A Slot-Asynchronous MAC Protocol Design for Blind Rendezvous in Cognitive Radio Networks

Globecom 04 - Wireless Networking Symposium A Slot-Asynchronous MAC Protocol Design for Blind Rendezvous in Cognitive Rdio Networks Xingy Liu nd Jing Xie Deprtment of Electricl nd Computer Engineering The University of North Crolin t Chrlotte Emil: {xliu33, Lind.Xie}@uncc.edu Abstrct In cognitive rdio networks (CRNs), two users hve to rendezvous on common vilble chnnel before communictions. Most existing rendezvous ppers focus on the chnnelhopping (CH) sequence design. However, rendezvous my suffer from the hndshke filure on the rendezvous chnnel, especilly in unsynchronized-slot scenrios. In this pper, the chllenge of slot-synchronous rendezvous in CRNs is ddressed for the first time. A protocol iming to improve the hndshke performnce during the CH process is proposed. By nlyzing the potentil fctors leding to the hndshke filure, we design novel MAC protocol with n optiml size of time slot which cn mitigte the effects of these fctors nd provide the shortest time for rendezvous. In ddition, we lso propose probbilistic model for estimting the verge rendezvous time under different CRNs. Simultion results vlidte our nlyticl model nd demonstrte tht our proposed protocol cn chieve the rendezvous time close to the theoreticl vlue under slot-synchronous scenrios. I. INTRODUCTION In order to solve the spectrum scrcity nd under-utiliztion problem, cognitive rdio emerges s promising technology which llows secondry user () to ccess the spectrum unoccupied by licensed users, or, primry users (PUs). It lso requires to vcte chnnels for the returning of PUs. In other words, the vilble chnnels for my chnge by time or its loction. Hence, unlike trditionl wireless networks, control chnnel tht is commonly vilble to ll s in cognitive rdio network (CRN) my not exist or cnnot lst for long time. It is lso imprcticl for to obtin other s chnnel informtion using such common control chnnel (CCC). Therefore, two s meeting ech other on common chnnel is bsic step before they cn estblish communictions in CRNs. This process is clled blind rendezvous. To chieve blind rendezvous, the sequence-bsed chnnelhopping (CH) technique cn be used. In this pproch, ech first senses the spectrum nd genertes set of vilble chnnels. Then, it hops onto these chnnels one by one following predefined sequence. Thus, two s cn rendezvous if they hop on sme chnnel t the sme time. The stte-of-thert CH [] cn gurntee the rendezvous between ny two s if they hve t lest one common vilble chnnel. It minly focuses on designing the CH sequence to chieve rendezvous in short time period which is clled time to rendezvous (TTR). TTR represents the number of chnnels needs to go through before hopping to sme chnnel with nother. However, in prcticl scenrios, successful hopping on sme chnnel does not necessrily led to successful hndshke which cn be ffected by mny fctors []. Only fter successful hndshke, cn two s truly estblish dt communictions. Thus, we define time to hndshke (TTH) s our performnce metric in this pper. In order to get shorter TTH, existing hopping lgorithms need to work under This work ws supported in prt by the US Ntionl Science Foundtion (NSF) under Grnt No. 0953644, 875, nd 343355. pproprite MAC protocols to gurntee successful hndshke in CRNs. In such MAC protocol, the key feture to support the CH is to mintin per-unit-length the sme in ll sequences, i.e., ech s sojourn durtion on ech chnnel should be the sme. This feture ccords with the time-slotted system where the stying time on ech chnnel cn be treted s one time slot. A time slot should be long enough for two s to complete hndshke process. It mens tht Requestto-Send (RTS) nd Cler-to-Send (CTS) cn be successfully exchnged by the sender nd the receiver, bsed on the Crrier Sense Multiple Access with Collision Avoidnce (CSMA/CA) mechnism in IEEE 80.. On the other hnd, the time slot of ech might need to be synchronized