Mutpe Rados for Fast Rendezvous n Cogntve Rado Networks Lu Yu, Ha Lu, Yu-Wng Leung, Xaowen Chu, and Zhyong Ln bstract Rendezvous s a fundaenta operaton n cogntve rado networks (CRNs) for estabshng a councaton nk on a coony-avaabe channe between cogntve users. The exstng work on pcty assues that each cogntve user s equpped wth one rado (.e., one wreess transcever). s the cost of wreess transcevers s droppng, ths feature can be expoted to sgnfcanty prove the perforance at ow cost. In ths study, we nvestgate the probe n CRNs where cogntve users are equpped wth utpe rados and dfferent users ay have dfferent nubers of rados. We frst study how the exstng agorths can be generazed to use utpe rados for faster. We then propose a new agorth, caed roe-based parae sequence (R), whch specfcay expots utpe rados for ore effcent. Our basc dea s to et the cogntve users stay n a specfc channe n one dedcated rado and hop on the avaabe channes wth parae sequences n the reanng genera rados. We prove that our agorth provdes guaranteed (.e., can be copeted wthn a fnte ) and derve the upper bounds on the axu -to- (TTR) and the expected TTR. The suaton resuts show that ) utpe rados can cost-effectvey prove the perforance, and ) the proposed R agorth perfors better than the ones generazed fro the exstng agorths. Index Ters cogntve rado, bnd, channe hoppng INTRODUCTION WITH the tradtona statc spectru anageent, a sgnfcant porton of the censed spectru s underutzed n ost of whe the uncensed spectru s over-crowded due to the growng deand for wreess rado spectru fro exponenta growth of varous wreess devces []. Dynac pectru ccess utzes the wreess spectru n a ore ntegent and fexbe way. Cogntve rados are a prosng enaber for Dynac pectru ccess because they can sense and access the de channes. Wth cogntve rados, the uncensed users (Us) can opportunstcay dentfy and access the vacant portons of the spectru of the censed users (Us). In cogntve rado networks (CRNs), utpe de channes ay be avaabe to Us. If two or ore Us want to councate wth each other, they ust seect a channe whch s avaabe to a of the. The process of two or ore Us to eet and estabsh a nk on a coony-avaabe channe s known as []. Rendezvous s a fundaenta and essenta operaton for estabshng councaton nks of Us. Channe-hoppng (CH) s one of the ost representatve technques for. Wth CH technque, each U seects a set of avaabe channes and hops aong these channes. s sad to be acheved f two Us hop on the sae channe sutaneousy. Many effectve agorths have been proposed n the terature and they are descrbed n ecton. To the best of our knowedge, a the exstng agorths pcty assue that L. Yu, H. Lu, Y.-W. Leung, X. Chu, and Z. Ln are wth the Departent of Coputer cence, Hong Kong Baptst Unversty, Kowoon Tong, Hong Kong R. E-a: yu, hu, yweung, chxw, zyn@cop.hkbu.edu.hk. prenary verson was presented at ICC. each user s equpped wth one rado (.e., one wreess transcever). s the cost of wreess transcevers s droppng, ths feature can be expoted to sgnfcanty prove the perforance at ow cost. In partcuar, when a U s equpped wth utpe rados, the -to- (TTR,.e., the requred by the operaton) can potentay be reduced by a arge aount whe the addtona cost (.e., cost of the extra rados) s ow. In addton, the energy consupton can aso be reduced (f the nuber of rados s ncreased fro to n, n rados woud consue energy but the spent on coud be reduced by ore than n s, so the tota energy consupton can be reduced). In ths paper, we study the probe n CRNs where each U s equpped wth utpe rados and dfferent Us ay have dfferent nubers of rados. We ake three contrbutons. ) We nvestgate a new approach (.e., expotng utpe rados per user) to sgnfcanty provng the perforance at ow cost. ) We generaze the Rando agorth and the exstng agorths n order to use utpe rados for faster. ) We propose a new agorth, caed roe-based parae sequence (R), whch specfcay expots utpe rados for ore effcent. We derve upper bounds on the axu TTR (MTTR) and the expected TTR (E(TTR)) of ths agorth. We conduct extensve suaton to deonstrate that ts MTTR and E(TTR) decrease sgnfcanty wth the ncrease of the nuber of rados. In the terature, there are two odes to descrbe the channe avaabty [, 9,, ]: ) syetrc ode n whch a users have the sae avaabe channes; and ) asyetrc ode n whch dfferent users ay have dfferent avaabe channes. Both the syetrc
TBLE COMRING THE UER BOUND OF MTTR Exstng agorths generazed to utpe rados gorths yetrc Mode syetrc Mode Rando Infnty Infnty J/Independent M( G)+ J/arae when = n Infnty when n New gorth R ax{,n} TBLE COMRING THE UER BOUND OF E(TTR) M( G)+ when = n Infnty when n ( ( G +))when = n ( ax{,n} ) + ( G) n{,n} when n Exstng agorths generazed to utpe rados New gorth gorths yetrc Mode syetrc Mode Rando J/Independent J/arae R n n +n + +n ( ++ ) /+ when = n Infnty when n ( ) when = n ax{,n} + ( ax{,n} ) n{,n} when n C n C C n C C G n C G + +n ( ++ ) + ( G) ( G)+(+ (G )/) when = n Infnty when n ( ( G +))when = n ax{,n} + ( ax{,n} ) n{,n} + ( G) when n n{,n} Rearks: ), n are the nubers of rados of two users; s the nuber of a channes; s the saest pre nuber whch s not saer than ; G s the nuber of coony-avaabe channes of two users; C, C are the nubers of avaabe channes of two users, respectvey; j s the nuber of possbe perutatons of objects fro a set of j. ) We seect the Jup-tay (J) agorth [9] for coparson snce t was recenty proposed and was shown to have a very good perforance [9]. The upper bounds of MTTR and E(TTR) of the exstng agorths are derved n ecton. ) In a CRN, dfferent Us ay be equpped wth dfferent nubers of rados. Centrazed Dedcated CCC Rendezvous Usng CCC Decentrazed Requre synchronzaton Wthout CCC Wthout synchronzaton No guaranteed Guaranteed Fg.. taxonoy of the exstng agorths. and asyetrc odes are portant n practce. For exape, the syetrc ode s sutabe for Us who are ocated n a reatvey sa area (copared wth ther dstance to Us) whe the asyetrc ode s appcabe f geographca ocatons of Us are far. Tabe and Tabe suarze the dfferences between: ) the proposed