Enhancing Downlink Performance in Wireless Networks by Simultaneous Multiple Packet Transmission

Transcription

1 Enhning Downlink Prormn in Wirlss Ntworks y Simultnous Multipl Pkt Trnsmission Zhngho Zhng n Yunyun Yng Dprtmnt o Eltril n Computr Enginring, Stt Univrsity o Nw York, Stony Brook, NY 11794, USA Astrt In this ppr w onsir using simultnous Multipl Pkt Trnsmission (MPT) to improv th ownlink prormn o wirlss ntworks. With MPT, th snr n sn two omptil pkts simultnously to two istint rivrs n n oul th throughput in th il s. W ormliz th prolm o ining shul to sn out ur pkts in minimum tim s ining mimum mthing prolm in grph. Sin mimum mthing lgorithms r rltivly ompl n my not mt th timing rquirmnts o rl tim pplitions, w giv st pproimtion lgorithm tht is pl o ining mthing t lst 3/4 o th siz o mimum mthing in O( E ) tim whr E is th numr o gs in th grph. W lso giv nlytil ouns or mimum llowl rrivl rt whih msurs th spup o th ownlink tr nhn with MPT n our rsults show tht th mimum rrivl rt inrss signiintly vn with vry smll omptiility proility. W lso us n pproimt nlytil mol n simultions to stuy th vrg pkt ly n our rsults show tht pkt ly n grtly ru vn with vry smll omptiility proility. In Trms: Multipl pkt trnsmission, wirlss LAN, mthing, pproimtion lgorithm, mimum llowl rrivl rt, ly. 1 Introution Wirlss ss ntworks hv n mor n mor wily us in rnt yrs sin ompring to th wir ntworks, wirlss ntworks r sir to instll n us. Du to th trmnous prtil intrsts, muh rsrh ort hs n vot to wirlss ss ntworks n grt improvmnts hv n hiv in th physil lyr y opting nwr n str signl prossing thniqus, or mpl, th t rt in wirlss LAN (Lol Ar Ntwork) hs inrs rom 1Mps in th rly vrsion o to 54Mps in [8]. W hv not tht Th rsrh work ws support in prt y NSF grnt numrs CCR n ECS n ARO grnt numr W911NF Ass Point Figur 1. Multipl pkt trnsmission Th ss point n sn two pkts to two usrs simultnously. in ition to inrsing th point to point pity, nw signl prossing thniqus hv lso m othr novl trnsmission shms possil whih n grtly improv th prormn o wirlss ntworks. In this ppr, w s- tuy Multipl Pkt Trnsmission (MPT), with whih th snr n sn mor thn on pkts to istint usrs simultnously. Tritionlly, in wirlss ntworks, it is ssum tht on vi n sn to only on othr vi t tim. Howvr, this rstrition is no longr tru i th snr hs mor thn on ntnns. By prossing th t oring to th hnnl stt, th snr n mk th t or on usr ppr s zro t othr usrs suh tht it n sn istint pkts to istint usrs simultnously. W ll it Multipl Pkt Trnsmission (MPT) n will plin th tils o it in Stion 2. For now, w wnt to point out th prooun impt th MPT thniqu hs on wirlss LANs. A wirlss LAN is usully ompos o n Ass Point (AP) whih is onnt to th wir ntwork n svrl usrs whih ommunit with th AP through wirlss hnnls. In wirlss LANs, th most ommon typ o tri is th ownlink tri, i.., rom th AP to th usrs whn th usrs r rowsing th Intrnt n ownloing t. In toy s wirlss LAN, th AP n sn on pkt to on usr t tim. Howvr, i th AP hs two ntnns n i MPT is us, th AP n sn two pkts to two usrs whnvr possil, thus ouling th throughout o th ownlink in th il s. MPT is sil or th ownlink us it is not - iiult to quip th AP with two ntnns, in t, mny wirlss routrs toy hv two ntnns. Anothr vn /06/$ IEEE

2 tg o MPT whih mks it vry ommrilly ppling is tht lthough MPT ns nw hrwr t th snr, it os not n ny nw hrwr t th rivr. This mns tht to us MPT in wirlss LAN, w n simply rpl th ss point n upgr sotwr protools in th usr vis without hving to hng thir wirlss rs, n thus inurring minimum ost. In this ppr w stuy prolms rlt to MPT n provi our solutions. W ormliz th prolm o sning out ur pkts in minimum tim s ining mimum mthing in grph. Sin mimum mthing lgorithms r rltivly ompl n my not mt th sp o rl tim pplitions, w onsir using pproimtion lgorithms n prsnt n lgorithm tht ins mthing with siz t lst 3/4 o th siz o th mimum mthing in O( E ) tim whr E is th numr o gs in th grph. W thn stuy th prormn o wirlss LAN nhn with MPT n giv nlytil ouns or mimum llowl rrivl rt. W lso us n nlytil mol n simultions to stuy th vrg pkt ly. Enhning wirlss LANs with MPT rquirs th Mi Ass Lyr (MAC) to hv mor knowlg out th stts o th physil lyr n is thror orm o rosslyr sign. In rnt yrs ross-lyr sign in wirlss ntworks hs ttrt muh ttntion us o th grt nits in rking th lyr ounry. For mpl, [5, 6] onsir pkt shuling n trnsmission powr ontrol in ross-lyr wirlss ntworks. Howvr, to th st o our knowlg, pkt shuling in wirlss ntworks in th ontt o multipl pkt trnsmission hs not n stui or. [3, 4] hv onsir Multipl Pkt Rption (MPR) whih mns th rivr n riv mor