Effective Carrier Sensing in CSMA Networks under Cumulative Interference

Ef fecti v e Carrier Sensing in CSMA Netw orks und er Cumulati v e Interfer ence Liqun Fu, Soung Chang Liew , Jianwei Huang Departmen t of I nform ation Engineering The Chinese University of Hon g Kong Shatin, New T erritories, Hong K ong Email: { lqfu 6,soung ,jwhuang } @ie.cuhk.edu. hk Abstract —This paper proposes and in vestigates the concept of a safe carrier-sensing range that can guarantee interference- safe (also termed hidden-node-free) transmissions in CS MA networks under th e cumulative interference model. Compared with the safe carrier -sensing range und er the commonly assumed but less realistic pairwise interference model, we show th at the safe carrier -sensing range r equired under th e cumulative interference model is large r by a constant multipli cativ e factor . For example, if th e SINR requirem ent is 10 dB and the path- loss exponent is 4 , the factor is 1 . 4 . T he concept of a safe carrier -sensing range, although am enable to elegant analytical results, is inherently n ot compatible with the con ventional p ower - threshold carrier -sensing mechanism (e.g., that used in IEEE 802.11). Specifically , the absolute power sensed by a node in the con ventional mechanism does not contain enough in fo rmation fo r it to deriv e its distances from other concurrent transmitter nodes. W e show that, f ortunately , a carrier -sensing mechanism called In cremental-P ower Carrier -S ensing (IP CS) can realize the carrier -sensing range concept in a simple way . Inst ead of monitoring the absolute detected power , the IPCS mechanism monitors every increment in the detected power . This means that IPCS can separate the detected power of eve ry concurrent transmitter , and map the power profile to the required d istance informa tion. Our extensive simulation results indicate that IPCS can boost spatial reuse and network throughput by more than 60% relativ e to the conv entional carrier -sen sing mechanism. Last but not least, IPCS n ot only allows us to i mplement our safe carrier -sensing range, it also ties up a loose end i n many oth er prior theoretica l works that implicitly assume the use of a carrier - sensing range (safe or otherwise) wi thout an explicit design to realize i t. Index T erms —carrier -sensi ng range, cu mulative in terference model, CSMA, WiFi, IEEE 802.11, SINR constraints, spatial reuse. I . I N T RO D U C T I O N A N D O V E RV I E W Due to th e broadcast nature of wireless ch annels, sig- nals tr ansmitted over wireless link s can mutu ally interfere with each othe r . How to o ptimize spatial reuse and n etwork throug hput under such mutual interf erences has been an in- tensely studied issue in wireless networking. In par ticular, it This work was supported by two Compet iti ve E armark ed Researc h Grants (Project Number 414507 and Project Number 412308) esta blished under the Uni versi ty Grant Committee of t he Hong K ong Specia l Administrati ve Re gion, China, the Direct Grant (Project Number C001-2050398) of The Chinese Uni versi ty of Hong Kong, and the National Key T echnolog y R&D Program (Project Number 2007BAH17B0 4) established by the Ministry of Science and T echnology of the People’ s Republic of China. is desirable to allo w as m any links as possible to tra nsmit together in an in terferenc e-safe (or collision-fr ee) manner . The problem of interfer ence-safe transmissions under the coo rdina- tion of a cen tralized T DMA (Time-Division Multiple-Access) scheduler has been well studied ( e.g., see [1] –[6]). Less well unde rstood is the issue of inte rference- safe tran smissions under the co ordinatio n of a distributed scheduling proto col. The CSMA (Carrier-Sense Mu ltiple-Access) protoco l, such as IEEE 802.1 1, is the most widely ad opted distributed scheduling pr otocol in practice. As the g rowth o f 802. 11 net- work deployments continues unaba ted, we are witnessing an increasing level of mutual inter ference among tran smissions in such networks. It is critical to establish a rig orous c onceptua l framework u pon which effective solu tions to in terferenc e-safe transmissions can b e construc ted. W ithin this context, this paper h as three major contributions listed as f ollows (more de tailed overview is g iv en in the succeeding paragr aphs): 1) W e propo se the concep t o f a safe carrier -sensing range that can guaran tee interference-safe transmissions in CSMA networks un der the cumulative in terfer en ce model . 