in order to ensure tht two s cn hop on the sme chnnel t the sme time. Although some CH ppers [], [3] clim tht their sequence design cn work under the synchronous scenrio, it differs from the synchronous cse in the protocol design. For exmple, Fig. () shows synchronous CH cse where nd strt to hop t the sme time; Fig. (b) shows the synchronous CH cse where two s strt to hop t different time slots; nd in Fig. (c), the slots re unsynchronized. We nme the ltter two cses s user-synchronous cse nd slotsynchronous cse, respectively. In this pper, we focus on the slot-synchronous cse. CH 3 CH 3 CH 3 CH 3 CH 3 () synchronous (b) user-synchronous (c) slot-synchronous Fig.. Synchronous cse nd synchronous cses in different designs. However, not ll MAC protocols in CRNs cn be used for blind rendezvous, such s the protocols under the single chnnel [4] nd the underly model [5], [6]. Other designs [7], [8] either directly use the synchronized time slot ssumption [9] [] or chieve synchroniztion by imprcticl methods, including employing CCC [3] [7], using multiple trnsceivers [], [3], [7] [9], or brodcsting becons on ll chnnels before rendezvous [0], which is less efficient due to high overhed nd collision. The protocol proposed in [] is climed to be robust enough under synchronous slots. However, it does not fully consider the potentil problems in such scenrio nd lcks detils. In summry, ll the existing MAC designs consider time synchroniztion s necessry condition for blind rendezvous nd chieve it in different imprcticl wys. In this pper, we consider chieving rendezvous from different perspective. We propose novel RTS/CTS hndshke mechnism to mitigte the effects cused by synchronous time slots. This mechnism cn chieve successful hndshke with high probbility when the sender nd the receiver rrive on sme chnnel t different moments due to their synchronous time slots. Menwhile, this mechnism cn lso solve the problems prohibiting successful hndshke in synchronous scenrios. The length of time slot plys crucil CH 3 978--4799-35-3/4/$3.00 04 IEEE 464

Globecom 04 - Wireless Networking Symposium role in this design. It is trdeoff prmeter. When time slot is long, the probbility of hving successful hndshke is high, but it tkes long time for two s to hop on sme chnnel. On the other hnd, if time slot is short, the hndshke process cnnot be gurnteed, which my cost more time to get successful rendezvous. Therefore, we lso obtin the optiml length of time slot for our design in terms of the shortest TTH. Simultion results vlidte the optimlity of the time slot nd indicte tht the synchronized slot ssumption is not necessry for our rendezvous protocol design in CRNs. To the best of our knowledge, this is the first work on MAC design for blind rendezvous in CRNs under the slot-synchronous scenrio. The rest of this pper is orgnized s follows. In Section II, we nlyze three problems tht my hppen during one time slot which cn potentilly ruin hndshke process. At the sme time, we design n pproprite protocol to mitigte the effect of these problems s much s possible. In Section III, we derive the expression of the optiml length of time slot which cn minimize the TTH. Simultion results re shown in Section IV, followed by the conclusions in Section V. II. PROBLEM ANALYSIS AND PROTOCOL DESIGN In this section, we first introduce the system model in our nlysis. Then, we nlyze three min problems tht my result in hndshke filure in one time slot nd t the sme time estblish our protocol design to solve these problems. A. System Model The system considered in this pper consists of finite number of rndomly distributed PUs nd s which cn operte over set of orthogonl chnnels. Pcket rrivls of both PUs nd s follow the Poisson distribution. Ech PU is rndomly ssigned chnnel when new pcket needs to be trnsmitted. Ech source rndomly chooses within its trnsmission rnge s its destintion when new pcket needs to be trnsmitted. Both the source s nd listening s sense