agorth, and ) the exstng agorths after they are generazed to use utpe rados. The rest of ths paper s organzed as foows. Reated works are revewed n ecton. yste ode and probe foruaton are presented n ecton. In ecton, we generaze the Rando agorth and the exstng agorths to use utpe rados for faster. Then we propose a new agorth whch specfcay expots utpe rados for ore effcent. We present suaton resuts n ecton for perforance evauaton and concude our work n ecton. RELTED WORK The exstng CH agorths can be cassfed nto two categores based on ther structures: ) centrazed systes where a centra server s preseected to aocate the spectru for a Us, and ) decentrazed systes where there s no centra server. The decentrazed systes can be further cassfed nto two subcategores: ) usng a coon contro channe (CCC), and ) not usng CCC. Fg. shows a possbe taxonoy of the exstng agorths. Centrazed systes: Under centrazed syste, such as D[] and DIMUMNet [], a centrazed server s operated to schedue the data exchanges aong users. Wth a centrazed server for goba coordnaton, ths approach eases the process but t nvoves the overhead of antanng the server and ths server s a snge pont of faure []. Decentrazed systes usng CCC: In decentrazed syste, a channe s preseected as a CCC. In []
and [], a goba CCC s preseected and known to a users. In [7] and [], a custer-based controchanne ethod was proposed, n whch a oca CCC s seected for each custer. However, the extra costs n estabshng and antanng the goba/oca CCCs are consderabe. Decentrazed systes wthout usng CCC: Ths approach does not use CCC and hence t s known as bnd [9]. Wth ths desrabe feature, ths approach has drawn sgnfcant attenton n the terature and soe effectve agorths for bnd have been proposed (e.g., Jup- tay [][9], M-/L-CH [] and YNC-ETCH []). Te-synchronzaton s a key pont we shoud pay attenton to n bnd. Based on t we can further dvde decentrazed systes wthout usng CCC nto two types (Fg. ). gorths that requre -synchronzaton: Many CH agorths whch requre synchronzaton have been proposed. Two CH agorths M-/L-CH were proposed n [] based on quoru systes whch can guarantee a of users under the syetrc ode. nother CH agorth caed -CH was proposed n [] for the asynchronous systes. Bah et a. desgned a nk-ayer protoco naed CH []. Each user can seect ore than one pars and generate the CH sequence based on these pars. It was desgned to ncrease the capacty of IEEE. networks. However, sar to M-/L-CH, they are not appcabe n the asyetrc ode. uthors n [] proposed a deternstc approach n whch each user s schedued to broadcast on every channe n an exhaustve anner. gorths that work wthout -synchronzaton: There are CH agorths whch do not requre -synchronzaton. In [], they presented a rng-wak (RW) agorth whch guarantees the under both odes. In RW, each channe s represented as a vertex n a rng. Users wak on the rng by vstng the vertces (channes) wth dfferent veoctes and s guaranteed snce users wth ower veoctes w be caught by users wth hgher veoctes. However, RW requres that each user has a u- nque ID and knows the upper bound of network sze. Recenty, a notabe work by Thes et a. presented two CH agorths: oduar cock agorth (MC) and ts odfed verson MMC for the syetrc ode and the asyetrc ode, respectvey []. The basc dea of MC and MMC s that each user pcks a proper pre nuber and randoy seects a rate ess than the pre nuber. Based on the two paraeters, the user generates ts CH sequence va predefned oduo operatons. though MC and MMC are shown to be effectve, both agorths cannot guarantee the f the seected rates or the pre nubers of two users are dentca. Yang et a. proposed two sgnfcant agorths, naey deternstc sequence (DRE) [] and channe sequence (CRE) [9], whch provde guaranteed for the syetrc ode and the asyetrc ode, respectvey. In CRE, the sequence s constructed based on trange nubers and oduar operatons. In ters of MTTR, CRE s qute good under the asyetrc ode but t does not perfor we under the syetrc ode. Ban et a. [] presented an asynchronous channe hoppng (CH) agorth whch as to axze dversty. It assues that each user has a unque ID and CH sequences are desgned based on the user ID. Though the ength of user ID s a constant, t ay resut n a ong TTR n practce gven that a typca MC address contans bts. In [7], an effcent agorth based Dsjont Reaxed Dfference et (DRD) was proposed whe DRD ony costs a near to construct. They proposed a dstrbuted asynchronous agorth that can acheve and guarantee fast under both the syetrc and the asyetrc odes. They derved the ower bounds of MTTR whch are and + under the syetrc ode and the asyetrc ode, respectvey. They showed that t s neary opta. In [], the authors addressed the parwse as we as the utcast probes under fast prary user (U) dynacs. They consdered the ssue of adaptvey adjustng the channe hoppng (CH) sequences to account for the spectru heterogenety and U dynacs, whch are the two an chaenges n D systes. Wu et a. proposed an effectve agorth caed Heterogeneous Hoppng (HH) whch s based on the avaabe channe set ony. There are other agorths n ths category such as synchronous CH [], YNCETCH and YNC-ETCH [], MRCC [], C-MC [7], and Mt-Drdv []. Due to ted space, these agorths are not revewed and readers ay refer to the survey n [] for detas. Most of the exstng agorths use one snge rado per user for. The authors n [9] studed the utnterface probes for CRNs. However, they assued that a nodes have the sae nuber of rados (say, n), the channes are dvded nto n parttons, and each rado perfors over the channes n one partton. YTEM MODEL ND ROBLEM FORMULTION We consder a CRN consstng of K (K ) users. Te s dvded nto sots of equa duraton. The censed spectru s dvded nto non-overappng channes c,c,, c, where c s caed channe. Let C be the channe set {c,c,, c }. Let C C be the set of avaabe channes of user ( =,,, K), where a channe s sad to be avaabe to a user f the user can councate on ths channe wthout causng nterference to any Us. The avaabe channes can be dentfed by any spectru sensng ethod (e.g., []). Wthout oss of generaty, we consder the of a par of users, say user and user j, j and, j =,,, K. s equpped wth
