Optimal Data Transmission and Channel Code Rate Allocation in Multi-path Wireless Networks

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1 Optmal Data Transmsson and Channl Cod Rat Allocaton n Mult-path Wrlss Ntwors Kvan Ronas, Amr-Hamd Mohsnan-Rad,VncntW.S.Wong, Sathsh Gopalarshnan, and Robrt Schobr Dpartmnt of Elctrcal and Computr Engnrng Th Unvrsty of Brtsh Columba, Vancouvr, Canada Txas Tch Unvrsty, Lubboc, TX E-mal: {vanr, vncntw, sathsh, rschobr}@c.ubc.ca, hamd.mohsnan-rad@ttu.du Abstract Wrlss lns ar oftn unrlabl and pron to transmsson rror du to varyng channl condtons. Ths can dgrad th prformanc n wrlss ntwors, partcularly for applcatons wth tght qualty-of-srvc rqurmnts. A common rmdy s to us channl codng whr th transmttr nod adds rdundant bts to th transmttd pacts n ordr to rduc th rror probablty at th rcvr. Howvr, ths prln soluton can comproms th ffctv ln data rat, ladng to undsrd nd-to-nd prformanc. In ths papr, w show that ths lattr shortcomng can b mtgatd f th nd-to-nd transmsson rats and channl cod rats ar slctd proprly ovr multpl routng paths. W formulat th jont channl codng and nd-to-nd data rat allocaton problm n multpath wrlss ntwors as a ntwor throughput maxmzaton problm, whch s non-convx. W tacl th non-convxty by usng functon approxmaton and tratv tchnqus from sgnomal programmng. Smulaton rsults confrm that by usng channl codng jontly wth mult-path routng, th nd-to-nd ntwor prformanc can b mprovd sgnfcantly. I. INTRODUCTION Rcnt advancs and tchnologcal dvlopmnts n communcaton, dgtal lctroncs, and rado frquncy systms hav placd wrlss ntwors at th forfront of today s data transmsson systms. Wrlss lns can, howvr, b unrlabl and pron to transmsson rrors du to bacground nos, nvronmntal obstacls, wathr condtons, and usr moblty. Unrlabl lns can dgrad ntwor prformanc partcularly for applcatons wth tght qualty-of-srvc rqurmnts such as voc-ovr-ip and vdo stramng. It s, thrfor, crucal to dvlop ffcnt stratgs n ordr to mprov th rlablty of data transmsson n wrlss ntwors [1]. Communcaton rlablty n a wrlss ntwor can b mprovd usng svral mthods ncludng channl codng [2], ntwor codng [3] [5], data rat allocaton [2], [6], and multpath routng [7] [9]. W propos jont applcaton of channl codng and mult-path routng to ralz gratr rlablty whn compard wth usng thr channl codng or multpath routng on an ndvdual bass. Th cntral contrbuton of our wor ls n surmountng som of th challngs posd by th optmal rsourc allocaton problm n ths sttng. Channl codng s commonly usd as a tool to lvrag rlabl transmssons ovr lossy wrlss lns. Wth channl codng, th transmttr nod of ach ln ncods th transmttd pacts by addng auxlary or rdundant bts. Addng th xtra bts ncrass th dstanc among th codwords and dcrass th pact rror probablty. If th numbr of xtra bts s th sam across all wrlss lns, thn th channl codng s non-adaptv. On th othr hand, f w chang th amount of rdundant bts for ach ln basd on ts currnt channl stat, thn th channl codng s adaptv. In gnral, channl codng ntroducs a trad-off btwn data transmsson rlablty and data transmsson rat [2], [9] [12]. In fact, by changng th cod rat,.., th rato of th data bts compard to data plus rdundant bts, w can chang th data rat at whch th nformaton s transmttd ovr ach wrlss ln. In partcular, th cod rat can b dcrasd n ordr to mprov rduc th probablty of rror at th cost of havng lowr data rats. Smlarly, w can ncras th cod rat to ncras th transmsson data rat, but at th cost of ncrasng th probablty of rror. Through ths artcl, w answr th followng quston: How should w slct nd-tond