On the Optimal Number of Hops in Infrastructure-based Fixed Relay Networks

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1 Ths full txt ar was r rvw at th rcto of IEEE Commucatos Socty subjct mattr xrts for ublcato th IEEE GLOBECO 5 rocgs. O th Otmal Numbr of Hos Ifrastructur-bas Fx Rlay Ntworks Ara Flora a Halm Yakomroglu Broaba Commucatos a Wrlss Systms (BCWS) Ctr t. of Systms a Comutr Egrg Carlto Uvrsty, Ottawa, Caaa E-al: {aflora, halm}@sc.carlto.ca Abstract * - I ths ar w aalyz th otmal umbr a locatos of fx rao rlay statos formg mult-ho lks frastructur-bas wrlss tworks. Ur th assumtos of usg orthogoal chals for th hos a all lks havg th sam avrag ath loss xot, w show usg th sctral ffccy as a mtrc that th otmal rlay locatos ar at qual trvals alog th straght l btw sourc a stato whvr multho s to b utlz. W show that a sgl-ho fx rao lk ca b ffctly (from a sctral ffccy rsctv) rlac wth th mult-ho lk oly f th sgl-ho SNR has a rlatvly low valu. W trouc a mult-ho crtro to quatfy th sgl-ho SNR valu blow whch a mor ffct mult-ho rlacmt xsts. W also trm th otmal umbr of hos for th mult-ho lks havg rlays sos straght l at qual trvals. I. INTROUCTION It s grally acct that th archtctur of th rst ay cllular tworks ca ot mt th strgt rqurmts vso for 4G cllular systms. Ecoomcally fasbl solutos ar lkly to b bas o som form of mult-ho rlayg allowg uform covrag at vry hgh ata rats a rucg th rqur umbr of xsv cll sts. ult-ho rlayg wth fx rlays s bas o fx rlay statos loy as art of th twork frastructur []. Thr crmtal cost s offst by ruc rqurmts o th mobl trmals, a by th smlcty a ffccy of th rao rotocols volv. Th fx rlay statos ar art of th cllular twork frastructur, thrfor thr loymt wll b a tgral art of th twork lag, sg a loymt rocss. It s cssary to stablsh stratgs a mthos for ffct loymt of fx rlay statos, such that th ovrall cost of th twork s mmz. Effct rao rsourc allocato to twork lmts s a crtcal art of th ovrall twork cost otmzato ffort. O of th o qustos rgarg th loymt of wrlss tworks usg fx rlays s th otmal umbr of hos btw th sourc a stato rao statos. It s mortat to b abl to c wth rasoabl accuracy what cotos t s mor avatagous to s a sgal rctly to stato (may b by crasg th allocat trasmt owr, bawth or tm) or rout th sam sgal ovr a umbr of rlay statos, ach usg lss rsourcs comar to th rlac lk. u to thr ambtous rqurmts, th 4G systms wll most lkly us solutos wth at last two hos (whvr cssary), o of th hos bg btw th fx rlay a th mobl trmal. Th rmag rao lk gog back to th bas stato, rfrr to as th fr, s comrs of a st of fx rlays trcoct through rao lks, a ca hav o or mor hos. Th urlyg sco of th followg scusso s to trm th otmal umbr of hos of th fr orto of th 4G wrlss systm. I [], th ottal gas of a mult-ho lk ar aalyz bas o th Shao caacty of th rao chal. Assumg that th rlays ar qually lac alog th mult-ho lk a that th ovrha crass larly wth th umbr of hos, t s coclu that a mult-ho lk offrs a ostv systm ga scfc rao cotos (low trasmt owr, larg stac, larg ath loss xot) oly for a ruc umbr of hos (u to four). I ths ar w assum that th trmat rlays ca b locat aywhr, ot oly alog straght ls at qual trvals. Also, w tak a ovl aroach by aalyzg th aggrgat sctral ffccy of th mult-ho commucato systm. Sctral