( ) rados and user j s equpped wth n (n ) rados. Note that ay not be equa to n n CRNs. Let G be the nuber of coony-avaabe channes of user and user j. The CH sequence of user s denoted by {,,, }, where vector t = {t, t, t,, t} represents that user hops on channes th on rado h n sot t. Fg. shows the sequence structure where user has rados and user j has rados. For any gven and fxed C and C j where C and C j have at east one coony avaabe channe, f a agorth can ensure that can be copeted wthn a fnte, we say that ths agorth provdes guaranteed. In ths study, we do not requre synchronzaton n the networks. Wthout synchronzaton, the sot coud be doubed so as to ensure that the overap of two sots s ong enough to copete a necessary steps of [] [9]. In ths sense, the CH sequences of two users are sot-agned even wthout -synchronzaton []. In each sot, user hops on channes and user j hops on n channes to attept. We say that a s acheved f user and user j hop on the sae channe on any of the rados n the sae sot. Typcay the -to- (TTR) s usuay n the order of tens of seconds, whch s very sa copared wth the U dynac. To ustrate, et us consder an exape. uppose t s necessary to send a handshakng packet of bytes (e.g., contanng nforaton of user IDs) for and the data rate of the wreess channe s Mbps. That s, The duraton of each frae s ( )/( ) =.s. uppose that the IF (hort Inter Frae pacng) s μs (e.g., n IEEE.b or IEEE.n []) and the three-way handshakng s adopted. We can estate the necessary for as (. + ) =.7s. Thus, the duraton of each sot s.s around (.e., the overap of two sot s no ess than.7s). If t takes to sots to acheve (see the nuerca resuts n our paper), the to- s ony.s to s. On the other hand, a coon type of prary users quoted n the terature s the TV staton whch ony uses ts TV channes at certan of each day (say, evenngs and nghts), and the actvty of ths U changes very sowy copared wth the -to-. Therefore, channes avaabtes are assued to be statc n the process of. We defne the probe as foows. Rendezvous probe for two users: uppose two users have and n (, n ) rados respectvey and they ay start the process at dfferent. The probe s to deterne a CH sequence for each rado of each user, such that these users w hop on a coony-avaabe channe n the sae sot. OLUTION In ths secton, we generaze the Rando agorth and the exstng agorths to use utpe rados for faster. Then we desgn a new agorth, caed roe-based parae sequence Rado Rado Rado Rado Rado j j j j j j j j Fg.. equence structure when user has rados and user j has rados. (R), whch specfcay expots utpe rados for ore effcent.. Generazed Rando gorth When there s a snge rado, the Rando agorth randoy seects an avaabe channe n each sot and attepts to acheve on ths channe n ths sot. When there are utpe rados, ths Rando agorth can be generazed as foows: each rado randoy and ndependenty seects an avaabe channe n each sot and attepts to acheve on ths channe n ths sot. When two or ore rados happen to seect the sae channe, these rados w randoy seect agan unt they seect dfferent channes (suppose that for each user the nuber of avaabe channes s no ess than that of rados). Obvousy, the Rando agorth cannot guarantee the wthn fnte and hence the MTTR s nfnty. The agorth s foray presented as foows. gorth : Rando gorth Requre:,, C //for user : t = : t = {t, t, t,, t} : whe not do : for k =to do : : tk = Randoeect(C ) for j =to (k ) do 7: f tk == tj then : 9: tk = Randoeect(C ) j = : end f : end for : end for : t = t + : ttept on t : end whe In ne, t = {t, t, t,, t} denotes the set of channes that user hops on respectve rados n sot t. In nes -, user randoy seects an avaabe channe for each rado. In ne, the s acheved when one channe n t s equa to one channe of another user. The foowng theore gves the perforance propertes of the generazed Rando agorth. Theore. The E(TTR) of the generazed Rando agorth s equa to n under the syetrc n +n ode where s the nuber of a channes and j s the nuber of possbe perutatons of objects fro a set of j.
roof: In sot k, user hops on dfferent channes k and user j on n dfferent channes j k (Fg. ). If any channe n k s the sae as one channe n j k, the s acheved. In any sot, each channe has choces. The probabty that a channes of user are not equa to any channe of user j s +n n.op(ttr = t) =( +n ) t ( n +n ). The E(TTR)= + t p(ttr = t) = + n = = ( +n n ) t ( +n n )= +n n = n n +n. t Theore. The E(TTR) of the generazed Rando agorth s equa to C n C under the C n C C G n C G asyetrc ode where C and C are the nubers of avaabe channes of two users and G s the nuber of coony-avaabe channes of the two users. roof: In sot k, the probabty that a channes of user are not equa to any channe of user j s C G n C G. o C n C p(ttr = t) =( C G n C G ) t ( C G n C G C n ). C C n C The E(TTR) = + t p(ttr = t) = + = t ( C G n C G C n C C G n C G C n C = = ) t ( C G n C G ) = C n C C n C C n C C G n C G. Generazed Exstng Rendezvous gorths In the terature, severa agorths have been proposed for CRNs. They pcty assue one rado per user. In ths subsecton, we generaze these exstng agorths such that each user can use utpe rados for faster. We consder two strateges to generaze these agorths to use utpe rados: ) Independent equence: ppy an exstng agorth to ndependenty generate a CH sequence for each rado. If ths agorth aways generates the sae CH sequence usng a deternstc ethod, then the sequence s rotated by x postons where x s a randoy generated nteger. For exape, suppose a user s equpped wth two rados and an exstng agorth generates two CH sequences {s,s, } and {r,r, }. In the frst sot, the two rados hop on channes s and r respectvey. In the second sot, the two rados hop on channes s and r respectvey. ) arae equence: ppy an exstng agorth to generate a CH sequence and appy ths CH sequence on a rados n parae. For exape, suppose a user s equpped wth three rados and an exstng agorth generates the CH sequence {s,s,s,s,s,s, }. In the frst sot, the three rados hop on channes s, s and s, respectvey. In the second sot,. the three rados hop on channes s, s and s, respectvey. We propose two schees (naey, ndependent sequence and parae sequence) to generaze the exstng snge-rado agorths to use utpe rados. Whe the deas of these schees are spe, ther theoretca anayss s not trva. When dfferent nodes have dfferent nuber of rados, ony the ndependent sequence schee works. Nevertheess, the parae sequence schee aso has ts own ert: when the nodes have the sae nuber of rados, the parae sequence schee works and t gves better perforance than the ndependent sequence schee. Therefore, we report both schees n our paper. When the Independent equence strategy s apped to generaze an exstng agorth to use utpe rados for faster, the steps are gven n gorth. gorth : Independent equence Requre:,,, C //an exstng agorth denoted by and user : t = : t = {t, t, t,, t} : for k =to do : f s a deternstc ethod and the sequence s t then : kt = t rotated by rando postons : ese 7: kt s generated by agorth : end f 9: end for : whe not do : for k =to do : tk = kt : end for : t = t + : ttept on t : end whe In ne, snce the user has rados, t generates ndependent CH sequences by an exstng agorth. In nes -, k-th sequences are generated by ths agorth. In ne, the k-th sequence s perfored n the k-th rado. Let MTTR be the MTTR of the exstng agorth usng one rado. The foowng theore gves the MTTR of gorth. Theore. If two users perfor an exstng agorth on utpe rados wth Independent equence, the axu -to- (MTTR) s equa to MTTR under both the syetrc ode and the asyetrc ode. roof: When the users run an exstng agorth ndependenty on utpe rados, each par of rados of the two users w acheve on or before MTTR. s a resut, the two users w acheve a guaranteed on or before MTTR by any par of rados. o MTTR MTTR. Next we prove MTTR MTTR. Consder a worst case n whch a the rados of a user use the sae sequence (e.g., the exstng agorth happens to ndependenty generate dentca sequences). In ths case, the s the sae as the orgna agorth. o
User User j Rado Rado Rado Rado Rado Fg.. Rendezvous acheved by Jup-tay wth Independent equence. r r r j r j (a), r r j ; TTR r r / r j r j (c) <, = ; TTR / r j r j r r (e) <,r r j ; TTR r r r j r j (b), r = r j ; TTR + r r r j r j (d) = ; TTR r j r j r r / (f) <,r = r j ; TTR Fg.. x cases under the syetrc ode. Rearks: We say r = r j (Fg. (b) and Fg. (f)) f a stepengths of CH sequences wth user are dentca and are equa to those wth user j, and r r j (Fg. (a) and Fg. (e)) otherwse. MTTR MTTR. Overa, the MTTR s equa to MTTR. ong the exstng agorths, the Jup-tay agorth perfors we n ters of MTTR and E(TTR) [9]. When the Jup-tay agorth s generazed by the Independent equence strategy, we have the foowng resuts. Coroary. Under the syetrc ode, any two users perforng the Jup-tay agorth on utpe rados wth Independent equence acheve n at ost sots whch s an upper bound of MTTR. The + E(TTR) s not greater than ( ++ +n ), where s the saest pre nuber whch s not saer than. roof: Fg. ustrates how the sequences are generated. uppose that = and = 7. has rados wth ( =,r =), ( =,r =), and ( =,r = ). has rados wth ( =,r =)and ( =,r =). The s acheved n sot when user hops on channes (,, ) whe user j hops on channes (, ). Fg. shows the sx cases when users perfor the Jup-tay agorth ndependenty on utpe rados. tep-ength r takes nteger vaue n [,]; two users seect dfferent step-engths, r k r k, wth probabty ( ) whe seect the sae stepength wth probabty +n. Thus, we copute +n the occurrence probabtes of sx cases when a users have utpe rados are + ( ), +n +, +n,, ( ) +n and, respectvey. o the upper +n bound of MTTR shoud be and E(TTR) shoud be: + ( ) + +n + +n ( +)+ ( )+ + ( ) ( ) + +n ( ) = +n + ( ++ +n ). ar to the proof n [9], we can prove that the upper bound of E(TTR) under the asyetrc ode s saer than + ( ++ +n )+ ( G). When the arae equence strategy s apped to generaze an exstng agorth (say, agorth ) to use utpe rados for faster, the steps are gven n gorth. In ne, an exstng agorth s run to generate one CH sequence t. In nes -7, ths CH sequence s apped n parae to the rados of the user. The foowng theore gves the MTTR of gorth. gorth : arae equence Requre:,,, C //an exstng agorth denoted by and user : t = : t = {t, t, t,, t} : Generate sequence t by agorth : whe not do : for k =to do : tk = ((t ) +k) 7: end for : t = t + 9: ttept on t : end whe Theore. uppose two users are equpped wth rados. Under the syetrc and the asyetrc odes, when the arae equence strategy s apped n conjuncton wth an exstng agorth wth MTTR = MTTR, can be acheved n at ost MTTR sots. roof: nce the CH sequence s apped n parae to the rados of the user, the user w fnsh the hoppng sequence n sot MTTR. Fg. shows the new acheved when the users are equpped wth the sae nuber of rados. The channe at sot T n the hoppng sequence when the user s equpped wth one rado w appear n sot T when the user s equpped wth rados. If user and user j acheve on channe k n sot TTR before MTTR n agorth, then the two users w hop on ths channe and acheve at TTR before sot. Therefore, can be acheved n MTTR