thn on pkts rom istint usrs simultnously. MPR is quit irnt rom MPT sin MPR is out riving multipl pkts t on no whil MPT is out sning multipl pkts rom on no to multipl nos. 2 Multipl Pkt Trnsmission In this stion w rily plin th MPT thniqu. As mntion rlir, to us MPT, th snr mks th t snt or on usr ppr s zro t othr usrs. This is possil i th snr hs mor thn on ntnns. With multipl ntnns, th snr n just th mplitu n phs o th trnsmitt signls on irnt ntnns suh tht th signls will up onstrutivly or strutivly s sir. Th ollowing is mor til plntion whih ollows Chptr 10 in [1]. Wirlss hnnls n mol s ompl sn hnnls, whih mns tht with on ntnn t th snr, th rivr will riv y = h, whr is th ompl t snt y th snr n h is th ompl hnnl oiint. Th rivr n rovr th t y iviing y y h. Not tht hr w onsir lt ing whih mns tht thr is no intr-symol-intrrn, n o not onsir nois so tht th or i o MPT n mor sily sn. I thr r two ntnns t th snr, th snr n sn two irnt symols not s 1 n 2 on ntnn 1 n ntnn 2, rsptivly. I thr r two rivrs, rivr 1 will riv y 1 = h h 12 2 n rivr 2 will riv y 2 = h h 22 2, whr h ij is th hnnl oiint rom ntnn j to usr i. For simpliity w will us h i to not [h i1,h i2 ] T n us to not [ 1, 2 ] T, n ll thm th hnnl oiint vtor n th trnsmitt vtor, rsptivly. In vtor orms, rivr i will riv y i = h i. Now lt 1 n 2 not th t tht shoul snt to rivr 1 n rivr 2, rsptivly. W nnot simply sn 1 vi ntnn 1 n 2 vi ntnn 2 us th t will mi up t th rivrs. Howvr, suppos thr r vtors u 1 =[u 11,u 12 ] T n u 2 =[u 21,u 22 ] T suh tht h 1u 2 =0n h 2u 1 =0. W n lt trnsmitt vtor = 1 u u 2, tht is, sn 1 u u 21 on ntnn 1 n 1 u u 22 on ntnn 2. Thus rivr 1 will riv h 1( 1 u u 2 )= 1 h 1u 1, n similrly rivr 2 will riv 2 h 2u 2, thus istint t is snt to h rivr. u 1 n ny vtor lis in V 1 whih is th sp orthogonl to h 2, howvr, to mimiz th riv signl strngth, u 1 shoul li in th sm irtion s th projtion o h 1 onto V 1. u 2 shoul similrly hosn. Sin th totl trnsmitt powr is limit, not ll pirs o rivrs r omptil, i.., n us MPT. Bsilly, th snr shoul hoos two rivrs i thir hnnl oiint vtors r lry nr orthogonl. To prorm MPT, th snr ns 4 mor ompl multiplirs. It lso ns to know th hnnl oiint vtors o th rivrs n run lgorithms to smrtly pir up th rivrs. Howvr, th rivrs ns no itionl hrwr n n riv th signl s i th snr is only sning to it. It is lso possil to sn to mor thn 2 rivrs t th sm tim i th snr hs mor thn 2 ntnns. In this ppr w ous on th mor prtil 2-ntnn s. Also not tht MPT rquirs wirlss hnnls to slow hnging s ompr to th t rt, whih is otn tru in wirlss LAN whr th wirlss vis r sttionry or most o th tim. 3 MAC Lyr Moiitions In this stion w sri th moiitions to MAC lyr protool, in prtiulr, , to support MPT. W sy two usrs U 1 n U 2 r omptil i thy n riv t th sm tim. I U 1 n U 2 r omptil, somtims w lso sy tht th pkts stin or U 1 n U 2 r omptil. Th AP kps th hnnl oiint vtors o ll nos tht hv n rport to it prviously. I, s on th pst

3 v 1 v 2 v 3 v 4 v v v v v v v v v 1 v 2 v 3 v 4 v 1 v 3 v 2 v 4 Figur 2. 4 pkts n irnt shuls. hnnl oiint vtors, U 1 n U 2 r omptil n thr r two pkts tht shoul snt to thm, th AP sns out RTS (Rquir To Sn) pkt, whih ontins, in ition to th tritionl RTS ontnts, it il initing tht th pkt out to sn is MPT pkt. I U 1 pprs rlir thn U 2 in th stintion il, upon riving th RTS pkt, U 1 will irst rply CTS (Clr To Sn) pkt ontining th tritionl CTS ontnts plus its ltst hnnl msurmnts. Atr short i mount o tim U 2 will lso rply CTS pkt. Atr riv th two CTS pkts, th AP will upt th hnnl oiint vtors. It will thn i whthr U 1 n U 2 r still omptil n most likly thy still r sin th nvironmnt is slow hnging. I in th rr s tht th hnnls hv hng signiintly suh tht thy r no longr omptil, th AP n hoos to sn to only on no. Thror, or sning th t pkts, th AP irst sns 2 its in whih it i is 1 mns th pkt or U i will snt or 1 i 2. Atr th t pkt is snt, U 1 n U 2 n rply n knowlgmnt pkt in turn. In this ppr w onsir mthing usr pkts o th sm siz. For simpliity, w onsir th s whn th it rts r lso th sm, suh tht ll pkts n th sm trnsmission tim whih w ll tim slot. W o not mk ny ssumption out th omptiilitis o usrs n trt thm s ritrry. 4 Shuling Algorithms or Ass Points Whil th i o MPT is simpl, th AP will nountr th prolm o how to mth th pkts with h othr to sn thm out s st s possil. For mpl, suppos in th ur o th AP thr r 4 pkts stin or 4 usrs not s v 1, v 2, v 3 n v 4, rsptivly. Assum pkt v i is omptil with v i+1 or 1 i 3, sshown t th top o Fig. 2 whr thr is n g twn two pkts i thy r omptil. I w mth v 2 with v 3,th 4 pkts hv to snt in 3 tim slots sin v 1 n v 4 r not omptil. Howvr, ttr hoi is to mth v 2 with v 1 n mth v 3 with v 4 n sn th 4 pkts in only 2 tim slots. Whn th numr o pkts grows th prolm o ining th st mthing strtgy will om hrr. In this stion w sri lgorithms tht solv this prolm. 