ffccy s mortat bcaus th most sgfcat challg aha of th 4G cllular systms s th rovso of cost-ffctv quas-ubqutous covrag wth vry hgh ata rats. A romsg soluto for surassg ths ffculty a xtg th covrag of th classcal cllular twork s by usg rlays. Th atoal rao owr trouc th twork by rlays coms from th wall lug a shoul ot b a to th rao rsourc costs. or sgfcat tha th rlay cosum owr s th ffct of th rlayg schms ovr th aggrgat -to- sctral ffccy. Hghr ovrall sctral ffccy allows bttr us of th avalabl sctrum lcs th most xsv asst of th cllular orator. II. SYSTE ESCRIPTION W cosr th mult-ho lk R -R th fr art of a fx rlay twork wth a -ho lk, as show Fg., whr R a R ar th sourc a th rct fx rlays, rsctvly. Th mssag ca b thr st rctly from R to R (sgl-ho orato th fr art), or th mssag ca b st va - trmat fx rlays ovr hos. sgl-ho * Ths work was suort art by th Natural Sccs & Egrg Rsarch Coucl of Caaa ur artcato rojct WINNER (Wrlss Worl Itatv Nw Rao) - R Fgur. ult-ho lk toology. R IEEE Globcom /5/$. 5 Crow Coyrght

2 Ths full txt ar was r rvw at th rcto of IEEE Commucatos Socty subjct mattr xrts for ublcato th IEEE GLOBECO 5 rocgs. Th rlays ar of th gtal rgratv ty. Each rlay uss oly th formato rcv from ts mmat rcssor th cha,.., thr s o vrsty combg of rcv sgals from multl uhll trasmttrs. Sc th rlays ar fx, th toology of th systm s cosr kow a o-yamc, so vry lttl ovrha s for ackt routg. As th ovrha mssagg rlat to mult-ho fuctoalty wll ot b sgfcat, t was ot cosr as a factor th scusso blow. As wll, th rocssg lays th rlays hav b cosr to b much smallr comar wth th trasmsso tm of rlay ata ackts. Th systm uss a tm-slott rsourc allocato schm, whr ach rao lk s assg a chal th frqucy tm oma. Each trmat rao lk aots a arorat moulato schm rsultg th bst ossbl sctral ffccy bas o th gv sgal-toos rato (SNR) cotos. W aot th followg assumtos as stat [3]:. All hos of th mult-ho lk, as wll as th sgl-ho R -R lk, ar allocat th sam amout of bawth B (Hz), accss a tm-vso mar. Th vual tm rqur to ass a mssag ovr a ho s t =, () Bη whr s th mssag sz bts a η s th sctral ffccy ( bts/sc/hz) of th rao lk ovr ho.. Th tmslots {t } ar cosr orthogoal to ach othr. Also for smlcty but wthout loss of gralty, w assum that all lks orat o th sam carrr frqucy. Although th orthogoalty coto sms cosrvatv, ths s a ralstc assumto; th umbr of hos a wrlss twork s ot xct to b xcssv u to a umbr of ractcal rasos, a such cass th multl accss trfrc may ot allow chal rus a multho cha. I cocluso, th total mssag trasfr tm (TTT), T, rqur to ass a mssag of sz from R to R s th sum of all trmat tmslots: T = t = B η. () = = 3. Sc th chals us by ach ho th mult-ho lk ar orthogoal, o co-chal trfrc s grat from wth th mult-ho lk tslf. Extral trfrc affcts all hos th sam way; wthout loss of gralty, w assum t gras th rcvr thrshols of all rlays by th sam amout. 