Design and Analysis of an Asynchronous Zero Collision MAC Protocol

Design and Analysis of an Asynchr onous Zer o Collision MA C Pr otocol Jiwoong Lee Depar tment of Electrical Engineering and Computer Sciences Unive rsity of Calif or nia at Berkele y Berkele y , Calif or nia 94720 porce@eecs .ber k eley .edu Jean W alrand Depar tment of Electrical Engineering and Computer Sciences Unive rsity of Calif or nia at Berkele y Berkele y , Calif or nia 94720 wlr@eecs.be rkele y .edu ABSTRA CT This pap er proposes and analyzes a distributed MAC pro- tocol that ac hieves zero collision with no con trol message exchange nor syn c hron ization. ZC (Zero Collisi on) is neither reserv ation-based nor d ynamic TDMA; the proto col su p - p orts v ariable-length pack ets and d oes not lose efficiency when some of the stations do not transmit. At the same time, ZC is not a CSMA; in its steady state, it is completely collisio n-free. The stations transmit repeatedly in a round - robin o rd er once the con vergence state is reac hed. If some stations sk ip their turn, their transmissions are replaced by idle 20 µ -second mini-slots that enable the oth er stations to kee p t rack of their order. T o claim the right to transmit, a station selects one of the idle mini-slots in one round and transmits in that mini-slot in t h e n ext roun d . If multiple stations select the same mini-slot, th ey collide and rep eat their rand om selec- tion in a subsequent round. A station loses its right to trans- mit after b eing idle in a given number of successive rounds. The number of transmissions p er round can be adapted to the num b er of active stations in the netw ork. Because of its short medium access delay and its efficiency , the protocol supp orts b oth real-time and elastic app lica- tions. The p roto col allo ws for no des leaving and joining the netw ork; it can allo cate more throughput to sp ecific no des (such as an access point). The proto col is robust against carrier sensing errors or clo c k drift. While collision a voidance is guaranteed in a single collision domain, it is n ot the case in a multiple collision one. How- ever, exp eriments show ZC supp orts a comparable amount of go odp ut to CSMA in a multiple colli sion domain environ- ment. The pap er presents an analysis and extensive simulations of the p rotocol, confirmin g that ZC outp erforms b oth CSMA and TDMA at h igh and low load. Categories and Subject Descriptors C.2.2 [ Netw ork Protocols ]: Multiple access; C.2.5 [ Local and Wide -Area Netw orks ]: Collision av oidance multiple access— c onver genc e analysis, p erformanc e c omp arison General T erms Algorithms, P erformance, Exp erimenta tion Keyw o rds Wireless Medium Access Control , Collision a voi dance 1. INTR ODUC TION Prolifera tion of mobile data devices, incessan t increase of In- ternet usage, and growing co existence of dela y sensitive and dela y toleran t traffic demand impro vemen ts of wireless com- municatio n p erformance. This p aper p roposes a new MAC protocol that is very efficien t and supports real-time and elastic applications. The mechanism is d istributed, which is desirable in terms of ov erhead, fault-resilience, and imple- mentatio n complexity . A ma jorit y of CSMA [17] typ e MAC proto cols use a limited leve l of memory for congestion relaxation purp ose by in- creasing backo ff range on collisions, but the propagation is reset on an y transmission success, which limits the mem- ory of the system (see [28]). On the other h and, some reserv ation-type MAC proto cols use strong memory . F or instance, a station might resp ect a series of perio dic time slots reserved for others. Due to th e random channel fad- ing or unpredictable hardware imp erfections how ever, the station can lose track of syn c hron ization, and p ossibly pro- duce colli sions. Therefore some self-stabilization process is indisp ensable for any robust distributed MAC proto col. In t h is research, we prop ose a d istribu ted, collision-free, self- stabilizing single channel MAC proto col requiring n o con- trol messages. W e call it t he ZeroCollis ion (ZC) p rotocol. In the steady state, the stations select a round -robin order in which they transmit. If a station h as a v ariable length pack et to send at its own turn, it accesses th e medium d ur- ing a va riable duration of transmission slot. Otherwise, it skips its turn and others in t h e netw ork see an empty mini- slot, which is fixed duration and m u ch shorter than a typical transmission slot. The existence of an empty mini-slot or a transmissio n slot is a sufficient statistic so that stations can count the order of transmissio n for its o wn access. No pack et decod in g is necessary for this op eration. In the tran - sien t state, any collision-experiencing station chooses one of the empty mini-slots in the nex t roun d. S tations rep eat t h e random min i-slot selection until they stop exp eriencing col- lisions; they keep their assigned order thereafter. A station loses its assignment if it stops transmitting for a num b er of consecutive rounds. The contribution of this research is as follo ws. • ZC results in short access delays, both for low and h igh load. This c haracteristic contrasts with the fact th at TDMA-based MA C p rotocols inevitably exhibit exces- sive access d ela ys at lo w load and that so do random access b ased MAC protocols at high load. ZC exhibits sup erior p erformance ov er a wide ran ge of active num- b er of stations in the system. • ZC does not inv olve an ex c h ange of con trol messages among adjacent no des. Therefore th e assignmen t p ro- cess is simple and robust, and do es not incur a signif- ican t control ov erh ead. It do es not requ ire stations to decod e any message for access decision purp ose. S o the gap b etw een in terference range and transmission range does not affect the correct op erations of ZC. Since the interf erence range is three or more times larger than the transmissio n range, the w orking cov erage of ZC is substantial ly larger th an th at of some proto cols re- quiring message deco ding for access contro l purp ose. The only necessary information for a station to d ecide when to access th e channel is the cumulativ e length of silence, where silence is technically defi ned by sub- threshold signal-to-noise ratio. • ZC is self-stabilizing: the n et work, b y follow in g the ZC algorithm, reco vers from d y namic even ts such as sud- den no de arriv als or departures, from erroneous even ts such as carrier sensing error, whic h are typically caused by channel fading or hardware defect, and