, h i th twk is t u t st ID ss, s u t ttiy st xusi ss, thi u t ttiy st iusi ss (i th ik hs t tsitt xusi ss yt). Th, s u PHY t vus th t st utis, h ID ss (iusi xusi ss) shu t u t st τ 1 (τ ). H, th vh Mxi is u t (τ 1 +τ )N wh N is th u s i th twk.

9 MMF ts. Hwv, this x iustts tht xi shui t is t shu siiit ut its whih shu usi WBN-MWIS. BWN MWIS / Mxi L D VIII. CONCLUSIONS I this, w itu istiut w-xity xiu-thuhut shui t Wiss Bkhu Ntwks (WBNs). W shw tht th MWIS th WBN stti sv i i ti usi v h-thti tis th WBN tti h. W s vi iths t iiiz th vh th shui hs. W hv stt tht u t uts xisti ts s xi shui th i ts vh s w s thuhut. ACKNOWLEDGEMENTS This sh ws sut y NSF ts ANI-0160 ANI y It Cti. Fi. 6. MWISTS-t-Mxi Ovh Rti i Si-h Ntwks F D-y ts with th L, th vh Mxi s (τ 1 +τ )(D L+1 1)/(D 1). Fi. 6 its th vh ti u si-h t (t y WBN-MWIS) v Mxi s uti D L ts. Th vh u t is ut usi th SEQ1 SEQ us i T III th ts us th iiu st utis τ 1 =17µs τ =9µs ut i [6] ssui PHY siitis. W sv tht th vh WBN-MWIS t st 75% Mxi (D=,L=4) t st 0.0% Mxi (D=8,L=8). I th ti uiky ss wh ith th th th th t iss. Wh D is ix L vis, th s is xti O(LD L ). Wh L is ix D is vi, th s is yi O(D 1 L ). H, WBNs ws Mxi t th WBN-MWIS. Stiity. Si-h xi shui ts hv ytiy shw t ut t st 1/8 th xiu thuhut i i ity wiss twks. W shw thuh siutis tht thi iittis h th s WBN tis. W ut xits th WBN shw i Fi. 1. Th ity h tsissi ik is iz t kt st kts u siz. Usi th thiu i [5], w ut th ik Mx-Mi Fi (MMF) ts, us th s iut CBR ti t th WBN iks. W it th v uu sizs vy 900,000 ti uits. Th tt ui ti th xit ws 17,100,000 ti uits. F WBN-MWIS, w sv tht th v uu siz h ik i stt, whih viis th stiity ty th shui ith th (si) MMF ti. Hwv, Mxi ws y u st y si w iut ts t 5% th MMF t ti. This ti s th tiu WBN ty tht th iv ts w s w s ti th REFERENCES [1] G. Zuss A. Bzziski E. Mi. Ei istiut thuhut xiizti i wiss sh twks - titii h. I P. ACM MiC, Ls As, CA, St 006. [] I. Ci M. Sii. Distiut ssit iths utih kt i twks. IEEE Tstis Cuts, 8(10):15 161, Ot [] V. Gusk F. Htzis. A Istutu Dyi WiMAX Systs. I P. IEEE Vhiu Thy C (VTC), Ds, TX, 005. [4] J. H. Si istiut wiht this. I it s.dc/ , Ot 004. [5] X. Hu B. Bsu. O x-i iss shui i wiss -h twks: Ayti wk itti. I Pis ACM MiH, L Bh, CA, Ot 001. [6] A. Ki, T. Siis, E. Kihty. Distiut Lw-xity Mxiu Thuhut Shui i Wiss Bkhu Ntwks. Thi Rt, Ri Uivsity, 006. [7] X. Li N.B. Sh. Th it it shui ssy t t i wiss twks. I P. IEEE INFOCOM, Mii, FL, Mh 005. [8] E. Mi, D. Shh, G. Zuss. Mxiizi thuhut i wiss twks vi ssii. I P. ACM SIGMETRICS, Sit- M, F, 006. [9] C. Piitiu K. Stiitz. Citi Otiizti, Aiths Cxity [10] B. R K. Chu. Dsi Evuti w MAC Pt L-Dist Msh Ntwks. I P. ACM MiC, C, Gy, Auust 005. [11] P. Chk S. Sk K. K. Fiss thuhut uts with xi shui i utih wiss twks. I P. WiOt06, Bst, MA, Ai 006. [1] T. Siis L. Tssius. Distiut Dyi Shui F Et- Rt Guts I Wiss A H Ntwks. I P. ACM MiH, U-Chi, IL, USA, My 005. [1] D. Shh M. Kik. Dy us xit xiu wiht thi iths iut uu swiths. I INFOCOM, Nw Yk, NY, Ju 00. [14] L. Tssius A. Ehis. Stiity Ptis Csti Quui Systs Shui Piis Mxiu Thuhut i Mutih Ri Ntwks. IEEE Tstis Autti Ct, 7(1): , D 199. [15] G. Wi, G. Nik, L. Zh. Dsii utih wiss khu twks with y uts. I P. IEEE INFOCOM, B, Si, 006. [16] X. Wu R. Sikt. Rut Mxi Mthi: A Distiut Shui Aith Muti-H Wiss Ntwks With N- Exusiv Stu Shi. I P. IEEE CDC-ECC05, Svi, Si, D 005.