4. All rao lks ar art of th fr systm a hav smlar rao roagato aramtrs. Sctral Effccy [bts/s/hz] SNR for BER= -5 [B] Sctral ffccy - ractcal valus Shao Lmt Fgur. Sctral ffccy vs. SNR (Th ata for th ractcal sctral ffccy lot assum Bt Itrlav Co oulato [4]. Th ata wr rov by r. Srkat Lk Aryavstakul.). W obsrv that th sctral ffccy for a gv lk has a aroxmat lar cy o th SNR masur B, as show Fg.. W ca aroxmat th sctral ffccy for a sgl ho, η, a for th ho of th mult-ho lk, η, as η K log γ, (3) η K log γ, (4) whr γ a γ ar th SNR for th rct lk R -R a lk, rsctvly, a K s a costat of roortoalty. Th xrssos (3) a (4) rma val for othr gtal moulato schms as log as th lar cy of sctral ffccy o SNR s rsrv as th o show Fg. (straght l assg through org). Although th xrssos (3) a (4) o rsmbl th Shao caacty formula C [bts/sc/hz] = log ( + γ ) 3.3log( +γ ), thy ar ot tcal. Howvr, t ca b obsrv Fg. that for hgh SNR valus (mor tha B), th Shao formula follows a smlar rogrsso law as (3), albt wth a hghr slo tha that of th l of th ractcal sctral ffccy valus. Assumg that all trascvrs at R, R, a th - fx rlays ar tcal ( trms of trasmt owr, trasmt a rcv ata gas, a rcvr os fgur), w ca xrss th ma SNR (xclug shaowg) for th sglho cas a for lk th -ho cas as γ = K ( ) / (5) γ ( ) = K / (6) rsctvly, whr: - = stac R -R - = lgth of th ho - = rfrc clos- stac rao roagato - K = a costat of roortoalty - = ath loss xot I gral th costat K caturs th rao lk systm gas (or th lk costs) trms of rao rsourcs: IEEE Globcom /5/$. 5 Crow Coyrght

3 Ths full txt ar was r rvw at th rcto of IEEE Commucatos Socty subjct mattr xrts for ublcato th IEEE GLOBECO 5 rocgs. P G G λ =. (7) T T R K ( 4π ) PN I (7), P T s th trasmt owr, G T a G R ar trasmt a rcv ata gas, rsctvly, P N s th os owr, a λ s th rao wavlgth; was trouc arlr as th rfrc clos stac for rao roagato. Th total tm rqur to ass th mssag ovr th mult-ho lk (TTT) ca th b xrss, usg (), (4) a (6), as T = = = = log γ log K. III. OPTIAL RELAY LOCATIONS Imrovg th aggrgat -to- sctral ffccy mas rucg th mssag trasfr tm. W ar lookg for th otmal st { } whch wll mmz th valu T (8). Thorm : Cosr a st of ralzatos, =,,, of a ral raom varabl,, wth ma. K a) If K, for ay =,,,, th, (9) = log K log K wth qualty f th raom varabl s costat. b) If K, for ay =,,,, th, () = log K log K wth qualty f th raom varabl s costat. Proof: W cosr th fucto f ( ) =. () log K By stuyg th sg of ts sco rvatv, t ca b show that th fucto f( ) s strctly covx for th trval (8) K a strctly cocav for K () K (3) for =,,,. For th trval whr f( ) s strctly covx, alyg Js s qualty [5], w obta = log K log wth qualty ff Rlacg = K (4) = =... =. (5) = / (4), w obta quato (9), thrfor art a) of Thorm s rov. If f( ) s strctly cocav, th, usg Js s qualty w gt,(6) = log = K K log also wth qualty ff (5) s tru. Rlacg = / (6), w obta quato (), thrfor also art b) of Thorm s rov. Th covxty a cocavty trvals () a (3) ca also b xrss SNR trms as γ (7) γ >, (8) rsctvly, for =,,,. It s to b ot that th mol aot works oly for SNR valus largr tha uty (.., K > ), whch cas th sctral ffccy a th corrsog mssag trasfr tm rma ostv. W cosr th scal cas of a -ho lk wth short trmat lks,.., all trmat ho lgths satsfy th coto (3), a as a cosquc all trmat ho SNRs {γ } satsfy (8). Accorg to (), a -ho lk rsctg th cotos abov woul hav a bttr aggrgat sctral ffccy (smallr TTT) comar to th scal cas of a -ho lk havg all hos of th sam lgth, qual wth th ma ho lgth = /. IEEE Globcom /5/$. 5 Crow Coyrght