7 at ost MTTR. Coroary. uppose a users are equpped wth rados. Under the syetrc ode, when the arae equence strategy s apped n conjuncton wth the Jup-tay agorth, can be acheved n at ost sots. The E(TTR) s upper bounded by ( + + )/, where s the saest pre nuber whch s not saer than. The proof of Coroary s very sar to the proof of Theore and we do not repeat the detas. Overa, the Independent equence strategy can guarantee even when the users have d- fferent nuber of rados, whe the arae equence strategy can guarantee ony when the users have the sae nuber of rados but t can better expot these rados to acheve saer MTTR.. New gorth In ths subsecton, we desgn and anayze a new agorth that specfcay expots utpe rados for ore effcent. Our basc dea s to assgn one of two possbe roes, caed genera rado and dedcated rado, to each rado. The s expected to be acheved between the genera rados of one user and the dedcated rado of the other. Our R agorth generates CH sequences n rounds and the ength of each round s n nversey proportona to the nuber of genera rados. The upper-bounds of MTTR and E(TTR) of R are the expresson of the ength of the round (ater shown n proof of Theore &). Therefore, arge nuber of genera rados eads to a shorter round whch consequenty gves saer upper-bounds of MTTR and E(TTR). In R, we use ony one rado as the dedcated rado and the reanng rados as the genera rados to optze the perforance. Users hop on avaabe channes n the genera rados whe stay on a specfc avaabe channe n the dedcated rado. uppose that a user s equpped wth rados. In our paper, each node s equpped wth utpe rados where each rado has a roe: ether dedcated or genera. o we ca the new agorth Roe-based arae equence (R). It s descrbed as foows. ) rados are dvded nto two groups, ( ) genera rados and one dedcated rado. ) startng ndex s randoy seected fro [, ] and a step-ength r s randoy seected fro [, ], where s the saest pre nuber whch s not saer than. ) ( ) genera rados hop on channes wth stepength r n the round-robn anner. For exape, n Fg., = 7, the startng ndex s and step-ength s. The frst channe n the channe hoppng sequence s and the k-th channe s ( + r k)% (( + k)%7 n ths exape). The CH sequence s {,,, 7,,,,,, } and two genera Rados and hop on ths sequence n parae as foows. Rado hops on subsequence {,,,, } and Rado hops on subsequence {, 7,,, }. v) Dedcated rado stays on one channe for sots and swtches to next channe for the CH sequence wth rados CH sequence wth rados j j j j New MTTR MTTR j MTTR Orgna Orgna CH sequence MTTR MTTR Orgna CH j j j j j sequence MTTR Fg.. Rendezvous of an exstng agorth wth arae equence when = n =. Rado Rado Rado 7 k 7 Genera rados Dedcated rado Fg.. CH sequence of a user wth rados and 7 channes. sae duraton, where the channe s taken fro [,] n a round-robn anner. For exape, n Fg., dedcated rado stays on channe for sots and then swtches to channe. v) If the channe seected n ) and v) s not avaabe to the user, randoy seect an avaabe channe. The agorth s foray presented as foows. gorth : R gorth Requre:,, C : t = : t = {t, t, t,, t} : =the saest pre nuber not saer than : = Randoeect(,) : r = Randoeect(,) : whe not do 7: for k =to do : tk =( +((t ) ( ) + k ) r ) od + 9: f tk then : tk = tk od : end f : f tk / C then : tk = Randoeect(C ) : end f : end for : t t =( ) od + 7: f t / C then : t = Randoeect(C ) 9: end f : t = t + : ttept on t : end whe
In ne, startng ndex and step-ength r are preseected randoy. In nes 7-, the ( ) genera rados w hop on contnuous ( ) channes wth and r. In ne, the dedcated rado w swtch to the next channe after sots. Lnes - and 7-9 ensure that the channes are avaabe to the user. Fg. 7 shows of two users by perforng the R agorth. uppose that C = =, =. s equpped wth rados and the avaabe channes are {, }. It starts wth channe. tep-ength s. Each round conssts of = sots. s equpped wth rados and the avaabe channes are {, }. It starts wth channe. tep-ength s. Each round conssts of = sots. In ths exape, there w be a rando avaabe channe n the poston wth underne snce the channe based on the agorth s not avaabe to the user. In Fg. 7(a), two users hop on channe n sot. Rendezvous s acheved by the dedcated rado of user j and the genera rados of the user. w stay on channe for sots. has a perutaton of a channes n any consecutve sots. nce there s at east one coony-avaabe channe between the, channe n ths exape, there ust be a between the dedcated rado of user j and the genera rados of user. However, before ths, there s an earer one n sot. It s acheved by the genera rados of user and the genera rado of the user j. In Fg. 7(b), user j starts ater than user for sot. When the dedcated rado of user (wth a shorter round) stays on the coony-avaabe channe (channe ) for sots, the genera rados of user j do not have a perutaton of a avaabe channes n consecutve sots. Therefore, t s possbe that channe s not n these sots. We cannot guarantee a between the dedcated rado of user and the genera rados of user j. The guaranteed reazed at sot. Now we theoretcay anayze the R agorth. pecfcay, we derve the upper bounds of MTTR of the R agorth n Theore and Theore under the syetrc ode and the asyetrc ode, respectvey. In addton, we derve the upper bounds of E(TTR) of the R agorth under both the syetrc and asyetrc odes. Theore. Under the syetrc ode, et and n denote the nubers of rados of two users, respectvey. If n, the MTTR of the R agorth s upper bounded by ( ax{,n} ) and the E(TTR) of the R agorth s upper bounded by ax{,n} + ( ax{,n} ) n{,n} ;f = n, the MTTR of the R agorth s upper bounded by ax{,n} and the E(TTR) of the R agorth s upper bounded by ax{,n}. roof: Let user be equpped wth rados whe user j be equpped wth n rados. Fg. sts the four cases of under syetrc ode. Fg. (a), (b), and (c) happen when n. nce the resuts depend on whch user starts hoppng frst, we assue Rado Rado Rado Rado Rado (a) offset= Rado Rado Rado Rado Rado (b) offset= Fg. 7. Rendezvous of two users by perforng R. ' (a) n, ; TTR n n (c) TTR n ' (b) > n, ; TTR n n (d) TTR n Fg.. Four cases of R under the syetrc ode. <n,.e., > n. Fg. (d) happens when = n. rearkabe dstngushent between the s whether the engths of each round of the two users are the sae. ) Case : Fg. (a). n pes that there s a perutaton of a channes before the dedcated rado of user transfers to next channe. The s acheved between genera rados of user j and dedcated rado of user durng the frst n sots. That s, TTR n. ) Case : Fg. (b). In ths case, < n pes that there s no enough sots for user j to have perutaton of a channes before the dedcated rado of user transfers to next channe. The can ony be guaranteed between genera rados of user and dedcated rado of user j durng the frst n sots. That s, TTR n. ) Case : Fg. (c). starts frsty. has a perutaton of a channes n any n consecutve sots. When user starts, t w stay on one channe for sots. > n. Thus, a s guaranteed before n sots.