data transmsson rats and pr-ln channl cod rats n ordr to achv th optmal rat-rlablty trad-off n multpath wrlss ntwors? Adaptv channl codng, wthout mult-path routng, has bn usd by L, Chang and Caldrban [2] to nhanc th communcaton rlablty. W ntroduc two y mprovmnts to th modl consdrd by L, Chang and Caldrban. Frst, L, Chang and Caldrban assumd that th lns n th ntwor ar thr wrd or ntrfrnc-fr wrlss. On th contrary, w fully ncorporat th mpact of wrlss ntrfrnc. Scond, w consdr th cas whr thr ar multpl nd-to-nd routng paths avalabl across th ntwor. Clarly, ths ncluds sngl-path routng as a spcal cas. Ths two modfcatons rqur non-trval tchnqus for optmal rsourc allocaton. In summary, th contrbutons w ma hr ar: W formulat ntwor aggrgat throughput maxmzaton n mult-path wrlss ntwors wth channl codng Scton II. Both adaptv and non-adaptv channl codng ar bng consdrd. W tacl th non-convxty of th formulatd optmzaton problm n two stps Scton IV. Frst, w ntroduc crtan functon approxmatons to rformulat

2 th problm as a sgnomal programmng problm. Nxt, w dvlop an tratv algorthm to solv th rsultng sgnomal programmng problm by solvng a chan of tractabl gomtrc programmng problms. Although our proposd algorthm s cntralzd, t shds lght on th xtnt to whch pr-ln channl codng can mprov nd-to-nd prformanc n wrlss ntwor wth multpath routng. It can also b usd as a bnchmar for prformanc comparsons wth othr hurstcs. Furthr, w ntroduc a non-adaptv channl codng schm wth lowr complxty as a sub-optmal practcal soluton. W nvstgat th convrgnc proprts of th proposd algorthm as wll as ts ffcncy. Th lattr s partcularly studd by valuatng th mpact of th approxmatons mad n th drvaton of th algorthm. In fact, w s that n spt of th approxmatons, th algorthm convrgs to th optmal soluton n most cass Scton V. Fnally, w show th advantags of an adaptv codng schm usng our dsgn ovr a non-adaptv codng schm wth lowr computatonal complxty Scton V. Through smulatons, w nvstgat th tmcomplxty tradoff btwn th adaptv and nonadaptv schms. II. SYSTEM MODEL Consdr an ad-hoc wrlss ntwor. W can modl th ntwor topology as a drctd graph GV, E, whr V = {1, 2,...,V} s th st of wrlss nods and E s th st of wrlss lns. Lt I = {1, 2,..., I} dnot th st of all uncast sssons n th ntwor. For ach ssson I, th sourc and dstnaton nods ar dnotd by s and t, rspctvly. Furthrmor, w dnot K = {1, 2,...,K } as th st of all avalabl routng paths from sourc nod s to dstnaton nod t across th ntwor. For ach ssson I, ach ln E, and ach K,whav =1f ln blongs to th th routng path for ssson, and = othrws. For ach ssson I,ltα dnot th data rat of sourc s on ts th routng path, K. Th aggrgat transmsson rat for ssson s obtand as K α. W dfn R as th cod rat of wrlss ln E,.., th rato of th data bts to data plus rdundant bts. Notc that f no channl codng s prformd, thn R =1as thr wll b no rdundant bts n th transmttd pact. Gvn th sourc transmsson rats α =α, I, K and th ln cod rats R =R, E, w can modl th aggrgat traffc load on wrlss ln E as u = 1 R α. 1 I K From 1, th smallr th cod rat R, th mor rdundant data s addd to th transmttd pacts on ln E ladng to mor rlabl transmsson.., transmsson wth lowr rror probablty. Howvr, ths wll b at th cost of ncrasng th pact transmsson traffc load on th ln. Lt R 1 dnot th cut-off rat on wrlss ln E. W assum that th rat R of th adoptd codng schms.g., convolutonal cods s lmtd by th cutoff rat [13]. Gvn cod rat R R, w can bound th pact rror probablty on wrlss ln as [13] P 2 T R R, 2 whr T s th codng bloc lngth. In gnral, th cut-off rat R dpnds on th sgnal-to-nos rato SNR