4 Ths full txt ar was r rvw at th rcto of IEEE Commucatos Socty subjct mattr xrts for ublcato th IEEE GLOBECO 5 rocgs. Furthr, th scal cas wh all rlays ar lac o th straght l R R, ay cofgurato has a bttr aggrgat sctral ffccy comar wth th stuato wh th - rlays ar strbut alog qual trvals o th l R R. I othr wors, f all rlays ar always lac o th straght l R R, vly sac rlays achv th worst rformac (lowr bou) trms of sctral ffccy. Nxt, cosrg th oost cas of a -ho lk wth log trmat lks, w assum that all ho lgths { } rsct th coto (), thrfor th trmat SNRs {γ } ar as (7). I ths cas, (9) stas tru, a th mmum trasfr tm s b achv wh = = /, for all. Gomtrcally, th smallst valu of th sum of all vual ho lgths s rach wh all rlays ar lac o th straght l R R, whch cas, th sum s th actual stac from sourc to stato: m =. (9) = As a rsult, th bst ossbl gograhcal locatos for th - trmat fx rlays whch woul mmz th sum trm (8) a th TTT T, ar alog qual trvals o th straght l R R. I such a cas, = for =,,,. () W hav rach th aartly cular cocluso that thr s o uqu otmal cofgurato for th locatos of th - rlays; rathr, th otmal cofgurato (th umbr a locato of rlays) o th systm aramtrs. For log trmat hos, th cofgurato wth vly strbut rlays s otmal, whl for short trmat hos, t s th worst. I orr to show ths bhavor grahcally, w cosr th xaml of a two-ho lk as show Fgur 3. W wat to chck f th rlay locato at qual stac from sourc a stato s otmal, a as such, Fgur 4, th rato of TTT for a two-ho lk wth ho lgths (, ) a th TTT for a magary comarso lk wth ho lgths (/, /), s lott (whr = + ). Th valu of th rato blow or abov uty - cats whch cass th rlay th ml scaro s otmal. R R R =/ =/ R R R Fgur 3. Two-ho lk. Rato of Total ssag Trasfr Tms / [m] 5. Fgur 4. Th rato btw th total mssag trasfr tms of a two-ho lk a th lk wth th two hos of lgth. Assumg th systm aramtrs of P T = 3 Bm, G T = B, G R = B, P N = -95 Bm, = m, λ =.6 m (carrr at 5 GHz) a = 3.5, a trmat ho s short f K 9.6 m. () Th stac R R = trms whthr th -ho lk R R has short trmat hos or ot (rfr to Fg. 4). For stacs R R smallr tha th valu () of about m, both trmat hos ar always short a th TTT s largst wh th rlay s lac th ml of th sgmt R R. W ca s from th lot that th bst rlay locato ths cas s as clos as ossbl to thr R or R, whch sms to suggst that for short trmat hos th sgl-ho lk outrforms th -ho lk (wth th framwork of th tal assumtos, th -ho lk wth th rlay lac at o s quvalt to a sgl-ho lk). For larg stacs R R, th stuato chags to th oost: th trmat hos R R a R R ar ow log, a th ml of th sgmt R R s th otmal locato for th rlay; th us of th rlay mrovs th ovrall rformac comar wth th sgl-ho. Somwhr btw ths two xtrm cass ls th curv whr th TTT for rlay locat at sourc or stato (quvalt wth a sgl-ho lk) s qual wth th TTT for th rlay locat at qual stac from lk s (-ho lk). To f out th thrshol valu wh th -ho lk s as ffct as th sgl-ho R R, w st th coto that th TTT s th sam for th sgl-ho a for th -ho lk wth th rlay lac th ml of th sgmt R R : =. () log K log K Smlfyg () w obta IEEE Globcom /5/$. 5 Crow Coyrght

