Efficient Asynchronous Channel Hopping Design for Cognitive Radio Networks

Transcription

1 Effcen Asynchronous Channel Hoppng Desgn for Cognve Rado Neworks Chh-Mn Chao, Chen-Yu Hsu, and Yun-ng Lng Absrac In a cognve rado nework (CRN), a necessary condon for nodes o communcae wh each oher s ha hey have a rendezvous (swchng o he same channel a he same me). Mos of he exsng rendezvous-guaraneed schemes have some undesred requremens, such as demandng role preassgnmen, a common conrol channel, or a synchronous envronmen. In hs paper, we frs defne he complee rendezvous problem and hen propose a saggered channel hoppng scheme (SCH) o solve he complee rendezvous problem. SCH ulzes he Chnese remander heorem o enable a node o have a rendezvous wh any of s neghbors. We have proved he correcness of SCH hrough Chnese remander heorem. We have also analyzed he performance of SCH n some sample scenaros. Smulaon resuls verfy he superory of SCH n erms of me o rendezvous (R), sandard devaon of R, and maxmum R. Index erms Cognve Rado Neworks, Dynamc Specrum Access, Rendezvous Guaranee, Channel Hoppng, Chnese Remander heorem I. INRODUCION Wreless specrum s a scarce resource bu mos lcensed specrum s sll underulzed []. o ncrease specrum ulzaon, Cognve Rado (CR) s a promsng echnology whch enables an unlcensed user (secondary user, SU ) o recognze specrum holes and change s CR ranscever parameers as needed o access an dle channel whou producng nruson o any lcensed user (prmary user, PU). A PU has prory o use s channel and when a PU uses s channel, SUs ha deec hs ransmsson wll mark he channel as occuped and swch o anoher avalable channel. Each SU perodcally monors all he channels o oban s avalable channel se. An SU can also esmae he probably of a parcular channel based on hsorcal monorng []. SUs may group ogeher o form a Cognve Rado Nework (CRN). o communcae wh ohers, a node and s nended recpen mus une o he same channel concurrenly. When hs happens, we say hese wo nodes have a rendezvous. How an SU can have a rendezvous wh any of s neghbors n a CRN s a challengng problem because he avalable channel se for each SU s changed over me and space. I s dffcul for a node n a CRN o mee s neghbors f her channel hoppng schedules are unavalable. he rendezvous problem has araced a lo of aenon recenly. Several mul-channel rendezvous-guaraneed proocols have been proposed for wreless ad hoc and sensor neworks [3] [5] However, hey do no work well n a CRN because he PU occupancy ssue s no consdered. here exs many Copyrgh (c) 3 IEEE. Personal use of hs maeral s permed. However, permsson o use hs maeral for any oher purposes mus be obaned from he IEEE by sendng a reques o pubs-permssons@eee.org. he erms SU and node may be used nerchangeably o represen a secondary user n hs paper. ABLE I CLASSIFICAION OF CRN RENDEZVOUS GUARANEE PROOCOLS synchronous asynchronous asymmerc [9](RRCH) [6], [7](A-ACH), [9](ARCH) symmerc [8], [], [4] [7](S-ACH), [9](SARCH), [4], [8], [], [3] [9], ours rendezvous soluons for CRNs [6] []. Some of hem use a leas one common conrol channel (CCC) [], [], [5], []. Snce he avalable channels for dfferen users may be dfferen, a CCC may no exs a all. Even here exss a CCC, suffers from he boleneck problem due o exensve conrol packe ransmssons. Some oher schemes avod usng a CCC o provde rendezvous among nodes, bu only a par of hem provde rendezvous guaranee [6] [], [4], [8] []. Among hem, some requre a node o equp wh wo or more rados, whch mpose hgher hardware cos. For he sngle-rado proocols, n general, some knd of channel hoppng scheme s used o acheve rendezvous guaranee. hese rendezvous-guaraneed proocols can be classfed by he followng wo facors: synchronous or asynchronous: Wheher SUs sar channel hoppng a he same me or no. symmerc or asymmerc: Wheher a node s assgned o a specfc role or no. A symmerc proocol does no requre role pre-assgnmen. Based on hs classfcaon, exsng rendezvous guaranee proocols and he one proposed n hs paper are lsed n able I. Several soluons requres role pre-assgnmen where dfferen roles have dfferen channel hoppng sequences [6], [7] (A-ACH), [9] (ARCH) 3. A node acs as eher a sender or a recever. A man ssue of hese asymmerc proocols s ha here s no rendezvous guaranee for nodes of he same role. Furhermore, he permanen role assgnmen mples a node canno ac as a forwarder (o receve a packe from one node and hen forward o anoher node). hs grealy lms he applcaons of hese proocols. here are some rendezvous proocols ha do no rely on role pre-assgnmen [7] (S-ACH), [8], [9] (SARCH), [], [4], [8], [9], [], [3]. Some of hem requre synchronzaon and can be appled o some wo asynchronous channel hoppng (ACH) proocols were proposed n Reference [7]: Asymmerc ACH and Symmerc ACH, denoed as A-ACH and S-ACH respecvely hereafer. Nodes runnng A-ACH requre role preassgnmen. 3 hree channel hoppng proocols were proposed n Reference [9]: asynchronous rendezvous channel hoppng scheme, rendezvous couple channel hoppng scheme, and symmerc asynchronous rendezvous channel hoppng scheme, denoed as ARCH, RRCH, and SARCH respecvely hereafer. Nodes runnng ARCH and RRCH requre role pre-assgnmen.