ll chnnels to find their own vilble chnnels nd utilize CH lgorithm to hop mong them. This process should be done periodiclly to void chnnel sttus chnge due to PU ctivities. We ssume tht ech works in hlf-duplex mode. When source wnts to communicte with its destintion, the sender sends n RTS messge on ech chnnel it hops on during ech time slot until receiving the correct CTS. We cll the in this process n ctive. On the other hnd, since pssive who hs no pcket to send my become potentil destintion of its neighbor, it keeps listening on ech chnnel it hops on during ech time slot until receiving correct RTS. The hndshke in our model is ssumed to be free of propgtion-interference loss. Compred with trditionl hndshke filures, more possible reson for hndshke filure in CRNs is tht the destintion is not on the sme chnnel in the current time slot. Hence, for the ske of quick rendezvous, the source should leve its current chnnel nd keep hopping on to other chnnels if the corresponding CTS is not received t the end of time slot. Only fter successful hndshke, cn the dt trnsmission tke plce. The totl time source spends before dt pcket being trnsmitted is the TTH. Since this pper focuses on the hndshke process, we denote the TTH s our service time in the queuing system nlysis. Moreover, even the destintion is on the sme chnnel with the source, they my still fil to hndshke. The first reson is tht n RTS cnnot be received completely due to synchronous time slots. The second reson is neighbor s interference, including chnnel contention nd the hidden terminl collision. The lst reson is clled both-shouting problem cused when the destintion is lso in n ctive mode. The ltter two issues lso exist under synchronous scenrios. B. Anlysis of the Filure Receiving Problem Fig.. Chnnel j Chnnel k RTS RTS RTS Chnnel k Chnnel h The RTS filure receiving cses. In n synchronous CRN, s illustrted in Fig., pssive my not receive complete RTS due to hopping onto potentil rendezvous chnnel lter thn the strting time of n RTS sending (on chnnel i), or leving erlier before the sending finishes (on chnnel k). RTS RTS t t + t +- t + t t + t t + i) the cse of the erliest possible t ii) the cse of the ltest possible t Fig. 3. The cses tht t lest one RTS cn be completely received. Let nd rrive on sme chnnel t moments t nd t, respectively. We normlize the length of sending n RTS/CTS to. If the length of time slot is, should be longer thn so tht t lest one pir of RTS nd CTS exchnge cn be completed in time slot. In ddition, we hve the constrint t t to ensure tht nd hve overlpping time on the common chnnel. There re lso the following constrints (see Fig. 3) to help her t lest one complete RTS from.ift t ( hops on the chnnel lter thn ), the leving time of should be t lest lter thn the end time of the first RTS sent by on the common chnnel, i.e., t + t +.Ift t ( hops on the chnnel erlier thn ), the rriving time of should be t lest erlier thn the strt time of the lst possible RTS sent by on the common chnnel, i.e., t t +. Note tht fter ech RTS is sent out, must wit for while for the potentil CTS. Thus, the lst RTS in the current time slot must be sent before. To sum up, we hve the following equivlent inequlities nd their corresponding grphic illustrtion: 0 t 0 t t + t t t t t + - t 0 - The bove shdow re represents the fesible rnges of t nd t to ensure the receiving of t lest one complete RTS. Therefore, we cn derive the probbility tht pssive t 464