from corre- sp on d ing clo c k d rifts. It is necessary t h at the pro cess should stabilize fast enough, relative to the occu rrence frequency of dynamic or erroneous even ts. I n section 4, we show the exp ected stabilization time in the worst case is upp er-b ounded, and t ypically around 2 to 3 sec- onds. In the nominal case where eac h individual sta- tion maintains some information about the netw ork, the system stabilizes even faster. In addition, while ZC works in a non- in frastrucure mo de by design, we provide a simpler w ay for the pow er saving no de to retrieve its access slot by listening to a b eacon slot of the access p oint, if any , in the infrastructure mo de operation. • ZC ac h iev es a collision-free state in a single collision domain. Although it is not guaranteed in a multi- ple collision d omain d u e to hidden/exp osed n ode syn- drome, exp erimen tal results show comparable p erfor- mance. Simulati ons in section 5 consider a few realistic mo deling as- sumptions in cluding tw o-wa y propagation time, captu re ef- fect, carrier sensing error, and clock drift. W e also consider the case where more num b er of stations are active than a defined netw ork capacity . By si mula tion, w e provide per- formance comparison b etw een ZC, CSMA, and TDMA, and confirms that ZC inh erits adv antages b oth from CSMA an d TDMA. W e also show th at ZC enables d esirable coexistence of tw o representativ e traffic patterns - delay sensitive p e- riodic t raffic (voice ov er Internet protocol, V oIP [12]) and dela y toleran t best effort t raffic (web transactions). In sec- tion 5.5 w e show th at ZC substantially increases the V oIP capacit y web traffic is present. A qualitativ e comparison of existing collis ion-free MAC pro- tocols is presented in Fig. 1 . 2. RELA TED WORKS Collisi on av oidance or collision-free distributed medium ac- cess control h as a long history that we briefly review b elo w. The ma jority of the protocols can b e classified into one of follo wing typ es: r eserv ation TDMA , dynamic TDM A , flo or ac quisition , and out-of-b and or mul ti channel signal ling . In a reserv ation or dynamic TDMA proto col, a device re- serve s some future ep ochs to transmit its packets. The ran- dom reserv ation TDMA p rotocols hav e many v ariations. R - ALOHA [7], PRMA [13] and their v ast number of deriv a- tives adopts “reserve -on-success.” There, time is divided into a seq u ence of frames and each frame consists of fixed- length slots. If a trial transmission is successful during a non-reserved slot within a frame, the corresp onding slot of the follo wing frames are regarded as reserv ed. [9] [21] ana- lyze p erformance and stabilit y of this class of protocols. Some protocols as in [8 ] and [27] are classified as dyn amic TDMA. Time is d ivided into a sequence of frames and each frame consists of at least tw o constant-length ph ases - on e contro l ph ase which is u sed for comp etition for slot alloca- tion v ia rand om access, auction [2] [18] [22], or distributed election [19] and the remaining phase(s) whic h is (are) d e- voted t o data [27]. Stations are sy n c h ronized at start of each phase, except for some cases as in [1] and [30]. TRAMA [19], MMA C [1] and their d eriva tives b elong to this class in the context of wireless sensor netw orks. While the slot is regarded as reserved in reserv ation TDMA until its reserv ation is revok ed, the reserv ation is typicall y v alid for a single frame time in dyn amic TDMA. An in- evitable disadv antage of TDMA frame based proto cols, such as random reserv ation TDMA or dyn amic TDMA, is th at they exhibit p o or u tilizatio n and excessive dela y s when only a few p ortion of stations in the netw ork are active. On the other h and, flo or acqu isition p rotocols are suitable for b urst y traffic. Floor means exclusive channel access righ t, and is designed to relieve collisions from hidden sta- tions. CSMA/CA [28], MACA [15], MACA W [3] [20], F AMA [10], CAR MA [11] and their many d escendan t s b elong to th is category . The main difference from dynamic TDMA is t hat transmission ep och needs to b e immediately after the flo or acquisition while in dynamic TDMA it is reserved for certain future ep o chs. Although these protocols enable use of sta- tistical multiplexing, they requ ire frequent contro l message exchange typically k no wn as R TS/CTS dialogue overhea d, whic h can b e of sub stan t ial consumption in channel resource. Besides, control messages are not protected to be collision- free since they still foll o w random access. This sc h eme en- ables med ium access in multiple collision domain at a cost of channel underut ilizati on of exp osed stations. SMAC [29], TMA C [26] and their many deriv atives for energy limited Figure 1: Quali tative comparison of Distributed MAC protocols wireless sensor netw orks b elong to this class. ZC is neither TDMA-b ased nor fl oor-acquisition based; it supp orts va riable transmission time and no control dialogue exchange is required. ZC is asynchronous in th e sense that it do es n ot require explicit global / lo cal sy nc hronization. A ZC station is only required to coun t th e elapsed time from the end of channel activity that it last observed. Because of these features, ZC exh ibits go od p erformance b oth at high and lo w traffic load. There are proto cols utilizing out-of-band signalling su ch as BTMA [24], DBTMA [14] or multic hannel control such as SRMA [25] BRO A DEN [23]. While those schemes en able distributed collision-free medium access, th ey demand multi- ple radios p er device so it incurs complexity and cost. There- fore in our research, we fo cus on only th e single radio case. The closest relative of Z C is MSAP [16]. In MSAP , the se- quence of mini slots plays a role analogous to a silen t p olling sequence and the station with toke n d oes not release the channel until it empties its b uffer. Moreo ver, in MSAP the access sequence is assumed to b e p red efined and shared by all stations ahead of time, and n o distributed self-stabilizing process is prop osed. So MSAP cann ot reco ver from any dy- namic ev ent or carrier sensing error. I n contrast, ZC is a simple but robust self-stabilizing, solving the time assign- ment p roblem in an effective w ay . BRAM [6] and SUP- BRAM [5] are extensions of MSAP with va rying length of idle mini slots, but these proto cols requ ire pack et deco ding and global synchronization at every no de. There are also centralized algorithms to solve the assignment problem, but they frequently face feasibility challenges in collecting and redistributing information. A s a result, such sc hemes are out of th e topic of th is research. The remaining of the p aper is structured as follo ws. Section 3 describes the op erations of the proto col. Section 4 pro- vides an analysis of the conv ergence time of the algorithm. Section 5 sho ws the p erformance of the ZC p rotocol in a v ariety of metrics based on N S -2 simulations with realistic assumptions. Section 6 summarizes the results of the pap er and discusses future work. 3. ZC PR O TOCOL W e describe the op erations of ZC and w e illustrate them on a simple example. W e th en commen t on a k ey design assumption. 