2 lmed applcaons. he soluons n he asynchronous, symmerc class can be appled o almos all applcaons n CRNs. In hs class, some provde paral rendezvous guaranee n ha rendezvous occurs only on some channels [9] (SARCH), [8]. Some ohers provde complee rendezvous guaranee n ha any par of nodes have a leas one rendezvous on all he channels [7](S-ACH), [4], [9], [], [3]. he complee rendezvous guaranee soluons are consdered more flexble and robus snce hey provde hgher communcaon probables when some channels are occuped by PUs. We found ha hese complee rendezvous guaranee proocols may suffer from long me o rendezvous (R), low channel ulzaon, and/or larger R sandard devaon beween any par of nodes. hs may produce a long delay for packe ransmssons. A more dealed revew of he complee rendezvous guaranee proocols n he asynchronous, symmerc class can be found n Secon II. he proocol proposed n hs paper, saggered channel hoppng scheme (SCH), belongs o he asynchronous, symmerc class. SCH s a complee rendezvous guaranee soluon. he man dea of SCH s o use he concep of rple o adjus he nsananeous rao of usng a channel n a saggered way. he rple concep was frs nroduced o solve he sleep/acve schedulng problem n a sngle-channel wreless sensor neworks [5]. Ulzng he Chnese remander heorem (CR), hs scheme provdes rendezvous guaranee for any par of nodes. A rple consss of wo prmes and a rao, denoed as (p, p, c c ). o mplemen he arge rao of usng a channel, a rple (p, p, c c ) s seleced such ha he rao of usng channel s eher p or p n a shor me duraon; however, he rao of usng channel s close o he arge rao of usng n a long perod of me. he acve/sleep schedulng scheme proposed n Reference [5] does no work n a mul-channel CRN. Drecly applyng hs scheme n a CRN canno generae a schedule for mulple channels. For example, schemes such as how o deermne he prmes used for dfferen channels and how o deermne he channel o be swched o for a slo denfed by mulple prmes canno be found. he PU occupancy ssue s also a unque feaure n CRNs. hus, some more effors are needed o apply he rple concep o a rendezvous soluon n a CRN. Usng he concep of rple, he man dfference of SCH from exsng rendezvous proocols usng CR s s flexbly. In SCH, he rao of usng a channel s no dencal all he me. hs enables a node runnng SCH o acheve rendezvous guaranee o any oher node wh lower R and lower sandard devaon of R. A premaure verson of he work has been publshed [6]. he exenson/dfference of he earler work s lsed below. he SCH scheme s revsed (usng hree nsead of wo canddae prmes and changng he sraegy of prme allocaon, ec.). Some more revews of recen rendezvous soluons are ncluded n hs verson. A complee proof of he correcness of SCH and a performance of SCH are also provded. We have also conduc more smulaons wh new mplemenaon of laes soluons o evaluae he performance of SCH. he man conrbuons of hs paper are summarzed as follows. ) Whou usng a CCC or neghbor nformaon, we desgn an asynchronous CRN rendezvous proocol provdng lower R and sandard devaon of R (Secon IV). ) Prove ha SCH provdes complee rendezvous guaranee (heorem 3 n Secon IV-C). 3) Provde heorecal analyss of SCH n a smplfed envronmen (Secon V). 4) Provde smulaon o evaluae he performance of SCH (Secon VI). II. RELAED WORKS In hs secon, we revew fve represenave complee rendezvous guaranee proocols belongng o he asynchronous, symmerc class: S-ACH [7], Jump-say (denoed as JS hereafer) [4], CRSEQ [9], a scheme based on relaxed dfferen se (RDS) [3], and E-AHW []. S-ACH s an exenson of A-ACH whch ulzes he nersecon propery of quorum sysems o provde rendezvous beween any wo channel hoppng sequences [7]. he A-ACH proocol operaes smlar o a grd quorum sysem. However, nsead of usng he nersecon beween a column and a row n a grd, nodes runnng A-ACH ulze he nersecon beween a column and a span n a grd where a span consss of one elemen from each column. In A-ACH, a node s assgned o be eher a sender or a recever and uses an M M grd o deermne s channel hoppng sequence, where M s he number of channels. A sender and a recever use dfferen grds. he grd beng used for sendng purpose s generaed by randomly assgnng M channels o M dfferen columns. Each column of he grd beng used for recevng purpose s generaed by randomly assgnng M channels o M dfferen spans. A node s channel hoppng sequence s obaned from he grd beng used, n a lef-o-rgh and opo-down way. S-ACH, bul on op of A-ACH, enables a node o swch beween sendng and recevng o provde complee rendezvous. A problem of S-ACH s he long R among nodes when M or he number of nodes s large. If he number of nodes s n, he R can be as long as 6M log n. A cycle of he channel hoppng sequence of a node runnng he JS proocol consss of a jump-paern and a say-paern [4]. Assume ha he number of avalable channels n he nework s M. he lengh of a jump-paern and a say-paern s P and P me slos respecvely where P s he smalles prme ha s greaer han M. In he frs slo of he jumppaern, a node randomly selecs an neger I ha s nongreaer han he prme number P as he channel o be swched o. A node also randomly selecs a rae R, whch s nongreaer han M, o deermne he channel hoppng sequence of he jump-paern. For he k-h slo n he jump-paern, he channel o be swched o s ((I + kr ) mod P ) +. he say-paern consss of P dencal values. A node says on channel R for he las P slos n he frs cycle. he channel o be swched o n he say-paern for he nex cycle s (R +) mod M. A downsde of he scheme s he hgher sandard devaon of he number of rendezvous. Smlar o JS, a channel hoppng sequence generaed by CRSEQ [9] also consss of a channel hoppng perod and a

3 3 channel sayng perod. Assume ha he number of avalable channel n he nework s M. A node selecs he smalles prme P whch s non-lesser han M and generaes P channel hoppng sequences whle each of whch has a cycle of 3P - me slos. he frs P - slos of a cycle form a channel hoppng perod and he las P slos buld a channel sayng perod. he j-h channel hoppng sequence sars wh channel j = j(j+) and he channel o be swched o n he (k+)-h slo of a channel hoppng perod s (( j + k) mod P ) +, k =... P. he channel o be swched o n he las P slos of a cycle for he j-h channel hoppng sequence s j (mod M)+. A undesrable feaure of CRSEQ, smlar o JS, s he hgher sandard devaon of he number of rendezvous. A scheme whch ulzes RDS o buld nodes channel hoppng sequences has been proposed recenly [3]. In hs scheme, he frs P me slos are n he lsenng sage where P s he leas prme greaer han or equal o he number of avalable channels. In lsenng sage, a node unes o s frs avalable channel. A lsenng sage s followed by an accessng sage conssng of 3P slos. he channel o be swched o n an accessng sage s deermned by a Dsjon RDS (DRDS). Specfcally, f here are m avalable channels for a node, a DRDS wh m dsjon RDSs s generaed. Each RDS corresponds o an ndvdual channel and denfes some slos n whch node wll swch o he correspondng channel. A flaw of hs scheme s hgher sandard devaon of he number of rendezvous. hs scheme also performs worse han some oher rendezvous guaranee proocols. E-AHW [] also has smlar concep as ha of JS. A channel hoppng cycle consss of m nner channel hoppng sequences, where m s he number of channels. A nner channel hoppng sequence comprses 49 elemenary channel hoppng sequences each of whch consss of hree subsequences. wo modes, WAI and HOP, are defned for subsequences n E- AHW. In HOP mode, a node u hop o channel ( u +r u k ) mod P + a slo k, where u and r u s he channel hoppng dsance and he nal channel of node