2) W e show that the co ncept is implemen table using a very simple In cremental-Power Carr ier-Sensing ( IPCS) mechanism. 3) W e d emonstrate that imp lementation of safe carrier- sensing range under IPCS can significantly imp rove spatial reu se and network throug hput as comp ared to th e conv entional absolute-power carrier sen sing m echanism. Regarding 1), this paper considers th e cumulative interfer- ence m odel (also te rmed physical interfer ence mod el in [7] ), in whic h the interf erence at a receiver nod e i consists of the cumulative power r eceiv ed fro m all the o ther nodes that are currently transmitting ( except its own transmitter) . Th is m odel is known to be more practical an d much m ore difficult to analyze than the wide ly studied pair wise in terferenc e model (also termed the p rotocol interferen ce mod el in [7]) in th e literature. Under the cumulative interference model, a set of simultaneou sly transmittin g link s are said to be interfer ence- safe if the SI NRs (Sig nal-to-In terferenc e-plus-Noise Ratios) at the receivers o f all th ese links are ab ove a th reshold. Given a 2 set of links L in the network, there are m any subsets of links, S ⊂ L , that are inter ference- safe. The set of all such sub sets F = {S | the SINR requ irements of all link s ar e satisfied } constitutes th e feasible inter ference-saf e state space. For cen- tralized TDMA, all subsets are av ailab le for schedulin g, an d a TDMA schedu le is b asically a seq uence ( S t ) n t =1 where each S t ∈ F . For CSMA, beca use of the rand om and distributed nature of the carrier-sensing o perations by in- dividual no des, the simultan eously transmittin g links S C S may or may not belong to F . L et F C S = {S C S | simultaneou s tr ansmissions o f links in S C S are allowed by the carrier-sensing opera tion } . The CSMA ne twork is said to be in terference- safe if F C S ⊆ F . This is also th e con dition for the so-called h idden- node free o peration [8]. Howe ver , this issue was studied under the context of an idealized pairwise interferen ce mo del [8] r ather than the practical cumulative interferen ce mod el of interest her e. In this p aper, we show that if the carrier-sensing mech anism can guarantee that the distance between e very pair of transmitters is separated by a safe carrier-sensing range , then F C S ⊆ F can be guaran teed and the CSMA network is interfer ence-safe even un der a cumulative interferenc e mode l. W e believe that the safe carrier- sensing r ange established in this paper is a tight upper bound and achieves good spatial r euse. Anoth er issue is how to im- plement the concep t of safe carrier-sensing ran ge in pr actice. This b rings u s to 2) above. I n tr aditional carrie r sensing based on po wer thresho ld (e.g., that of the basic mode in IEEE 802.1 1), the absolute power received is being mo nitored. This power consists of the sum to tal of powers receiv ed from all the other transmitters. It is im possible to infer from th is ab solute power the exact separation o f the n ode from each of the other transmitters. This leads to subp ar spatial reuse. Fortunately , we show th at a simple mech anism that monitors the incremen tal power chan ges over time, IPCS, will enab le u s to map the power p rofile to the req uired distance in formatio n. W e be lie ve that this contr ibution, a lthough simp le, is significant in that it shows that th e theor etical concept of safe ca rrier -sensing range c an be implem ented rather easily in practice. It also ties up a loose end in many other prior theo retical works that implicitly assume the use of a carrier-sensing r ange (safe or otherwise) withou t an explicit design to r ealize it. That is, IPCS can be used to implement the required carrier-sensing range in these works, no t just our safe carrier -sensing range here. W ithou t IPCS, and usin g on ly the conventional carrier- sensing mechan ism, the resu lts in th ese prior works would have b een overly