3.1 Protocol Description When using th e Z C protocol, in the steady state, the sta- tions transmit in a roun d-robin order. The total num b er of transmissions in one round is upp er-b ounded, but can b e adapted later. In the b asic op eration, eac h station can trans- mit a single v ariable-length p ac ket (typically 200 µ − 2200 µ seconds long) d u ring one roun d (extension of this is ex - plained shortly). If a station skips its tu rn, the medium is left idle for an empty 20 µ second- long mini-slot duration and the next station can access th e mediu m after that. The length of a mini-slot is chose n to b e substantially longer than a round-trip p rop agation across a few kilometer-wide netw ork. T o select its order of transmission, a station ob- serve s the sequen ce of idle mini-slots or busy transmission slots. Either one idle mini-slot or one transmission slot is called a virtual slot. V irtual slots correspond to transmis- sion opportunities. F or instance, sa y that the number of transmissions p er round is 8 and that a station observes the sequence T E T T E E T E where T designates a transmission of a vari able-length pack et and E an empty mini-slot. In steady state, a station is able to count the sequ ence of virtual slots, and therefore find its own reserv ation. I n t he transient state, when not all the stations hav e th eir own reserv ation yet, eac h station tries and se es if a random choice of virtual slot works for itself . S upp ose a station c ho oses the third empty mini-slot as its virtu al slot after observing one round of medium access. If the station collides with another station that happ ened to select the same slot, the colliding stations rep eat th eir random selectio n of an empty mini-slot in the next round. Once a station has selected a successful order, it keeps it. If a station stops transmitting for a given num b er of roun ds, the station loses its reserv ation. The num b er of transmissions in a round can be adapted automatically to the num b er of active stations. Note that the time length of one round vari es as the number of activ e stations and pac ket lengths v ary . Any collision caused by a new n ode arriv al, carrier sensing error or clo c k drift triggers a new virtual slot selection pro- cess. This process is a robust self-stabili zation mec hanism: it converge s to a zero collision state within at a few seconds, in the w orst case for a reasonably big net w ork and typi- cal sy stem parameters. After the conv ergence, the netw ork operates without collisio n u n til another even t triggers the selection pro cess. Due to its collision-free prop erty and the relativ ely small size of mini-slots compared to transmission times, ZC ac hieves almost the maximum p ossible channel utilization, irrespec- tive of th e fraction of activ e stations. Using an alysis and sim u lation we show th at self-stabilization is fast enough to reco ver from collisions, c h annel rand omness and carrier sens- ing error, and th at ZC supp orts desirable co existence of de- la y sensitive and delay toleran t traffic, which is n ot the case for WiFi netw orks. On p o wer up or arriv al, station C randomly and uniformly selects one of the unreserved virtu al slots for its own trans- mission. (The case in which stations ma y sele ct m ultiple slots will b e exp lained later.) Since th e station initially has no knowledge ab out the netw ork, it is p ossible that the se- lected slot is in fact already reserved by another, implying p oten t ial collisions later. That station transmits during the selected virtual slot if it has a p ac ket ready to transmit. If the t ransmissio n is successful, as the station detects when it gets an ackno wledgement within half of a mini slot time from the end of its transmission, the station remem b ers that the slot is reserved for itself. A ssume for now that there are neither random carrier sensing errors nor capture effects. Then the successful transmission within a virtual slot im- plies (a) there was no other transmitting station within the interf erence region in that slot, (b) all the other stations in the interference region sensed t h e slot to b e busy if they we re a w ake, and (c) they mark ed that slot as reserv ed and will not trespass in future as long as they already hav e not re- serve d it for themselves. Therefore the station is guaranteed to b e collision-free in an y future channel access as long as there is no other o wner of th e slot and a p erturbation even t does not o ccur. If th e transmission w as not successful, the station gives up that slot and rand omly and u n iformly selects one of the un- reserv ed v irtual slots again. This process repeats un til it finds a successful slot. Even though t here are more than one o wner of a v irtual slot, as long as they d o not exp erience a collision due to sparse traffic, they may enjo y statistical multiplexing o ver the same slot. Reserv ation is not p erma- nent. If it w ere, the p rotocol could not be stabilized up on dynamic even ts of the n et work. If a virtual slot is sensed not to b e used for a certain amount of time T r , th e slot b ecomes unreserved again. If a carrier sensing error or clo c k drift o ccurs, it is p ossible that a station unintentional ly t respass by mistaking other’s reserv ation for its o wn one. If this happ ens, the colliding stations simply follo w the ab ov e d escrib ed proto col to re- solv e th e issue: searc h again for an exclusive slot. It is imp ortant that this self-stabilization pro cess is fast enough to deal with randomly occurring p erturbations. Section 4 sho ws the worst case bound of mean con vergence time in analysis and simulation. While ZC is designed for ad ho c mo de op eration, it provides more features und er the infrastructure mo de op eration. If an access p oin t(A P) exists, it ma y designate a slot as an an- chor slot during which only b eacon messages are b roadcast. The anchor slot is not changed