4.1 Algorithm or Optiml Shul W ll shul y whih pkts n snt out in minimum tim n optiml shul. Clrly, in n optiml shul mimum numr o pkts r snt out in pirs, thror th prolm o ining n optiml shul is - quivlnt to ining mimum numr o omptil pirs mong th pkts. To solv this prolm, s shown in Fig. 2, w rw grph G whr h vrt rprsnts pkt n two vrtis r jnt i th two pkts r omptil. In grph, mthing M is in s st o vrt isjoint gs, tht is, no g in M hs ommon vrt with nothr g in M. Thror th prolm rus to ining mimum mthing in G. For mpl, th son mthing in Fig. 2 is mimum mthing whil th irst on is not. Mimum mthing in grph n oun in polynomil tim y lgorithms suh s th Emons Blossom Algorithm whih tks O(N 4 ) tim, whr N is th numr o vrtis in th grph [10, 2]. Bor ontinuing our isussion, w irst giv initions o som trms. Lt M mthing in grph G. W ll gs in M th mthing gs. I vrt is inint to n g in M, w sy it is sturt; othrwis, it is unsturt or r or singl. An M-ugmnting pth is in s pth with gs ltrnting twn gs in M n gs not in M, n with oth ns ing unsturt vrtis. For mpl, with rgring to th irst mthinginfig.2,v 1 v 2 v 3 v 4 is n ugmnting pth. It is wll known in grph thory tht th siz o mthing n inrmnt y on i n only i thr n oun n ugmnting pth. Th ur o th AP my stor mny pkts, s rsult, th grph n quit lrg. Howvr, th siz o th grph n ru y tking vntg o th t tht vrtis rprsnt pkts or th sm usr hv tly th sm st o nighors in th grph. Mor spiilly, in th grph, w sy vrt u n v long to th sm quivlnt group, or simply th sm group, i th pkts thy rprsnt r or th sm usr. Vrtis tht long to th sm group hv th sm nighors n r not jnt to h othr. Lt A = { 1, 2, 3 } n B = { 1, 2, 3 } two groups o vrtis n suppos i is mth to i or 1 i 3. Whv Lmm 1 I thr is n ugmnting pth trvrsing ll 3 mthing gs twn A n B, thr must ist n ugmnting pth trvrsing only 1 mthing g twn A n B. Proo. This n st plin with th hlp o Fig. 3, whr gs in th mthing r shown s hvy lins n gs not in th mthing r shown s sh lins. As in th igur, suppos n ugmnting pth trvrsing ll thr mthing gs twn A n B is y. Howvr, i is jnt

4 Figur 3. shortut ists. Th mthing gs r shown s hvy lins n th non-mthing gs r shown s sh lins. to 1, it must lso jnt to 3 sin 1 n 3 long to th sm group, thus thr is shortr ugmnting pth trvrsing only to th lst mthing g twn A n B whih is 3 3 y. (Not tht th sm proo lso hols i in th ugmnting pth, th sgmnt twn, sy, 1 n 2, is longr.) As rsult o this lmm, i thr ists n ugmnting pth, thr must lso ist n ugmnting pth trvrsing no mor thn two mthing gs twn ny two groups o vrtis. This is us i th pth trvrss mor thn two mthing gs twn two groups o vrtis, s w hv shown in th lmm, thr must shortut y whih w n only to trvrs th lst o th irst thr mthing gs, n w n kp on ining suh shortuts n ruing th numr o trvrs mthing gs until it is lss thn 3. Thror, or ny two groups o vrtis, only two mthing gs twn thm n to kpt n othr runnt mthing gs n rmov. Atr tht thr will O(n 2 ) sturt vrtis lt whr n is th numr o usrs. Also not tht or th purpos o ining ugmnting pths, only on o th unsturt vrtis longing to h group ns to onsir. Thror, th grph w work on ontins O(n 2 ) numr o vrtis whih os not pn on th siz o th ur. 4.2 Prtil Consirtions Although th optiml shul n oun or givn st o pkts y th mimum mthing lgorithm, in prti, th pkts o not rriv ll t on ut rriv on y on. It is not sil to run th mimum mthing lgorithm vry tim nw pkt rrivs u to th rltivly high omplity o th lgorithm. Thror, tr nw pkt rrivs, w n mth it oring to th ollowing simpl strtgy: A nw vrt is mth i n only i it n in n unsturt nighor. In this wy w lwys mintin miml mthing, whr mthing M is miml in G i no g not longing to M is vrt isjoint with ll gs in M. For mpl, th two mthings in Fig. 2 r ll miml mthings. Th mimum mthing lgorithm n ll only on whil to ugmnt th isting miml mthing. Anothr prolm is tht th pkts o not sty in th ur orvr n must snt out. W will hv to mk th isions o whih pkt(s) shoul snt out on y th AP hs gin ss to th mi n thr is lit tro twn throughput n ly. To improv th throughput, w shoul lwys sn out pkts in pirs; howvr, this poliy vors th pkts tht n mth ovr th pkts tht nnot mth, n will inrs th ly o th lttr. To prvnt ssiv ly o th singl pkts, in prti, w n kp tim stmp or h pkt n i th pkt hs sty in th ur or tim longr thn thrshol, it will snt out th nt tim th AP hs gin ss to th mi. I thr r multipl suh pkts, th AP n