Closing the Gap of Multicast Capacity for Hybrid Wireless Networks

Transcription

1 Closg the Gp of Multcst Cpcty fo Hybd Weless Netwos Xg-Yg L, Xufe Mo d Shoje Tg Abstct We study the ultcst cpcty of do hybd weless etwo cosstg of weless tels d bse sttos. Assue tht weless tels odes e doly deployed sque ego d ll odes hve the ufo tssso ge d ufo tefeece ge R = Θ; ech weless ode c tst/eceve t W -bps. I ddto, thee e bse sttos ethe souce odes o eceve odes tht e plced ufoly ths sque ego; ech bse stto c coucte wth djcet bse sttos dectly wth dt te W B-bps d the tssso te betwee bse stto d weless ode s W c-bps. Assue tht thee s set of doly selected odes tht wll seve s the souce odes of ultcst flows ech flow hs doly selected eceves. We foud tht the ultcst cpcty fo hybd etwos hs thee eges d fo ech of eges, we deve the tchg syptotc uppe d lowe bouds of ultcst cpcty. Idex Tes Hybd etwos, cpcty, ultcst, bodcst. I. INTRODUCTION The syptotc cpcty of lge scle do weless etwos hs bee wdely studed. It s well ow tht the cpcty of weless etwo depeds o y spects of the etwo, le etwo chtectue, outg stteges, powe costts, tefeeces d ode desty, etc. A good udestdg of the cpcty of dffeet etwos wll help the uses to use cuet etwo esouces oe effectvely wth espect to dffeet evoet d codtos, especlly fo stutos le bttlefelds, dgeous volco es. I pue weless d hoc etwos, weless odes y coopete outg ech othes pcets. Howeve, lc of cetlzed cotol of the fuctolty d possble ode oblty gve se to y chllegg ssues t the etwo lye, the edu ccess lye, d physcl lye of weless d hoc etwo. Aothe well ow d studed etwo s cellul etwos, whch ll weless odes coucte wth the bse sttos wth oe hop. I ddto, the fstuctue of bse sttos cellul etwo s eltve stble, thus gutee hgh pefoce. Howeve, oe cses, to deploy bse sttos oe e s ot oly expesve, but lost possble. I ths ppe, we study the ultcst cpcty of hybd etwos whe we choose the best potocols fo ll lyes. A hybd weless etwo cossts of two types of etwo devces: bse sttos d weless tels. We ssue tht ll bse sttos e egully plced s gd sque ego wth sde-legth etes, d ech bse stto s coected wth djcet bse sttos by wed les o weless chels. Hee two bse sttos e sd to be djcet f the Vooo egos Deptet of Copute Scece, Illos Isttute of Techology, Chcgo, IL, USA e-l: xl@cs.t.edu; xo3@t.edu; stg7@t.edu. she coo boudy seget. Assue tht ech l tht coects two bse sttos hs te W B -bps d bse stto s ethe dt souce o dt eceve; t sply seves s ely gtewy. Futhe we ssue tht weless tels wth coo coucto ge d tefeece ge R = Θ e doly plced ths sque ego. Whe the coucto s successful, the dt te betwee two weless tels s W -bps. The dt te betwee bse stto d y weless tel s W c -bps. Gve ll bse sttos Z, the Vooo ego, deoted s Voz, Z, of bse stto z s clled the sevce cell of bse stto z. We study the cpcty of gve hybd etwo, d how the cpcty of hybd etwos scle wth the ube of odes, o wth the ube of bse sttos the etwos whe fxed deployet ego s gve. Fo ost esults peseted ths ppe, we ssue tht the ubes,,, d e selected such tht the esultg hybd etwo s coected wth hgh pobblty. Due to sptl septo, sevel weless odes c tst sulteously povded tht these tsssos wll ot cuse destuctve weless tefeeces to y of othe tsssos, d ll tsssos betwee two bse sttos e cosdeed wed ls, thus thee e o tefeece to othe sulteous tsssos. Notce tht the tssso betwee weless ode d bse stto s weless tssso, thus bes tefeece costts. Fo ll doly dstbuted odes, ech ode v hs doly chose destto odes fo othe weless odes, to whch t wshes to sed dt t bty dt te λ. The u pe-flow ultcst cpcty of do etwo s defed s s = λ whe thee s schedule of tsssos such tht ll ultcst flows wll be eceved by the destto odes successfully. Fo pesettoplcty, we ssue tht thee s oly oe chel the weless etwos. As lwys, we ssue tht weless ode hs eough eoy to buffe ll the pcets t geetes o ely fo othes such tht o pcets wll be lost though oe- o ult-hop tssso. Fo ost of the esults peseted hee, the dely of the outg s ot cosdeed,.e., the dely the wost cse could be lge fo soe esults. Bsclly, fo ultcst, thee e thee dffeet outg stteges hybd etwo. The fst oe s ed Ad Hoc Routg: gve the souce ode d eceves, ultcst tee usg oly weless tel s costucted d the outg s pefoed o ths tee. Ths ppoch hs the se cpcty s d hoc weless etwo. The secod oe s bsed oevce cell. Fo ultcst flow, fo ech cell tht cots t lest oe eceve o souce ode sde, we costuct tee tht sps the eceves o souce ode cludg the bse stto

2 ths cell. The the foest coposed of tees bult fo ech cell wll be coected by ls og bse sttos. Ths s sl to the outg cellul etwos. We cll ths outg sttegy s Cellul Routg. The thd outg ethod c use y subgph of the ogl coucto gph tht sps the eceves d the souce ode fo outg. We cll ths outg sttegy s Hybd Routg. Thus, hybd etwos ctully peset tdeoff og tdtol BS-oeted etwo d d hoc weless etwo. Coped wth the sl wo by Mo et l., [9], we study oe geel cses the th oly studyg Cellul Routg sttegy d futhe close the gp betwee the uppe boud d lowe boud ultcst cpcty fo hybd etwos. Supsgly, ou esults show tht we c ethe use Ad Hoc Routg sttegy o Cellul Routg sttegy to bet y othe outg sttegy syptotclly. The ultcst cpcty of hybd weless etwos hs bee studed [9]. They ssue tht ll ls ls betwee bse sttos, ls betwee bse sttos d ody odes, d ls betwee ody odes hve the se cpcty W -bps. They deve syptotc uppe bouds d lowe bouds o ultcst cpcty of the hybd weless etwos. The totl ultcst cpcty s O log W whe = O = O, ϑ = log, d = o ; the totl ultcst cpcty s Θ log W whe = O log, = Ω d 0. Whe = O log d = O, the uppe boud fo the u ultcst cpcty s t ost O W d s t lest ΩW espectvely. Whe = Ω log, the ultcst cpcty s ΘW. Coped wth the esults, we ssue oe geel heteogeeous l cpctes d close the gp betwee lowe bouds d uppe bouds fo ll possble cses. Ou M Cotbutos: I ths ppe we deve tchg uppe bouds d lowe bouds o ultcst cpcty of hybd weless etwo, whch bse sttos e dstbuted egully gd llustted by Fgue. Assue tht the deployet ego d the tssso ge e selected such tht the etwo s coected w.h.p. I othe wods, π = Θlog [3]. We lwys ssue tht = O /. We show tht Theoe : The syptotc pe-flow cpcty of ultcst sessos by Cellul Routg s { Θ W B, Wc, W f = O Θ W B, Wc, W f = Ω Whe the Ad Hoc Routg sttegy s used, t ws poved [] tht the u pe-flow ultcst cpcty s { Θ λ = W f = O Θ W f = Ω We the poved tht the Hybd Routg sttegy wll cheve etwo cpcty t ost the lge oe of the syptotc cpcty cheved by Cellul Routg sttegy d the syptotc cpcty cheved by the Ad Hoc Routg sttegy. Cobg