and th modulaton schm bng usd. For xampl, for a bnary phas shft yng BPSK wavform [13], w hav R =1 log γ, 3 whr γ dnots th SNR at th rcvr nod of wrlss ln E. In partcular, w hav γ =Γ d σ f 2, E, 4 whr Γ dpnds only on th transmsson and nos powrs, d s th dstanc btwn th transmttr and rcvr nods of wrlss ln, σ s th path loss xponnt.g., btwn 2 and 5, and f s th small-scal fadng gan. Gvn th pr-ln falur modl n 2, for ach ssson I, th probablty that a pact s transmttd succssfully along th th routng path, for K, s obtand as E, =1 1 P = E P. 5 From 5, for ach ssson I, th aggrgat rcvng rat at dstnaton nod t bcoms P. 6 α K E W can modl th mutual ntrfrnc among th wrlss lns n a ntwor by usng a contnton graph G C V C, E C.In th contnton graph G C, th st of vrtcs V C rprsnts th st of all wrlss lns E n th ntwor graph G. Thr xsts an dg btwn any two vrtcs n st V C f th wrlss lns corrspondng to th two vrtcs mutually ntrfr wth ach othr.., th rcvr nod of on ln s wthn th ntrfrnc rang of th sndr nod of th othr ln. Gvn th contnton graph, ach complt subgraph.., a subgraph n whch all vrtcs ar connctd to all othr vrtcs s calld a clqu. Amaxmal clqu s thn dfnd as a clqu whch s not a subgraph of any othr clqu [14]. W dnot th st of all maxmal clqus n contnton graph G C by Q C. Only on ln among all th lns corrspondng to th vrtcs of a maxmal clqu Q Q C can b actv at a tm. Lt c dnot th nomnal data rat of ln E. Th rato u c dnots th proporton of tm that ln Es actv whn t s bng usd at a data rat of c. It s rqurd that Q u c ν, Q Q C, whr ν, 1] s calld th clqu capacty. 7

3 III. JOINT OPTIMAL TRANSMISSION RATE AND CHANNEL CODE RATE ALLOCATION Consdrng 1, 2, 6, and 7, th rat-rlablty tradoff can b dscrbd as follows. For ach ln E,byncrasng th cod rat R w can rduc th traffc load on ach ln as w hav alrady shown n 1. Thus, hghr transmsson rats would b allowd wth th sam clqu capacty. On th othr hand, by dcrasng th cod rat R, w can rduc th rror probablty n 2 whch lads to hghr probablty of succssful transmsson along ach routng path as shown n 5. Thrfor, w may slct thr hghr transmsson rats, but wth mor pacts bng pron to rror, or lowr transmsson rats, but wth hghr prcntag of corrctly rcvd pacts. Th y quston to b answrd s: What transmsson rats α and cod rats R should b slctd to achv optmal prformanc? To answr th abov quston, w formulat th optmzaton problm as follows: maxmz α, R R I Q K α E P subjct to P 2 T R R, E, 1 R c I K α ν, Q Q, 8 whr R =R, E dnots th vctor of cut-off rats for all lns n th ntwor. Th objctv functon n 8 s th summaton of rcvng rats for all sssons, whr for ach ssson I, th rcvng rat s as n 6. By solvng 8, w can fnd α and R such that th total numbr of succssfully rcvd pacts across all sssons s maxmzd. IV. SOLUTION APPROACH In gnral, problm 8 s non-convx and can b dffcult to solv. Not that th non-convxts n problm 8 com from th followng thr sourcs: a Th product forms n th objctv functons. b Th xponntal forms n th nqualty constrants wth rspct to rror probablts. c Th fractonal forms n th nqualty constrants wth rspct to clqu capacts. In ths scton, w propos an tratv algorthm to solv th ntwor aggrgat throughput maxmzaton problm to achv optmal allocaton of sourc transmsson rats α as wll as optmal channl cod rats R n th ntwor. A. Problm Rformulaton to Tacl Non-convxts Consdr th thr sourcs of non-convxts for problm 8 whch w mntond arlr. Lt us frst consdr th ssu wth havng xponntal forms n th nqualty constrants n th rror probablty modls. By usng th Taylor srs xpanson, w can rwrt th rror probablty 2 as N L R n P X, E. 9 n= whr X = 2 