u, respecvely. In WAI mode, a node u always says a he channel w u whch s deermned by wang marx of u. A node u generaes a 49- elemen ID sequence by concaenang and node u s MAC address. Each elemen of hs ID sequence s used o generae an elemenary channel hoppng sequence. Specfcally, wh an elemen of,, and, he correspondng elemenary channel hoppng sequence s WAI-WAI-HOP, HOP-HOP-HOP, and WAI-HOP-HOP, respecvely. E-AHW s an sophscaed desgn and performs well. However, sll suffers from he hgh sandard devaon of he number of rendezvous. III. PRELIMINARY AND PROBLEM DEFINIION In hs secon, we frs nroduce he Chnese remander heorem (CR) and hen defne he problem o be solved n hs paper. A. Chnese Remander heorem he CR can be descrbed as follows: heorem. Le p,p,..., p n be parwse relavely prme posve negers. For any gven neger a, a,..., a n, here exss a leas one soluon X for he followng lnear congruence sysem. X a (mod p ) X a (mod p ) (). X a n (mod p n ) he CR has been wdely ulzed o desgn awake/sleepng schedule for wreless sensor neworks [5], [7]. I has also been used for desgnng channel hoppng sequences for CRNs [4], [9], []. When used for desgnng channel hoppng sequences, for example, a smple way s o le a node swches o a parcular channel x every p slos. Le a denoe he slo number offse (mod p ) beween hose ha are labeled by node and he reference ones 4. Anoher node j has he smlar parameers of p j and a j. he soluon X s he reference slo number wheren wo sensor nodes swch o channel x. B. Problem Defnon In hs paper, we consder a CRN, denoed as G(V, E) where V s he se of SU n he CRN and E s he se of lnks connecng SUs. An SU s equpped wh one cognve rado and perodcally monors all he channels n he CRN. Each node keeps rack of he occupancy condon of all he channels o esmae he avalably probably of each channel. he probably of occupancy for a channel j, esmaed by node, s denoed as po (j). Nodes hop among he avalable channels whch are usually a small poron of he whole channel se n pracce [8]. We assume ha a cycle me for node s dvded no a seres of equalszed slos 5. he whole nework has he same reference slo numbers (ermed global slo numbers) whch can be arbrarly labeled wh consecuve negers. Each node can also assgn slo numbers (ermed local slo numbers) wh consecuve negers ndvdually. A node runnng SCH uses s local slo number o deermne he channel o be swched o a each slo. For he same slo, he global slo number and he local slo number may be dfferen and we need wo varables o dsngush hem. We use varable o presen he global slo number and varable o represen he local slo number of node. We also defne a funcon LG( ) o ransform o. Each node has a unque channel hoppng sequence, CHS, whch conans node s channel hoppng schedule for each slo n me nerval. CHS can be represened by a ls (s, ) where s, s he channel ha node swches o a slo. Le r,j () represen he rendezvous saus for nodes and j a global slo number. We have {, f s, = s r,j () = j,j, L G( ) = L G( j ) =,, oherwse. 4 he whole nework has he same reference slo numbers whch can be arbrarly labeled wh consecuve negers. 5 How o deermne he number of slos n a cycle wll be descrbed n Secon IV-B, afer Equ. (7). ()

4 4 Varable M po (j) CHS s, r,j () R,j R,j rc,j w F ar F j sub N p P j p j,k P cb P P C F j prac F p ns ABLE II VARIABLE LISING Defnon he number of avalable channels occupancy probably of node for channel j node s channel hoppng sequence he channel node wll swch o a local slo number he rendezvous saus for nodes and j a global slo number he number of rendezvous for nodes and j whn me nerval me o rendezvous for nodes and j whn me nerval he se of channels ha have experenced a rendezvous already beween nodes and j n me. he number of slos n a he arge rao of channel he j-h sub-rao of channel he number of paerns of he rple he j-h rple of channel he k-h prme of he j-h rple of channel he combne rple of channel he -h canddae prme lss he canddae prme pool he j-h praccal sub-rao of channel olerance hreshold for F j selecon prac he smalles prme no beng used by SCH he number of rendezvous for nodes and j whn me nerval, denoed as R,j, can be obaned by R,j = r,j (k). (3) k= he average me o rendezvous for nodes and j whn me nerval, denoed as R,j, s hus gven by R,j = R,j. (4) Denoe rc,j as he se of channels ha have experenced a rendezvous already beween nodes and j n me. ha s, rc,j = {s, s, = s j,j, L G( ) = L G( j ) =, }. Some mporan noaons beng used n hs paper are lsed n able II. he rendezvous problem o be solved n hs paper can be defned as follows. Defnon. he Complee Rendezvous Problem Gven a CRN G(V, E) wh M avalable channels and a me duraon, fnd a CHS for each node such ha SU can have a rendezvous wh any neghbor n any channel and he average me o rendezvous s mnmzed. ha s, gven he number of communcaon pars np, he goal s o mnmze,j V, j R,j (5) np under he consran rc,j = M. IV. PROPOSED SOLUION he proposed SCH scheme s an asynchronous rendezvous proocol. We consder he asynchronous envronmen defned recenly where me skews beween wo nodes are neger mulples of a me slo [9]. he SCH proocol ulzes he CR o arrange each SU s channel hoppng sequence. In he followng, we frs nroduce he concep of rple whch Fg.. s f f+ f+ f f + f+ cycle cycle + slo 3 w An example me paronng n SCH where a cycle conssng of f s used n he channel hoppng mechansm of SCH. hen, we presen how a rple s generaed o solve he complee rendezvous problem. A. he rple Concep In SCH, me s dvded no a seres of cycles, each of whch consss of several s. A s furher paroned no w slos, as shown n Fg. (how o deermne he value of w wll be descrbed laer). Noe ha wo nodes do no need o have algned s. In a nework wh M avalable channels, an SU ams o swch among M channels wh equal rao of M. We defne he rao of swchng o channel o be he arge rao of channel, denoed as F ar. Each arge rao F ar can be mplemened by a rple conssng of wo prmes and a rao, denoed as (p, p, c c ), where p < F ar and p > F ar. When a rple (p, p, c c ) s chosen, a cycle correspondng o hs rple s composed of c + c s. For each cycle, an SU swches o channel every p slos (swches o channel when a slo s labelled as a mulple of p ) n c s. In he oher c s of a cycle, he SU swches o channel every p slos. Afer deermnng he rple, an SU needs o decde he paern o be used. A paern s a permuaon of s usng p and p and s represened by an ordered sequence of c + c elemens. For nsance, (,3,3,3,3) represens he paern of he rple (3,, 4 ) where he prme s used n he frs. A rple (p, p, c c ) has a oal of N p = Cc c+c paerns. o mplemen he arge rao of usng a channel, F ar, a rple (p, p, c c ) s seleced such ha he rao for channel s eher p or p n a ; however, he rao of usng channel s close o F ar n a cycle. An example of he rple mplemenaon of he frs wo channels n a nework wh M = 4 s shown n Fg.. Assume ha node A selecs (, 5, 5 ) and (3, 7, 4 3 ) for channels and, respecvely. he randomly seleced paern s (5,5,,5,5,5) and (3,3,3,3,7,7,7) for channels and, respecvely. In, node A swches o channel every fve slos and swches o channel every hree slos. he lengh of a cycle for channels and s sx and seven, respecvely. Noe ha F ar s mplemened by and F ar s mplemened by Usng a rple o deermne a node s channel hoppng schedule, a node runnng SCH has he followng propery. heorem. Nodes A and B use (p, p, c c ) and (p 3, p 4, c 3 c 4 ) o deermne her channel hoppng schedule for a parcular channel, respecvely. If p, p, p 3, and p 4 are all dfferen, nodes A and B are guaraneed o have a rendezvous a channel n any.