the pecedg esults, we futhe pove tht Theoe : The u pe-flow cpcty ϕ by Hybd Routg sttegy whe = O s of ode Θx Θ [ WB, W c, W, W f = Ω, = O Θ W f = Ω W ] f = O 3 The ultcst cpcty of hybd etwos whe usg Cellul outg d Ad Hoc Routg s show Fg.. A pott fdg of ou ppe s tht we pove tht the syptotc ultcst cpcty cheved hybd etwos s the uppe evelop of those two cuves Cellul outg d Ad Hoc Routg. So we c ethe use Cellul outg o Ad Hoc Routg to bet y othe outg sttegy syptotclly. Cpcty MBts/s W W B = 0000 = 000 Metes = 0 Metes s = 00 = 000 WB = 5 MBts/s Wb = 3 MBts/s W = MBts/s log Cellul Routg Ad Hoc Routg Fg.. The cpcty bouds cuves fo Cellul Routg d Ad Hoc Routg. The uppe evelop of two cuves s the cpcty boud fo Hybd Routg. Note tht whe the tssso ge s slle, the chevble syptotc cpcty wll be lge. Howeve, o the othe hd, the tssso ge should be t lest cet vlue such tht the etwo foed by bse sttos d tels wll be coected w.h.p. It hs beehow [9] tht whe cπ logc +β fo β, the esultg etwo G = V Z, E s coected wth pobblty t lest e e β, whe β d c s costt. The est of the ppe s ogzed s follows. I Secto II we dscuss detl the etwo odel used ths ppe. I Secto III d IV we peset the tchg uppe bouds d lowe bouds fo ultcst cpcty espectvely fo the hybd etwos whe Cellul Routg sttegy s used. I Secto V, we gve the ultcst cpcty boud whe Hybd Routg sttegy s used. We evew the elted esults o etwo cpctes Secto VI d coclude the ppe Secto VII wth the dscusso of soe possble futue wos. II. NETWORK MODEL We ssue tht thee s set V = {v, v,, v } of ody weless tels doly deployed sque ego wth sde-legth. Ech weless ode hs tssso ge such tht odes v d v j c coucte successfully ff the Euclde dstce v v j. The dt te of evey l v v j s W -bps whe o tefeece occus. A coucto fo v to v j s tefeece-fee f thee s o

3 3 y othe ode u tht s tsttg d wth dstce R of ecevg ode v j. We futhe ssue tht thee e bse sttos Z = {z, z,, z } egully plced the ego. Fo exple, the bse sttos e plced egully t postos, + j + wth 0, d 0 j. We geelly ssue tht s sque of soe tege. Fg. llusttes sple exple of hybd etwos. Clely, these egully plced bse sttos dvde the ogl sque ego to cells s Vooo dgs wth se sde legth. We use S to deote the cell defed by bse stto z, d fo splcty, by busg the otto lttle bt, we sy the cell S s the sevce ego of bse stto z,.e., z seves s fuctol gtewy fo ll weless odes cell S whe Cellul Routg sttegy s used. The tssso ge of bse stto s lso ssued to be. I othe wods, bse stto c oly dectly seve odes wth dstce. The totl dt te tht bse stto ceve ll ody weless odes s t ost W c -bps wth W c W. I othe wods, bse stto ceve t ost W c λ flows f ech flow eques dt te λ. Ech bse stto s coected to ts djcet bse sttos t ost 4 by wed les o weless chels usg fequecy dffeet fo the fequecy used betwee ody weless odes. The ls betwee bse sttos hve lge cpcty W B to suppot tffcs. We futhe ssue tht = o thoughout the ppe due to the followg obsevto: whe the ube of bse sttos, ll these egully dstbuted bse sttos wll cove the whole sque, thus hybd etwo wll ct s cellul etwo. A ody ode d bse stto c coucte wth ech othe oly f the Euclde dstce betwee the s t ost. I othe wods, the weless coucto ge of y bse stto s lso ssued to be. The coplete coucto etwo s gph G = V Z, E, whee V = {v, v,, v } s the set of ody weless odes d Z = {z, z,, z } s the set of bse sttos, d E = E E B E c s the set of ll possble coucto ls E s the set of d hoc ls uv whee u V, v V, d u v. Ech l E hs dt te W -bps. E B s the set of bcboe ls z z j whee z Z, z j Z, d z z j =. The dt te of ech l E B s W B -bps. 3 E c s the set of cellul ls z v j whee z Z, v j V, d z v j. The dt te both up-l d dowl of ech l E c s W c -bps. Fo splcty, we use E d to deote the set of cossg d hoc ls: E d = {v, v j v d v j e fo dffeet cells}. We ssue tht W W c W B. Gve ultcst flow wth souce v d the set of eceves U, the outg stuctue ust be subgph of G. Thee dffeet outg stteges tht wll be studed hee c be ctegozed s follows Ad Hoc Routg sttegy wll use oly the ls E. We use λ to deote the syptotc ultcst cpcty chevble by d hoc outg sttegy. Cellul Routg sttegy wll ot use ls E d,.e., uv E such tht u d v e fo dffeet cells. We use ϑ to deote the syptotc ultcst cpcty chevble by cellul outg sttegy. 3 Hybd Routg sttegy c use y ls G. We use ϕ to deote the syptotc ultcst cpcty chevble by hybd outg sttegy. Plese see Fg fo llustto. I ths ppe, we ly s- Ogl etwo c Cellul outg sttegy b Ad Hoc outg sttegy d Hybd outg sttegy Fg.. Illustto of Thee Routg Stteges, we use the ed ode to deote souce ode d blues odes to deote ts eceves sue tht the tssso ge s fxed d thus olzed to oe ut thoughout the ppe. Rdo Multcst Flows: I ths ppe, we wll cocette o the ultcst cpcty of do hybd etwo, whch geelzes both the ucst cpcty [6] d bodcst cpcty [9, 4] fo do etwos whe = 0. Assue tht subset S V of = S do odes wll seve s the souce odes of ultcst sessos. We doly d depedetly choose ultcst sessos. To geete the - th ultcst sesso, pots p,j j e doly d depedetly chose fo the deployet ego. Let v,j be the eest weless ode fo p,j tes e boe doly. Obseve tht dog ths, t s possble tht soe odes wll seve s eceve of ultple ultcst flows, d ultcst flow y hve less the eceves. It s ot dffcult to show tht wth hgh pobblty, ech flow wll hve t lest ɛ eceves fo sll vlue 0 < ɛ <. Thus, fo splcty, we lwys ssue tht ech flow hs eceves. I the -th ultcst sesso, v, wll be chose s souce ode d ultcst dt to odes U = {v,j j } t bty dt te λ. I ths ppe, we ly focus o the potocol tefeece odel duced [6]. We ssue tht ech ode v hs fxed tefeece ge R whch s wth sll costt fcto of the tssso ge,.e., ϱ R ϱ fo soe costts < ϱ ϱ. Ude the potocol tefeece odel, y ode v j wll be tefeed by the sgl fo v f v v j R whee ode v s sedg sgl to soe ode