9 ) Case : Fg. (d). = n. The s acheved before the frst (or n ) sots. When n, n the above anayss, s repaced by n{,n} and n by ax{,n}. ccordng to the anayss of these cases, we prove that the MTTR s ( ax{,n} ). Cobnng wth the occurrence probabtes we derve an upper bound of E(TTR) under the syetrc ode. The E(TTR) [ n{,n} ax{,n} + n{,n} ax{,n} + ax{,n} ( n{,n} ax{,n} )]+ ax{,n} ax{,n} + ( ax{,n} ). n{,n} There s ony one case when = n. the MTTR and the upper bound of E(TTR) are both or n. Theore. Under the asyetrc ode, et and n denote the nubers of rados of two users, respectvey. If n, the MTTR of the R agorth s upper bounded by ( ax{,n} ) + n{,n} ( G) and the E(TTR) of the R agorth s upper bounded by ( ax{,n} ax{,n} + ) + n{,n} n{,n} ( G); f = n, the MTTR of the R agorth s upper bounded by ( G +) and the E(TTR) of the R agorth s upper bounded by ( G +). roof: Under the asyetrc ode, snce the avaabe channe sets of two users are dfferent fro each other, the users ay acheve any potenta ( to ( G) n Fg. 9). In Fg. 9(a), user j has a perutaton of a channes before n and the dedcated rado of user stays on one channe durng ths perod. There s a potenta before n ; ths channe ay not be a coony-avaabe channe to a users. The next potenta can be guaranteed n the next round of user ( to n Fg. 9) because ony after these sots the dedcated rado of user w transfer to the next channe. We can say, under asyetrc ode, we expect a between the dedcated rado of the user wth ess rados (user ) and the genera rados of the user wth ore rados (user j). The worst case s repeatng the under syetrc ode for ( G) s. bove a, the MTTR shoud be equa to or saer than ax{,n} + n{,n} ( G). We assue a unfor dstrbuton that a coonyavaabe channe appears on the st to ( G)-th potenta. The upper bound of E(TTR) when n extend for n{,n} ( G) n a cases. In ths way, the E(TTR) ax{,n} + ( ax{,n} ) + n{,n} n{,n} ( G). nd sary, the MTTR and the upper bound of E(TTR) s ( G +) when = n. IMULTION We but a suator n Vsua tudo to evauate the effectveness of the proposed approach (.e., usng ' (a) n, ( G) ' ( G) ; TTR n n + ( G) (b) > n, < ; TTR n n + ( G) Fg. 9. ode. ( G) (c) TTR n + ( G) ( G) (d) TTR n + ( G) Four cases of R under the asyetrc utpe rados for ) and the proposed agorth (.e., the Roe-based arae equence (R) agorth). When each user has a snge rado, we consder the foowng agorths for coparson: ) the Rando agorth [] (t s the ost spe agorth), and ) the Jup-tay agorth [, 9], MMC agorth [], HH [] (They are recenty proposed and they have good perforance). When each user has utpe rados, we consder the foowng agorths for coparson: ) the generazed Rando agorth(ecton.), - ) the generazed Jup-tay (MMC, HH) agorth wth the Independent equence strategy (ecton.), ) the generazed Jup-tay (MMC, HH) agorth wth the arae equence strategy (ecton.), and v) the R agorth (ecton.). The perforance s easured n ters of the average TTR and the axu TTR, where TTR s counted as the nuber of sots requred to acheve. We consder both the syetrc ode and the asyetrc ode. We use the notaton (, n) to denote the case that two users are equpped wth and n rados respectvey. We consder the foowng key paraeters: the nuber of channes n the whoe channe set s vared fro to. Under the syetrc ode, a channes are avaabe to a users. Under the asy-
9 7 of Mut rado (syetrc) (,) (,) (,) of Mut rado (syetrc) (,) (,) (,) Energy Consupton (U) Energy Consupton of Mut rado (syetrc) (,) (,) (,) of Mut rado (syetrc) (a) V. (b) V. (c) Energy Consupton V. (,) (,) (,) (,) (,) (,) Nubers of rados of two users (,n) (d) V. (, n) Fg.. Coparson of ut-rado and snge-rado under the syetrc ode (Rando). etrc ode, we ntroduce a paraeter θ ( <θ ) and randoy seect channes fro the channe set, such that the average nuber of channes avaabe to a user s equa to θ. Ifθ <, we reset θ to. For each set of paraeter vaues, we perfor, ndependent runs and then copute the average TTR and the axu TTR.. Effectveness of roposed pproach In ths subsecton, we deonstrate that utpe rados can effectvey prove the perforance... Under the yetrc Mode Fg. shows the perforance of the Rando agorth under the syetrc ode. It can be seen that: ) utpe rados can sgnfcanty reduce both the average TTR and the axu TTR, and ) the perforance proveent s especay sgnfcant when the nuber of rados per user s ncreased fro a sa vaue. For exape, we suppose that each sot has duraton of s [9]. When there are channes and the nuber of rados per user s ncreased fro to, the average TTR s decreased fro. (.s) to. (.s) (.e., 7.7% reducton) whe the axu TTR s decreased fro (.s) to 7 (.s) (.e., 7.% reducton). When the nuber of rados per user s ncreased fro to, the average TTR s decreased fro. (.s) to. (.s) (.e.,.% reducton) whe the axu TTR s decreased fro 7 (.s) to (.7s) (.e., 9.% reducton). Therefore, a cost-effectve tradeoff between cost of the rados and perforance s to seect to rados per user. In addton, we fnd that adopton of utpe rados can reduce the overa energy cost on. For exape, when the nuber of rados s ncreased fro to, the average spent on (E(TTR)) s reduced by 7.9% (ore than s). We assue that one rado perforng n one sot costs one unt of energy, denoted by U. Then the expected energy consupton s E(TTR) U ( + n) when the two users are equpped wth and n rados, respectvey. Fg. (c) shows the expected consupton when each user s equpped wth snge rado and utpe rados. For exape, when there are channes, the energy consupton of users wth,, and rados are.u,.77u,.u and.u, respectvey. Fg. (d) shows the average TTR under dfferent (, n) of Mut rado (syetrc) (,) (,) (,) (a) V. of Mut rado (syetrc) (,) (,) (,) (b) V. Fg.. Coparson of ut-rado and snge-rado under the syetrc ode (Jup-tay). of Mut rado (asyetrc) (,) (,) (,) (a) V. of Mut rado (asyetrc) (,) (,) (,) (b) V. Fg.. Coparson of ut-rado and snge-rado under the asyetrc ode (Rando). when M =. We fnd that the average TTR drops sgnfcanty wth the ncrease of nuber of rados. Fg. shows the perforance of the Jup-tay agorth under the syetrc ode. We observe sar propertes as those n Fg.. For exape, when the nuber of channes s and the nuber of rados s ncreased fro to, the average TTR s reduced fro.99 to. (.e,.7% reducton) whe the axu TTR s decreased fro to (.e,.% reducton). ccordng to Theore and Coroary, however, the upper bound of E(TTR) s decreased fro 9. to. whe the MTTR reans. The theoretca resuts are uch bgger than experenta resuts. It s because that Jup- tay agorth randoy seects avaabe channes to repace the unavaabe channes on the generated CH sequence, whch practcay owers TTR n suaton.