hoos on rnomly. Th thrshol n trmin ptivly s on th msur lys o th pkts tht wr snt out in pirs. Finlly, lthough mimum mthing n oun in polynomil tim, mimum mthing lgorithms r in gnrl ompl [11] n my not mt th timing rquirmnts o rl tim pplitions, onsiring tht th prossors in th AP r usully hp n not powrul. Thror in som ss, st pproimtion lgorithm whih is pl o ining irly goo mthing my usul, whih will isuss nt. 4.3 A Linr Tim 3/4 Approimtion Algorithm or Fining Mimum Mthing Th simplst n most wll known pproimtion lgorithm or mimum mthing simply rturns miml mthing. It is known tht this simpl lgorithm hs O( E ) tim omplity whr E is th numr o gs in th grph n hs prormn rtio o 1/2, whih mns tht th mthing it ins hs siz t lst hl o M whr M nots th mimum mthing. In this stion w giv simpl O( E ) pproimtion lgorithm or mimum mthing with n improv rtio o 3/4. To th st o our knowlg it is th irst linr tim pproimtion lgorithm or mimum mthing with 3/4 rtio. Th i o our lgorithm is to limint ll ugmnting pths o lngth no mor thn 5. Not tht ny M- ugmnting pth must hv i gs in M n i+1 gs not in M or som intgr i 0. Thror i th shortst M- M M ugmnting pth hs lngth t lst 7, > 3/4, sin to inrmnt th siz o th mthing y on, th tr rtio is t lst 3/4, i.., th st w n o is to tk out 3 gs in M n in 4 gs not in M. A miml mthing os not hv ugmnting pths o lngth 1, whih is why th siz o miml mthing is t lst hl o th siz o mimum mthing. In our lgorithm, w will strt with miml mthing n thn limint ugmnting pths o lngth 3 n thn o lngth 5. As n sn, th lgorithms thmslvs r simpl n strightorwr. Howvr, it is intrsting n somwht surprising tht thy n implmnt to run in linr tim.

5 Tl 1. Fining Augmnting Pths o Lngth 3 Initilly, M = S whr S is miml mthing. or i =1to S o Lt (u, v) th i th g in S. Chk i u n v hv istint unsturt nighors. i ys Lt th nighors o u n v n y, rsptivly. M M {(, u), (v, y)}\{(u, v)}. n i n or Eliminting Augmnting Pths o Lngth 3 W strt with miml mthing not y S n th output o our lgorithm is not y M. For h vrt, list is us to stor its nighors. An rry is us to stor th mthing, tht is, th i th lmnt in th rry is th vrt mth to th i th vrt. Not tht with this rry, it tks onstnt tim to ugmnt th mthing with i lngth ugmnting pths or to hk whthr prtiulr vrt is sturt or not. Th lgorithm is summriz in Tl 1. Initilly, lt M = S. W will hk gs in S rom th irst to th lst to ugmnt M. Whn hking g (u, v), w hk whthr oth u n v r jnt to som istint unsturt vrtis. I thr r suh vrtis, sy, u is jnt to n v is jnt to y, thr is n M-ugmnting pth o lngth 3 involving (u, v) whih is u v y. W n - limint this ugmnting pth n ugmnt M y rmoving (u, v) rom M n ing (u, ) n (v, y) to M. W ll (u, ) n (v, y) th nw mthing gs. Th lgorithm trmints whn ll gs in S hv n hk this wy. Nt w prov th orrtnss n riv th omplity o this lgorithm. It is importnt to not tht i th mthing is lwys ugmnt oring to ugmnting pths, th ollowing two ts lwys hol. First, ll sturt vrtis will rmin sturt tr h ugmnttion. Son, s rsult o th irst t, i vrt is unsturt tr n ugmnttion, it must unsturt or th ugmnttion n thror throughout th pross M rmins miml mthing. Lmm 2 Th nw mthing gs n not to hk us thr nnot ugmnting pths o lngth 3 involving thm. Proo. To s this, suppos u v y is lngth-3 ugmnting pth, s shown in Fig. 4. I thr is lngth-3 ugmnting pth involving on o th nw mthing gs, sy, (, u), lt it s u t, s shown in th right prt o Fig. 4. Not tht s n r not sturt or th mthing is ugmnt oring to u v y, whih ontrits th t tht th mthing is lwys miml. Th lt prt o Fig. 4 shows this sitution. u v s t y y Figur 4. Augmnting M oring to u v y. s n r oth unsturt whih ontrits th t tht M is miml mthing. Corollry 1 Whn th lgorithm trmints, thr is no lngth-3 ugmnting pth. Proo. By ontrition. I thr is still lngth-3 ugmnting pth, lt it u v y whr u n v r sturt. By Lmm 2, (u, v) nnot nw mthing g, thror it is in S. But this nnot hppn sin i suh n ugmnting pth ists tr th lgorithm trmints, it must lso ist whn (u, v) ws hk n shoul hv n oun. Lmm 3 Th lgorithm runs in O( E ) tim whr E is th numr o gs. Proo. Not tht whn hking g (u, v), th gs inint to u n v wr hk t most on. Sin th gs in S r vrt isjoint, th lgorithm hks n g in G no mor thn twi. Comining ths isussions, w onlu tht Thorm 1 Th lgorithm in Tl 1 limints ll lngth- 3 ugmnting pths in O( E ) tim Eliminting Augmnting Pths o Lngth 5 Atr liminting ugmnting pths o lngth 3, w srh or ugmnting pths o lngth 5. W irst hk ll gs in th urrnt mthing to onstrut st T. A vrt v is to st T i v is mth to som vrt u n u is jnt to t lst on unsturt vrt. W ll v n outr vrt n u n innr vrt. Not tht v n oth n outr vrt n n innr vrt whn v n u r oth jnt to th sm unsturt vrt n r not jnt to ny othr unsturt vrtis. Clrly, to in ugmnting pths o lngth 5 is to in jnt outr vrtis. Also not tht T n onstrut in O( E ) tim. Th lgorithm is summriz in Tl 2 n works s ollows. W hk th vrtis in T rom th irst to th lst. Whn hking vrt v, ltu th innr vrt mth to v. W irst gt or upt l(u) whih is th list o unsturt nighors o u: Il(u) hs not n stlish rlir, w srh th nighor list o u to gt l(u); othrwis, w hk th vrt in l(u) (in this s, thr n only on vrt in l(u), or rsons to sn shortly) n rmov it u v s t