4 4 othe th v j. Cpcty Defto: We ssue tht y ode v could seve s the souce ode fo soe ultcst, hee. Ad fo ech souce ode v, we doly select eceve odes fo othe odes, sy U V {v }. Assue tht ode v wll sed dt to these eceves U wth dt te λ. Let λ = λ, λ,, λ, λ be the te vecto of the ultcst dt te of ll ultcst sessos. Whe gve fxed etwo G = V Z, E, whee the ode postos of ll odes V, the posto of ll bse sttos Z, the set of eceves U fo ech souce ode v, d the ultcst dt te λ fo ech souce ode v e ll fxed, Defto : Gve etwo, ultcst te vecto λ = λ, λ,, λ, λ bts/sec s fesble f thee s sptl d tepol schee fo schedulg tsssos such tht by opetg the etwo ult-hop fsho d buffeg t teedte odes whe wtg tssso, evey ode v ced λ bts/sec vege to ts chose destto odes. Tht s, thee s T < such tht evey te tevl wth ut secods [ T, T ], evey ode ced T λ bts to ts coespodg eceves. The vege pe flow ultcst thoughput cpcty s defed = s α = λ, whee s the ube of ultcst sessos, d s the totl ube of odes ech ultcst sesso, cludg the souce ode. Slly, gve ultcst sessos wth S s souce odes, the u pe-flow ultcst cpcty s defed s ϕ = v S λ. I ths ppe, we wll focus o the u pe-flow cpcty. Defto : We sy tht the ultcst cpcty pe flow of clss of do etwos s of ode Θf bts/sec f thee e detestc costts c > 0 d c < c < + such tht l = cf s fesble = l f = c f s fesble < Thoughout ths ppe, we wll focus otudyg the u pe-flow ultcst cpcty whch s defed s ϕ = v S λ. III. UPPER BOUNDS IN MULTICAST CAPACITY BY CELLULAR ROUTING Whe Cellul Routg s used, the cpcty fo hybd etwo c be costed due to thee dffeet cogestoceos: the bcboe foed by the ls E B s cogested; the cellul ls E c e cogested; d 3 the d hoc ls E \ E d oe cell e cogested. We wll deve uppe boud septely o u pe-flow ultcst cpcty fo ech of the foeetoed thee codtos. TECHNIQUE LEMMAS: Thoughout ths ppe, we wll epetedly use these les. Le 3: Fo the -th flow, let,j be the ube of tels tht wll fll sde the sevce cell of the jth bse stto z j. The,,j s do vble wth e E,j = d vce V,j =. Note tht P,j = t s t t t. Le 4: Let vble,j deotes the ube of tels of the -th flow tht fll sde the cell of the bse stto z j, but ot sde the coucto ds of z j ceteed t z j wth dus. The P,j = t t = t t + The ts e s E,j = d vce V,j = +. Recll tht ths ppe, we ssued tht the ube of bse sttos c costt 0 < c <. Thus, c c fo soe. Ths ples tht E,j c d vce c c V,j. Le 5: Let vble X,j {0, } deote whethe the j- cell defed by bse stto z j cots soe tels fo the th flow,.e., X,j = f,j > 0, d X,j = 0 f,j = 0. Thus, P X,j = =. I ddto, VX,j = EX,j EX,j =. Le 6: Let vble f j deote the ube of flows, ech of whch hs t lest tel ode sde the j-th cell. The f j = = X,j. I ddto, Ef j = d vce Vf j = VX,j =. Le 7: Let vble X,j {0, } deote whethe soe tels fo the th flow fll to the j th -cell, but ot sde the coucto ds ceteed t z, X,j = f,j > 0, d X,j = 0 f,j = 0. Thus, P X,j = =. I ddto, VX,j = EX,j EX,j =. Le 8: Let vble f j deote the ube of flows, ech of whch hs t lest tel ode sde the j-th cell, but ot sde the coucto ds ceteed t z j. The f j = s = X,j. I ddto, Ef j = d vce Vf j = VX,j =. Le 9: Let vble deote the ube of cells tht hs t lest oe tel fo flow sde. Clely, = j= X,j. The E = d vce V = VX,j =. Futheoe, whe, E,. Whe >, E. Gve outg sttegy A, let T A be the tee used to oute the -th flow. Whe A s cle fo the cotext, we wll splfy t s T by doppg A. The followg le ws show []. Le 0: Gve pots Q doly plced sque of sde legth, the Euclde u spg tee, deoted s EMSTQ, hs expected totl edge legth Θ d ts vce VEMSTQ log. It ws poved [, ] tht y outg tee T fo set Q of do pots the sque of sde-legth, ts totl edge legth s t lest tes the totl edge legth of EMSTQ.

5 5 A. Uppe Boud Due to Ls E B The uppe boud o ultcst cpcty due to ls E B hs two eges: = O d = Ω. Whe = O: I ths cse, fo ech flow, we let B be the set of bse sttos whose sevce cell cots t lest oe tel fo the -th flow. The we eed buld coected stuctue usg oly ls E B to sp B. Let TB be the tee coveg ll bse sttos B costucted by gve outg ethod. The we ow tht TB EMSTB /. Heefte, f S s set, we use S to deote the cdlty of S; f S s tee, we use S to deote the totl Euclde legth of tee S. Notce tht the set B s do vble d B =, whee do vble s s defed befoe. Sl to [], we c pove the followg le: Le : Gve bse sttos B doly selected d ll bse sttos e plced sque ego of sde-legth, the Euclde u spg tee EMSTB hs expected totl edge legth c fo costt c 0, ] d ts vce V EMSTB log. Theoe : Whe θ 0 fo soe costt θ 0, thee s costt c 3 such tht, wth pobblty t lest e s/8, the u dt te tht c be suppoted usg cellul outg sttegy s t ost W B c 3 fo y outg sttegy due to the cogesto bcboe ls. Poof: Let CTB deote the ube of cells tht the outg tee TB wll use,.e., the ube of bse sttos used TB. Obvously, CT B, the ube of cells tht cot the eceves of the -th flow. Notce tht ech bse stto s coected to t ost 4 djcet bse sttos. The TB /4 CTB = T B /. Let vble L = = CT B, deotg the totl lod of ll cells. Hee the lod of cell by outg ethod s the ube of flows pssg the cell fo the ultcst tee costucted. The L = T B /4 = EMSTB /8. Notce tht E = EMSTB = c E d V = EMSTB s log. Thus EL c E /8. We the copute the vlue E. Recll tht vble X,j deotes whethe the j-th cell cots y tel fo the -th flow d = j= X,j. By defto, E = E j= P X,j = =. The,, / E, Whe θ 0, we hve E c fo costt c = θ 0,. Defe do vbles X q = q j= EMSTB j E EMSTB j. The EX q+ X,, X q = X q,.e., vbles X e tgle. I ddto, X q X q = EMSTB q E EMSTB q EMSTB q, whch s. Fo Azu s Iequlty, we hve t P X s X 0 t exp s. = 8 Let t = ɛ = E EMSTB. Clely, ɛ c c t ɛ. Note tht X 0 = 0. The, s P EMSTB E EMSTB t = = P X s t exp t = 8 exp ɛ sc c = exp sɛ c c 8 8 Cosequetly, fo costt ɛ 0,, we hve s P EMSTB ɛ c E e sɛ c c 8, = s P EMSTB c E / e sc c /3. The, = It ples = P L c E /6 e sc c /3. P L c c /6 e sc c /3 f θ 0. W B sc c = W B Recll tht L deotes the totl lod of ll cells. The by Pgeohole pcple, wth pobblty t lest e sc c /3, thee s t lest oe cell, tht wll be used by t lest sc c flows. Thus, wth pobblty t lest e sc c /3, the u dt te tht c be suppoted usg cellul outg sttegy s t ost c c fo y outg sttegy due to the cogesto bcboe ls. By lettg c 3 = c c fshes the poof of the theoe. Whe = Ω: Recll tht ths cse, we hve show tht E /,.e., fo ech flow, the expected ube of cells tht wll cot ts tels s t lest /. Moe pecsely, t s esy to show tht, fo y cell j, the pobblty, P X,j =, tht t wll cot tel fo flow s t lest /e > /. The usg Azu s Iequlty, we c pove tht, wth pobblty t lest e s/8, the totl lod L /4. Thus, by Pgeohole pcple, thee s oe cell such tht ts lod the ube of flows usg ts bse-stto s t lest /4. Cosequetly, we hve the followg theoe. Theoe 3: Whe θ 0 fo soe costt θ 0 >, wth pobblty t lest e s/8, the u dt te tht c be suppoted usg cellul outg sttegy s t ost 4W B fo y outg sttegy due to the cogesto bcboe. B. Uppe Boud Due to Ls E c I ths subsecto, we study the u pe-flow dt te due to the cogesto whe ody weless odes ccess the bse-sttos the cells. Recll tht we ssue tht both the upl te d the dow-l te betwee the bse-stto d the ody weless odes ts cell s W c -bps. We wll study uppe bouds bsed o two subcses, whethe = O o ot. We essetlly wll study the ube of flows f j sde of j th cell tht wll pss though bse-stto z j.