5 5 channel slo channel slo cycle cycle Fg.. An example of rple operaon. Node A uses rple (, 5, 5 ) and paern (5,5,,5,5,5) o deermne he schedule for channel and rple (3, 7, 4 ) and paern (3,3,3,3,7,7,7) for channel. he number n each 3 s he prme beng used whle he number n a slo represens he channel o be swched o n he slo. Fg. 3. channel of A slo channel of B slo Nodes A and B selec he same rple and dfferen paerns Proof: Snce he prme numbers seleced by nodes A and B are dfferen (hey are coprmes), wh he help of he CR, hese wo nodes mee a slo X whch s he soluon o he Equ. (6). { X aa (mod p A ), p A = p or p X a B (mod p B ), p B = p 3 or p 4 (6) Usng a sngle randomly seleced rple o mplemen he hoppng of a parcular channel may produce poor rendezvous f wo asynchronous nodes use he same rple. For example, consder a 4-channel CRN n whch wo asynchronous nodes A and B (A sars a cycle one slo before B) use an dencal rple bu dfferen paerns o deermne he me hey should swch o channel, as shown n Fg. 3. wo nodes use he same prme n a do no have a rendezvous f hey are asynchronous. hs mples nodes A and B can communcae only n wo (s hree and fve) ou of sx s. o provde more rendezvous, n SCH, a arge rao of channel s paroned no wo sub-raos, F sub and F sub. Dfferen channels have he same sub-raos. ha s, F sub = F sub =... = F sub M and F sub = F sub =... = F sub M. Sub-raos F sub and F sub are mplemened by wo rples, denoed as P and P, respecvely. A combned rple, denoed as P cb, consss of P and P. Usng wo rples P and P o mplemen a arge rao of a channel enables a node o use wo prmes o deermne he schedule of a channel n a. hs guaranees ha any par of nodes wll always have a rendezvous (wll be shown n heorem 3). Le a rple P j ha deermnes he hoppng schedule of channel be represened by (p j,, p j,, c j, /c j, ), = o M, j= o. For each P, one of he prmes s a dedcaed prme whch s dedcaed o channel and he oher prme s a shared prme whch can be used by channel and he oher channels. he wo prmes used for each P are boh dedcaed prmes. Snce any F ar, = o M, s mplemened by wo rples, a oal of four prmes wll be ulzed o deermne when channel should be swched o. Among hese four prmes, hree of hem are dedcaed prmes. In SCH, we assume ha p,, p, and p, are dedcaed prmes. B. Combned rple Generaon he essence of SCH s he generaon of he combned rple for each channel. o generae a combned rple for channel, wo man asks need o be handled: deermne he wo sub-raos, F sub and F sub, and selec four prmes for each P cb. hese wo asks correlae closely. In SCH, he laer s handled frs. Four canddae prme lss, P, P, P 3, and P 4, are generaed n advance and he four prmes p,, p,, p,, and p, n each P cb, = o M, are assgned from P, P, P 3, and P 4, respecvely. Whou loss of generaly, we assume F sub s no less han F sub and p, < p, < p, < p,. Snce p,, p, and p,, = o M, are dedcaed prmes, a oal of 3M prmes are needed for P, P 3 and P 4 wh M prmes for each of hem. he number of prmes n P depends on he value of M. o generae four canddae prme lss, a canddae prme pool, P C, conssng of connuous prmes sarng from wo s bul frs. he prmes n P C s sored n ncreasng order and we use P C[] o represen he -h elemen of he pool, N. he generaon of canddae prme lss mus sasfy he arge rao consran F sub + F sub = M. he consran wll be volaed when boh F sub and F sub are oo large. In each rple, he larger prme deermnes he lower bound of he rao mplemened. ha s, he consran wll be volaed f he sum of he lower bounds of wo rples s larger han M. In SCH, P C[] s assgned o P nally snce P mus have a leas one prme. hen we verfy f sum of he lower bound, P C[] + P C[+M], s larger han M. If so, P C[] s assgned o P and P C[3] + P C[3+M] s verfed. hs process s repeaed unl a prme P C[] sasfyng P C[] + P C[+M] M s found. hen, he prmes P C[] o P C[ + M ], P C[ + M] o P C[ + M ], and P C[ + M] o P C[ + 3M ] are assgned ls P, P 3, and P 4, respecvely. Wh hese four canddae prme lss, he four prmes for each P cb, = o M, are assgned as follows: One of he prmes n P s randomly assgned o p, ;, he prmes n P, P 3 and P 4 are sequenally assgned o p, p, and p,, respecvely. Afer deermnng he four prmes for all he combned rples, he nex job s o decde he wo sub-raos, F sub and F sub for each channel. In SCH, he value of F sub s chosen frs snce he prmes used o generae F sub are larger and he range of he rao generaed s much smaller han ha of F sub. he value of F sub s deermned from he larges prme n P 3. ha s, F sub =. he value of F sub s hus p, M

6 6 M F sub. When he sub-raos are obaned, we need o fnd ou a rao c j, /c j, o use wo prmes p j, and p j, such ha he praccal j-h sub-rao, F prac j, s equal o he sub-rao F sub j, = o M, j = o. he calculaon of F prac j can be found n Equ. (7). F prac j = c j, p j, + p j, c j, + c j, c j, he lengh of should be large enough o mplemen he arge rao for each channel. Specfcally, s se o conan u s where u =max {(c j, + c j, ) = o M, j = o }. o guaranee rendezvous a a channel n a beween wo nodes, here mus be a rendezvous slo ha s no shared by any oher channel. Such a rendezvous slo can be found by usng he wo prmes beng used for channel n he and a prme ha s no used by SCH. Because he nerval beween wo such rendezvous slos s he produc of hese hree prmes [3], n SCH, we se he value of w (he number of slos n a ) o be larger han he produc of he larges prmes beng used for any channel and he smalles prme no beng used by SCH. Consderng an asynchronous envronmen, he value of w s furher mulpled by wo o guaranee a rendezvous. ha s, we se w o be larger han p, M p, M p ns, where p ns s he smalles prme no beng used by SCH. Such a seng ensures ha suffcen overlappng beween s of wo nodes o guaranee a rendezvous n any channel n an asynchronous envronmen. In SCH, each sub-rao for a channel s mplemened by a rple whch uses wo prmes o realze he sub-rao. In general, s no easy for he praccal sub-rao (mplemened by wo prmes) o be exacly equal o he sub-rao. We use a hreshold F o lm he dfference beween he praccal sub-rao and he arge sub-rao as shown n Equ. (8). F sub j F prac j F sub j F. (8) F prac j s consdered o be equal o F sub j f he rao of he dfference beween he sub-rao and he praccal sub-rao (he numeraor of Equ. (8)) o he sub-rao s no larger han he hreshold F. he seleced rao c j, /c j, mus sasfy Equ. (8). If here are mulple canddaes, we randomly selec one. Wh such a hreshold F, he SCH can be consdered as a flexble scheme. If usng a precse dencal rao among dfferen channels s a concern, a smaller hreshold can be used. If no, a larger hreshold can be used o allow dfferen channels use a lle dfferen raos. he process of he combned rple selecon s he same for M channels. he paern for each rple s randomly seleced. he pseudo code for he combned rple selecon s shown n Algorhm. In lnes o 5, four prmes for each P cb are seleced. he wo sub-raos, F sub and F sub, are deermned n lnes 7 and 8. he rple o mplemen a sub-rao s assgned n lne. Each combned rple s used o denfy he slos o be swched o a parcular channel. I should be noed ha here may exs some shared slos whch are denfed by mulple combned rples and some empy slos whch are denfed by no combned rple. A node should swch o a sngle Algorhm : Combned rple Selecon npu : M, P C, F oupu: P j, = o M and j =, P [] P C[]; ; 3 whle P C[] P C[+M] M 4 P [],.., P [ ] P C[],.., P C[ ];, = o M, j = o,. (7) 5 + ; 6 end 7 