5 Ths full txt ar was r rvw at th rcto of IEEE Commucatos Socty subjct mattr xrts for ublcato th IEEE GLOBECO 5 rocgs. = K, (3) or, = 44 m for th sam systm aramtrs as bfor. As s Fgur 4, for R R stacs = 44 m a largr, th bst rlay locato s th ml of th sgmt R R. I sgl ho SNR trms (3) ca b xrss as γ = (4) W ca coclu that, for th lk R R, f th SNR of th sgl ho s gratr tha, ay -ho lk woul hav a lowr rformac tha th sgl ho. If th SNR s blow, th bst aggrgat sctral ffccy s achv by th -ho lk wth th rlay lac th ml of R R. IV. ULTI-HOP CRITERION Thorm Th ult-ho Crtro Cosr a mult-ho lk rlacmt of a sgl-ho lk, whr all hos hav th sam ath-loss xot, a sam mssag sz a bawth B ar us all hos. A -ho rlacmt lk whch ca achv a bttr aggrgat sctral ffccy tha th rct lk xsts oly f γ, (5) whr γ s th SNR of th rct lk. Proof: Wth th mssag sz a bawth B bg th sam, th mult-ho lk has a bttr aggrgat sctral ffccy tha th rct lk f = η η. (6) Th coto (6) smly stats that orr for th mult-ho lk to b mor ffct, th tm rqur to ass a mssag of a gv sz from R to R ovr th rct lk must b logr tha th tm rqur for th sam orato ovr th mult-ho lk. Usg Thorm a) a (9), w ca xrss th lowr bou o th TTT for th -ho as T = log K ( γ ). (7) log Th xrsso (7) shows th smallst ossbl TTT that ca b achv usg hos, gv that (7) s tru. Usg (7), th qualty (6) ca b rwrtt as (8) log( γ ) log γ whch ca b smlf to th xrsso (5). If for a gv sgl-ho lk, th coto (5) s ot mt, a -ho lk rlacmt wth a bttr ovrall sctral ffccy os ot xst, o mattr whr th rlays ar locat. Th qualty (5) rrsts a quattatv crtro that ca b us to c whch stuato a mult-ho lk coul b cosr. Th brak-v SNR valus (5) ar lott Fgur 5 for varous valus of th ath loss xot. Th lots show th varato th thrshol (or brak-v ) SNR valu at whch a mor ffct mult-ho altratv to th gv sgl-ho lk bcoms fasbl. W obsrv Fgur 5, for stac, that f th ath loss xot of all trmat hos has to b = 3.6, th accorg to (5), thr xsts a ossbl 3-ho cofgurato (th rlays vly strbut alog th l R R 3 ) wth a bttr aggrgat sctral ffccy, as log as th sgl-ho lk SNR s lss tha 8.6 B. Lt us cosr a -ho wth th - rlays lac at qual trvals o th straght l btw th sourc a stato. Comarg th TTT for a -ho lk vrsus a (+)-ho lk, usg a smlar ratoal as bfor, w f a qualty whch ca b us to trm th otmal umbr of hos from sourc to stato: ( + ) γ ( + ). (9) If th SNR of th sgl-ho lk rscts (9), th (+)-ho lk wll achv a lowr TTT comar wth th -ho lk,.., ag th (+) th ho woul mrov th aggrgat -to- sctral ffccy. Th xrsso (9) s lott Fgur 6, for varous valus of th ath-loss xot. Plottg (9) for valus of =,, 3, rsults curvs whch lmt th SNR valus abov whch hos ar otmal, a blow whch + hos ar otmal. Ths vs th lot ara rgos whr,, 3, hos ar otmal, as ct Fgur 6 (show u to 4 (or mor)-ho rgo). Ths ca b xrss aalytcally as ( ) ( ) ( + ) γ ( + ). (3) If (3) s tru, th th otmal umbr of hos s +. Sgl-ho SNR [B] = 4 =3 = Path Loss Exot Fgur 5. SNR valus abov whch th sgl-ho has bttr sctral ffccy comar to ay -ho. IEEE Globcom /5/$. 5 Crow Coyrght

6 Ths full txt ar was r rvw at th rcto of IEEE Commucatos Socty subjct mattr xrts for ublcato th IEEE GLOBECO 5 rocgs. Sgl-ho SNR [B] > ->3 3->4 O-ho rgo Four(or mor) -ho rgo Two-ho rgo Path Loss Exot Thr-ho rgo Fgur 6. SNR valus blow whch a (+) ho s mor ffct tha a -ho lk. Rmarks:. Th crtro vlo abov uss as th rformac mtrc th cosumto th frqucy-tm la for trasfrrg mssags from sourc to stato. Th rgy rqur for th trasfr s ot cosr as a mtrc; that s, th atoal owr srt th systm by trmat rlays s cosr fr.. Sc th vlomts ths ar ar bas o ma SNR valus ot clug shaowg, ths rsults ar val as a statstcal avrag ovr a st of