6 rom l(u) i it hs n mth. Atr gtting l(u), i l(u) is mpty, w quit hking v, rmov v rom T n go on to th nt vrt in T. Othrwis, w hk th nighors o v to in n outr vrt. I n outr vrt w is oun to jnt to v, ltz th innr vrt mth to w. W gt l(z) whih is th unsturt nighor list o z in th sm wy s or u. Il(z) is mpty, w rmov w rom T n go on to th nt nighor o v. Othrwis, w hk i thr is n ugmnting pth o lngth 5 involving (u, v) n (w, z), n not tht this n in onstnt tim. This is us (1) i l(z) ontins t lst 2 vrtis, thr must suh pth; (2) i l(z) ontins tly 1 vrt, thr is suh pth i n only i l(u) is irnt rom l(z). I n ugmnting pth is oun, w ugmnt M oring to this pth n rmov oth v n w rom T ; othrwis, w ontinu to hk th nt outr vrt nighor o v. Ill nighors o v hv n hk n no ugmnting pth is oun, w rmov v rom T n ontinu to th nt vrt in T. Now w n s why i n outr vrt is still in T tr it hs n hk, th unsturt nighor list o th innr vrt mth to it must ontin tly on vrt. This is us i it ontins mor thn 1 vrtis, n ugmnting pth must hv n oun whn hking this outr vrt n it woul hv n rmov rom T. Th lgorithm trmints whn T is mpty. Not tht this lgorithm mks sur tht it will in n ugmnting pth o lngth 5 involving (u, v) i suh pth ists whn hking outr vrt v. Also not tht rmoving n lmnt in st is quivlnt to mrking this lmnt whih tks onstnt tim. Rll tht i M is ugmnt y ugmnting pths, M rmins to miml mthing whih mns tht it os not hv ny ugmnting pth o lngth 1. Th nt lmm shows tht y ugmnting M y lngth-5 ugmnting pths, thr will nvr nw ugmnting pths o lngth 3. Lmm 4 Throughout th ution o th lgorithm no ugmnting pths o lngth 3 will rt. Proo. By ontrition. Sin M os not hv lngth-3 ugmnting pths t th ginning, suppos th irst suh pth ws rt tr ugmnting M with lngth-5 ugmnting pth. Th lngth-3 ugmnting pth must involv on o th nw mthing gs whih r (, ), (, ) n (, ). I(, ) rts n ugmnting pth o lngth 3, sy, u v, s shown in th right prt o Fig. 5(), thr must ist ugmnting pth u whih hs lngth 3, s shown in th lt prt o Fig. 5(), whih ontrits th t tht thr is no suh pth or th mthing ws ugmnt. Thus (, ) nnot rt n ugmnting pth o lngth 3. I (, ) rts n ugmnting pth o lngth 3, sy, y, s shown in th right prt o Fig. 5(), n must oth unsturt or th mthing ws ugmnt, s shown in th lt prt o Tl 2. Fining Augmnting Pths o Lngth 5 Construt T, th st o outr vrtis. whil T is not mpty Lt v vrt in T tht hs not n hk. Suppos v is mth to u. Gt or upt l(u), th unsturt nighor list o u. i l(u) is mpty Rmov v rom T n ontinu to th nt outr vrt in T. n i whil not ll nighors o v hv n hk Lt w n outr vrt nighor o v n suppos w is mth to z. Gt or upt l(z), th unsturt nighor list o z. i l(z) is mpty Rmov w rom T n ontinu to th nt nighor o v. n i Bs on l(u) n l(z), trmin i thr is lngth-5 ugmnting pth. i n ugmnting pth is oun Augmnt M oring to this pth n rmov oth v n w rom T ; rk rom th innr whil loop. n i n whil i no ugmnting pth is oun Rmov v rom T n i n whil Fig. 5(), whih ontrits th t tht th mthing is lwys miml. Hn (, ) nnot rt ugmnting pth o lngth 3 n or th sm rson nithr n (, ). Lmm 5 Throughout th ution o th lgorithm th nw mthing gs nnot involv in ny ugmnting pth o lngth 5. Proo. Suppos uring th ution o th lgorithm, th irst tim tht nw mthing g oms involv in lngth-5 ugmnting pth is tr w ugmnt th mthing oring to ugmnting pth. W irst show tht (, ) nnot involv in ugmnting pth o lngth 5 y ontrition. I thr is suh n ugmnting u v u v () () Figur 5. No nw ugmnting pths o lngth 3 will rt. y y