P [],.., P [M] P C[],.., P C[ + M ]; 8 P 3 [],.., P 3 [M] P C[ + M],.., P C[ + M ]; 9 P 4 [],.., P 4 [M] P C[ + M],.., P C[ + 3M ]; for = o M do p, a randomly seleced prme from P ; p, P []; 3 p, P 3 []; 4 p, P 4 []; 5 end 6 for = o M do 7 F sub ; p, M 8 F sub M F sub ; 9 for j = o do Fnd a rao cj, c j, such ha Equ. (8) s sasfed; P j (p j,, p j,, cj, ); c j, end 3 end 4 reurn P j, = o M and j =, channel n any slo. For a shared slo, among all he channels ha share he slo, a node swches o he channel ha has he leas probably of PU occupancy. For a empy slo, he probably of occupancy of all he channels s consdered and wll be swched o he channel wh he leas probably of PU occupancy. he pseudo code o deermne he channel o be swched o for a node s shown n Algorhm. In lne, he ordnal n he curren cycle where he curren slo locaes s obaned. We use a varable c o denfy f he curren slo s a shared slo or an empy slo. Afer runnng hs Algorhm, he curren slo s an empy slo or a shared slo f he value of c s or larger han, respecvely. In lne, varable c s nalzed. In lnes 3 o, he channel ha may be swched s deermned. In lnes 4 and 5, he prmes used n he curren for channel are obaned. In he curren slo, he channel s a canddae channel ha he node may swch o f he curren slo number s a mulple of he prmes beng used for channel (lnes 6 o ). If he curren slo number s no a mulple of any prme beng used for any channel, he channel wh he leas PU occupancy among he M channels wll be seleced o be swched o (lnes o 4). Smlarly, f he curren slo number s a mulple of wo or more prmes used for mulple channels, he channel wh he leas PU occupancy among hese channels wll be seleced (lnes 5 o 7). A node k runnng SCH execues Algorhm 3 o oban s channel hoppng sequence. In lne, a node frs calls Algorhm o generae he needed combned rples. In lnes o 6, a paern s randomly seleced for each rple. he

7 7 Algorhm : Channel Deermnaon npu : M, curren slo, P aern j, = o M and j =, oupu: ch curren curren slo/w ; c ; 3 for o M do 4 prme he prme used for channel wh P aern n he curren 5 prme he prme used for channel wh P aern n he curren 6 f (curren slo (mod prme or prme )) hen 7 ch ; 8 CanddaeCHSe[c] ; 9 c c + ; end end f c = hen 3 ch he channel wh he leas PU occupancy probably among M avalable channels; 4 end 5 f c > hen 6 ch he channel n CanddaeCHSe wh he leas PU occupancy probably 7 end 8 reurn ch channel channel channel 3 F sub 3 slo F channel number sub Fsub Fsub Fsub Fsub 3 Fsub 3 F sub F sub,,, p p p ( a ) p, ( b ) cycle shared slo ( c ) number of slos n a s decded n lne 7. In lnes 9 o, Algorhm s execued o oban he channel o be swched o for each slo. Algorhm 3: SCH npu : M, P C, k, F Call Combned rple Selecon(M, P C, F ); for = o M do 3 for j = o do 4 Randomly assgn a P aern j for P j 5 end 6 end 7 w a value larger han (p, M p, M p ns ); 8 k ; 9 whle do s k,k Call Channel Deermnaon(M, k, P aern); k k + ; end We use a nework wh hree avalable channels o llusrae he operaon of SCH. Consder he channel hoppng assgnmen for node A. Snce M = 3, he arge rao for each channel s 3. he canddae prme ls P consss of wo prmes and 3 snce 5 s he frs prme ha sasfes he arge rao consran. hen, prmes 5, 7, and are assgned o P ; prmes 3, 7, and 9 are assgned o P 3 ; prme 3, 9, and 3 are assgned o P 4. he dedcaed prmes n each combned rple are sequenally assgned from P, P 3 and P 4 and he shared prmes s randomly assgned from P, as shown n Fg. 4(a). he value of F sub and F sub, = o 3, s 9 and 9, respecvely. he rples o mplemen dfferen sub-raos are hus P = (, 5, /5), P = (3, 3, /8), P = (, 7, /3), P = (7, 9, 3/), P3 = (3,, 4/), Fg. 4. An example of SCH operaon, where each node uses sx rples o deermne s channel hoppng schedule, wh (a) he prme assgnmen (b) sub-rao assgnmens, and (c) dealed channel hoppng for F sub. In (b), he 3 number n a represens he prme beng used n ha. In (c), he number n a slo represens he channel o be swched o n he slo. and P 3 = (9, 3, 4/5). Assume ha he randomly assgned paern for P o P 3 s (5, 5, 5,, 5, 5, ), (3, 3, 3, 3, 3, 3, 3, 3, 3), (7, 7, 7,, ), (7, 7, 9, 7), (3, 3, 3, 3, ), and (9, 9, 3, 9, 9, 3, 3, 3, 3), respecvely, as shown n Fg. 4(b). Noe ha he lengh of he cycle for each sub-rao s dfferen. For he frs, node A wll swch o channel,, and 3 f he slo number s a mulple of 5 or 3, 7 or 7, and 3 or 9, respecvely. More specfcally, consder he frs of F sub 3, as shown n Fg. 4(c), node A swches o channel 3 n a slo wh slo number a mulple of 3. A slo wh slo number a mulple of 5 (he leas common mulple of prmes 3 and 5) s a shared slo, shared by channels and 3. If we assume po A () =.3 and po A (3) =.5, node A wll swch o channel. In fac, a slo wh slo number a mulple of he leas common mulple of 3 and any of he prmes used for he oher wo channels (5, 3, 7, 7) s also a shared slo. ha s, a slo wh slo number a mulple of 5, 69,, or 5 s a shared slo. C. Rendezvous Guaranee of SCH o help verfy he rendezvous guaranee propery of SCH, we provde wo examples frs. Example. wo synchronous nodes A and B use prmes 5 and 7 o deermne he slos n whch channel should be swched o n R. If boh nodes swch o channel a shared slos, he slo X ha boh nodes have a rendezvous n

8 8 channel sasfes { X (mod pa ), p A = 5 or 7 X (mod p B ), p B = 5 or 7 he soluon of X s 5k or 7k, k and k N. Snce boh nodes may no swch o channel a shared slos, nodes A and B are guaraneed o have a rendezvous only a slo X = 5k and X = 7k, where k and k N S, S s he se of mulples of he prmes beng used by oher channels n R. Example. Consder a nework wh wo channels, and j, wheren boh nodes A and B use prmes fve (p ) and seven (p ) o deermne he slos n whch channel should be swched o n a. Node B s wo slos ahead of node A. he slo X ha boh nodes have a rendezvous n channel sasfes { X (mod pa ), p A = 5 or 7 () X (mod p B ), p B = 5 or 7 Usng he soluon formula of CR [3], we have he soluon X = (a A Y A p + a B Y B p ) (mod p p ) and X = (a A Y A p + a BY B p ) (mod p p ) where varables Y A, Y A, Y B, and Y B sasfy p Y A (mod p ), p Y A (mod p ), p Y B (mod p ), and p Y B (mod p ). hs mples Y A = 3, Y A = 3, Y B = 3, and Y B = 3 n hs example. he rendezvous slo beween nodes A and B usng channel s X = ( ) (mod 35) = 3 (mod 35) = k and X = ( ) (mod 35) = 4 (mod 35) = k, k and k N. Now consder node A uses prme hree (p j ) o deermne he slos channel j should be swched o n he same. he slo X whch channel and j share sasfes { X (mod p A ), p A = 5 or 7 or 3 X () (mod p B ), p B = 5 or 7 or 3 Usng he soluon formula of CR, we can fnd he soluon X = (a A Y A p p j + a AY A3 p p + a B Y B p p j ) (mod p p p j ) and X = (a A Y A p p j + a AY A3 p p + a B Y B p p j ) (mod p p p j ) where Y A, Y A, Y A3, Y B, Y B, and Y B3 s,,,,, and, respecvely. hs mples X = 3 (mod 5) = 3 + 5k and X = 4 + 5k, k and k N. he rendezvous slo beween nodes A and B usng channel, consderng prme hree used n channel j, s hus equal o { X X k, k + 3n, n N = k, k + 3n, n N { k, k = + 3n, + 3n, n N = k, k = + 3n, + 3n, n N () Smlar shared slo removal process should be appled for he prme hree beng used for nodes B n channel j. Afer such shared slo removal, he slos ha nodes A and B have a rendezvous n channel are gven by { k, k = + 3n, n N k, k (3) = + 3n, n N Now we are ready o presen an mporan propery of SCH as descrbed n he followng heorem. (9) heorem 3. SCH s a complee rendezvous guaranee proocol. Proof: : Snce each node uses hree dedcaed prmes for a channel, here mus exs a R n whch a node A uses wo dedcaed prmes. We prove he heorem by showng ha node A can have a rendezvous wh any oher node B n R for any channel. Suppose ha node A uses wo prmes, denoed as p A, and p A,, o deermne he schedules of channel n R. Smlarly, node B uses p B, and p B, for channel n R. Accordng o wheher node B uses wo dedcaed prmes n R, here exs wo cases: : Node B uses wo dedcaed prmes. In hs case, based on wheher he prmes beng used by nodes A and B are dencal or no, here are wo subcases:.: p A, = p B, and p A, = p B,. Le p and p denoe p A, and p A,, respecvely. Frs we consder he suaon ha nodes A and B are synchronzed. In such a suaon, fndng he rendezvous slo beween nodes A and B s equvalen o fndng he soluon X of he followng lnear congruence sysem (refer o Example ) { X (mod pa ), p A = p or p X (mod p B ), p B = p or (4) p If boh nodes swch o channel a shared slos, nodes A and B have a rendezvous a slo X wh X = p k and X = p k, where k and k N. However, snce nodes may no swch o channel n shared slos, nodes A and B are guaraneed o have a rendezvous only a slo X = p k and X = p k, where k and k N S, S s he se of mulples of he prmes beng used by oher channels n R. Afer such a shared slo removal process, we need o prove ha N S. Noe ha N S can be rewren as pm S (N pm ) n whch each em, N pm, s acually equal o (pm k +δ ) (mod pm ), where k, δ N (refer o Example ). hus, pm S (N pm ) can be represened as he soluon of he followng lnear congruence sysem X δ (mod pm ) X δ (mod pm ). X δ n (mod pm n ) (5) Accordng o CR, here mus be a soluon X, whch verfes ha N S = pm S N S. hs proves ha nodes A and B have a rendezvous usng channel n R f hey use he same prmes and are synchronzed. Nex, we consder he suaon ha nodes A and B are no synchronzed. Agan, consder boh nodes swches o channel a shared slos frs. Nodes A and B have a rendezvous slo usng channel when he slo s a soluon X of he followng lnear congruence sysem { X aa (mod p A ), p A = p or p X a B (mod p B ), p B = p or (6) p

9 9 he soluon X when node A uses p and node B uses p s gven by X = (a A Y A p p j + a AY A3 p p + a B Y B p p j ) (mod p p p j ). he soluon X when node A uses p and node B uses p s gven by X = (a A Y A p p j +a AY A3 p p +a BY B p p j ) (mod p p p j ). When akng no accoun he prmes beng used o deermne he schedules of oher channels, he shared slo removal process s appled. And he saemen ha nodes A and B have a rendezvous usng channel n R can be proved n he same way as nodes A and B are synchronzed.. p A, = p B, = p and p A, p B, : he mehod o prove ha nodes A and B have a rendezvous usng channel n R s smlar o ha n case.: Smply replacng p by p A, or p B,. : Node B uses one dedcaed prme and one shared prme. In hs case, nodes A and B have a rendezvous usng channel n R can be proved n he same way as n. When fndng he soluons ha boh nodes have a rendezvous, only he dedcaed prmes used n boh nodes are ulzed. he slos seleced by he shared prme beng used by node B are consdered as shared slos. V. PERFORMANCE ANALYSIS In SCH, he R beween wo nodes A and B n a depends on he dedcaed prme(s) used by hem. In mos cases, he R s abou O(prme prme) where prme and prme s a dedcaed prme used by A and B n he, respecvely. o calculae he exac R, we need o oban he exac number of rendezvous where he shared/empy slos mus be consdered. However, wheher here s a rendezvous n a shared/empy slo s unceran. o analyze he average R of SCH s complex and lenghy. o hs end, we analyze he leas number of rendezvous for a sngle channel whou consderng he possble rendezvous of shared/empy slos. hs can be consdered as he mos conservave way o oban he maxmum R (MR) beween wo nodes usng a sngle channel. Because MR can be calculaed by he leas number of rendezvous, n he followng, we focus on how o oban he leas number of rendezvous on a channel. Snce he analyss s que complex, we consder a very smple scenaro where only wo nodes A and B are n he nework. Recall ha a node runnng SCH uses four prmes (from wo rples) o deermne s channel hoppng schedule n a cycle. Le p k (A) be he k-h prme seleced from P k beng used by node A o deermne he schedule for channel, k = o 4. here are four dfferen prme combnaons can be used by A o deermne he schedule of channel n a : p (A)+p3 (A), p (A) + p4 (A), p (A) + p3 (A), and p (A) + p4 (A), whch s denoed as A, A, A 3, and A 4, respecvely. Smlarly, here are four dfferen prme combnaons can be used by B (denoed as B, B, B 3, and B 4, respecvely) o deermne he schedule of channel n a, whch makes a oal of 6 possble combnaons when consderng he prmes beng used by nodes A and B n a. Recall ha boh nodes use wo prmes n a and le p (A) and p (A) denoe he wo prmes used by node A o deermne he schedule for channel n a where p (A) represens he shared prme f one of he wo prmes s a shared prme. Accordng o he prmes beng used by A and B, he 6 combnaons can be classfed no he followng sx caegores: : Boh nodes use wo dedcaed prmes and hese dedcaed prmes are he same. ha s, p (A) = p (B) and p (A) = p (B)6. : Boh nodes use wo dedcaed prmes and one of he dedcaed prmes beng used by each node s he same. Whou loss of generaly, le p (A) = p (B) and p (A) p (B)7. 3: Boh nodes use one dedcaed prme and one shared prme, and he dedcaed prme s he same. ha s, p (A) = p (B)8. 4: Boh nodes use one dedcaed prme and one shared prme and he dedcaed prmes are dfferen. ha s, p (A) p (B)9. 5: One of wo nodes uses wo dedcaed prmes whle he oher uses one dedcaed prme and one shared prme. he dedcaed prme s he same. ha s, p (A) = p (B). 6: One of wo nodes uses wo dedcaed prmes whle he oher uses one dedcaed prme and one shared prme. he dedcaed prmes are dfferen. ha s, p (A) p (B). Consderng he possble dfferen slo number offses and prmes beng used for oher channels, here are many subcases. For example, f he prmes beng used for one more channel j are consdered, here are a oal of 36, 8, 6, 8, 36, and 8 subcases n,, 3, 4, 5, and 6, respecvely. A oal of 43 more subcases exs when one more channel s beng consdered. I s me-consumng o compleely analyze SCH. herefore, nsead of evaluang all he dfferen cases, we analyze 5 whch domnaes he oher cases. We use se heory and he ncluson-excluson prncple o calculae he leas number of rendezvous. Whou loss of generaly, assume ha A uses wo dedcaed prmes, p (A) and p (A) whle B uses one shared prme p (B) and one dedcaed prme p (B) o deermne he schedule of channel. Also, we assume p (A) = p (B) = k, and p (A) p (B). Accordng o wheher he shared prme p (B) s seleced by B o deermne he schedule of anoher channel, wo subcases exs: 5. where p (B) s only used o deermne he schedule of channel and 5. where p s used o deermne he schedule of anoher channel. Here (B) we analyze 5. frs. Consderng he possble dfferen slo number offses, wo more cases can be found n 5.: 5.. where a A a B (mod k ) and 5.. where a A a B = (mod k ). In 5.., several subcases 6 Combnaons belongng o hs caegory nclude A 3 + B 3 and A 4 + B 4. 7 Combnaons belongng o hs caegory nclude A 3 + B 4 and A 4 + B 3. 8 Combnaons belongng o hs caegory nclude A + B and A + B. 9 Combnaons belongng o hs caegory nclude A + B and A + B. Combnaons belongng o hs caegory nclude A + B 3, A + B 4, A 3 + B, and A 4 + B. Combnaons belongng o hs caegory nclude A + B 4, A + B 3, A 3 + B, and A 4 + B.