TR, L = T log 2, and N s a postv ntgr wth N 1. Th modl n 9 s n polynomal form. Furthrmor, w notc that f th rror probablts ar rlatvly small whch s usually th cas n practc, w can approxmat th rcvng rat for ach ssson I as α K E P α 1 P. K E 1 Consdrng th worst-cas pact rror scnaro n 9 and by rplacng t, along wth 1 n problm 8, w can rwrt th ntwor aggrgat throughput maxmzaton problm as maxmz α,r subjct to I Q I K α K 1 E N X n= L R n α R 1 c 1 ν, Q Q α, R R. 11 Th objctv functon and constrants n problm 11 ar sgnomals,.., polynomals wth both postv and ngatv trms. Thrfor, w can apply sgnomal programmng tchnqus [15] to solv problm 11 as w xplan nxt. Lt t b an auxlary varabl. Problm 11 bcoms mnmz t, α, R subjct to t 1 t I Q I K α K 1 E α R 1 c 1 t>, α, R R. W can rwrt th frst constrant n 12 as N X n= L R n ν, Q Q 12 t + N α a X L R n α. I K E n= I K 13 Nxt, w follow th sgnomal programmng tchnqus ntroducd n [15] to approxmat th polynomal on th rght-hand sd of 13, whch s a functon of only α, asamonomal,.., a polynomal wth only on trm and postv multplr. Ths approxmaton can b prformd around som ntal pont ˆα. For a paramtr f s > 1, whch s clos to 1, whav I I, K α ˆα K α I K ˆα ˆα / I, K ˆα, α [ ˆα/f s,f s ˆα], 14 whr [ ˆα/f s,f s ˆα] s a small nghborhood around ntal pont ˆα. For notatonal smplcty, w dfn a nw trm ˆΛ, whch only dpnds on th ntal pont ˆα, as ˆΛ 1 = ˆα. 15 I K From 14 and 15, nqualty 13 can b approxmatd

4 around th ntal pont ˆα as ˆΛ t + N I K E n= α ˆα I K α a ˆα ˆΛ 1. X L R n 16 Th abov constrant s a posynomal,.., a polynomal wth only postv trms. Rplacng 16 n 12, th ntwor throughput maxmzaton problm bcoms mnmz t, α, R subjct to t 1 ˆΛ t + N I K E n= α ˆα ˆΛ ˆα 1 I K Q I K α R 1 α a c 1 X L R n ν, Q Q t>, ˆα/f s α f s ˆα, R R. 17 Th abov optmzaton problm s a standard gomtrc program, whch can b asly convrtd nto a convx problm cf. [15], [16]. Thrfor, problm 17 can b solvd by th ntror pont mthod [17]. Thus, w can solv th sgnomal programmng problm 12 by tratvly solvng 17. W ar now rady to ntroduc Algorthm 1 for solvng problm 8. Algorthm 1 starts by ntalzng varous systm paramtrs. Th ntal nd-to-nd transmsson rats ˆα ar slctd such that optmzaton problm 17 s fasbl. Svral tratons ar prformd, whr n ach traton, w solv th gomtrc programmng problm 17 n Ln 5 by usng th ntror pont mthod [17]. Gvn th optmal transmsson rats α opt n ach traton, w updat paramtr ˆΛ accordng to 15 and corrspondngly rformulat optmzaton problm 17 to b solvd agan n th nxt traton. Th tratons contnu untl th changs n optmal objctv valu t opt s suffcntly small compard to th prvous traton. Algorthm 1 : Algorthm to solv optmzaton problm 8. 1: Intalz f s, T, N, R, X, L, c, ν, and ˆα for ach E, I, and K. 2: St t opt := ; ɛ := : rpat 4: t old := t opt. 5: Solv problm 17 to obtan α opt, R opt, and t opt. 6: Updat ˆΛ as n 15, gvn ˆα := α opt. 7: untl t opt t old ɛ. 8: Optmal nd-to-nd data rats := α opt ; Optmal pr-ln cod rats := R opt. Th convrgnc of th algorthm n ach traton s guarantd snc th ntror pont mthod s usd [18]. Accordng to [15], although thr s no proof that ths algorthm always convrgs to th optmal soluton of 17, t s obsrvd Fg. 1. A sampl ntwor wth 2 nods randomly locatd n a 5 5 grd and wth fv sssons: 1 16, 3 13, 2 8, 14 17, and6 2. Thr ar 4, 2, 2, 1, and 3 routng paths avalabl for ths sssons, rspctvly. through smulatons that Algorthm 1 always convrgs to th optmal soluton aftr a numbr of tratons. B. Non-adaptv Channl Codng In ths subscton, w smplfy th systm modl n Scton IV-A and assum that th channl cod rat s fxd and s no longr