7 () u v () Figur 6. No nw mthing g will involv in ugmnting pths o lngth 5. pth, thr must n unsturt vrt jnt to ithr or. Lt th unsturt vrt n without loss o gnrlity u to symmtry, suppos is jnt to, s shown in Fig. 6(). Thn thr is n ugmnting pth o lngth 3 or th lgorithm strt:, whih ontrits th t tht thr r no suh pths. W now show tht (, ) nnot involv in ugmnting pth o lngth 5. Th sm proo n us or (, ) u to symmtry. First not tht sin th mthing is miml, nnot jnt to n unsturt vrt. Thror th ugmnting pth must in th orm o u v w, s shown in Fig. 6(). W lim tht g (v, w) nnot n ol mthing g, i.., nnot in th mthing or th lgorithm strt. Sin i so, v w is n ugmnting mthing o lngth 3 or th lgorithm strt. Thror (v, w) is lso nw mthing g. Suppos g (v, w) ws irst to th mthing y ugmnting th mthing oring to lngth-5 ugmnting pth P. Sin no nw mthing g ws involv in ny lngth-5 ugmnting pth or th mthing ws ugmnt oring to, g (v, w) ws not involv in ny lngth-5 ugmnting pth pt P. Not tht u to th sm rson, sin (, ) nnot in ny lngth-5 ugmnting pth, (v, w) nnot th g in th ntr o P, n thus (v, w) must t th n o P. Sin w is jnt to n unsturt vrt, w nnot th n vrt o P, thus v must th n vrt o P. Howvr, this mns tht oth n v wr unsturt or th lgorithm strt, whih nnot hppn sin th mthing is miml. (Not tht th proo lso hols i (v, w) is (, ) or (,).) Corollry 2 Whn th lgorithm trmints, thr is no ugmnting pth o lngth 5. Proo. Suppos it is not tru, tht is, thr still ists n ugmnting pth o lngth 5, sy,. By Lmm 5, nithr (, ) nor (, ) is nw mthing g. Howvr, in this s th lgorithm must hv oun this ugmnting pth. Lmm 6 Th lgorithm runs in O( E ) tim. w Proo. Consir hking n outr vrt v in st T.Not tht ll tim n or hking v is O((v)) whr (v) is th gr o v pt or gtting th unsturt nighor lists or som innr vrtis. Thror ovrll th lgorithm runs in O( E ) tim plus th tim n or gtting th unsturt nighor lists or innr vrtis whih lso tks O( E ) tim sin it ns to on or h innr vrt no mor thn on. Thus, th lmm ollows. Comining th ov isussions, w onlu tht Thorm 2 Th lgorithm in Tl 2 limints ll lngth- 5 ugmnting pths in O( E ) tim. 5 Prormn Stuy In this stion w stuy th prormn o th wirlss LAN tr nhn y MPT. W irst riv th mimum rrivl rt o th ownlink n thn stuy th vrg pkt ly y n nlytil mol n simultions. Th prormn o wirlss ntwork pns on mny tors, or mpl, th physil nvironmnt, th lotions o th wirlss nos, t., suh tht th prormn o on ntwork oul irnt rom tht o nothr vn whn thy r using th sm vis. In mny ss th prormn o th sm ntwork my lso hnging u to th osionl movmnts o th wirlss nos. This mks th prormn vlution in gnrl iiult jo. Howvr, w not tht th prormn gin o opting MPT is minly trmin y th proility o two nos ing omptil, n this proility shoul th sm in ntworks unr similr nvironmnts n with sm vis. It is thus mor insightul to us th omptiility proility p s th prmtr or prormn vlution. For simpliity, w ssum tht th proility tht two usrs r omptil is inpnnt o othr usrs. 5.1 Mimum Arrivl Rt Th irst n th most importnt qustion is: Atr using MPT, how muh str os th ownlink om? This n msur y th mimum llowl rrivl rt, whr n rrivl rt is llowl i it os not us th ur o th AP to ovrlow. Mor spiilly, suppos on th AP hs got ss to th mi, on vrg it hs to wit T sons to l to gt ss to th mi gin. In th ollowing, or onvnin, w rr to T s tim slot. Th normliz rrivl rt λ is in s th vrg numr o pkts rriv in tim slot. Without MPT, lrly, λ m =1whr λ m nots th mimum llowl rrivl rt. Nt w riv th vlu o λ m whn MPT is us. Suppos thr r n usrs mong whih usrs r omptil with som othr usrs. Ths usrs r ll th non-isolt usrs n th rst r ll th isolt usrs. Consir W rriv pkts. Assuming pkts hv rnom stintions, thr will W (/n) pkts or

8 th non-isolt usrs n W (1 /n) pkts or th isolt usrs. Th stst wy to sn out ths W pkts is to lwys sn out th pkts or th non-isolt usrs in pirs, thus th minimum tim n to sn out ll th pkts is W (1 /2n) tim slots. In othr wors, W pkts shoul rriv in t lst W (1 /2n) tim slots. Thus, th mimum rrivl rt or givn n n is (1 /2n) 1. Th numr o non-isolt usrs is rnom vril. Lt P n (l) th proility tht out o n usrs, thr r l isolt usrs. Th vrg mimum rrivl rt is n λ m = P n (n )(1 /2n) 1 =0 Thror in th ollowing w ous on ining P n (l). Apprntly, whn n =1, P 1 (0) = 0 n P 1 (1) = 1; whn n =2, P 2 (0) = p, P 2 (1) = 0 n P 2 (2) = 1 p whr p is th omptiility proility. To in P n (l) or lrgr n, w onition on th numr o isolt usrs mong th irst n 1 usrs. Lt E,y th vnt tht in th usrs, y r isolt n lt L rnom vril noting th numr o isolt usrs mong n usrs. n 1 P n (l) =P (L = l) = P n 1 (i)p (L = l E n 1,i ) i=0 Clrly, or i<l 1, P n (L = l E n 1,i )=0, sin y ing usr, w n t most on isolt usr. For i = l 1, P n (L = l E n 1,l 1 )=(1 p) n 1, sin givn thr r l 1 isolt usrs mong th n 1 usrs, thr r l isolt usrs in th n usrs i n only i th n th usr is isolt, whih ours with proility (1 p) n 1.Fori = l, whv P n (L = l E n 1,l )=[1 (1 p) n 1 l ](1 p) l, sin i thr r lry l isolt usrs in th n 1 usrs, th n th usr must not isolt, i.., must omptil with som usr in th irst n 1 usrs. Howvr, it nnot omptil with ny o th isolt mong