Globecom 04 - Wireless Networking Symposium receives complete RTS from nother on chnnel in the synchronous scenrio, P, size of the shdow re P = size of the squre =.5,. () In synchronous CRNs, it is nturl to define the size of one time slot to be the length of n RTS nd CTS exchnge for the ske of quick rendezvous, i.e., =. However, ccording to (), this design leds to probbility of 0.375 to hve successful RTS reception even fter two s hop on sme chnnel in the synchronous scenrio. Since P is monotoniclly incresing function of, from this point of view, we should design the length of time slot s long s possible nd let keep sending RTS until the current time slot ends. C. Anlysis of the Neighboring Interference Problem In CRNs, especilly in cognitive rdio d hoc networks (CRAHNs), severl other s my be within s sensing rnge. Hence, three or more s my hop on sme chnnel in one time slot during their rendezvous processes. They my interfere with ech other in two scenrios. One is the presence of RTS collisions in the hidden terminl cse. The other is the continuous contention for sending RTS between ctive neighboring s in one time slot. In trditionl wireless networks, one reson tht n ctive node cnnot receive the correct CTS is the RTS collision from hidden terminl. Thus, 80. CSMA/CA requests node to perform binry exponentil bckoff when experiencing the bsence of CTS. However, this mechnism my not increse the successful rte of hndshke when pplied to CRNs, since more possible reson for the bsence of CTS is tht the destintion node is not on the sme chnnel. In ddition, to support bckoff, the size of time slot needs to be uncceptble long. Moreover, the bckoff my still collide with new who just hops on this chnnel fter the bckoff under the synchronous scenrio. On the other hnd, ech time when resends n RTS, it is n dditionl contender for other s. If the destintion is not on the sme chnnel, the source keeps rejoining the contention, which ffects other s opportunities to send the RTS. For exmple, in Fig. 4, hs successfully sent n RTS severl times in time slot (gry RTS/CTS mens the supposed sending/receiving but not chieved). If is hidden terminl of, it will collide with s jth resend. If is neighbor of, it will lose the opportunity to send its RTS becuse of s kth resend. If s destintion is bsent on this chnnel during this time slot, this contention keeps hppening till leves the chnnel. Fig. 4. RTS CTS RTS CTS j th resend k th resend RTS RTS The neighboring interference cses. Therefore, the trditionl method for resolving the RTS collision is not desirble for synchronous rendezvous in CRNs. A better mechnism is required in our protocol. From Section II-B, note tht P is only ffected by three fctors: the sending moments of the first nd the lst RTS in time slot, nd the length of time slot. Bsed on this observtion, we redesign the protocol which cn solve the neighboring interference problem nd menwhile hs n equivlent effect s the design in Section II-B. We propose tht n ctive only sends n RTS twice in time slot: one t the beginning nd one t the end of time slot if chnnel is idle, nd listens to the chnnel during other periods, s the illustrted in Fig. 5. However, if the length of time slot,, is not long enough for sending the second RTS (with CTS receiving), gives up the resend nd listens to the chnnel until the current time slot ends. Thus, we design the length of time slot to be either >4 (neglect the lengths of DIFS, SIFS, nd the contention-window in 80.) or = (without resending nd contention mechnisms in time slot). On the other hnd, if senses chnnel busy, it wits to send the first RTS until the chnnel idle long enough for CTS time, s the in Fig. 5. The gp between the dshed line nd the solid line is the time for DIFS nd the bckoff time for contention. We neglect these obligtory frmes in our nlysis. RTSCTS listening period RTSCTS t t + t + - t + RTSCTS RTSCTS Fig. 5. t The revised resending mechnism. Let be source nd be its neighbor. Let t nd t be their rrivl times on sme chnnel. Assume tht their destintion s re not on the sme chnnel. If t <t, cn send its RTS in this time slot. If t t, in the = cse, cn send its RTS if rriving fter finishing its RTS sending, i.e., t t +. Using the sme constrint t t nd solving the inequlities the sme wy s P, the probbility tht cn successfully send n RTS with neighbor on the sme chnnel is 0.65 when =. In the >4 cse, even rrives during s first RTS sending time s in Fig. 5, cn still send its RTS s long