optimistic. Given the impleme ntability o f safe carrie r-sensing range, the n ext issue is how tight th e simultaneou sly transmitting n odes can be packed. This brin gs u s to 3) above. In the co n ventional carrier sensing mech anism, in order that th e de tected absolu te power is below the carr ier-sensing power threshold , the separ ation between a ne wly active transmitter and other existing acti ve transmitters mu st incr ease prog ressiv ely as the n umber of concur rent transmissions increases. T hat is, the co st of en- suring interferen ce-safe transmissions beco mes pr ogressively higher an d high er in the “packing p rocess”. This reduce s spatial reu se and the overall network thro ughpu t. Fortunately , with I PCS, th e r equired separation betwe en a ny pair of active transmitters rem ains constant as the safe ca rrier -sensing range which is indep endent of th e number of concurrent transmis- sions. Indeed , ou r simulation results indicate that compared to the conv entional carrier-sensing mechanism, IPCS mechanism improves the spatial reuse and the netw ork through put by more than 60% . A. R elated W ork In the literature, most stud ies on carr ier sensing (e. g., [8] – [13]) are based o n the pairwise interfer ence model. For a link under the pairwise interfer ence mod el, the inter ferences fr om the other links are co nsidered one b y one. If the in terference from each o f the other links on the link co ncerned does no t cause a collision, the n it is assumed that th ere is n o co llision overall. Ref. [8] estab lished the carrier-sensing rang e required to prevent hidden-no de co llisions in CSMA network s under the p airwise inter ference m odel. Th e resu lting carrier-sensing range is too optimistic and can not eliminate hidden -node collisions if the more accurate cum ulativ e interfer ence model is ad opted in stead. A nu mber of recen t papers studied the CSMA networks under the c umulative interferen ce mod el (e.g., [1 4]–[1 7]). An earlier unpub lished technical report of ours [ 17] deri ved the safe carrier-sensing range under th e cumu lativ e interfe rence model. The technical report, ho wever , did no t include the IPCS realization presen ted in this paper . Neither did Ref. [ 14]–[ 16] address the implementatio n o f a carrier-sensing range based on power detec tion. Ref. [ 14] stud ied the asympto tic capacity of large- scale CSMA networks with h idden- node-f ree design s. The focus of [1 4] is on “or der” result rath er than “tight” result. For example, if γ 0 = 10 dB and α = 4 , th e safe carrier-sensing range der i ved in [14] is 8 . 7 5 d max . In this paper, we show that setting the saf e carrier-sensing range to 5 . 27 d max is enoug h to p rev ent hidd en-no de collisions. The authors in [15], [1 6] attempted to imp rove spatial reuse and cap acity by tu ning th e transmit p ower and the carrier- sensing range. Althou gh the cumula ti ve interferen ce m odel is con sidered in [15], [ 16], spatial reuse and capacity are analyzed based on ca rrier-sensing rang e. In par ticular , they assumed tha t the transmitters of concu rrent transmission link s can be u niformly packed in th e network. As discussed in this paper, such u niform packin g can not be r ealized using the cu rrent 802.11 carr ier-sensing m echanism. Th erefore , the results in [1 5], [16 ] are overly o ptimistic withou t an appr opri- ate carrier-sensing mechanism. IPCS fills this gap so that the theoretical r esults o f [15] , [16] remain valid. W e summarize the key related m odels and results in the literature in T able I ∗ . The rest of this paper is organized as follows. Section II presents the c umulative interf erence model and the carr ier sensing mec hanism in the curren t 802.11 proto col. Section III derives the safe car rier-sensing rang e tha t s uccessfully pre vents ∗ This paper focuses on the incrementa l-po w er carrier -sensing (IPCS) mech- anism under the cumulati ve interferen ce model. But IPCS proposed in this paper can also deal with the pairwise interference model. 