6 6 Whe = O: We fst study the cse whe the ube of tels pe-flow s = O. Notce tht whe, Ef j = > d Vf j <. Le 4: Whe stsfes the codto 4, the vble x j= f j s Θ wth pobblty t lest. Poof: We use the VC-Theoe to pove ths le. Let the set C = {Voz j, Z j } be the clss of cells defed by ll bse-sttos. Let F be the -th flow d F s sd to belog to the j-th cell f soe of ts tels s coted sde the j-th cell Voz j, Z, whch s deoted s F Voz j, Z. The f j = F IF Voz j, Z, whee IF Voz j, Z = f F Voz j, Z d IF Voz j, Z = 0 othewse. Obvously, VC-dC log sce the cdlty of C s. I ddto, the pobblty P A tht flow belogs to cell A s P A =. It s esy to show tht, whe 0 < <, we hve < P A <. The by VC-Theoe, we ow tht fo evey ɛ, δ > 0, s = P sup IF A A C P A ɛ > δ wheeve > x 8 VC-dC { } log 3, 4 log ɛ ɛ ɛ δ. Whe we choose the petes ɛ = 4, δ =, d 3 log > x log 5, 6 log, 4 The P sup = f P A > 4. Hece, P [, ], 4 f 9 4 >. Bsed o the pecedg le, we coclude tht, Theoe 5: Whe, the te due to the cogesto of ccessg the bse-sttos, wth pobblty t lest, s t W ost c Wc 4 = O W c. x f Whe = Ω: We thetudy uppeboud o the te chevble due to the cogesto of ccessg bse-sttos whe >. I ths cse, > Ef j = > e. Le 6: Whe stsfes the codto 5, the vble x j= f j s lso Θ wth pobblty t lest. Poof: Sl s the poof fo Le 4, except tht whe, we hve e < P A <. Bsed o VC- Theoe, by choosg the petes ɛ = e, δ =, we ow tht whe The > x 8e log log3e, 4e log, 5 P sup = f P A > e. Hece, we hve P [, ], e f >. Ths fshes the poof. Obvously, we hve the followg theoe. Theoe 7: Whe, wth pobblty t lest, the u pe-flow te by y Cellul Routg sttegy s t ost W c e. C. Uppe Boud Due to Ls E \ E d I pevous subsectos, we study uppe bouds o the ultcst cpcty hybd etwos due to the cogesto t the bcboe ls coectg ps of bse-sttos, d due to the cogesto ccessg the bse-sttos. We ow focus o studyg the cpcty uppe bouds due to the cogesto d hoc ls E \ E d. A tvl uppe boud fo totl ultcst cpcty s W sce thee e ouce odes totl d ech ced dt t W bts/sec. Howeve, we c e the uppe bouds oe tght due to the followg obsevtos. Fo ech souce ode v, whe we ultcst the dt fo oe souce ode v to ll ts eceves et U = {v, v,, v }, the esultg ultcst tee wll cot t lest odes, d possbly oe. Moe possbly, whe o-lef ode v the ultcst tee seds dt to ts chlde, ll odes tht e wth ts tssso ge wll eceve the dt o t lest they cot tst successfully t the se te o tte these odes e teded eceves o ot. I ths cse, we sy ll these odes e chged copy of the dt. To study the ultcst cpcty, we ptto the deployet sque to gds of sze. Clely, thee e t ost = Θ such gd cells. Notce tht og such gd cells, soe of the c be dectly eched by soe bse-sttos. Let g be the totl ube of gd cells tht s dsjot fo the uo of dss j= Dz j,. The obvously g 9 = Θ whe 0. Hee, the costt 9 coes fo the fct tht y bse stto oly c ove t ost 9 gds of sze t the se te due to ou pevous ssupto tht the tssso ge of ech bse stto s lso. Thus, thoughout ths ppe, we ssue tht 0. Recll tht we ssue tht the tefeece ge R > ϱ. The t y te stce, the dstce betwee two ctve sedes v d v s t lest R ϱ. Cosequetly, we hve Le 8: Fo y gd of sde-legth, thee e t ost costt ube deoted s κ < + ϱ of odes sde the gd tht ced dt sulteously wthout cusg tefeece to eceves. Ths le ples tht the totl dt tht c be set out fo y gd dug y te tevl t s t ost W κt fo costt κ. To pove uppe boud o the cpcty, we wll oly cosde the gd cells tht e dsjot fo the dss defed by bse-sttos. I othe wods, fo odes locted sde these gd-cells, t cot ech the bse-sttos dectly d ts dt hve to be elyed by soe othe odes to ech the bse sttos. Gve outg sttegy, fo the -th flow d j-th gd cell, let Y,j be the vble deote whethe the -th flow wll be outed though the j-th gd cell by ths outg sttegy. Let Y j = = Y,j the totl ube of flows tht wll be outed though the j-th gd cell by ths outg sttegy. The fo Le 8, we c coclude tht the u pe-flow dt te s t ost W κ x g j= Y j The est of the subsecto s devoted to gve bette lowe boud o x g j= Y j, thus tghte uppe boud of ultcst cpcty of the hybd weless etwo. Notce tht the boud o x g j= Y j depeds o the outg sttegy. We ctully wll 6

7 7 pove tht, egdless of outg stteges used, x g j= Y j wll be t lest cet vlue w.h.p. Fo splcty, heefte whe we sy eceves, we e tht oe souce ode pluses ll ts eceves. Whe = O: As we ow, ude the Cellul Routg sttegy, ll flows sde of cell wth sde-legth wll fstly go to the closest bse stto by oe- o ult-hop. Befoe the tffc ech the bse stto, the lst hop tssso s cellul l d the secod to the lst hop l s d hoc l. Thus, the potetl cogesto wll hppe o those secod to lst hop d hoc ls fo the followg two esos: Ech bse stto oly hs tssso ge d c cove touch eltvely sll e t ost 9 djcet cells wth sde legth oud the bse stto. b Itutvely, the cell closed to the bse stto wll hve uch bude to ely d hoc tffc to the bse stto. Clely, t s equvlet to study the ube of flows f j sde of jth cell but ot sde of the coucto ds ceteed t z. Le 9: Whe stsfes the codto 7, the vble x j= f j s Θ wth pobblty t lest. Poof: We use the sl poof used Le 4 to pove ths. Assue the set C = {Voz j, Z j } be the clss of egos ech ego s the cell wthout the coucto ds of the bse stto defed by ll bsesttos. The th flow F s sd to belog to the jt h cell f soe of ts tels s coted sde the j-th cell, but ot sde the coucto ds ceteed t z j, Voz j, Z, whch s deoted s F Voz j, Z. I ddto, the pobblty P A tht flow belogs to ego A s P A =. It s esy to show tht, whe 0 < < 0 9 d 0 we hve 9 0 < P A < 9 5. By the se guet Le 4 d VC-Theoe, we hve s ɛ, δ > 0, P sup = IF A P A A C by Pgeohole pcple. Theefoe, by equto 6, the flow cpcty cot exceed W