s the moment it strts to send is erlier thn s second RTS sending, i.e., t +<t +, which lwys stnds since >4. Moreover, if rrives during s second RTS sending (t + <t <t + ), we hve t >t + when >4, or,t + (t +) >. In other words, hs enough time for sending its RTS fter leves the chnnel. Hence, the probbility tht cn successfully send n RTS, P, increses to 00% when >4. Therefore, { 0.65, = P =, > 4. () This design lso reduces the RTS collision rte due to the low RTS sending frequency. Compred with the trditionl design, the listening period in the middle of time slot provides the opportunity for nother to send n RTS without collision. Suppose tht is hidden terminl of. Then, the cse tht cn successfully send t lest one RTS without collision is when the RTS from hs no overlp with the RTS from, i.e., t + < t, or t + >t +. We nme this probbility P 3.We derive it in similr wy s P, ( ) P 3 =, =or >4. (3) It lso requires the to be s long s possible to chieve the collision-free sending. 4643

Globecom 04 - Wireless Networking Symposium D. Anlysis of the Both-Shouting Problem In Section II-B, we consider the filure receiving problem when the destintion is pssive. However, it my lso be nother ctive. Consequently, when is hopping nd serching for its destintion, the trget my lso be serching for nother. An extreme cse is tht they re serching for ech other, which is dedlock when =. RTS RTS dt t t + - Cse : RTS CTS dt t Cse : RTS CTS dt t t + t + Fig. 6. Two successful cses when t <t under the both-shouting scenrio. Let be source nd be the trget which is lso n ctive when it rrives on the sme chnnel. If t <t, there re two cses in which cn her complete RTS from. Cse is tht rrives during s sending its first RTS. As illustrted in Fig. 6, bsed on our designed protocol from Section II-C, fter witing for CTS-long idle period, strts to send its own RTS. If s trget is, replies CTS nd they begin to trnsmit dt. In other words, the dedlock cse cn be esily solved to hve successful hndshke when >4. If s trget is not, then wits until the moment t +. If the chnnel is idle, sends its lst RTS nd will her the lst RTS from. In this cse, it requires tht t +4 t + (enough time to send the lst RTS), or, 6. Cse is tht rrives fter s first RTS (t t +). senses the chnnel idle nd sends its first RTS fter rriving. Similrly, overhers this RTS nd wits until t + if it still hs time to send its lst RTS, i.e., t + t +. It requires tht 5 in this cse. RTSCTS RTSCTS t t + t + - t + Cse : RTS dt t Cse : t RTS dt Fig. 7. Two successful cses when t >t under the both-shouting scenrio. If t >t, we lso nlyze two cses, s illustrted in Fig. 7. In cse, t <t <t +, strts to send t t +, which should before s second RTS sending (t + t + ). Then, >4 is required for this cse to ensure successful RTS reception t. In cse, t + t, cn send its first RTS fter rriving. will her this RTS s long s it is sent before s lst RTS (t t + ). This cse requires tht 3. In both cses, s end their current time slot immeditely once the hndshke is finished. To sum up, we hve the following equivlent inequlities nd their corresponding grphic illustrtion: 0 t 0 t t t t + t + t t + 4 t t t + -4 t 4 0 - The bove shdow re represents the fesible rnges of t nd t under different vlues of. Therefore, we cn t derive the probbility tht s RTS cn be herd by its destintion ctive on the sme chnnel, P 4, which is lso monotoniclly incresing function of. 0, = ( P 4 = 4)/, 4 <<5 ( 9)/. (4), 5 <6 ( 0)/, 6 III. PROPOSED MAC PROTOCOL We elborte the complete protocol in detils nd derive n optiml in terms of the minimum TTH in this section. A. Protocol Detils The overll flow chrt for our proposed MAC protocol is presented in Fig. 8. In the initil period, senses ll chnnels nd collects its vilble chnnel set. Existing sequence/probbilistic-bsed CH designs cn be employed in this step to generte the CH sequence. Then, the tunes its rdio to the ordered chnnel nd begins new time slot. During time slot, ny cn become destintion