3 T ABLE I S U M M AR Y O F T H E R E LAT E D W O R K Interfer ence Models Pairwi se Interfer ence Model Cumulati ve Interfer ence Model Absolute po wer carrie r sensing many (e.g., [8], [10]) [15], [16] Increment al power carrie r sensing This paper This paper the hidden-n ode c ollisions u nder the cumu lati ve interferen ce model. Sectio n IV presents th e IPCS mechan ism. Section V ev alu ates the per forman ce of IPCS in terms of spatial r euse and network thr oughp ut. Section VI conc ludes this pap er . I I . S Y S T E M M O D E L A. Cumu lative I nterfer ence Mod el W e represen t links in a wireless network by a set of distinct and dire cted transmitter-receiver pair s L = { l i , 1 ≤ i ≤ | L|} . Let T = { T i , 1 ≤ i ≤ |L|} an d R = { R i , 1 ≤ i ≤ |L|} denote the set of transmitter nodes an d the set of receiver nodes, respectively . A r eceiv er dec odes its signal successfully if a nd on ly if the received Signal-to-In terferen ce-plus-Noise Ratio (SINR) is above a certain thre shold. W e ado pt th e cumulative interferen ce mode l, wher e the inter ference is the sum of t he recei ved powers from all transmitters except its own transmitter . W e a ssume that rad io signal propa gation follows the lo g-distance path model with path loss exponent α > 2 . The path gain G ( T i , R j ) from transmitter T i to recei ver R j follows a geometr ic model: G ( T i , R j ) = d ( T i , R j ) − α , where d ( T i , R j ) is the Euclidean distance between nodes T i and R j . In 802.1 1, each packet transmission on a link l i consists of a D A T A frame in the forward direc tion (from T i to R i ) fo llowed by an A CK frame in the reverse direction (fro m R i to T i ). T he packet transmission is said to b e successful if and only if b oth the D A T A frame and the A CK frame are received correc tly . Let L ′ ( L ′′ ) de note the set of lin ks that transmit concu rrently with the DA T A (AC K) frame on link l i . Under th e cumu lati ve interferen ce model, a successful transm ission o n lin k l i needs to satisfy th e following cond itions: P t · G ( T i , R i ) N + P l j ∈L ′ P t · G ( S j , R i ) ≥ γ 0 , (D A T A fram e) (1) and P t · G ( R i , T i ) N + P l j ∈L ′′ P t · G ( S j , T i ) ≥ γ 0 , (A CK fra me) (2) where P t is the transmit power , N is the a verage noise po wer , and γ 0 is th e SINR threshold for correct rec eption. W e assume that all nod es in th e network use the same tr ansmit power P t and ado pt the same SINR thre shold γ 0 . For a lin k l j in L ′ or L ′′ , S j represents the sender of link l j , which can be either T j or R j . Th is is because e ither DA T A or A CK tran smission on link l j will c ause interfer ence to link l i . B. E xisting Carrier S ensing Mechan ism in 802. 11 If th ere exists a lin k l i ∈ L suc h that no t both (1) an d (2) are satisfied, this means there is collision in the network. In 802.1 1, carrier sensing is d esigned to pr ev ent collision du e to simultaneou s transmissions th at cause the violation o f either (1) or (2). In this pap er , we assume carrie r sensin g by en ergy detection. Consider a lin k l i . If tran smitter T i senses a power P C S ( T i ) th at exceeds a power th reshold P th , i.e., P C S ( T i ) > P th , (3) then T i will not transmit and its backoff countdown p rocess will b e fr ozen. This will prevent the DA T A frame transmission on l i . In most stud ies of 80 2.11 networks, the co ncept of a c arrier- sensing range C S R is introdu ced. The carrier-sensing range C S R is mapped fro m the carrier-sensing power threshold P th : C S R =  P t P th  1 α . Consider two links, l i and l j . If the distance betwee n transmitters T i and T j is no less than the carrier-sensing range, i.e., d ( T i , T j ) ≥ C S R, (4) then T i and T j can not carrier sense ea ch other, and thus c an initiate co ncurren t transmissions between them. The p airwise relationship can be g eneralized to a set of links S C S ⊆ L . If the cond ition in (4) is satisfied by a ll pairs of transmitters in set S C S , then all links in S C S can tr ansmit co ncurre ntly . Setting a n appropriate carr ier-sensing range is cru cial to the perf ormance of 8 02.11 n etworks. If C S R is too large, spatial reuse will b e unnecessarily limited. If C S R is no t large enough , then hidd en-nod e co llisions may oc cur . T he underly ing cau se of hidd en-nod e collisions are as f ollows. A numb er of transmitters transmit simu ltaneously because