on collision b ecause the AP does not rely on ackno wledgement anyw ay . Then a pow er sa v ing station can retrieve its previously reserved slot on w ake up , without losing synchronization with the net w ork and without in cu rring collisions. Secondly , the AP may re- serve d multiple slots since it typically requires multiple times of channel access to serve its asso ciated stations. By allo w- ing asymmetric assi gnment, the netw ork performance can impro ve. In order to complete th e d escription, we discuss a few sp e- cial cases. It is possible th at the transmission is still success- ful even though a collisio n occurs. Define ( · , · ) b e the dis- tance metric. Consider tw o transmitters A and B and their receiv ers a and b respectively . Supp ose ( A, a ) << ( A, B ) and ( B , b ) << ( A, B ). Then although A and B access the channel sim ultaneously , a regards A ’s transmission as signal and B ’s as background noise, d eco ding A ’s message success- fully . a will return ac k no wledgement. Similarly so does b . In this case ZC still prop erly funct ions. This spatial reu se phenomenon is confi rmed via simula tion. One interesti ng interpretatio n of this ph enomenon is the sum of individual throughputs is larger than th e naive channel capacity . If there are more activ e stations th an the n et work capac- it y , that is M > N , the collis ion-free channel access is not guaran teed any more. If all stations are backlog ged, all vir- tual slots wi ll b e alw a y s u sed an d some stations will fa ce collisio ns while th e rest will n ot. F or those of colliding sta- tions, since there is no unreserved slot, they will not mo ve around and stic k to their colliding slots. This makes t h e rest non-colliding stations not b othered and enables p ositive net- w ork throughp ut. It is not immediately clear that ZC in this case still outp erforms CSMA. Simulations in 5 sh o w that ZC still outperforms CSMA and even the netw ork throughpu t is lo wer b ound ed. If stations are not backlog ged and have in- dep endent pack et arriv al, some level of statistical multiplex- ing will b e taken part in, together with ZC self-stabilization process on collisions. Before netw ork congestion, ZC, CSMA and TDMA do es not show big d ifference in throughpu t and mean inter-access delay . Afterwards, p erformance vari es de- p ending on how sparse th e traffic is and how big the n et work is. How ever, we d o not stu dy this issue in d etail. 3.2 Comments on Bounded Number of Sta- tions ZC assumes that the number of activ e stations is b ounded in order to guarantee its complete collision a voi dance op er- ation. This assumption can b e justified b ecause • Physical systems hav e hard capacity limits b ecause of implemen tation issues. As an example, t h e IEEE 802.11 MAC cannot supp ort more than 2008 stations, whic h is the maximum length of the Partial Virtual Bitmap of T raffic In dication Message in the Beacon frame. P artial Virtual Bitmap is u sed to wak e up p o wer-sa ving stations. • Every n etw ork technology has its own covera ge limit and it is unnatural to pack more th an a certain num b er of stations in to th e co verage of a single netw ork. F or example, IEEE 802.3 100BASE-T and IEEE 802.3ab 1000BAST-T hav e a limited cable distance up to 100m. Also, the t y pical op erating ranges of I EEE 802.11b, 802.11g and 802.11a are 100m, 50m and 20m respec- tively . Given the technolog y , the op erator of the net- w ork already hav e d ecided the netw ork capacity . • Per-user performance b ecomes unusable after a cer- tain threshold of th e net w ork size. As the netw ork size increases, severe p erformance d egradation in terms of netw ork throughput, p er-user th roughput, and trans- mission delay is induced. After a certain threshold of the netw ork size, infin ite capacity loses its meaning. • A wireless user’s mobilit y is quasi-static. Think of a conference ro om in which stations sporadically arriv e or depart. Inter-ev ent(arri v al or departure) time is on a vera ge exp ected to b e more than tens of seconds at least. During t his p eriod , the netw ork is static and we can exploit this feature. Therefore in t h e follow ing section, w e analyze the conv er- gence time to a zero collision state with a hard capacity limit in netw ork size. I n fact, if the netw ork size is unbounded, conv ergence do es not make sense. 4. ZC ANAL YSIS In th is section we show th at ZC conv erges to a zero collision state in finite time. W e then analyze the av erage conv er- gence time and w e derive a simple upp er b ound on that time. W e conclude the section by show ing that th e conv er- gence holds for an arbitrary sequence of w ake up times of stations. 4.1 Con vergen ce It is fairly immediate to show that, for a single collision d o- main n et work with M ≤ N , the ZC algorithm is guaranteed to conv erge to a zero collision state defined nex t . Definition 1. Zero collision state W e sa y t h at the netw ork has reached a zer o c ol lision state when all M stations have reserved a different slot. Definition 2. Conv ergence time The c onver genc e time of ZC is the first time when the net- w ork reaches a zero collision state. Theorem 1. Conver genc e the or em Assume that T r is fini te and that M ≤ N . Then the ZC algorithm r e aches a zer o c ol lisi on state in finite time. Pr oof. Let x n b e the num b er of stations with a reserved slot after n idle slots. If x n < M , th ere is p ositive probabilit y that one station will b e alone in transmitting in the next idle slot and will consequently reserve a slot. Consequently , M is an absorbing state for the finite Marko v c hain { x n , n ≥ 0 } . The follow ing section derives an upp er boun d on the a verage conv ergence time. W e th en provide a simpler up per b ound. 4.2 Con vergen ce Time Analysis It is imp ortan t to sho w th at th e algorithm con verges fast to a zero colli sion state. Indeed, in practice, stations are alw ays joining and lea ving the n et work and the algori thm should quickly conv erge to a new zero-collision state for it to ha ve a high throu gh p ut. In this section we study the conv ergence time of ZC and d erive an upp er b ound on the mean con vergence time. This upp er b ou n d assumes that a station attempts to transmit only once in N consecutive slots instead of trying in eac h unreserved slot with some p ositiv e p robabilit y . W e consider the number z n of stations with reserved slots after the n -th cycle of N slots. It is clear that { z n , n ≥ 0 } is a Marko v chain. First, we derive t he transition matrix of that Marko v chain. Second, w e calculate th e av erage n u m b er of cycles until z n = M . Third, we calculate the av erage total du ration of th ose cycles. Theorem 2. Pr ob ability of Re servation Consider a set of M stations that sele ct i ndep endent ly and uniformly one of N slots. The pr ob ability that exactly k stations among M sele ct a slot that no other station sele cts is gi ven by p N,M ( k ) = M X j = k ( − 1) j − k M j ! j k ! N !