th n 1 usrs, sin this will ru th numr o isolt usrs, thus it must omptil with t lst on o th usrs mong th n 1 l non-isolt usrs, whih is n vnt tht ours with proility [1 (1 p) n 1 l ](1 p) l.fori>l, P n (L = l E n 1,i )= ( i i l ) p i l (1 p) l, sin i i>l, th ition o th n th usr rus th numr o isolt usrs y i l, thus it must omptil with tly i l prvious isolt usrs. Fig. 7 shows th mimum rrivl rt or ntworks o irnt sizs unr irnt omptiility proilitis. Mimum Arrivl Rt n=5 n=10 n=20 n= Comptiility Proility Figur 7. Mimum rrivl rt or ntworks o irnt sizs unr irnt omptiility proilitis. It is rmrkl to s tht signiint improvmnt n hiv vn with vry smll omptiility proility. For mpl, or n =10, whn p =0.04, th mimum rrivl rt is 1.2, whih is 20% inrs. Finlly, w wnt to rgu tht th mimum rrivl rt is pproimtly hivl, lthough it is t th ost o ssiv ly or th isolt usrs. Not tht s mntion rlir, th mimum rrivl rt is hiv i pkts stin or non-isolt usrs r lwys snt out in pirs n i no tim slot is wst, i.., thr is lwys t lst on pkt snt out in tim slot. Thror, i thr r omptil pkt pirs in th ur, w sn th pir; othrwis, w sn pkts stin or th isolt usrs n kp on oing so until nw pir hs orm tr som nw pkts hv rriv. Sin t high rrivl rt th quus or th isolt usrs r most likly quit long, it is highly likly tht w n wit until pir pprs or th quus or th isolt usrs r hust. 5.2 Avrg Pkt Dly As w hv sn, opting MPT n grtly inrs th mimum llowl rrivl rt. Not tht MPT n lso ru th quuing ly o th pkts ompring to Singl Pkt Trnsmission (SPT). In this stion w us n nlytil mol long with simultions to s how pkt ly n ru An Approimtion Anlytil Mol W irst sri our nlytil mol. Th mol is vlop or th purpos o ompring MPT with SPT n thror only onsirs rrivl rts lss thn 1. W ssum tht th AP mintins n quus in its ur, on or h usr. Not tht to tly mol th hvior o th quus in th AP, mny MAC lyr rlt issus hv to onsir, or mpl, how otn n th AP gin ss to th mi n how mny pkts will rriv t th AP in givn tim prio, t. All suh issus r intrting with h othr whih mks t nlytil moling vry iiult. W thror us som pproimtions

9 to simpliy th mol. As th simultions show, our mol is vry urt whn λ<1. Th mol is s on Mrkov hins. W tk th totl numr o pkts stor in th ur or th AP hs gin ss to th mi, whih w will ltr rr to s th AP sning or onvnin, s th stt o th Mrkov hin. Th vntg o oing so is tht th Mrkov hin oms isrt-tim sin w r only looking t th ur t som isrt tim instnts. W will ssum tht twn two AP snings, th numr o pkts rriv t th AP ollows th wily us Poisson istriution, tht is, P (K = k) = λ λ k /k!, whr K is th rnom vril noting th numr o rriv pkts. It shoul not tht our mol is not limit to Poisson istriution n n lso us i th rrivl ollows othr istriutions. W lso ssum tht th rriv pkts hv rnom stintions. Not tht w hv voi pliitly ling with th ompl issu o how otn n th AP gin ss to th mi, us it hs n npsult in th ssumption o th rrivl istriution. Tht is, i th AP hs to wit longr to ss th mi, w n hoos lrg λ sin mor pkts n pt to rriv n othrwis w n hoos smll λ sin lss pkts n pt to rriv. To mor urtly mol th quus, w lso onsir whthr thr ists pir o omptil pkts in th ur. Thror in our mol, w us (, r) s th stt o th Mrkov hin, whr is th totl numr o pkts, n r =0mns tht thr is no omptil pir n r =1 othrwis. W ssum tht i thr ists omptil pir th AP will lwys sn it, sin whn th rrivl rt is s- mll, most likly th pkts will not ly longr thn th thrshol. Thus, th trnsition proility or (, 0) is (, 0) ( 1+k, 0) : P 1 P (k) whr P 1 is th proility tht givn thr is no omptil pir in th 1 pkts lt in th ur, thr is no omptil pir tr k nw pkts hv rriv, n, lrly, (, 0) ( 1+k, 1) : (1 P 1 )P (k). Similrly, th trnsition proility or (, 1) is (, 1) ( 2+k, 0) : P 2 P (k) whr P 2 is th proility tht givn thr ws omptil pir, tr sning out th pir n tr riving k nw pkts, thr is no omptil pir. Also, (, 1) ( 2+k, 1) : (1 P 2 )P (k). Thror in th ollowing w n only to ous on ining P 1 n P 2. Not tht sin w hv us only two rnom vrils to mol n quus, som inormtion is lost, n th mol is only n pproimtion mol in th sns tht th Mrkovin proprty only hols pproimtly. Howvr, this is nssry sin i th quus r onsir sprtly, th omplity o th mol will ponntil. To in P 1 n P 2, w will mk th ssumption tht th pkts stor in th ur hv rnom stintions. Not tht this is not tru sin th AP vors pkts stin non-isolt usrs, n s rsult, in th ur, thr will mor pkts stin or isolt usrs thn or th non-isolt usrs. Howvr, this ssumption mks th nlytil moling trtl n yils rmrkly urt rsults whn λ 1. Lt F th vnt tht mong pkts thr is no omptil pir. W hv P 1 = P (F 1+k F 1 )= P (F 1+k,F 