condition (4) is satisfied by all pair s of the transmitters. Howe ver, there is at least o ne of the links does no t satisfy either (1) or (2). As a result, collision s happen a nd the carr ier sensing mech anism is said to ha ve failed in pre venting such collisions. W e now de fine a safe carrier -sen sing range that always prevents the hidd en-no de collisions in 80 2.11 networks under the cu mulative interf erence mode l. Definition 1 (Safe- C S R cumulative ): L et S C S ⊆ L denote a subset of links that are allowed to transmit concurre ntly und er a carrier-sensing range C S R . Let F C S = {S C S } deno te all such su bsets of links in the n etwork. A C S R is said to b e a Safe- C S R cumulative if for any S C S ∈ F C S and for any link l i ∈ S C S , both conditions ( 1) a nd (2), w ith L ′ = L ′′ = S C S \ { l i } , are satisfied . For ana lysis simplicity , we assume tha t the backg round noise power N is small com pared with interfer ence a nd thus can be igno red. W e will co nsider Signal-to -Interfer ence Ratio (SIR) in stead of SINR. 4 I I I . S A F E C A R R I E R - S E N S IN G R A N G E U N D E R C U M U L A T I V E I N T E R F E R E N C E M O D E L In this section, we derive a sufficient thresho ld for Safe- C S R cumulative . Whe n discussing the h idden-n ode free design [8], it is req uired that the receivers are operated with the “RS (Re-Start) m ode” (see Appen dix A for details). I n the following discussion, we also make th e same assumption . Ref. [8 ] studied the safe carrier-sensing ran ge u nder th e pairwise interfer ence model . Th e threshold is given as follo ws: Safe- C S R pairwise =  γ 0 1 α + 2  d max , (5) where d max = max l i ∈L d ( T i , R i ) is the maximu m link len gth in the network. Howev er , the pairwise interference model does not take into acco unt the cumu lativ e nature of interference s from other links. T he threshold gi ven in (5) i s overly optimistic and no t large enoug h to prevent hid den-no de collisions un der the cu mulative interfer ence mode l , as illustrated by the three- link exam ple in Fig. 1. 1 T 1 R ma x d 2 T 2 R ma x d m a x 2 d 3 T 3 R ma x d m a x 4 d DATA DATA ACK 3 l 2 l 1 l Fig. 1. Settin g the carrier -sensing range as Safe- C S R pairwise is insuf ficient to pre vent hidden-nod e collisions under the cumulati ve interference model In Fig. 1, suppo se that the SIR requ irement γ 0 = 8 and the path-loss expon ent α = 3 . Acco rding to (5), it is enough to set the c arrier-sensing ra nge as  γ 0 1 α + 2  d max = 4 d max and the carr ier sensing power thresho ld P th = P t (4 d max ) − 3 = 0 . 0156 P t d − 3 max . In Fig. 1, there are thr ee links: l 1 , l 2 , and l 3 with the same link length d max . The distance d ( R 1 , R 2 ) e quals 2 d max and the distance d ( T 1 , R 3 ) equals 4 d max . Since the distance d ( T 1 , T 2 ) = 4 d max =  γ 0 1 α + 2  d max , f rom (4), we find that T 1 and T 2 can simultaneo usly in itiate tran smissions since they can not carrier sense each o ther . W e can verify that the SIR requ irements o f both D A T A and ACK tran smissions on l 1 and l 2 are satisfied. This means l 1 and l 2 can indeed successfully transmit simultaneou sly . Suppose that l 3 wants to initiate a transmission when T 1 is sending a DA T A frame to R 1 and R 2 is sending an ACK frame to T 2 . T ran smitter T 3 senses a power P C S ( T 3 ) g iv en by P C S ( T 3 ) = P t · (5 d max ) − 3 + P t · (8 d max ) − 3 = 0 . 00995 · P t d − 3 max < P th . This means th at T 3 can not sense the tran smissions on l 1 and l 2 , and can initiate a D A T A transmission. However , when all these three links are active simultaneously , the SIR a t R 1 is P t ( d max ) − 3 P t (6 d max ) − 3 + P t (2 d max ) − 3 = 7 . 