( N − j ) M − j ( N − j )! N M (1) for 0 ≤ k ≤ M . Pr oof. Let I i b e the even t { Station i selects a slot th at no other station chooses } . Fix a set Λ with | Λ | = j . Then P ( \ λ ∈ Λ I λ ) = N j ! j ! N j „ N − j N « M − j (2) = N !( N − j ) M − j ( N − j )! N M . (3) Also consider any set Γ with | Γ | = k . Then p N,M ( k ) = X Γ: | Γ | = k P ( \ γ ∈ Γ I γ ∩ \ ¯ γ ∈ Γ C I C ¯ γ ) (4) = M k ! P ( \ γ ∈ Γ I γ ∩ \ ¯ γ ∈ Γ C I C ¯ γ ) . (5) By t he inclusion-exclusion principle, P ( \ γ ∈ Γ I γ ∩ \ ¯ γ ∈ Γ C I C ¯ γ ) = X Λ:Γ ⊂ Λ ( − 1) | Λ |−| Γ | P ( \ λ ∈ Λ I λ ) (6) = M X j = k X Λ:Γ ⊂ Λ , | Λ | = j ( − 1) | Λ |−| Γ | P ( \ λ ∈ Λ I λ ) (7) = M X j = k ( − 1) j − k k k ! M − k j − k ! P ( \ λ ∈ Λ I λ ) . (8) Putting these expressions together p roduces the result. Note t hat p m,m + k := P [ z n +1 = m + k | z n = m ] = p M − m,N − m ( k ) 0 16 32 48 64 80 96 112 128 1 2 3 4 5 6 7 8 9 10 11 N=32 N=64 N=128 E[L] (cycle) # Nodes (M) Expected Convergence Cycle Figure 2: Exp ected Con vergence Cy cle since, when z n = m , there are M − m stations left with unreserved slots and N − m remaining slots to choose from. Next we analyze t h e mean time u n til z n reac h es the state M . That is, let L = min { n ≥ 0 | z n = M } . W e wan t t o calculate E [ L | z 0 = 0]. Define β ( i ) = E [ L | z 0 = i ]. Then one step equations are obtained and can b e solved algebraically: β ( i ) =  1 + P M j = i p i,j β ( j ) , 0 ≤ i ≤ M − 1 0 , i = M . (9) Solving these equations yields E [ L | z 0 = 0] = β ( 0). Finally , we d eriv e the exp ected conv ergence time. Let π n ( k ) := P [ z n = k ], so that π n = π 0 P n where P is the transition matrix of th e Marko v chain z n that we derived earlier and π 0 ( m ) = 1 { m = 0 } . Also, P ( L ≤ n ) = π n ( M ). Note that π n | L ( k ) := P ( x n = k | x L = M ) = π n ( k ) ˆ π k n ( M ) π L ( M ) , ( 10) where ˆ π k n := ˜ π k P L − n and ˜ π k ( j ) := 1 { j = k } for all n ≤ L . The second equ ality comes from Bay es’s rule. Assume that cy cle n has G n successful transmissions, V n idle slots, and B n = N − G n − V n slots with colli sions. Designate by t g , t v , and t b the du ration of a su ccessful transmission, of an idle slot, and of a collision, resp ectiv ely . t s denotes th e inter-slo t gap. (See App endix) Then the length of cycle n , T n , is given by T n = t g G n + t v V n + t b B n + t s N (11) = ( t s + t b ) N + ( t v − t b ) V n + ( t g − t b ) G n . ( 12) T y pically t b ≈ t g ≫ t v ≥ t s holds. Given the stopp ing p e- riod L , th e t otal conv ergence time is T = P L n =1 T n . There- fore E [ T ] = ∞ X l =1 π l ( M ) l X n =1 E [ T n | L = l ] , (13) where E [ T n | L ] = ( t s + t b ) N + ( t g − t b ) E [ G n | L ] + ( t v − t b ) E [ V n | L ] , E [ G n | L ] = M X k =0 kπ n | L ( k ) , 0 16 32 48 64 80 96 112 128 0 500 1000 1500 2000 2500 3000 N=32 N=64 N=128 Convergence Time (msec) # Nodes (M) Upperbound Expected Convergence Time Figure 3: Upp er b ound of Exp ected Conv ergence Time E [ V n | L ] = M X k =0 ( N − k )(1 − 1 N − k ) M − k π n − 1 | L ( k ) . Putting the ab ove ex pressions together yields the follo wing result. Theorem 3. Exact Exp e cte d C onver genc e Time ZC’s c onver genc e time c an b e c ompute d as E [ T ] = ( t s + t b ) N E [ L ] + ∞ X l =1 π l ( M ) l X n =1 M X k =0 { ( t g − t b ) k π n | l ( k ) + ( t v − t b )( N − k )(1 − 1 N − k ) M − k π n − 1 | l ( k ) } . 4.3 Simpler Upper Bound While exact and computab le, t h e foregoing equation is less tractable and provides little physical sense. Instead, we de- rive a simpler up per b ound. In (13), t g − t b can b e either p ositive or n egativ e. Since 0 ≤ E [ G n | L ] ≤ M , N − M ≤ E [ V n | L ] ≤ N and ( t v − t b ) < 0, if ( t g − t b ) is p ositive, we find E [ T n | L ] ≤ ( t s + t b ) N + ( t g − t b ) M + ( t v − t b )( N − M ) = ( t s + t v ) N + ( t g − t v ) M or if ( t g − t b ) is negative, E [ T n | L ] ≤ ( t s + t b ) N + ( t g − t b ) · 0 + ( t v − t b )( N − M ) = ( t s + t v ) N + ( t b − t v ) M . Therefore, Theorem 4. Upp erb ound of Exp e cte d C onver genc e Time When stations ar e b acklo gge d and p ower e d up synchr onously, the exp e cte d c onver genc e time i s b ounde d by E [ T ] ≤ { ( t s + t v ) N + (max( t g , t b ) − t v ) M } E [ L ] (14) wher e E [ L ] is c ompute d fr om (9) and p ortr aye d in Fig. 2. The forego ing equation provides us an upp er b ound of the exp ectation of th e con vergence time. R ecalling that t g is of order of a few milliseco nds, that N or M are typically less th an a few hundreds, and that E [ L ] is around 10 (Fig. 2), E [ T ] is roughly less than a few seconds. Indeed, Fig. 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120 128 0 500 1000 1500 2000 2500 3000 Number of Nodes Convergence Time [ms] Average Convergence Time for 2 Different Slot Reselections N=32 N=64 N=128 Immediate Reselection Reselection at the end of each cycle Figure 4: Conv ergence Time of Immediate Rese lec- tion and Resel ection at the end of eac h cycle 3 shows the upp er b ound of exp ected conv ergence time in IEEE 802.11b DS SS Long preamble MAC/PHY n etw ork with parameters t g = 2150 µsec, t v = 20 µsec, t b = 2266 µsec , and t s = 0 µsec . F or example, fo r the n et work of N = 128 , M = 128, the upp er b ound of exp ectation of con vergence time is 2 . 92 sec whic h is notably close to th e simulation result. 