10 Analyss Smulaon 5 Number of rendezvous (a) (b) 5 Fg. 5. Illusraon of he leas number of rendezvous for (a) 5.. and (b) (consderng anoher channel j)). Marked area sands for he leas number of rendezvous of channel. exs. If no oher channel s consdered n he nework, he calculaon of he number of rendezvous can be llusraed by Fg. 5(a) where R, and R, sands for he se of rendezvous slos n channel ha are mulples of k and mulples of k respecvely n me duraon, where k = p (A) p he leas number of rendezvous n channel s equal (B). o R, R, = (R, +R, ) (R, R, ) = k + k k k. When anoher channel j comes no play, each of he wo prmes used by A and B o deermne he schedule of channel j n a has hree dfferen possbles and hus here are nne subcases: 5...: p j (A) p j (B) and p j (A) p j (B). 5...: p j (A) p j (B), p j (A) = p j (B) = k 4, and a A a B (mod k 4 ) : p j (A) p j (B), p j (A) = p j (B) = k 4, and a A a B (mod k 4 ) : p j (A) = p j (B) = k 3, a A a B (mod k 3 ), and p j (A) p j (B) : p j (A) = p j (B) = k 3, a A a B (mod k 3 ), p j (A) = p j (B) = k 4, and a A a B (mod k 4 ) : p j (A) = p j (B) = k 3, a A a B (mod k 3 ), p j (A) = p j (B) = k 4, and a A a B = (mod k 4 ) : p j (A) = p j (B) = k 3, a A a B = (mod k 3 ), and p j (A) p j (B) : p j (A) = p j (B) = k 3, a A a B = (mod k 3 ), p j (A) = p j (B) = k 4, and a A a B (mod k 4 ) : p j (A) = p j (B) = k 3, a A a B = (mod k 3 ), p j (A) = p j (B) = k 4, and a A a B = (mod k 4 ). We ake as an example o llusrae how he calculae he leas number of rendezvous. As shown n Fg. 5(b), he leas number of rendezvous n channel s equal o he number of elemens of he unon of R, and R, mnus ha of he nersecon of R j, and R 3 j,. ha s, he leas number of rendezvous n channel s (R, R, ) (R j, Rj, ) = k + k k k 3 k k k k 3 k k 4 k k 4 + k k k 3 + k k k 4 + k k 3 k 4 + k k 3 k 4 k k k 3 k 4. he oher subcases can be obaned lkewse 4. Followng he same way, we can fnd he leas number of rendezvous of 5.. Smlarly, he number of rendezvous for he oher fve cases (,, 3, 4, and 6) can also be derved. o verfy he correcness of our analyss, we have wren a compuer program o smulae a node s channel hoppng n 3, slos ( = 3, ) wh M = 5 and F = %. Our analyss can be exended o consder he shared slos generaed by all he channels. However, hs makes he analyss oo lenghy. o smplfy our analyss, we demonsrae how he leas number of rendezvous s obaned n a scenaro where he shared slos s generaed by one more channel j. 3 In oher words, regons I o I 7 n Fg. 5(b) should be excluded. 4 he complee calculaon of he leas number of rendezvous for 5. can be found a hp://wreless.cs.nou.edu.w/hsccl/sch/analyss case5..pdf Fg. 6. he analyss and smulaon resuls for he leas number for rendezvous beween wo nodes n a channel We gahered he number of rendezvous for wo nodes n a randomly seleced channel consderng he shared slos generaed by one more channel. he resuls of nne nsances for nne subcases n 5.. gahered from he smulaon and he ones calculaed from he analyss can be found n Fg. 6. We can see ha he analyss resuls concde wh he smulaon resuls. We have gahered/calculaed many nsances for dfferen subcases and smlar concdence can be found. hs verfes he correcness of our analyss VI. SIMULAION We have mplemened a smulaor usng Java o evaluae he performance of SCH. We have also mplemened S-ACH, JS and E-AHW for comparson purposes. hese proocols belong o he asynchronous, symmerc class and provde complee rendezvous guaranee. he CRSEQ proocol, anoher asynchronous, symmerc, and complee rendezvous guaranee soluon, was no mplemened because performs worse han JS [4]. he scheme proposed n [3] was no mplemened eher because performs worse han E-AHW []. A varan of SCH n whch he channel o be swched o n a shared or an empy slo s randomly seleced was also mplemened, denoed as SCH-R n he fgures. We mplemened he asynchronous envronmen defned recenly [9]. A oal of 4 SUs and PUs were dsrbued n an area of square meers. he lengh of user ID for E-AHW s se o 49 bs (he same as n he smulaons of E-AHW). We do no fnd he smulaon seng of he user ID lengh for S-ACH n [7]. Because wo sequences resulng from cyclc roaons are no consdered o be dsnc n S-ACH, we se he lengh of user ID for S-ACH o be bs o generae up o 5 dsnc ID sequences [3]. A PU s randomly assgned a fxed channel a he nework nalzaon phase. I s possble ha a channel s no used by any PU whle anoher channel s shared by wo or more PUs (usng mechansms such as me dvson mulple access, DMA). We se F = % n he frs wo expermens and s vared o observe s nfluence n he hrd expermen. he ransmsson range of an SU and a PU s and 3 meers, respecvely [3]. he PU occupancy model s based on he Cognve Rado Cognve Nework Smulaor where he ncumben raffc follows a busy/dle paern [9]. he dle perod follows an exponenal dsrbuon wh a mean of slos. he busy perod also follows an exponenal dsrbuon wh a mean of dfferen slos o generae dfferen

11 me o Rendezvous me o Rendezvous R Sandard Devaon Occupancy Probably of PUs (a) R Sandard Devaon Number of Channels (a) Occupancy Probably of PUs Number of Channels (b) (b) Maxmum me o Rendezvous Maxmum me o Rendezvous Occupancy Probably of PUs Number of Channels (c) (c) Fg. 7. Impac of dfferen occupancy probables of PUs Fg. 8. Impac of dfferen numbers of channels PU occupancy probables. A plo n he followng fgures s he average of 3 smulaon runs, each of whch smulang 3,, slos. In he followng, we observe from hree aspecs. A) Impac of occupancy probably of PUs: Frsly, we nvesgae he performance of dfferen proocols wh dfferen PU occupancy probables. he number of channels s se o 5. he resuls are shown n Fg. 7. Fg. 7(a) shows he R for dfferen proocols. Here R beween wo nodes s defned as he average me nerval of her wo consecuve rendezvous, consderng he PU occupancy ssue. We can see ha SCH always has he leas R. S-ACH and JS have smlar performance whle E-AHW produces he larges R. A feaure of JS s ha some of he avalable channels are beng ulzed more frequenly. A hgher R s produced when hese channels are occuped by PUs. Buldng on op of JS and usng channels even more unevenly, E-AHW has he hghes R. When he PU occupancy probably ncreases, he R ncreases for all he proocols; however, mpacs SCH he leas. For example, when he PU occupancy probably s ncreased from.5 o.75, he R produced by SCH s ncreased by 3% (from 3.7 and 4.). For SHC-R, S-ACH, JS, and E-AHW, when he PU occupancy probably s ncreased from.5 o.75, he R ncremen s 7%, 8%, 8%, and 3%, respecvely. he sandard devaon of R for dfferen proocols can be found n Fg. 7(b). I s obvous ha SCH has he bes performance. he larger sandard devaon of S-ACH and E-AHW comes from he wde range of SUs IDs. Boh S- ACH and E-AHW use a node s ID o deermne he node s channel hoppng sequence. he R for wo nodes wh smlar IDs s much lesser han ha of wo nodes wh very dfferen IDs. For JS, he selecon of R and I plays an mporan role for he number of rendezvous. wo nodes wh he same R bu dfferen values of I may have eher a very large or a very small number of rendezvous, depends on he correlaon beween seleced values of I. When he PU occupancy probably ncreases, he R sandard devaon ncreases for all he proocols. Agan, he ncrease of PU occupancy probably mpacs SCH he leas. he resuls of hs expermen verfy ha he SCH produces more number of rendezvous for dfferen SUs n a far way. Fg. 7(c) shows he MR for dfferen proocols. We can see ha SCH has he leas MR when PU occupancy probably s.5 and.75 whle has a slghly hgher MR han SCH-R and JS when PU occupancy probably s.5. he S-ACH generaes he hghes MR because of he varey of nodes IDs. B) Impac of number of channels: Nex, we vared he number of avalable channels o observe he effec. he maxmum