κ θ 9 s = O W s whee κ d θ e costts. Whe = Ω d = O : We e gog to study uppe boud o the te chevble due to the cogesto of ll d hoc flows ls whch tgets to o fo the bse stto whe >. Recll tht we use f j to deote ll the d hoc flows whch exst j th cell d s we hve show befoe, the expected vlue of E,j c fo soe costt c. It s ot dffcult to show tht > Ef j = > s e c. Next, we show tht the xu ube of d hoc flows sde soe cell s the costt fcto of totl ultcst flows by the followg Le. Le : Whe stsfes the codto 8, the vble x j= f j s lso Θ wth pobblty t lest. Poof: Sl s the poof fo Le 9 except tht whe we hve e c < P A <. Bsed o VC-Theoe, by choosg petes ɛ = e c, δ =, we ow tht whe > x 8e c log log3e c, 4e c log, 8 P sup = f P A e c >. P [, ], e f c s >. Hece, ɛ > δ, Fg. 3. The sceo whee ultple ody odes e close to the bse stto z j the j th gd cell. Zj { } wheeve > x 8 VC-dC log 3, 4 log ɛ ɛ ɛ δ. Whe we choose the petes ɛ = 4, δ =, d 3 log > x we hve P sup = log 5, 6 log f P A > 4., 7 >. Hece, P [, ], 5 f 4 0 Bsed o the pecedg le, we coclude tht Theoe 0: Whe 0 9 d stsfes the codto 7, the te due to the cogesto of d hoc ls, wth pobblty t lest, s O W. s Poof: By Le 9, fo j th cell, the ube of d hoc flows f j whch wll covege to bse stto z j, x j= f j = θ s wth pobblty t lest whee stsfes the codto 7 d θ s soe postve costt. I ddto, thee e t ost 9 cells wth sde-legth to ely these flows such tht thee exst t lest oe cell hs to ely t lest θ 9 flows Becuse ll the d hoc flows sde of cell wll flly go to o coe fo the bse stto bsed o the Cellul Routg sttegy, the potetl cogesto wll hppe whe the d hoc flows covege to the coucto ds of cetl bse stto. See Fg. 3 fo llustto. Becuse the bse stto z j hs tssso ge, z j s sevce ge c cove t ost 9 djcet sll gds wth sde legth. I othe wods, ll the d hoc flows sde the j th cell hve to go though these 9 sll cells. Fo Le 8, ech cell wth sde legth c oly hve κ cocuet tsttes. Theefoe, by Equto 6 d Le, we hve Theoe : Wth pobblty t lest, the u pe-flow te fo ultcst sessos s bouded by O W whee = Ω d = O. due to the cpcty costts of d hoc ls E \ E d. 3 Whe = Ω : I the pevous subsecto, we showed uppe boud of the ultcst cpcty of hybd etwo whe < θ /. I ths subsecto we wll peset uppe boud o ultcst cpcty whe θ /. I [], L hs poved tht whe = Ω, the uo of the tssso dss of these eceve odes ultcst

8 8 sesso wll cove t lest costt fcto, sy 0 < ρ, of the deployet ego. Thus the u pe-flow cpcty of hybd etwo due to the cogesto of d hoc l wll ppoxtely be equl to the bodcst cpcty,.e., O W. Cobed wth Theoe 3 d Theoe 7, we hve the followg theoe: Theoe 3: Whe θ / fo costt θ, the u pe-flow ultcst cpcty fo the hybd etwo s bouded by O W B, W c, W wth hgh pobblty. IV. LOWER BOUNDS IN MULTICAST CAPACITY BY CELLULAR ROUTING I ths secto, we wll deve syptotclly lowe boud the ultcst cpcty by pesetg ultcst schee. A. Ipleet of Routg Stteges We poposed the followg ultcst outg sttegy fo Cellul Routg Algoth. As we hve expled befoe, bsed o the Cellul Routg sttegy, ech eceve ode wll ty to ech o be eched by the closest bse stto by oe- o ult-hop. Assue set U = {v, v,, v } s the uoet of souce ode v d ts doly selected eceves fo the th ultcst flow, hee we ssue v = v fo splcty. Assue Uj s the ode set cotg ll eceves of the th ultcst flow whch e fllg to the j th cell. Obvously, U = j= U j. We futhe ssue set Z = {z, z,, zt} cots ll the bse sttos, ech of whose cell cots t lest oe eceve of the th ultcst flow. Clely, t. Algoth Cellul Routg sttegy fo th ultcst flow Iput: U : Copute Z bsed o U, the costuct Mu Spg Tee MST whch cots ll odes Z y eed othe bse stto s tel odes by bcboe ls oly. Assue the oot of the costucted MST s the bse stto sy z s whch flls the se cell s the souce ode v does. The do bodcstg fo z s to the othe bse sttos o the MST. : fo ech cell S j whch cots t lest oe eceve sde do 3: f S cots the souce ode v, the v fd shotest pth coectg to z 4: f S cots t lest oe eceve fo eceves, costuct BFS tee fo the oot z whch coves ll eceves sde. Ths y eed othe o-eceve odes s tel odes o the BFS tee. The do bodcstg fo bse stto z j to ll weless odes o the costucted BFS tee. 5: ed fo I the followg secto, we wll lyze the lowe boud ultcst cpcty whee = O d = Ω septely s we dd the pevous sectos. Whe the ube of eceves, plus the souce ode, s t ost θ, we wll costuct ultcst tee ech cell S whch spg eceves sde d thus obt ultcst foest whch sps ll eceves. Next, we wll show the lowe boud cpcty chevble by the Algoth ude dffeet cses. B. Whe = O Whe the ube of eceves of ech ultcst sessotsfes = O, we lyze the u pe-flow lowe boud cpcty chevble by bcboe ls, cellul ls d d hoc ls oe by oe. Fo ech ultcst flow, we use Algoth to do outg. We fst toduce the lowe boud cpcty chevble by the bcboe ls. We ow fo ech ultcst flow, the bodcst cpcty o the MST tee costucted Algoth s ΘW B due to the esult [9]. I ddto, ccodg to the esult Theoe 3 [] we ow tht f thee e do ultcst flows sque ego wth sde-legth, thee s sequece of δ 0 such tht fo y sque cell s wth sde-legth sde of the sque ego, P # of flows usg s 3δ 3 = 3δ 3 whee δ 3 s soe costt. Hece, w.h.p, the ube of flows eeded to be elyed by y bse stto s o oe th 3δ3s. Theefoe, the lowe boud cpcty fo bcboe ls s t lest Ω W B by Algoth wth TDMA schedule. Next, we show the lowe boud cpcty chevble by cellul ls whe = O. By Le 4, we ow fo ll cells, whe stsfes the codto 4, the vble x j= f j s Θ wth pobblty t lest. Hee, f j deotes the ube of flows sde of j th cell tht wll pss though the bse stto z j. Thus, fo y bse stto, by sple TDMA schedule, the chevble lowe boud cpcty fo cellul ls s Ω W c. The eg pt of