4.4 More general cases Theorem 4 h olds in more general cases. So far w e hav e considered the case where all the stations are p ow ered up sim u ltaneously . In a more typical case, stations join the netw ork and b ecome active in an asynchronous manner. In that case, when a later station arrives, a p ositiv e exp ected num b er of stations hav e already acquired their o wn access slot. By making a station scan t he channel for one p erio d b efore access, a station th at arrives later competes only with the stations t hat h ave not yet reserved a slot. Therefore, the exp ected con vergence time after the station arrive s is alw ays less than that of the syn c h ronous arriv al case. W e hav e developed and d iscussed the con vergence time of the ZC algori thm under a saturated queue traffic model in whic h stations alw ays transmit pack ets whenever they are allo wed to do so, which is no realistic. Supp ose again th at all th e stations join th e netw ork at the same time. Du e to the intermitten t traffic generation, some stations skip t h eir access chances, effectively redu cing th e number of ‘activ e’ stations for the same netw ork capacit y N . Those stations that do not transmit du ring the initial converg ing perio d are automatically giv en additional scanning perio ds, whic h results in a reduction of the subsequent con verg ence time as discussed in the previous p aragra ph. In t h e preceding analysis, we implicitly assumed that slot re- selection after collision is done at the end of every cycle. In practice, slot reselection is done immediately after collision. While these tw o schemes are obviously different, they effec- tively sh o w the same conv ergence time in sim ulation (Fig. 4). 5. PROOF OF CONCEPTS AND EXPERI- MENT AL RESUL TS 5.1 Proof of Conce pts T o verify the viabilit y of the ZC algorithm, w e implemented an NS- 2 ZC simulator on t op of the IEEE 802.11 PHY an d MAC modules. The mod ules supp ort propagation dela y , tw o-w a y ground c hann el mo del, signal-to-noise ratio compu- tation and thresholding, capture effect, carrier-sensing and collisio ns. The NS- 2 ZC sim u lator mo difies the b eha v ior of the access time d ecisions and rep laces t he exp onentia l ran- dom b ac koff. T o maximize the backw ard compatibilit y , the other features of IEEE 802.11 MA C are still used. V arious types of traffic models are used in the sim ulations: saturated queue traffic, V oIP Constant Bit Rate t raffic, and w eb traffic on top of full TCP . Except for the result of conv ergence time and netw ork go od put, th e base station reserv es M − 1 slots when it serves M − 1 stations. R TS/CTS wa s not used. In the remainder of the pap er, CSMA refers to I EEE 802.11b WiFi. T able 1: Exp erime n tal configuration P a rameter Notation ZC CSMA PHY protocol IEEE 802.11b/DSSS Long Preamble Data TX rate R T X 11 Mbps Slot time T S LO T 20 µ sec SIFS T S I F S 10 µ sec DIFS T D I F S 50 µ sec Con tention N Fixed 128 Dynamic Windo w (se e detail) 32-1024 Net work si ze M 4 - 128 T raffic mo del Saturated queue/V oIP/W eb F rame size x 2346 or v arying bytes ACK si ze L AC K 14 byt es PHY Preamble L P r e 18 byt es PHY PLCP L P LC P 6 byt es PHY TX rate R P H Y 1 M bps Recycle T i mer T r 10 0 5.2 Con vergen ce time Con verg ence t ime is d efi ned as the time to reach a zero col- lision state from the momen t of p erturbation. The largest p erturbation is induced when all the stations h a ve no history of th e netw ork and are p ow ered up si multaneousl y . This leads to the w orst case of converg ence time. U sing bac k - logged traffic wi th large p ac ket size (2346 bytes), Fig. 5 sho ws the conv ergence time for a netw ork with capacity 16, 32, 64 and 128 resp ectiv ely . The largest netw ork (128 no des) reac h es a zero collision state within 3 seconds. After the con- verge nce, the netw ork exp eriences no collision and netw ork p erformance including throughput and delay is enhanced compared to the stand ard 802. 11 p rotocols. Obviously , if M > N , the algorithm do es not conv erge. 0 16 32 48 64 80 96 112 128 0 500 1000 1500 2000 2500 3000 N=16 N=32 N=64 N=128 Convergence Time (msec) # Nodes (M) Zero collision Convergence Time Figure 5: Conv ergence time 5.3 Network Goodput Fig. 6 and 7 sho w the fundamental difference of ZC, CSMA, and TDMA in terms of netw ork goo dput and mean interac- cess time when the stations are backlo gged. ZC and TDMA’s netw ork capacit y is set to 64. F or a fair baseline compari- son, each station is allo w ed to take a single access chance in a cycle. Several observa tions are notew orthy: a) The fun- damental p erformance difference of ZC from CSMA is ZC’s goo dput actually improv es as the netw ork size gro ws until it reac h es the capacity and afterwards it starts to decrease. b) F or a substantially large range of netw ork size (6 - 192+), ZC outp erforms CSMA even when M > N . c) While b oth ZC and TDMA do not exp erience collisions when M ≤ N , ZC al w ays outp erforms TDMA. Especially when the net- w ork size is small, ZC’s p erformance is remark ably b et t er. d) When t he netw ork size is very small ( M < 6), ZC’s mini slot size is not n egligi ble any more and plays a role in con- tributing to p erformance degradation. This h appen s when the n et work capacit y is too ov erestimated than the actual netw ork size. When the stations are not backlogged, ‘re- 0 50 100 150 0 2 4 6 8 10 # Active stations Goodput [Mbps] System Goodput (Backlogged traffic) ZC (Capacity 64) CSMA TDMA (Capacity 64) Figure 6: Netw ork Goo dput (Backlogged traffic) 0 50 100 150 0 100 200 300 400 500 600 700 800 # Active stations Mean Interaccess Time [msec] Mean Interaccess Time (Backlogged traffic) ZC (Capacity 64) CSMA TDMA (Capacity 64) Figure 7: Me a n int e raccess Delay (Backlogged traf- fic) serving th e same slot’ do es not necessarily mean a collision. Indeed, stations can enjo y statistical multiplexing without generating collisions. I n Fig. 8 , a station generates a 2346 Byte pac ket at every 300 msec. Although the netw ork ca- pacit y is set to N = 64, stations do not exp erience collisions almost until M ≈ 2 N . In that range ZC, CSMA, and TDMA sho w virtually identical go odp ut except that TDMA is n ot defined for M > N . After a certain threshold of t he n et work size ( here 2 N ), collision effects are more pronounced in ZC than CSMA. 5.4 Carrier sensing error effect By design, ZC is sensitive t o carrier sensing and asynchronous clock. Thus it is important to verif y whether ZC’s goo d 0 50 100 150 0 2 4 6 8 10 # Active stations Goodput [Mbps] System Goodput (Sparse traffic) ZC (Capacity 64) CSMA TDMA (Capacity 64) Figure 8: Netw ork Goo dput (Sparse traffic) 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 0 1 2 3 4 5 6 7 8 9 10 11 Carrier Sensing Error Rate Goodput [Mbps] Carrier Sensing Sensitivity on Goodput ZC CSMA 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 0 50 100 150 200 250 300 350 400 450 500 Carrier Sensing Error Rate Mean Inter−access Delay [ms] Carrier Sensing Sensitivity on Delay ZC CSMA Figure 9: Carrier sensing and Cl ock drift error effect p erformance sho wn in the prev ious section can be robustly main tained u pon errors or clock drifts. The error mo del is as follo ws: eac h station p erforms carrier sensing and u nder- goes errors indepen dently . An idle mini slot can b e sensed to be bu sy with probabilit y