12 Fg. 9. arge Rao Sandard Devaon arge Rao Sandard Devaon SCH SCH-R SCH(R) SCH-R(R)..5. olerance hreshold ( ).3.3 (a) 4.E Channel Hoppng Proocols (b) Impac of dfferen values of F value of M s se o 5. he occupancy probably of PUs s se o.5. he resuls are shown n Fg. 8. From Fg. 8(a), we can see ha he R ncreases n proporonal o he number of channels for all proocols and SCH performs he bes. he resuls of R sandard devaon can be found n Fg. 8(b). We can see ha SCH sll performs well. he MR produced by dfferen proocols are shown n Fg. 8(c). Agan, SCH has he leas MR n mos cases. SCH has he larges MR when usng fve channels. We consder such a hghes MR s a specal case because, as shown n Fg. 8(a) and (b), SCH has he leas R and prey good R sandard devaon. Besdes, when usng more channels, he MR s reduced sgnfcanly. hese resuls verfy ha SCH also performs well n erms of MR. C) Impac of olerance hreshold (F ): In hs expermen, we vared he value of F o observe he sandard devaon of arge rao and R. he occupancy probably of PUs s se o.5 whle he number of channels s se o 5. he resuls of sandard devaon of arge rao for dfferen channels and R are shown n Fg. 9(a). For each value of F, four bars are presened n he fgure. he wo bars on he lef/rgh sde are he sandard devaon of he arge rao/r. Parenheses are also added o he wo bars on he rgh o ndcae ha hey are for R comparsons. For SCH-R, smaller sandard devaons of he arge rao can be observed wh smaller values of F. Because he channel o be swched o s randomly seleced, he resuls verfy ha a smaller value of F make dfferen channels usng closer arge raos. For SCH, snce he channel wh he leas PU occupancy s used n shared/empy slos, he channel has he larges arge rao. hs makes he sandard devaon of he arge rao hgher han ha of SCH-R. On he oher hand, dfferen values of F make no dfference n R for boh SCH and SCH-R. We have also observed he praccal arge rao of dfferen proocols. he resuls can be found n Fg. 9(b) where F s se o. for SCH and SCH-R. me o Rendezvous We can see ha SCH and SCH-R use dfferen channels n an even way when compared o JS and E-AHW. I s neresng o fnd ha S-ACH produce almos dencal arge rao for dfferen channels. Unforunaely, S-ACH does no perform well n erms of R, R sandard devaon, and MR. VII. CONCLUSIONS AND FUURE WORK In hs paper, we propose an asynchronous and symmerc saggered channel hoppng scheme (SCH) for CRNs. Usng he concep of rple and arge rao paronng mechansm, SCH solves he rendezvous problem. Nodes runnng SCH can have more number of rendezvous and smaller sandard devaon of R. Smulaon resuls verfy ha he superory of SCH. We beleve SCH s a praccal and effcen channel hoppng proocol for CRNs. SCH works when nodes have only lmed number of commonly-avalable channels. In fac, a node runnng SCH can exclude channels ha are no commonly-avalable o oher nodes.hs dea can be generalzed o a channel schedulng problem: How a node properly deermnes he raos o swch o dfferen channels. Inuvely, for each node, he rao of swchng o a channel ha s no commonly-avalable o any oher node can be se o zero. he rao of swchng o a channel ha s commonly-avalable o many nodes can be se o a lager value. he channel schedulng problem s no a rval one. Issues such as how o deermne he crera of rao adjusmen and how o oban he number of nodes ha s capable o use a parcular channel mus be solved. Solvng hs channel schedulng problem s our fuure work. ACKNOWLEDGEMENS hs research was sponsored by Mnsry of Scence and echnology, R. O. C., under gran NSC --E-9-7- MY3. REFERENCES [] I. Akyldz, W. Lee, M. Vuran, and S. Mohany. NeX Generaon/Dynamc Specrum Access/Cognve Rado Wreless Neworks:A Survey. Compuer Neworks, 5(3):7 59, 6. [] H. Km and K. G. Shn. Efffcen Dscovery of Specrum Opporunes wh MAC-Layer Sensng n Cognve Rado Neworks. IEEE ransacons on Moble Compuung, 7: , May 8. [3] C.-M. Chao and H.-C. sa. A New Channel Hoppng MAC Proocol for Moble Ad Hoc Neworks. he Inernaonal Conference on Wreless Communcaons and Sgnal Processng (WCSP 9), Nov. 9. [4] C.-M. Chao, H.-C. sa, and C.-Y. Huang. Load-Aware Channel Hoppng Proocol Desgn for Moble Ad Hoc Neworks. 7h Inernaonal Symposum on Wreless Pervasve Compung (ISWPC), July. [5] C.-M. Chao, Y.-Z. Wang, and M.-W. Lu. Mulple Rendezvous Mulchannel MAC Proocol Desgn for Underwaer Sensor Neworks. IEEE ransacons on Sysems, Man, and Cybernecs: Sysems, 43():8 38, Jan. 3. [6] C. Arachchge, S. Venkaesan, and N. Mal. An Asynchronous Neghbor Dscovery Algorhm for Cognve Rado Neworks. IEEE In l Symp. New Froners n Dynamc Specrum Access Neworks (DySPAN 8), pages 5, Oc. 8. [7] Kagu Ban and Jung-Mn Park. Maxmzng Rendezvous Dversy n Rendezvous Proocols for Decenralzed Cognve Rado Neworks. IEEE ransacons on Moble Compung, (7):94 37, 3. [8] Kagu Ban, Jung-Mn Jerry Park, and Rulang Chen. A Quorum-Based Framework for Esablshng Conrol Channels n Dynamc Specrum Access Neworks. 5h Annual Inernaonal Conference on Moble Compung and Neworkng (ACM MobCom), 9.

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Chh-Mn Chao receved he B.S. and M.S. degrees n compuer scence from Fu-Jen Caholc Unversy and Naonal sng-hua Unversy n 99 and 996, respecvely, and he Ph.D. degree n compuer scence and nformaon engneerng from Naonal Cenral Unversy n January of 4. He was an asssan professor a he amkang Unversy, awan, from 4 o 5. He joned he Deparmen of Compuer Scence and Engneerng, Naonal awan Ocean Unversy where he worked as an asssan professor (5-8), an assocae professor (84), and a professor (4-presen. Hs research neress nclude moble compung and wreless communcaon. Chen-Yu Hsu receved her BS degree n Compuer Scence from ape Muncpal Unversy of Educaon, ape, awan, n, and receved M.S. degree n Compuer Scence and Engneerng from Naonal awan Ocean Unversy, Keelung, awan, n 3. Her research neress nclude wreless sensor neworks and proocols desgn. Yun-ng Lng receved her B.S. degree n Compuer Scence and Informaon Engneerng from Fu Jen Caholc Unversy, New ape Cy, awan, n 3, and receved M.S. degree n Compuer Scence and Engneerng from Naonal awan Ocean Unversy, Keelung, awan, n 5. Her research neress nclude wreless sensor neworks and cognve rado neworks.