an optmzaton varabl n our dsgn. That s, R = R, E. 18 Th mpact of such an assumpton s two-fold. Frst, t smplfs th clqu capacty constrants n problm 8 as for ach maxmal clqu Q Q,whav Q 1 R c I K α = 1 a R Q I K c α Ths mpls Q I K ν 19 a α Rν, 2 c whch s smply a lnar nqualty constrant. Scond, snc w ar addng xtra qualty constrants to problm 8, any soluton w achv s a sub-optmal soluton for problm 8. V. PERFORMANCE EVALUATION In ths scton, w assss th prformanc of our proposd jont channl codng and transmsson data rat allocaton algorthm Algorthm 1. In our smulaton modl, ach ntwor topology s an m m squar grd wth V = V = mm 1 wrlss nods postond n randomly slctd grd locatons. As an xampl, for th ntwor n Fg. 1, w hav m =5and V =2. Th ntwor ncluds m sourc and dstnaton pars, wth potntally many avalabl routng paths from th sourc nod to th dstnaton nod. Unlss statd othrws, th rst of th systm paramtrs ar slctd as follows: T =1, N = 15, f s = 1.1, R = 1, ν = 2 3 [19]. Wthout loss of gnralty, w choos th ln capacty, c for ach ln E, to b qual to 1. Thrfor, th transmsson data rats, α, obtand n th optmal pont can b ntrprtd as th vctor of normalzd transmsson rats. W also st th ntal

5 Normalzd Aggrgat Ntwor Throughput Itratons Fg. 2. Convrgnc of Algorthm 1 wth rspct to solvng problm 8..8 Normalzd Aggrgat Throughput Among All Sssons Fg Adaptv channl codng Non adaptv channl codng Channl Cod Rat R Comparson btwn adaptv vs non-adaptv channl codng..7 1 Optmal Non adaptv Fxd Cod Rats Optmal Adaptv Cod Rats for Each Ln Optmalty Approxmaton Error.5.3 Cod Rat on Ln, R Dsgn Paramtr N Fg. 3. Th mpact of paramtr N n approxmaton 9. Th optmalty rror dcrass as N ncrass. It bcoms almost zro for N > 1. data rats to b small,.., ˆα =.1 for all Iand any =1,...,K, n ordr to guarant a fasbl startng pont for Algorthm 1, as w alrady dscussd n Scton IV-A. Algorthm 1 s tratv and ach traton ncluds a functon approxmaton stp and a gomtrc programmng stp. Consdrng th topology n Fg. 1, th convrgnc of th objctv valu for problm 8, whn Algorthm 1 s usd, s shown n Fg. 2. Rcall that th objctv valu for problm 8 s th ntwor aggrgat throughput. From th rsults n Fg. 2, Algorthm 1 convrgs aftr around 3 tratons. Smlar rsults can b obtand for othr ntwor topologs. Rcall from Scton IV that w us approxmaton 9 n ordr to convrt optmzaton problm 8 nto a tractabl gomtrc programmng problm as n 11. W can mprov th accuracy of th approxmaton n 9 by ncrasng th valu of N. Howvr, ths would b at th cost of mang optmzaton problm 11 mor complcatd. Thrfor, w ar ntrstd n choosng N to obtan a rasonabl accuracy wth low computatonal complxty. Consdrng 5 random topologs, th smulaton rsults, whn N vars from 1 to 2, ar shown n Fg. 3, whr ach pont ndcats th avrag optmalty rror obsrvd among all topologs. W dfn th optmalty rror as th dffrnc btwn th achvd throughput at a partcular choc of N and that at suffcntly Wrlss Ln Indx Fg. 5. Optmal adaptv vrsus non-adaptv cod rat dstrbuton among all lns for th ntwor n Fg. 1. larg choc of N,.., N =2. Rsults n Fg. 3 show that th optmalty rror approachs zro whn N s 1 or hghr. W now prsnt how choosng th cod rat for ach ln ndvdually.., adaptv channl codng can lad to dffrnt optmalty and computatonal complxty rsults, compard to th cas whn channl codng s non-adaptv. Rcall from Scton IV-B that n a non-adaptv channl codng scnaro, all wrlss lns adopt th sam cod rat R as xprssd n 18. In ths cas, for ach fxd R, optmzaton problm 8 bcoms a lnar program. As w xpland n Scton IV-B, ths can sgnfcantly rduc th computatonal complxty, but rsults n sub-optmal