p 1. Also an idle mini slot can b e sensed to b e idle but counted tw o times with probabil- it y p 2 due to sligh t timing mismatc h. A busy virtual slot can b e sensed to be idle as w ell with probabilit y p 3, but it is unlikel y to detect th e whole p ortion of a busy virtual slot, which is in fact long in time, is sensed to b e idle in t he spatial range of consideration. Therefore in fact p 1 and p 2 provides enough p erturb ation for sim ulation purp ose and we consider p 1 = p 2 = p for simplicit y and v ary p from 10 − 6 to 1. Netw ork size and capacity is fixed to 64 and all stations are backlog ged. Results are shown in 9. As exp ected, CSMA is relativel y insensitive to carrier sens- ing and clo c k drift errors while ZC is not. How ever interest- ingly enough, irresp ectiv e of the error rate, ZC’s go odpu t is lo wer b ound ed and do es n ot drop to zero. Considering that ZC never achiev es zero collision state, this is an unexp ected result. Intuitive ly , more p erturbation leads to more col li- sions. S ince the num b er of virtual slots and active stations are fixed, more collisio ns imply more stations are concen- trated in certain virtual slots. Then there are more non - conflicting slots, whic h contributes to goo dput. Fig. 10 p ortrays the system dyn amics when a station fre- quently comes and go es. Initially there are 31 stations and one AP , rand omly lo cated within an area of 25 × 25 square meters, with backlogged t raffic and the stations are pow ered up sim ultaneously . After the initial conv ergence, the net- w ork achiev es the maximal go od put. At 5, 10, 15 seconds in the simulation, a 32th station comes to the n etw ork with no prior information on the netw ork, access the c hannel for 1 second with backlogg ed t raffic, and th en leav es. W e can see that although a n ew arriv al of a station momentarily affects the p erformance of the netw ork, its collisio ns are qu ic kly re- solv ed and the netw ork conv erges to a zero collision state. 0 2 4 6 8 10 12 14 0 5 10 15 20 Collisions Time [s] Effect of Network Dynamics on Throughput with ZeroCollision/802.11b 1 AP 31 nodes on 25x25 sqr meter Grid A station arrives at 5.0 and leaves at 6.0 (Collision) # of Collision Figure 10: Coll isions on station arriv als 5.5 Per f ormance for Delay sensitive and De- lay tolerant traffic 0 0.2 0.4 0.6 0.8 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 CDF Delay [s] Voip G.711 Access Delay with ZeroCollision/802.11b 15 voip pairs 16 voip pairs 17 voip pairs 18 voip pairs 19 voip pairs 20 voip pairs 21 voip pairs 22 voip pairs 23 voip pairs 24 voip pairs 25 voip pairs (a) ZC 0 0.2 0.4 0.6 0.8 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 CDF Delay [s] VoIP G.711 Access Delay with CSMA/802.11b 15 voip pairs 16 voip pairs 17 voip pairs 18 voip pairs 19 voip pairs 20 voip pairs 21 voip pairs 22 voip pairs 23 voip pairs 24 voip pairs 25 voip pairs (b) CSMA Figure 11: G.711 V oIP D elay W e compare the cumulativ e distributions of V oIP access de- la y for ZC/IEEE 802.11b and CSMA/IEEE 802.11b. Using a G.711 co dec, each source of a V oIP conversa tion p air gen- erates a 240-byte long application pack et every 30msec. W e assume that t he p erformance is acceptable if 99th p ercent of the pack ets exp erience an access delay less th an or eq ual to 30msec. The simulatio n sho ws that ZC sup port up to 21 conv ersation pairs while CSMA supp orts ab out 17. Ind eed by simple compu tation, it is easily shown that 21 conv ersa- tion p airs actually fill up the 30msec time frame and are the maximum num b er of V oIP capacit y with stable queue size and dela y . Additional exp erimen t is found more interesting in case of the co existence with d ela y toleran t traffic. ZC is shown to supp ort 18 times more V oIP sessions than CSMA when 5 backgro und web sessions([4] mo del) are ongoing. It h as b een kn own that the V oIP capacity of CSMA n etw orks is significan tly impaired by the existence of a few TCP flow. That p h enomenon can b e observed in Fig. 12(b) where one V oIP session is barely supported. Different from CSMA, dela y sensitive traffic and d ela y toleran t traffic mingle eas- ily . With the same background W eb traffic, the ZC netw ork can support ab out 18 V oIP conn ections. W e also ha ve 0 0.2 0.4 0.6 0.8 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 CDF Delay [s] Voip G.711 Access Delay with ZeroCollision/802.11b 15 voip pairs 16 voip pairs 17 voip pairs 18 voip pairs 19 voip pairs 20 voip pairs 21 voip pairs 22 voip pairs 23 voip pairs 24 voip pairs 25 voip pairs (a) ZC 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 CDF Delay [s] VoIP G.711 Access Delay with 5 Standing Web in CSMA/802.11b 1 voip pairs 2 voip pairs 3 voip pairs 4 voip pairs 5 voip pairs 6 voip pairs 7 voip pairs 8 voip pairs 9 voip pairs 10 voip pairs (b) C SM A Figure 12: G.711 V oIP D elay wi th 5 Standing W eb T able 2: V oIP Capacity in IEEE 802.11b T raffic ZC CSMA V oIP only 21 17 V oIP + 5 W eb 18 1 w eb transaction dela y distribution result under both ZC and CSMA. With 25 sim ultaneous w eb users and nominal sim- ulation parameters, 90% delay is achiev ed within 5 seconds under CSMA, while ZC ac hieves it within 2 seconds. T o sa ve the space, we omit t h e exp erimental CDF. 5.6 Multiple Collision Domain Goodput W e hav e seen ZC outp erforms b oth TDMA and CSMA in a single collisio n domain in the foregoing sections. I n a multi- ple collision domain, neither ZC, CSMA nor self-regularizing distributed TDMA is collis ion-free. While ZC is original ly designed for the operation in a single collision domain, it does not mean it is not operable in a m u ltiple collision do- main environmen t. Rath er, sim u lations in complicated mul- tiple collision domain top ologies show the evidence th at ZC is commensurate with CSMA in terms of go od p ut. T o em- ulate a complex wireles s sensor netw ork environment with signal-blocking walls and buildings, we generate rand om top olo- gies in a follo wing manner: in a 200 × 200 sq. meter area, w e place N no des at random lo cation. There are N 2 distinct and indep endent flow pairs, eac h of which has appro x imately 600 K bps source rate. Eac h no de has a random connectivity with another with probability γ irrespective of its geographi- cal lo cation. H ere the connectivity betw een tw o no des means they can hear eac h other’s transmission. No connectivity im- plies th ere is a wall so that t wo no des cannot hear eac h other at all. A flow pair is alw a ys set to hav e a connectiv ity . On a vera ge, each no de has conn ectivities with γ fraction of N nod es indep endently from oth ers’ connectivity state, and it is hidden from 1 − γ fractio n of N no des. Connectivity is assumed to b e reciprocal. 