ths subsecto, we show the lowe boud cpcty chevble by d hoc ls usg Cellul Routg whe = O. Recll tht fte pplyg Algoth, ech ultcst flow wll hve BFS tee dow-l decto ooted t the bse stto o shotest pth up-l decto coectg the souce ode to the bse stto ech cell f ths cell cots t lest oe eceve of ths flow. Due to the esult [9], we ow tht fo ech flow, the bodcst cpcty cheved by the BFS tee costucted Algoth s ΘW d t s ot dffcult to show tht the up-l dectohotest pth whch coects the souce ode to the bse stto c cheve te ΘW s well wthout cosdeg ll othe oelted sulteously tssso. I ddto, by Le 9, we ow whe the totl ube of ultcst flow stsfes the codto 7, the xu ube of d hoc flows sde of y cell stsfes x j= f j s Θ. Assue x j= f j = c 8 fo soe costt c 8. I othe wods, w.h.p, we hve t ost hve c 8 up-l flows o c 8 dow-l flows exstg ech cell. We sply cosde the dow-l flows d up-l flows septely. Clely, by TDMA schedule, the u pe-flow te fo both up-l flows d dow-l flows c ech t lest. Hece, we W c 8 hve Theoe 4: Whe = O d stsfes the codto 7, the lowe boud cpcty fo d hoc ls cheved by

9 9 pplyg Algoth d TDMA schedule s Ω W Wth pobblty t lest. C. Whe = O / d = Ω We stll use Algoth to do outg. The chevble lowe boud cpcty fo bcboe ls d cellul ls e esy to get sl lyss s we dd whe = O. The oly dffeece ths cse s tht the ube of ultcst flows whch wll go though soe bse stto could be up to but o oe th flows due to Le. The fte pplyg Algoth d TDMA schedulg, the chevble lowe boud cpcty by bcboe ls d cellul ls e Ω W B d Ω Wc espectvely. The lowe boud cpcty chevble by ll d hoc ls usg Cellul Routg Algoth whe = Ω c be get by the sl poof s we dd ubsecto IV-B. The dffeece s tht the possble up-l flows d dow-l flows ech cell could be up to but o oe th flows due to Le. By the se guet, we hve the followg theoe. Theoe 5: Wth pobblty t lest, the u pe-flow te fo d hoc ls chevble by pplyg Algoth s Ω W whe = Ω d = O. D. Whe = Ω I ths cse, we hve poved tht the uppe boud o the totl ultcst cpcty s oly ΘW. Obvously, the totl ultcst cpcty fo hybd etwo s t lest the lowe boud of the cpcty fo bodcst o tte we use ethe Cellul Routg o Ad Hoc Routg. I [9], Keshvz-Hddd et l.. peset bodcst schee to cheve cpcty ΘW. Thus, we hve Theoe 6: The u pe-flow ultcst cpcty W chevble by ll d hoc ls s t lest c 7, whee c 7 + s costt. Obvously, the u pe-flow ultcst cpcty chevble by bcboe ls d cellul ls e Ω W B d Ω Wc by the sl lyss we used IV-C. V. CAPACITY BOUND FOR HYBRID ROUTING I ths secto, we wll gve syptotc uppe bouds fo y Hybd Routg sttegy. The supsg plcto of ths esults s tht f we choose the oe who c g lge cpcty betwee Ad Hoc Routg d Cellul Routg s ou outg sttegy, the ttble cpcty s the se ode of the uppe bouds of y Hybd Routg sttegy. It ples tht the uppe bouds e tght d ou outg sttegy s syptotclly optl. The foeetoed esult s bsed o the followg obsevto. Fst, whe Hybd Routg s ppled, y l G could be used, othe wods, fo y ultcst flow f, the coespodg esultt ultcst tee T fo Hybd Routg y cot t ost thee types of ls, the ls E, E B o E c. Assue T = E T, T B = E B T d T c = E c T,.e., T T B d T c cots ll d hoc ls bcboe ls d cellul ls used by tee T. Futheoe, we use sets T, T B d T c to deote the uo of ll d hoc ls, bcboe ls d cellul ls used by ll ultcst tees. Notce, hee the eso tht we quote the wod uo s becuse the l whch s belog to dffeet ultcst tees wll be couted ultple tes T, T B d T c,.e., f we use S to deote the suto of the legth of ll ls belog to l set S, the T = = T, T B = = T B, T c = = T c. A. Whe = O Isted of studyg the uppe boud fo y gvg outg sttegy dectly, we y vew ths poble ltetve wy: Fo y gve outg sttegy, f we c lwys costuct ew outg tee bsed o t such tht the uppe boud of ultcst cpcty by usg ou ew outg sttegy s o slle th the ogl oe, the the uppe bouds fo the ew outg sttegy ust be oe of the vld uppe bouds fo the ogl outg sttegy. Next, we fst gve llustto of ou costucto ethod, the uppe boud fo the ew costucted outg tee wll be deved, flly, we use the bove uppe bouds s oe desed uppe bouds. I the followg cotets, we wll use D to deote set of bse sttos used by flow such tht ech bse stto ths set hs t lest oe cellul l djcet to t. We fst gve llustto of ou costucto ppoch bsed o the gve outg sttegy T fo flow s follows: Use the u legth tee spg D to eplce TB, we use TB to deote ew tee. Adjust the ls coted Tc d T such tht thee e o oe th 9 cellul ls o the esultt tee fte costucto to ech bse stto, deotg the ew tees foests by Tc d T espectvely. Next we wll expl d lyze these stges detls, the followg cotets, we wll use λ, λ, λ B, λ B, λ c d λ c to deote the chevble dt te o T, T, T B, TB, T c d Tc espectvely. Fst, we hve TB T B, t s stght fowd fo the fct tht TB s u legth tee spg D. Secod, we hve λ c c λ c fo soe costt c, ths s bsed o the followg obsevto: Due to the esults [3], we ow we c fd t ost 9 odes s coectos oe hop wy fo the bse stto usg cellul ls whch c coect to ube of weless odes sy dotos, two hop wy fo the bse stto such tht these dotos c cove ll odes whch e two hop wy fo bse stto. Next, we let ll eceves extg the coucto ds of the bse stto whch e ot selected to the 9 bse sttos coect to the closest coecto. O the oe hd, fo dow-l fo the bse stto, ou costucto wll ot decese the cellul l te. O the othe hd, fo uppe l, we ow tht oly costt ube of weless ode ou cse t ost 4κ odes c tst t the se te such tht ech of 9 coectos wll hve ddto bude t ost 4κ tes th befoe the costucto. Obvously, copg wth the ogl schedulg peod T, fte costucto, 4κT te s eough fo schedulg peod. Hece, we hve λ c c λ c fo soe costt c. See Fg. 4 fo llustto. Thd, λ c0 λ fo soe costt c 0. We gutee ths pot by the followg esevto. Afte we get coectos dug the secod step fte