dsgn solutons. Consdr th ntwor n Fg. 1. Th corrspondng smulaton rsults ar shown n Fg. 4. Hr, w xamn varous chocs of th non-adaptv cod rat R wthn th fasbl rang [,R ]. W can s that by usng non-adaptv channl codng, th hghst throughput s achvd whn th cod rat on all lns s 5. At ths pont, w rach 89% of th optmal valu that s achvabl by usng adaptv channl codng. It s ntrstng to also loo at th dstrbuton of th optmal adaptv cod rats at all lns, compard to th optmal non-adaptv cod rat. In th adaptv channl codng cas, th optmal cod rats for varous lns can b sgnfcantly dffrnt. Th corrspondng

6 Normalzd Aggrgat Throughput Among All Sssons Cod Rats Ar Updatd For Each Snapshot Avrag Cod Rats Ar Not Updatd For Each Snapshot Avrag Channl Snapshot t Fg. 6. Prformanc n a fadng channl for 5 channl snapshots. W can s that t s mor ffcnt to updat th cod rats at ach snapshot compard to th cas that cod rats ar obtand onc for all snapshots. rsults ar shown n Fg. 5. Fnally, w study th mpact of fadng on th systm prformanc whn Algorthm 1 s usd. Rcall from Scton IV-A that w can ncorporat th mpact of fadng by sparatly solvng optmzaton problm 8 for ach wrlss channl ralzaton wth fadng gans f and corrspondng cut-off rats as n 3 and 4. In ths cas, Algorthm 1 s nvod vry tm nw channl masurmnt data bcoms avalabl. W rfr to ach channl masurmnt data as on channl snapshot. Smulaton rsults for th ntwor topology n Fg. 1 for 5 dffrnt channl snapshots ar shown n Fg. 6. In our smulaton modl, w gnrat th fadng gans at ach channl snapshot basd on a random ralzaton of th Raylgh fadng dstrbuton. For th rsults n Fg. 6, w compar th prformanc n two dsgn scnaros. Th frst dsgn s an adaptv channl codng schm basd on th avrag fadng nformaton. That s, solvng optmzaton problm 8 only onc by assumng that th fadng gans ta thr avrag valus wthn th Raylgh fadng dstrbuton. On th othr hand, n our scond dsgn, w solv problm 8 onc for ach channl snapshot. W can s that on avrag, th lattr cas sold ln can mprov th aggrgat throughput among all nd-to-nd sssons compard to th formr on dash ln. Th achvd prformanc mprovmnt s at th cost of a sgnfcantly hghr computatonal complxty du to th rqurmnt of solvng optmzaton problm 8 for ach snapshot, whch may not always b dsrd n practc. VI. CONCLUSION In ths papr, w consdrd th problm of usng pr-ln channl codng n wrlss ntwors wth mult-path routng. W focusd on pr-ln channl cod rat slcton and nd-tond transmsson data rat allocaton through ntwor throughput maxmzaton, whch rsults n a non-convx problm. W tacld th non-convxty by usng approprat functon approxmatons and tratv tchnqus from sgnomal programmng. Morovr, w studd dffrnt varatons of our proposd pr-ln channl cod rat slcton and nd-to-nd data rat allocaton algorthms to addrss both adaptv and non-adaptv channl codng and also th mpact of fadng. Smulaton rsults confrmd that by usng channl codng jontly wth mult-path routng, w can sgnfcantly mprov th nd-to-nd ntwor prformanc. W also showd through smulatons that as a sub-optmal approach wth lss complxty, non-adaptv channl codng achvs a hgh dgr of optmalty compard to adaptv channl codng. As for futur wor, w wll consdr th problm of fndng th optmal data transmsson and channl cod rats n a dstrbutd mannr. ACKNOWLEDGMENT Ths rsarch s supportd by th Natural Scncs and Engnrng Rsarch Councl NSERC of Canada. REFERENCES [1] A. Bljadd, A. S. Hafd, and A. Gndrau, Dsgn of wrlss msh ntwors: Expanson and rlablty studs, n Proc. of IEEE Globcom, Nw Orlans, LA, Dc. 28. [2] J. W. L, M. Chang, and A. R. 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