of Mut rado (asyetrc) (,) (,) (,) of Mut rado (asyetrc) (,) (,) (,) of dfferent (,n) when +n= (syetrc) (,) of dfferent (,n) when +n= (syetrc) (,) (a) V. (b) V. (a) V. (b) V. Fg.. Coparson of ut-rado and snge-rado under the asyetrc ode (Jup-tay)... Under the syetrc Mode Fg. shows the perforance of Rando agorth under the asyetrc ode. We et θ =. and G =. (.e., each user has. channes and each par of users have. coony-avaabe channes). It can be seen that utpe rados can reduce both the average TTR and the axu TTR sgnfcanty under the asyetrc ode. For exape, when there are channes and the nuber of rados per user s ncreased fro to, the average TTR s decreased fro. to.7 (.e.,.9% reducton) whe the axu TTR s decreased fro to (.e., 7.7% reducton). When the nuber of rados per user s ncreased fro to, the average TTR s decreased fro.7 to. (.e.,.% reducton) whe the axu TTR s decreased fro to (.e., % reducton). Fg. shows the perforance of the Jup-tay agorth wth Independent equence under the asyetrc ode. For exape, when there are channes and the nuber of rados per user s ncreased fro to, the average TTR s decreased fro. to. (.e.,.% reducton) whe the axu TTR s decreased fro 777 to (.e., 9.97% reducton).. roperty of roposed pproach: Infuence of Rado ocaton In ths subsecton, we study a basc property of the proposed approach: when the tota nuber of rados for two users s fxed (.e., + n s a constant), how does the aocaton (, n) for dfferent and n affect the perforance? For exape, f there are four rados, does (, ) gve better perforance than (, ) or (, )? In theory, the MTTR of the R agorth s decded by the nu vaue of and n. The MTTR of Rando agorth [] and other exstng agorths s aost ndependent of and n. resuts show that even aocaton has the best perforance. Under the syetrc ode, we et + n =and G =. Fg. shows the coparson of the average TTR and the axu TTR of Rando agorth between the three cobnatons of (, n) whch are (, ), (, ) and (, ). We can see that the cobnaton (, ) has the best perforance on both the average TTR and the axu TTR. For exape, when there are channes, the average TTR of (, ), (, ) and (, ) are.,.7 and. whe the axu Fg.. Coparson of dfferent aocaton of (, n) when +n =under the syetrc ode (Rando). of dfferent (,n) when +n= (syetrc) (,) (a) V. of dfferent (,n) when +n= (syetrc) (,) (b) V. Fg.. Coparson of dfferent aocaton of (, n) when + n =under the syetrc ode (Jup- tay). TTR are, and 9, respectvey. It s consstent wth the theoretca resuts. We aso do the sae coparson when we appy other exstng agorths on utpe rados. Fg. shows the coparson of the average TTR and the axu TTR of the Jup- tay agorth wth Independent equence between the three cobnatons of (, n) whch are (, ), (, ) and (, ). Fg. shows the coparson of the average TTR and the axu TTR of the R agorth between the two cobnatons of (, n) whch are (, ) and (, ). Under the asyetrc ode, we et + n =, θ =.7 and G =.. Fg. 7 shows the resuts of Rando agorth between the three cobnatons of (, n). The cobnaton (, ) has the best perforance on both the average TTR and the axu TTR. For exape, when there are channes, the average TTR of (, ), (, ) and (, ) are., of dfferent (,n) when +n=(syetrc) (,) Nuber of a Channes () (a) V. of dfferent (,n) when +n=(syetrc) 7 (,) Nuber of a Channes () (b) V. Fg.. Coparson of dfferent aocaton of (, n) when + n =under the syetrc ode (R).
of dfferent (,n) when +n= (asyetrc) (,) of dfferent (,n) when +n= (asyetrc) (,) of Mut rado (Jup tay) G=. G=. G=. of Mut rado (Jup tay) G=. G=. G=. (a) V. (b) V. (a) V. (b) V. Fg. 7. Coparson of dfferent aocaton of (, n) when + n =under the asyetrc ode (Rando). of dfferent (,n) when +n= (asyetrc) (,) (a) V. of dfferent (,n) when +n= (asyetrc) (,) (b) V. Fg.. Coparson of dfferent aocaton of (, n) when + n = under the asyetrc ode (Jup- tay). 7. and. whe the axu TTR are, and, respectvey. Fg. shows the resuts of the average TTR and the axu TTR of the Jup-tay agorth wth Independent equence between the three cobnatons of (, n). Fg. 9 shows the coparson of the average TTR and the axu TTR of the R agorth between the three cobnatons.. roperty of roposed pproach: Infuence of G for Fxed θ In ths subsecton, we study another basc property of the proposed approach: how does the nuber of coony-avaabe channes between two users affect the perforance? We et θ =., G =., G =. and G =., respectvey. = and n =. Fg. shows the resuts when of dfferent (,n) when +n=(asyetrc) (,) Nuber of a Channes () (a) V. of dfferent (,n) when +n=(asyetrc) 9 7 (,) Nuber of a Channes () (b) V. Fg. 9. Coparson of dfferent aocaton of (, n) when + n =under the asyetrc ode (R). Fg.. Coparson of dfferent G when θ =. (Jup-tay). of Mut rado (R) G=. G=. G=. (a) V. of Mut rado (R) G=. G=. G=. (b) V. Fg.. Coparson of dfferent G when θ =. (R). we appy the Jup-tay agorth to utpe rados wth Independent equence. When there are channes, the average TTR of three scenaros are.9,. and. and the axu TTR are, 9 and. Fg. shows the resuts of the R agorth. When there are channes, the average TTR of three scenaros are.,. (.e..% reducton) and. and the axu TTR are 7, (.e..% reducton) and. ccordng to Theore, however, when G s ncreased fro. to., the upper bound of E(TTR) s decreased fro.7 to. (.e.,.% reducton) and the MTTR s decreased fro. to. (.e.,.% reducton). It reveas that both the average TTR and the axu TTR drop faster than theoretca resuts when G ncreases.. erforance of Generazed gorths and roposed gorth We now study the perforance of the proposed agorth and the generazed versons of the exstng agorths. We want to ephasze that a foowng suatons are based on utpe rados. of the have sgnfcant proveent copared wth exstng agorths. In ths part, we copare the average TTR and the axu TTR under dfferent vaues of, n, θ and G. We copare Generazed Rando gorth (descrbed n ecton,), Generazed Exstng gorths (ecton.) and R gorth(ecton.) under the scenaro when a users are equpped wth utpe rados. For the exstng agorths, we appy the Jup-tay [], MMC [] and HH [] agorth wth both Independent equence and arae equence. HH s