node once it receives n RTS crrying its ID s the receiver. If the lso hs dt to send, it postpones its own dt in queue nd receives other s trnsmission first. On the other hnd, pssive cn become source node once it hs dt to send. In synchronous environments, it hs to wit till the next time slot to chnge its role. However, in our design, it sends n RTS immeditely if there is still time left in the current time slot since there is no need for slot-synchroniztion. Once pir of s completes the hndshke in time slot, they sty on the sme chnnel trnsmitting dt until they finish the communiction. When the pir detects PU presence on the chnnel, spectrum hndoff [] is performed for resuming the trnsmission on nother chnnel. Fig. 8. Initil Hndoff Interrupt Time up Listen Receive RTS ID mtch Send CTS Dt trnsmit Tune to next chnnel Dt to send Chnnel Busy Idle time> Left time= Left time> Sent RTS once Idle from rriving The flow chrt of the proposed MAC protocol. Send RTS Receive CTS 4644

Globecom 04 - Wireless Networking Symposium In the figure, Left time = refers to the moment pproching. The bckoff for the lst RTS sending is counted in reverse time from. For exmple, if rndom number 0. is generted for the bckoff time, the lst RTS will be sent from the moment 0.. The use of SIFS nd DIFS in our protocol is the sme s in 80. MAC. B. Optiml Time Slot When considering the whole rendezvous process, mny time slots re needed before pir of s hop on sme chnnel nd hve successful hndshke. We denote P s the probbility tht source successfully hndshkes with its destintion in the next time slot. Use X to represent the verge service time (TTH) of. Then, X = i( P ) i P = P. (5) i= From the nlysis in Section II, long time slot cn improve the successful hndshke rte in one time slot. Thus, nd P re positive correlted. Therefore, n optiml is needed for (5) in terms of the shortest X. We derive the optiml s follows. First, let ρ represent the ctive rte of, i.e., the probbility tht is in n ctive mode. Assume tht the dt trffic is homogeneous in the secondry network, i.e., the ctive rte of is the sme everywhere in the network. Let λ be the verge pcket rrivl rte of (in the unit of one RTS sending time). Then, ρ = λx. We further denote P 0 s the probbility tht source successfully hops on sme chnnel with its destintion in time slot. P 0 vries under different CH designs [3]. If we do not consider the neighboring interference problem, we cn derive P s P I (P in n idle environment): P I = P 0 ( ρ)p + P 0 ρp 4, (6) where P nd P 4 re the sme probbilities defined in Section II-B nd II-D when the destintion is pssive or ctive, respectively. However, the neighboring-inference problem cnnot be ignored when is in dense network where the number of its neighbors is lrge. Assume tht there re n verge of K neighbors of. Excluding the destintion, the number of the potentil contenders of source is K = K. We denote K s the verge number of its hidden terminl s. Then, P =Pr(K =0, K =0)P I + Pr(K =, K =0)P P I + Pr(K =0, K =)P 3P I +... where Pr(K = 0, K = 0) is the probbility tht no neighbor nd hidden terminl exists on the sme chnnel during the source s one time slot, i.e., ( P 0 ρ) K+K. We cn further derive other probbilities regrding different vlues of K nd K. In this wy, P cn be written s: P ( P 0ρ) K +K P I ( ) K + P 0ρ( P 0ρ) K +K P P I ( ) K + P 0ρ( P 0ρ) K +K P 3P I. We do not consider the cses when K + K > due to two resons. One is tht the probbilities of K + K > re (7) negligible due to the P 0 ρ prt. P 0 is usully on the order of M, where M is the totl number of chnnels in CRN [], [3], [3]. Menwhile, ρ should be smll enough in CRNs to void network congestion []. Then, P 0 ρ is quite smll vlue. Moreover, the probbility tht K + K involving (P 0 ρ) or higher power cn be neglected. The other reson is tht the probbilities of successful hndshke under K + K > re lso negligible, referring the derivtion prt of P nd P 3 in Section II-C. From (5)-(7), we cn get (8) XP I( P 0λX) K +K [ P 0λX( K P K P 3) ]. (8) It is trnscendentl eqution becuse of independent vribles K nd K. Once the network prmeters K, K, M, nd λ re given, the expression of X in terms of cn be derived nd the optiml tht minimizes X cn be obtined. Detiled exmples re given in Section IV. IV. PERFORMANCE EVALUATION In this section, we evlute the performnce of our proposed MAC protocol under different scenrios by compring simultion results with the nlyticl vlues. In our simultion, we ssume tht the pcket rrivl of ech follows the Poisson distribution. Moreover, since P 0 is vrible independent of our nlysis, we dopt the rndom CH lgorithm under which P 0 is exctly M in order to esily djust the vlue of P 0. Additionlly, we consider grid network where K =3nd K =3. More importntly, ech in the simultion hs its own clock nd is not required to be synchronized with others. Other prmeters used in our simultion re listed in Tble I. TABLE I. SIMULATION PARAMETERS Number of s 64 (8 8 grid) Chnnel dt rte Mbps The size of (RTS+CTS) 60 + bits (80.b/g) Simultion time 0000 Fig. 9 illustrtes the expected TTH (ETTH) of the whole network under different numbers of chnnels. The simultion results mtch the nlyticl results very well with mximum difference of 5%. Fig. 9() shows tht =4is the optiml size of time slot when the verge pcket rrivl rte is low, or, the ctive rte of is low (λ =50pkt/s, ρ 0. 0.). Since ρ is smll, there re more idle time slots during rendezvous. Consequently, the dvntges of lrge ( >4) when deling with complicted cses (P, P 3, nd P 4 ) cnnot be fully utilized. Therefore, the incresing rte of the ETTH fter =4is higher when there re more chnnels to hop (M =0). ETTH 40 0 00 80 60 40 Fig. 9. M=6, simultion M=6, nlyticl M=0, simultion M=0, nlyticl 0 4 6 8 () λ =50pkt/s. ETTH 0 00 ETTH vs. in different trffic conditions 80 60 40 0 4 5 6 7 8 (b) λ = 00 pkt/s. Fig. 9(b) shows the impct of different in nerly sturted network. When M is smll, =4still holds the 4645

Globecom 04 - Wireless Networking Symposium optiml size of time slot. However, note tht the ETTH when =6is lredy bit lower thn when =5. This mens tht the dvntges of lrge become dominnt in the results. In the M =0cse, the optiml size is when =6(even when =7hs the sme effect s when =4). Furthermore, the design of =cnnot stnd under this scenrio. This is becuse when =, the low probbility of the hndshke successful rte increses the TTH. Then, the long TTH leds to high ρ which further results in P 4 =0nd n infinite TTH. On the other hnd, from ()-(4), the improvement of ech probbility becomes less nd less when is lrger thn 6. It is lso shown in Fig. 9 nd 0 tht the ETTH monotoniclly increses fter =6. Fig. 0. ETTH 0 00 80 60 40 0 Asyn. (=) Proposed Syn. (=) 6 7 8 9 0 M Compre with the MAC without our design in different scenrios. Fig. 0 compres the performnce of different MAC protocols under the sme trffic condition (λ =50pkt/s). Since we lredy derive the optiml size of time slot under such scenrio, the proposed line is the performnce equipped with our MAC with =4over different M. The synchronous line belongs to the rndom CH protocol with the trditionl MAC under synchronous scenrios. The performnce of this trditionl MAC under the synchronous slot scenrio is shown s the squre-line. From Fig. 0 we cn see tht the proposed MAC performs much better thn trditionl MAC nd closer to the synchronous one (the idel cse). TABLE II. TTH VS. TTR M(λ) 6 (50) 6 (00) 0 (50) 0 (00) ETTH (in unit of slots) 7.75 8.95 4.3 3.50 ETTR (theoreticl) 6 6 0 0 To obtin the TTH in the unit of slots, we divide the minimum TTH using its corresponding. For exmple, the minimum TTH for the cse where M =0nd λ = 00, is 80.99/6 = 3.50 slots. Then, the verge numbers of time slots spent under different scenrios re shown in Tble II. It is shown tht our proposed MAC protocol under slot-synchronous scenrios cn mintin the ETTH with the theoreticl ETTR. 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