mult-ho lks; th rsults abov ar ot bg o a gv artcular ralzato of a mult-ho lk. 3. As xct, th cas of = (o fx rlay btw R a R ), th qualty (5) bcoms γ, cocorac wth (4). For xaml, f th roagato xot has a valu of 3, o rlay lac th al locato (rght th ml of th lk R -R ) woul b ffct oly f th SNR of th lk R -R s lss tha 8 (9 B). Furthrmor, valuatg (5) for =, a (9) for = (that s, comarg sgl-ho wth two-ho) rsult th sam coto: γ. 4. For th artcular cas of rlays lac straght l at qual trvals, (5) a (9)-(3), a thrfor Fg.s 5 a 6, ar comlmtary. For stac, w obsrv from (5) a Fg. 5 that, for = 3.6, thr xsts a - ho soluto bttr tha th sgl-ho ty wh γ.8 B; a thr xsts a 3-ho soluto bttr tha th sgl-ho ty wh γ 8.6 B; a thr xsts a 4- ho soluto bttr tha th sgl-ho ty wh γ 7. B. Bas o ths obsrvatos, o ca uc that th otmal umbr of hos s wh 8.6 B γ.8 B. But, t s uclar whthr th otmal umbr of hos s or 3 wh 7. B γ 8.6 B. Th aswr of ths qusto s obta from (3) a Fg. 6: th -ho soluto s otmal wh.8 B γ.8 B. 5. Th mult-ho crtro s gral alcabl to mult-ho lks wth all vual hos havg smlar rao roagato charactrstcs,.., qual ath loss xots. I th scal cas wh th mobl accss lk (th last ho of a 4G cllular lk usg fx rlays) has th sam ath loss xot as th fr systm, th mult-ho crtro ca b xt to covr th tr lk btw th bas stato a th mobl trmal. 6. Oc aga, (9) s val oly for statstcal avrags ovr sts of mult-ho lks, wth rlays lac straght l at qual trvals. For a gv lk, kowg th valus for th sgl-ho SNR a th ath loss xot, w ca stmat th otmal umbr of hos achvg th bst -to- aggrgat sctral ffccy. Howvr, ractc, th trmat hos of th mult-ho lk may hav ffrt ath loss xots, a atoal raom ath losss u to shaowg, whch cas th rcto gv by (9) may ot b accurat. V. CONCLUSIONS Usg Js s qualty, w hav show that for rlatvly hgh SNR valus, a sgl ho lk has bttr sctral ffccy comar wth a -ho rlacmt; for rlatvly small SNR valus, o th othr ha, a mor ffct -ho lk s ossbl, wth th otmal locatos of mult-ho gtal rlays bg at qual trvals alog th straght l btw th sourc a stato. A ovl quattatv crtro s vlo, offrg thrshol ma SNR valus blow whch a -ho rlacmt shoul b cosr ovr a sgl-ho lk. orovr, usg th qualty (9), th otmal umbr of rlays a mult-ho lk ca b trm, ur th assumtos that all lks hav th sam ath loss xot a th rlays ar locat at qual trvals. Atoal rsarch to th statstcal strbuto of sctral ffccs for mult-ho lks may brg furthr clarfcatos o th rorts of cllular systms usg fx rlays for mult-ho commucatos. REFERENCES. R. Pabst, B.H. Walk,.C. Schultz, P. Hrhol, H. Yakomroglu, S. ukhrj, H. Vswaatha,. Lott, W. Zrwas,. ohlr, H. Aghvam,.. Falcor, a G.P. Fttws, Rlay-bas loymt cocts for wrlss a mobl broaba rao, IEEE Commucatos agaz, vol. 4, o. 9,. 8-89, St Lott,. Wckrl, a. Sbrt, Schulg wrlss multho tworks, Proc. of Itratoal Symosum of Prformac Evaluato of Comutr a Tlcommucato Systms (SPECTS 4), Sa Jos, Calfora, 5-9 July H. Bolukbas, H. Yakomroglu,. Falcor, a S. Pryalwar, "Fasblty of rovg hgh ata covrag cllular fx rlay tworks", Worl Wrlss Rsarch Forum mtg o. (WWRF- ), 3-4 Novmbr 4, Toroto, Caaa. 4. G. Car, G. Tarcco, E. Bglr, Bt-trlav co moulato, IEEE Tras. Ifo. Thory, vol. 44, o. 3, , ay T.. Covr a J. A. Thomas, Elmts of Iformato Thory, Joh Wly a Sos, 99. IEEE Globcom /5/$. 5 Crow Coyrght