10 0 djust cellul ls. Fo soe odes whch e two hop wy fo the bse stto the outg tee befoe ou costucto, they could lost the coecto to the bse stto whe the elyg odes to the bse stto e ot selected s coectos the secod step. If so, we sply let these odes coect to the closest doto. See Fg. 4 fo llustto. Let us te u s exple. Fst, fo ech tel ode u doto closed to the bse stto. Afte ecostucto, soe othe odes who lost coecto to the bse stto wll tu to u d s u to help to ely tffc to the bse stto. Howeve, these odes ust stsfy two codtos.. They e the coucto ge of u.. They c tst sulteously bsed o the ogl outg d schedulg sttegy. Clely, u c t ost cove 4 closest sque ego wth sde legth d fo ech sque cell wth sde legth, thee e t ost κ odes c tst sulteously s we hve poved befoe. Hece, we c gutee, thee e t ost 4κ odes wll tu to u oe te slot fte costucto, othe wods, fte costucto, ode u c cheve t lest 4κ te of the ogl te befoe costucto by TDMA schedulg,.e., λ 4κ λ. These thee pott obsevtos gutee tht the uppe bouds fo the ew costucted outg sttegy deved fo the followg lyss e lso vld uppe bouds fo the ogl outg sttegy. See Fg. 4 fo llustto. 00 Zj u Befoe ecostucto. Zj u b Afte ecostucto. Fg. 4. Pt of the Hybd Netwo whch s e the bse stto. The dsh le fo ed odes to u e ew dded bude fo ode u. z j s the bse stto fo j th cell. Blc odes deote tel odes. Red odes e odes who wll chged the outg sttegy fte costucto. Blue ode s oe of eceves whch s the coucto ds of the bse stto z j. Fo ow o, we focus otudyg the uppe bouds fo the ew costucted outg tee: Due to the esult [], we ow tht the totl legth of tel edges of ultcst tees spg eceves stsfes s = T = = T T + T B + T c c 9 + s = T B s + = T c = fo soe costt c 9. The we dscuss the followg two cses espectvely: If T T B + T c : Sce the totl legth of T, T B d T c s o slle th c 9, we hve T c 9. As show [], the totl e coveed by ll of these d hoc tees s t lest η fo soe costt η, the ube of odes coveed by ll d hoc tees s t lest η wth hgh pobblty. Bsed o the dt copy guet, t follows tht: λ c 0 λ c 0 W η. If T < T B + T c : We hve T B + T c c 9 Accodg ou costucto ppoch, we ow tht fo y outg tee T, ech bse stto hs t ost 9 djcet cellul ls, the togethe wth the fct tht the legth of ech cellul l s t ost, we hve the followg equlty: T c T B / 9. Sce = O, we hve T c T B /9, t follows tht T B c 3 fo soe costt c 3. Ths ples tht thee s t lest oe bse stto whch s used by t lest c 3 flows. Due to the cogesto o ths bse stto, we g the followg uppe boud: λ B λ B W B c 3. Futheoe, becuse TB sps D bse sttos, we hve TB D ccodg to the esults []. It follows tht c 3 T B = = T B s = D. Togethe wth the fct tht p p b q q b, we hve the followg equlty: = s D η = D s η3s = η3 s s We coclude tht thee s t lest oe bsed stto whch s used by t lest η 3 flows to coect weless odes dectly. Due to the cogesto o both d hoc ls d cellul ls ccessg the bse stto, we futhe hve the followg two uppe bouds fo ths cse: { λ c 0 λ W λ c c λ c η3 / = W η3 W c η3 / = W c η3 It cocludes tht the uppe boud ths cse s O{ WB, W, W b } The fl uppe boud s ged by choosg the xu oe betwee cse d cse : Le 7: The cpcty boud fo y Hybd Routg sttegy s [ Ox { W B, W, W b }, ] W whe = Ω d = O. It ples tht whe = Ω, ts syptotc optl to choose the lge oe betwee Ad Hoc Routg d Cellul Routg s ou outg sttegy bsed o the clculted lowe boud fo ech outg sttegy. B. Whe = Ω d = O Se s the poof fo the pevous cse, we fst costuct ew outg tee bsed o y gve outg sttegy. Sce the uppe boud fo the ew costucted outg tee c lso be cosdeed s vld uppe boud fo the ogl outg sttegy, we wll focus otudyg the uppe boud fo the ew costucted outg tee. Slly, we hve two possble cses eed to ddess: If T T B + T c : The poof s exctly se s the oe show befoe fo the se cse, we g followg uppe boud: λ c 0 λ c 0 W η. 9

11 If T < T B + T c : We wll pove tht ths cse s possble whe = Ω. Sce TB s tee spg t ost bse sttos usg oly bcboe ls, we get T B / <, we lso ow tht T c 9 becuse ech bse stto hs t ost 9 djcet cellul ls. We edtely hve whe > Ω, t s possble tht T B + T c > c 9, othe wods, T T B + T c. We flly hve the followg le: Le 8: The cpcty boud fo y Hybd Routg sttegy s O W whe = Ω, = O d = O. Ths esult ples tht whe = Ω d = O, usg Ad Hoc Routg s ledy syptotc optu. C. Whe = Ω Ag, becuse whe = Ω, the uo of the tssso dss of these eceve odes ultcst sesso wll cove t lest costt fcto, sy 0 < ρ, of the deployet ego wth hgh pobblty whe = Ω. The bsed o the dt copy guet stted [], we hve the followg le: Le 9: The cpcty bouds fo y Hybd Routg sttegy s O W whe = Ω, = O. It s ot hd to fd tht ts syptotc optu to choose Ad Hoc Routg s ou outg sttegy whe = O. D. Put It All Togethe By suzg these esults, we hve Theoe 30: The uppe bouds of the ultcst cpcty fo y Hybd Routg sttegy s Ox O [ WB, W b, W, W f = Ω, = O O W f = Ω ] W f = O 0 We the gve geel outg sttegy tht c cheve the syptotc uppe boud fo hybd etwo N,, : If = O, we choose the oe who c g lge dt te betwee Ad Hoc Routg sttegy d Cellul Routg sttegy. If = Ω, we use Ad Hoc Routg sttegy. The togethe wth the lowe boud fo Ad Hoc Routg sttegy d Cellul Routg sttegy, we get the theoe. VI. LITERATURE REVIEW Gupt d Ku [6] studed the syptotc ucst cpcty of ult-hop weless etwos. Whe ech weless ode s cpble of tsttg t W -bps usg costt tssso ge, the thoughput chevble by ech ode fo doly W chose destto s Θ log bts pe secod. Gossgluse d Tse ecetly showed tht the ucst cpcty c be poved by the oblty of weless odes egdless of dely. Gstp d Vettel studed the cpcty of do etwos usg ely []. Chuh et l. [] studed the cpcty sclg MIMO weless systes ude coelted fdg. The cpcty sclg dely tolet etwos wth heteogeeous oble devces ws studed by Getto et l. [3]. Keshvz- Hddd et l. studed the bouds fo the cpcty of weless etwos posed by topology d ded [8]. The techques c be used to study ucst, ultcst d bodcst cpcty. Bodcst cpcty of both bty etwos d do etwos hs beetuded [9, 4]. Keshvz-Hddd et l. [4] studed the bodcst cpcty wth dyc powe djustet fo physcl tefeece odel. Multcst cpcty ws lso studed the ltetue. Jcquet d Rodols [5] studed the sclg popetes of ultcst fo do weless etwos. They cled tht the xu W te t whch ode c tst ultcst dt s O log. Recetly, goous poofs of the ultcst cpcty wee gve [, 5]. L et l. [] studed syptotc ultcst cpcty fo lge-scle do weless etwos. They showed the totl ultcst cpcty s Θ log W whe = O log d whe = Ω log, the totl ultcst cpcty s equl to the bodcst cpcty,.e., ΘW. L et l. [8] studed the lowe boud of ultcst cpcty fo lge scle weless etwos ude Guss Chel odel by pesetg soe ovel ethods. Ths esult ws ecetly poved by Wg et l. [4]. Hu et l. [0] ecetly studed the cpcty d dely tdeoffs of ultcst cpcty whe the oblty odel s..d. They show tht oblty d edudcy do pove the ultcst cpcty whe the ube of eceves pe flow s sll. Lee et l. [0] studed the sclblty of DTN ultcst outg. They popose RelyCst, outg schee tht exteds the two-hop ely lgoth [7] the ultcst sceo. Lu et l. [7] studed the ucst cpcty of hybd etwo weless d hoc etwo wth fstuctue. They essetlly studed the ucst cpcty of hybd weless etwos ude the oe-desol etwo odel d twodesol stp odel espectvely. Kozt d Tssuls [6] lso studed the ucst cpcty of d hoc etwos wth do flt topology ude the peset suppot of fte cpcty fstuctue etwo. They showed tht the pe souce ode cpcty of ΘW/ log. I [9], Mo et l., studed the ultcst cpcty fo hybd etwos by usg Cellul Routg sttegy. VII. CONCLUSIONS I ths ppe, we essetlly studed the ultcst cpcty tht c be cheved by hybd etwos wth doly dstbuted weless odes d egully dstbuted bse sttos. We deved lytcl uppe bouds d lowe bouds o ultcst cpcty of hybd etwos. Obseve tht ll ou esults e poved whe the deployet ego s sque wth sde-legth d the tssso ge of ll odes s ufo wth vlue. It s ot dffcult to show tht ll ou esults stll pply whe the deployet ego s fxed sque wth sde legth =, whle the tssso ge s selected ppoptely,.e., = Θ log c fo soe

12 costt c. I ddto, ou esults stll hold whe = whle the deployet ego hs bouded spect to such s ds o ectgul e whe to wdth/heght s bouded. A ube of teestg questos e chllegg. The fst s to study the cpcty whe othe tefeece odels e ppled such s physcl tefeece odel d Guss chel odel. The secod s to vestgte the cpcty egos whe oppotustc spectu usge s dopted by soe weless tels. The lst, but ot the lest, s to study the cpcty ego fo dely tolet etwos whee the weless tels e oble followg soe oblty odel d we wt to study the chevble cpcty whe we c tolete cet dely. REFERENCES [] GASTPAR, M., AND VETTERLI, M. O the cpcty of weless etwos: the ely cse. I IEEE INFOCOM 00. [] CHUAH, C.-N., TSE, D. N. C., KAHN, J. M., AND VALENZUELA, R. A. Cpcty sclg o weless systes ude coelted fdg. I IEEE Tsctos O Ifoto Theoy 00, vol. 48. [3] GARETTO, M., GIACCONE, P., AND LEONARDI, E. Cpcty sclg dely tolet etwos wth heteogeeous oble odes. I ACM MobHoc 007, pp [4] KESHAVARZ-HADDAD, A., AND RIEDI, R. O the bodcst cpcty of ulthop weless etwos: Iteply of powe, desty d tefeece. I IEEE SECON 007. [5] JACQUET, P., AND RODOLAKIS, G. Multcst sclg popetes ssvely dese d hoc etwos. I ICPADS 005, pp [6] GUPTA, P., AND KUMAR, P. Cpcty of weless etwos. IEEE Tsctos o Ifoto Theoy IT , [7] GROSSGLAUSER, M., AND TSE, D. Moblty ceses the cpcty of d hoc weless etwos. IEEE/ACM TON 0, 4 00, [8] KESHAVARZ-HADDAD, A., AND RIEDI, R. H. Bouds fo the cpcty of weless ulthop etwos posed by topology d ded. I ACM MobHoc 007, pp [9] KESHAVARZ-HADDAD, A., RIBEIRO, V., AND RIEDI, R. Bodcst cpcty ulthop weless etwos. I ACM MobCo 006, pp [0] CHENHUI HU, XINBING WANG, AND FENG WU MotoCst: O the Cpcty d Dely Tdeoffs ACM Mobhoc 009 [] LI, X.-Y., Multcst Cpcty of Weless Ad Hoc Netwos, I IEEE/ACM Tscto o Netwog 008, to ppe. [] LI, X.-Y., TANG, S.-J., AND OPHIR, F. Multcst cpcty fo lge scle weless d hoc etwos. I ACM Mobco 007. [3] PENROSE, M. The logest edge of the do l spg tee. Als of Appled Pobblty 7 997, [4] TAVLI, B. Bodcst cpcty of weless etwos. IEEE Coucto Lettes 0, Febuy 006. [5] SHAKKOTTAI S., LIU X. AND SRIKANT R., The ultcst cpcty of d hoc etwos, Poc. ACM Mobhoc, 007. [6] ULA C. KOZAT AND LEANDROS TASSIULAS, Thoughput cpcty of do d hoc etwos wth fstuctue suppot, Poc. ACM Mobhoc, 003. [7] BENYUAN LIU, PATRICK THIRAN, DON TOWSLEY, Cpcty of Weless Ad Hoc Netwo wth Ifstuctue, ACM Mobhoc, 007. [8] SHI LI, YUNHAO LIU, XIANG-YANG LI, Cpcty of Lge Scle Weless Netwos Ude Guss Chel Model Poc. ACM MobCo, 008. [9] XUFEI MAO, XIANG-YANG LI, AND SHAOJIE TANG, Multcst Cpcty fo Hybd Weless etwos Poc. ACM MobHoc, 008. [0] UICHIN LEE, S.Y. OH, K.-W. LEE, AND M. GERLA RelyCst: Sclble Multcst Routg Dely Tolet Netwos, IEEE ICNP 008. [] ABU-MOSTAFA,YASER S, The Vp-Chevoes Deso: Ifoto vesus Coplexty Leg Neul Coputto, 989. [] J. M. STEELE, Gowth tes of euclde l spg tees wth powe weghted edges. The Als of Pobblty, 64:767787, 988 [3] SCOTT C.-H. HUANG, PENG-JUN WAN, CHINH T. VU, YINGSHU LI, F. FRANCES YAO, Nely Costt Appoxto fo Dt Aggegto Schedulg Weless Seso Netwos. INFOCOM , 007 [4] CHEN WANG, XIANG-YANG LI, CHANGJUN JIANG, SHAO-JIE TANG, YUNHAO LIU, AND JIZHONG ZHAO Sclg lws of etwogtheoetc bouds o cpcty fo weless etwos, IEEE INFOCOM 009