of dfferent agorths(syetrc) R J/ndependent J/parae Rando MMC/ndependent MMC/parae (a) V. of dfferent agorths(syetrc) R J/ndependent J/parae Rando MMC/ndependent (b) V. 9 7 of dfferent agorths(syetrc) R J/ndependent J/parae Rando MMC/ndependent MMC/parae (a) V. of dfferent agorths(syetrc) R J/ndependent J/parae Rando MMC/ndependent (b) V. Fg.. Coparson of dfferent agorths under the syetrc ode when (, n) =(, ). for heterogeneous syste. We ony appy t under the asyetrc ode and we expand t wth rando repaceent. When the channe n the sequence s not avaabe, we randoy seect an avaabe one to repace t... Under the yetrc Mode = n: Under the syetrc ode, we et θ =and G =. Fg. shows the coparson of the average TTR and the axu TTR between the sx dfferent agorths when = n =. s shown n Fg. (a), when there are channes, the average TTR of R, J/Independent, J/arae, Rando, MM- C/Independent and MMC/arae are.,.,.,.,.9 and.9 respectvey. There s no arge gap between dfferent agorths. However, the axu TTR are 7,,,, and 9 respectvey. nce the of MMC/arae s too arge, we show the dfference between other fve agorths n Fg. (b). We fnd that the R agorth has no advantage on the average TTR but has sgnfcant proveent on the axu TTR. Its axu TTR s aost a thrd of others. n: In ths case, users are equpped wth dfferent nubers of rados. We et =and n =. Fg. shows the coparson of the average TTR and the axu TTR between the sx dfferent agorths. For exape, when there are channes, the average TTR of R, J/Independent, J/arae, Rando, MMC/Independent and MMC/arae are.,.,.,.,.9 and.9 respectvey. However, the axu TTR are,, 7,, 7 and respectvey. ae as the scenaro = n, Fg. (b) shows the dfference between fve of the agorths. The R agorth has no advantage on the average TTR but has sgnfcant proveent on the axu TTR. nother portant pont s that J/arae has the worst perforance when n, whch s consstent wth the theoretca resut... Under the syetrc Mode Large θ: Frsty we study the scenaro when θ s arge, that s, ost channes are avaabe to users. We et θ =., G =., = and n =. Fg. shows the coparson of the average TTR between eght dfferent agorths. For exape, when there are channes, the average TTR of R, J/Independent, J/arae, Rando, MMC/Independent, MMC/arae, H- H/Independent and HH/arae are.,.,.9, Fg.. Coparson of dfferent agorths under the syetrc ode when (, n) =(, ). 9 7 of dfferent agorths(asyetrc θ=.) R J/ndependent J/parae Rando MMC/ndependent MMC/parae HH/Independent HH/arae (a) V. of dfferent agorths(asyetrc θ=.) R J/ndependent J/parae Rando MMC/ndependent HH/Independent HH/arae (b) V. Fg.. Coparson of dfferent agorths under the asyetrc ode when θ =...,.77,.,. and. respectvey. However, the axu TTR are 9, 9,,,, 9, 97 and respectvey. HH agorth wth utpe rados has dstnct advantage on the. However, both MMC and HH has very arge. The reason s that ther guaranteed s based on soe speca condtons. Therefore, n ters of the average TTR, the generazed HH agorth s the best; n ter of the axu TTR, the R agorth s the best when θ s arge. a θ: Then we study the scenaro when θ s sa, that s, ony a sa part of channes are avaabe to users. We et θ =., G =., = and n =. For exape, when there are channes, the average TTR of R, J/Independent, J/arae, Rando, MMC/Independent, MMC/arae, H- H/Independent and HH/arae are.,.,.,.,.,.7,. and. whe the axu TTR are,, 7,,,, and respectvey. Fg. shows the coparson between the seven dfferent agorths (Except MMC/arae). Therefore, n ters of the average TTR, the generazed HH agorth s the best; n ter of the axu TTR, the R agorth s the best when θ s sa. CONCLUION ND FUTURE WORK We nvestgated a new approach (usng utpe rados per user) to sgnfcanty speedng up the process n cogntve rado networks, generazed the Rando agorth and the exstng agorths n order to use utpe rados for faster, and desgned a new agorth (caed roe-based parae sequence (R)) to specfcay
.... of dfferent agorths(asyetrc θ=.) R J/ndependent J/parae Rando MMC/ndependent HH/Independent HH/arae (a) V. of dfferent agorths(asyetrc θ=.) R J/ndependent J/parae Rando MMC/ndependent HH/Independent HH/arae (b) V. Fg.. Coparson of dfferent agorths under the asyetrc ode when θ =.. expot utpe rados for fast. We theoretcay derved the upper bounds of E(TTR) and MTTR of these agorths, and conducted extensve suaton studes to evauate ther perforance. We observed the foowng propertes: Mutpe rados can sgnfcanty speed up, especay when the nuber of rados per user s ncreased fro a sa vaue. For exape, when there are channes and the nuber of rados per user s ncreased fro to, the average TTR of the generazed Jup-tay agorth wth Independent equence s reduced by.7% whe ts axu TTR s reduced by 7.% under the syetrc ode. Gven a fxed nuber of rados for two users, the perforance woud be better f the rados are eveny aocated on the two users. 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