0 20 40 60 80 100 0 2 4 6 8 10 # Active stations Goodput [Mbps] Multi−Collision Domain (20% connectivity) ZC (Capacity 64) CSMA (a) Connectivity γ = 0 . 2 0 20 40 60 80 100 0 2 4 6 8 10 # Active stations Goodput [Mbps] Multi−Collision Domain (50% connectivity) ZC (Capacity 64) CSMA (b) C onnectivity γ = 0 . 5 Figure 13: Multiple col l ision domain go odput com- parison: ZC and CSMA Obviously the goo dput degradation is mainly caused by hid- den no de/exp osed node synd rome in the m u ltiple collision domain. Exp erimental results show when th e fraction of random connectivity is lo w with, say , γ ≤ 0 . 2, Z C is com- parable to CSMA. When γ = 0 . 5, note that this top ology is ex cessiv ely complex, and carrier sensing based medium access is more error-prone. The more the MAC relies on carrier-sensing, th e more colli sions are likely to o ccur, which is the case of ZC. 6. CONCLUSION W e prop osed a zero collision achieving asynchronous d is- tributed medium access control, called ZC, which provides sup erior performance compared to CSMA and TDMA in terms of goo dput and mean interaccess dela y . By design, ZC is sensitive to n et work dynamics, carrier sensing error, or corresponding clock drift. An alytically w e show ed that ZC’s mean converge nce time to zero collisio n state is up p er- b ounded and 2-3 seconds in worst case for a reasonably big netw ork size. Empirical results sh o w that even at a sev ere carrier sensing error rate, ZC robustly maintains sup erior p erformance to CSMA. ZC can b e easily implemented using 802.11 hardwa re follow in g t he same PHY and most of MAC sp ecification. Although all the p erformance figures are based on I EEE 802.11 PHY families in this pap er, its application can b e easily ex tended to other w ell-kn o wn wireless or wired technolo gies. 7. REFERENCES [1] M. Ali, T. Suleman, and Z. A. Uzmi. 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DEFINITIONS OF P ARAMETERS Based on IEEE 802.11 MAC/PHY sp ecification, timing pa- rameters used in the previous formula is defined as follo ws: t g = 2 × ( T P LC P P reamble + T P LC P H eader ) (15) + T M P D U + T S I F S + T Ack (16) t b = T P LC P P reamble + T P LC P H eader (17) + T M P D U + T E I F S (18) t v = T S LO T (19) t s = 0 (20) 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 0 1 2 3 4 5 6 7 8 9 10 11 Error Rate Goodput [Mbps] Sensitivity on Goodput ZC CSMA 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 0 50 100 150 200 250 300 350 400 450 500 Error Rate Mean Inter−access Delay [ms] Sensitivity on Delay ZC CSMA 0 2e+06 4e+06 6e+06 8e+06 1e+07 0 10 20 30 40 50 60 Bps Number of Nodes Network Throughput with ZeroCollision/802.11b Under ZeroCollision 0 2 4 6 8 10 0 10 20 30 40 50 60 MBps Number of Nodes Network Goodput in 802.11b Under ZeroCollision Under CSMA 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120 128 0 5 10 15 Germs M Periods Mean Work Period for Cat and Puppy N=32 N=64 N=128 Cat Work Period Puppy Work Period 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120 128 0 500 1000 1500 2000 2500 3000 Germs M Convergence Time [ms] Convergence Time for Cat and Puppy N=32 N=64 N=128 Cat Convergence Time Puppy Convergence Time 0 2 4 6 8 10 0 5 10 15 20 Goodput [MBps] Time [s] Effect of Network Dynamics on Throughput with ZeroCollision/802.11b 1 AP 31 nodes on 25x25 sqr meter Grid A station arrives at 5.0 and leaves at 6.0 (Collision) Instantaneous Goodput (Step 10msec) Smoothed Goodput (Step 1000msec) 0 2 4 6 8 10 12 14 0 5 10 15 20 Collisions Time [s] Effect of Network Dynamics on Throughput with ZeroCollision/802.11b 1 AP 32 nodes on 25x25 sqr meter Grid A station arrives at 5.0 and leaves at 6.0 (Collision) # of Collision 0 2 4 6 8 10 0 5 10 15 20 Goodput [MBps] Time [s] Effect of Network Dynamics on Throughput with ZeroCollision/802.11b 1 AP 32 nodes on 25x25 sqr meter Grid A station arrives at 5.0 and leaves at 6.0 (Collision) Instantaneous Goodput (Step size = 10msec) Smoothed Goodput (Step size = 1000msec) 0 1 2 3 4 5 6 7 8 9 10 11 0 10 20 30 40 50 60 M=8 M=16 M=32 M=64 Random walk Day Random walk approaching M at hitting time, with 64 cheeses 0 1 2 3 4 5 6 7 8 9 10 11 12 0 20 40 60 80 100 120 M=16 M=32 M=64 M=128 Random walk Day Random walk approaching M at hitting time, with 128 cheeses 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 pmf Good cheeses pmf of number of good cheeses(64 Cheeses M germs) M=8 M=16 M=32 M=64 0 16 32 48 64 80 96 112 128 0 5 10 15 N=32 Cheeses N=64 Cheeses N=128 Cheeses E[convergence periods] M Germs Expectation of Convergence Periods (or barking times + 1) 1 2 3 4 5 1 2 3 2 5 1 3 Packet arrival times at nodes Idle mini -slots Packet transmissions idle 5;p 2;n idle 3;p 5;e 1;n idle 3;p 1;p 2;n col 3;e 1;p col 4;n Time Time 0 20 40 60 80 100 120 140 160 180 0 100 200 300 400 500 600 700 800 # Active stations Mean Interaccess Time [msec] Mean Interaccess Time (Sparse traffic) ZC (Capacity 64) CSMA 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Expectation Number of germs Expectations given 64 cheeses and varying germs Vacancy Good cheeses Bad cheeses 0 200000 400000 600000 800000 1e+06 1.2e+06 1.4e+06 0 5 10 15 20 25 30 bps Voice Pairs VoIP G.729a Goodput with ZeroCollision/802.11b Under ZeroCollision 0 200000 400000 600000 800000 1e+06 1.2e+06 1.4e+06 0 5 10 15 20 25 30 bps Voice Pairs VoIP G.729a Goodput with CSMA/802.11b Under CSMA 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 CDF Delay [s] Voip G.711 Access Delay with 5 Standing Web in ZeroCollision/802.11b 15 voip pairs 16 voip pairs 17 voip pairs 18 voip pairs 19 voip pairs 20 voip pairs 21 voip pairs 0 10 20 30 40 50 60 0 0.05 0.1 0.15 0.2 0.25 0.3 pmf Vacancies pmf of number of vacancies out of 64 Cheeses M=64 M=48 M=32 M=16 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 CDF Delay [s] Web Request-Response Delay with ZeroCollision/802.11b 5 web pairs 10 web pairs 15 web pairs 20 web pairs 25 web pairs 30 web pairs 0 100000 200000 300000 400000 500000 600000 700000 5 10 15 20 25 30 bps Web Users Web Goodput with ZeroCollision/802.11b Under ZeroCollision 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 CDF Delay [s] Web Request-Response Delay with CSMA/802.11b 5 web pairs 10 web pairs 15 web pairs 20 web pairs 25 web pairs 30 web pairs 0 100000 200000 300000 400000 500000 600000 700000 5 10 15 20 25 30 bps Web Users Web Goodput with CSMA/802.11b Under CSMA a b:2 c:1, 3 d a:3 b:2 c d:1 Idle 1,3 Idle Idle Time