1 ) P (F 1 ) = P (F 1+k) P (F 1 ) sin i thr is no omptil pir in th 1+k pkts, thr nnot omptil pir in ny sust o it, in prtiulr, th 1 pkts. P 2 is hrr to in thn P 1, sin w o not know tr sning out omptil pir, whthr thr is still omptil pir in th 2 pkts. W mk th ssumption tht thr is no omptil pir in th 2 pkts, sin whn λ is not lrg, th pkts n snt out rthr switly, n w n sly ssum tht on omptil pir is orm, it will immitly snt out. With this ssumption, similr to P 1,whv P 2 = P (F 2+k )/P (F 2 ). To in P (F ), w irst in P U (, s) whih is th proility tht knowing tht pkts r or s usrs, thr is t lst on pkt or h o th s usrs, i.., non o th quus or th s usrs is mpty. Clrly, P U (, 1) = 1 or ll n P U (2, 2) = 1/2. For lrgr n s, osrv tht i givn t pkts r or th irst usr, th vnt tht non o th s quus is mpty ours i n only i th rst t pkts mk th rst s 1 quus ll non-mpty, whih ours with proility P U ( t, s 1). Thus, w hv th rursiv rltion P U (, s) = s+1 t=1 ( P U ( t, s 1) t ) (1/s) t (1 1/s) t Lt P W (, s) th proility tht i thr r totlly pkts in th ur, thr r tly s non-mpty quus mong th totl n quus. W hv P W (, s) = ( n s ) (s/n) P U (, s), ( ) n sin thr r wys to hoos s quus rom n s quus n or ny givn s quus, this vnt ours i n only i th pkts r ll or th s usrs whih ours with proility (s/n) n i th pkts mk th s quus ll non-mpty whih ours with proility P U (, s). Atr otining P W (, s), P F () is simply

10 P (F )= n P W (, s)(1 p) s(s 1)/2, s=1 sin th proility tht thr is no omptil pir mong s usrs is (1 p) s(s 1)/ Anlytil n Simultion Rsults W lso onut simultions to vriy our nlytil mol. In our simultions, h point is otin y running on 100 rnom topologis n h topology is run or 100, 000 rouns. Fig. 8() shows th vrg pkt ly s untion o omptiility proility whn λ<1 or n =10otin y simultions n th nlytil mol. First w osrv tht th nlytil rsults r vry los to th simultion rsults. Son w osrv tht MPT grtly rus th vrg ly vn whn p is vry smll. W lso osrv tht th vrg ly rss str whn p is smllr n will tn to onvrg to vlu whn p urthr inrss. Fig. 8() shows th vrg pkt ly otin y simultions whn λ>1. Similrly, w n osrv tht inrsing p will lwys ru th ly. W hv us lrg p in Fig. 8() thn in Fig. 8() us whn λ > 1, smllp osionlly rsults in too sprt topologis whih uss th ur tht hs limit siz in our simultions to unstl. 6 Conlusions In this ppr w onsir using Multipl Pkt Trnsmission (MPT) to improv th ownlink prormn o th wirlss LANs. With MPT, th ss point n sn t- wo omptil pkts simultnously to two istint usrs. W hv ormliz th prolm o ining minimum tim shul s mthing prolm, n hv givn prtil linr tim lgorithm tht ins mthing t lst 3/4 th siz o mimum mthing. W stui th prormn o wirlss LAN tr nhn with MPT. W gv nlytil ouns or mimum llowl rrivl rt whih msurs th spup o th ownlink n our rsults show tht th mimum rrivl rt inrss signiintly vn with vry smll omptiility proility. W lso us n pproimt nlytil mol n simultions to stuy th vrg pkt ly n our rsults show tht pkt ly n grtly ru vn with vry smll omptiility proility. Rrns [1] D. Ts n P. Viswnth, Funmntls o wirlss ommunition, Cmrig Univrsity Prss, My [2] D.B. Wst, Introution to grph thory, Prnti-Hll, [3] T. Lng, V. Nwr n P. Vnkitsurmnim, Signl prossing in rnom ss, IEEE Signl Prossing Mgzin, vol.21, no.5, pp.29-39, Avrg Pkt Dly Avrg Pkt Dly n=10 λ=0.75, n λ=0.85, n λ=0.95, n λ=0.75, simu λ=0.85, simu λ=0.95, simu Comptiility Proility () λ<1 n=10 λ=1.05, simu λ=1.15, simu λ=1.25, simu Comptiility Proility () λ>1 Figur 8. Avrg pkt ly s untion o omptiility proility unr irnt rrivl rts whn thr r 10 usrs. n n simu stn or nlytil n simultion, rsptivly. [4] G. Dimi, N. D. Siiropoulos n R. Zhng; Mium ss ontrol - physil ross-lyr sign, IEEE Signl Prossing Mgzin, vol.21, no.5, pp.40-50, [5] Q. Liu, S. Zhou n G.B. Ginnkis, Cross-lyr shuling with prsri QoS gurnts in ptiv wirlss ntworks, JSAC, vol.23, no.5, pp , [6] V. Kwi n P.R. Kumr, Prinipls n protools or powr ontrol in wirlss ho ntworks, JSAC, vol.23, no.1, pp.76-88, [7] A Czygrinow, M. Hnkowik n E. Szymnsk, A st istriut lgorithm or pproimting th mimum mthing, Algorithms - ESA 2004, Ltur Nots in Computr Sin 3221, pp , [8] [9] W. Xing; T. Prtt n X. Wng; A sotwr rio tst or two-trnsmittr two-rivr sp-tim oing OFDM wirlss LAN, IEEE Communitions Mgzin, vol.42, no.6, pp.s20-s28, [10] J. Emons, Pths, trs, n lowrs, Cn. J. Mth, 17: , [11] H. N. Gow, An iint implmnttion o Emons lgorithm or mimum mthing on grphs, Journl o th ACM (JACM), vol.23, no.2, pp , [12] J. Mgun, Gry mthing lgorithms: An primntl stuy, Pro. 1st Workshop on Algorithm Enginring, pp.22-31, 1997.