7 14 < γ 0 . This means the cu mulative interfe rence po wers from l 2 and l 3 will corrupt the DA T A tra nsmission on l 1 due to the insufficient SI R at R 1 . This example shows that setting the carrier-sensing range as in (5 ) is not sufficient to prevent collisions u nder th e cumulative interfer ence mo del. W e next establish a thresho ld for Safe- C S R cumulative so th at the system will rem ain safe under cu mulative interferenc e. Theor em 1: The setting Safe- C S R cumulative = ( K + 2) d max , (6) where K =  6 γ 0  1 +  2 √ 3  α 1 α − 2  1 α . (7) is sufficient to en sure interference-safe transmissions under the cumulative interfer ence mod el. Pr oof: Th e pr oof is given in Append ix B. Condition (6) provides a sufficiently large carrier-sensing range that prevents th e hidden -node collisions in CSMA networks. Theref ore, th ere is no n eed to set a C S R larger than the value given in ( 6). Let us comp are Safe- C S R cumulative with Safe- C S R pairwise with dif ferent v alues o f γ 0 and α . F or example, if γ 0 = 1 0 and α = 4 , which are ty pical f or wireless communicatio ns, Safe- C S R pairwise = 3 . 78 · d max , Safe- C S R cumulative = 5 . 27 · d max . Compared with Safe- C S R pairwise , Sa fe- C S R cumulative needs to be increased by a factor of 1 . 4 to ensur e successful tran smis- sions under the cumulative in terference mo del. Giv en a fixed path-loss expo nent α , both Safe- C S R pairwise and Sa fe- C S R cumulative increase in the SIR requirement γ 0 . This is because the separation among links must b e enlarged to me et a larger SIR target. For example, if α = 4 , we have Safe- C S R pairwise =  2 + γ 1 4 0  d max , Safe- C S R cumulative = 2 +  34 3 γ 0  1 4 ! d max . The ratio o f S afe- C S R cumulative to Safe- C S R pairwise is Safe- C S R cumulative Safe- C S R pairwise = 2 +  34 3 γ 0  1 4 2 + γ 1 4 0 , which is an in creasing fu nction of γ 0 , an d co n verges to a constant as γ 0 goes to infinity: lim γ 0 →∞ Safe- C S R cumulative Safe- C S R pairwise = lim γ 0 →∞ 2 +  34 3 γ 0  1 4 2 + γ 1 4 0 =  34 3  1 4 ≈ 1 . 8 348 . Fig. 2 sho ws the ratio Safe- C SR cumulative Safe- C S R pairwise as a function of the SIR requirem ents γ 0 . Dif ferent cur ves represent different ch oices of th e path-loss e xpone nt α . The ratio Safe- C S R cumulative Safe- C SR pairwise increases when γ 0 increases or α decre ases. For each choice of α , the 5 0 200 400 600 800 1000 1 1.5 2 2.5 SIR Requirement γ 0 The ratio Safe−CSR cumulative /Safe−CSR pairwise α =3 α =4 α =5 Fig. 2. The ratio of Safe- C S R cumulative to Safe - C S R pairwise ratio c on verges to a constant as γ 0 goes to infinity . T his sho ws that, c ompared with the pairwise interfer ence model, th e safe carrier-sensing range un der the cumulative inter ference mo del will n ot in crease ar bitrarily . I V . A N OV E L C A R R I E R S E N S I N G M E C H A N I S M W e now discuss the im plementation of Safe- C S R cumulative . W e fir st describe th e d ifficulty o f im plementing th e safe carrier-sensing r ange in (6) u sing the existing phy sical ca rrier- sensing mechanism in the cu rrent 8 02.11 pr otocol. The n, we pr opose a new I ncremen tal-Power Carrier-Sensing (IPCS) mechanism to resolve this implemen tation issue. A. Limitatio n o f Con ventiona l Carrier-Sensing Mechan ism In the curren t 802.11 MAC protocol, given the safe ca rrier- sensing ra nge S afe- C S R cumulative , th e c arrier-sensing power threshold P th is set as P th = P t · ( Sa fe- C S R cumulative ) − α . ( 8) Before tra nsmitting, a tr ansmitter T i compare s the power it senses, P C S ( T i ) , with th e power thr eshold P th . A key disad- vantage of this approach is th at P C S ( T i ) is a cumulati ve power from all the other nodes that are concurrently transmitting. The cumulative n ature makes it impo ssible to tell whether P C S ( T i ) is from one particu lar near by tran smitter o r a grou p of far-off transmitters [18 ]. This reduces spatial reuse, as illustrated by the example in Fig. 3. There are four link s in Fig. 3, with S afe- C S R cumulative set as in (6). In Fig. 3, the distance d ( T 1 , T 2 ) is equal to Safe- C S R cumulative . Fro m (4), we find that T 1 and T 2 can n ot ca rrier sense each other, thu s th ey can transmit simultaneo usly . First, con sider the lo cation requir ement o f the third link l ′ 3 that can ha ve a concu rrent tran smission with bo th l 1 and l 2 , assuming that each transmitter can p erfectly differentiate the d istances fr om the other tran smitters. Sup pose that the - c u m ul at iv e S a f e C S R 1 T max d max d max d 1 R 2 T 2 R 3 ' T 3 ' R max d 3 T 3 R 2 - c u m ul a ti v e S a fe C S R B ¸ 2 - c u m ula tiv e S af e C S R B ¸ - cu mu lati ve Sa fe CSR - cum ula tive Safe CSR 3 ' l 2 l 1 l 3 l Fig. 3. Con ventiona l carrier- sensing mechanism will reduce the spatial reuse in 802.11 networks. Link l 3 is place d based on the absolute po wer sensing mechani sm in current 802.11, and link l ′ 3 is place d base d on the Safe- C S R cumulative as enab led by our IPCS mechani sm. third link is lo cated on the m iddle line be tween l 1 and l 2 . Based on the car rier-sensing r ange analy sis, the req uire- ments are d ( T ′ 3 , T 1 ) ≥ Safe- C S R cumulative and d ( T ′ 3 , T 2 ) ≥ Safe- C S R cumulative . So the third link can b e located in the posi- tion of l ′ 3 , shown i n Fig. 3. Furthermo re, as the nu mber of links increases, a tight p acking of the con current transmitters will result in a regular equilater al trian gle packing with side length Safe- C S R cumulative . The “consumed area” o f each tran smitter is a constant g i ven by A = √ 3 2 Safe- C S R 2 cumulative . Now , let us consider the location req uirement of the third link l 3 under the carrier-sensing mechan ism of the c urrent 802.1 1 p rotocol. In order to have concu rrent transmissions with both l 1 and l 2 , the cumulative power sensed by T 3 due to transmissions of both links l 1 and l 2 should be no larger than P th , i.e., P C S ( T 3 ) = P t · d ( T 3 , T 1 ) − α + P t · d ( T 3 , T 2 ) − α = 2 · P t d ( T 3 , T 1 ) − α ≤ P th , where P th is given in equation (8). So th e minimum d istance requirem ent o n d ( T 3 , T 1 ) a nd d ( T 3 , T 2 ) is d ( T 3 , T 1 ) = d ( T 3 , T 2 ) ≥  2 P t P th  1 α = 2 1 α · Safe- C S R cumulative , as shown in Fig. 3. Since 2 1 α is alw ays g reater th an 1 , the requiremen t of the separatio n between tran smit- ters is in creased from S afe- C S R cumulative (i.e., d ( T 1 , T 2 ) ) to 2 1 α Safe- C S R cumulative (i.e., d ( T 1 , T 3 ) a nd d ( T 2 , T 3 ) ). The re- quiremen t on the separation between transmitters will incre ase progr essi vely as the number of concurrent lin ks increases, and the cor respond ing p acking of tr ansmitters will be m ore and more sparse. As a result, s patial reuse is reduced as the number of lin ks increases. 6 Another thing to notice is th at the o rder o f the tran smissions of links also a ffects spatial reuse in the con ventional carrier- sensing mech anism. Consider the three lin ks, l 1 , l 2 and l 3 in Fig. 3 again. If the sequence of transmissions is { l 1 , l 2 , l 3 } , as discussed above, T 1 , T 2 and T 3 sense a power no greater than P th , an d thu s l 1 , l 2 and l 3 can b e active simultaneo usly . If th e seque nce of transmissions on these link s is { l 2 , l 3 , l 1 } , howe ver , bo th T 2 and T 3 sense a power no larger than P th . But the cumu lati ve power sensed b y T 1 in this case is P C S ( T 1 ) = P t · d ( T 3 , T 1 ) − α + P t · d ( T 2 , T 1 ) − α = P t  2 1 α Safe- C S R cumulative  − α + P t ( Safe- C S R cumulative ) − α = 3 2 P th > P th . Therefo re, T 1 will sense the chan nel b usy and will no t initiate the transmission on l 1 . The spatial reuse is un necessarily reduced bec ause ther e would h av e been no co llisions had T 1 decide to tr ansmit †† . B. I ncr emental-P o wer Ca rrier - Sensing (IPCS) Mechan ism W e prop ose an en hanced ph ysical carrier-sensing mech- anism called In cremental-Power Carrier-Sensing (IPCS) to solve the issues identifie d in section IV -A. Spec ifically , the IPCS m echanism can implement the safe carrier-sensing rang e accurately by separatin g th e d etected powers from mu ltiple concur rent transmitters. There are two fundamental causes for collision s in a CSMA network. Besides hidden nodes, co llisions can also happen when the backoff mec hanisms of two transmitters count down to zero simultaneously , causing them to tra nsmit together . Note that for the latter , each of the two tra nsmitters is not aw are that the other transmitter will begin transmission at the sam e time. Based o n the power that it d etects, it co uld perfectly be safe f or it to transmit together with the existing active transmitters, o nly if the other transmitter did not d ecide to join in at the same tim e. Ther e is no way tha t the car rier- sensing mechanism can prevent th is kind of collisions. This paper add resses the hid den-no de phen omeno n o nly . T o isolate the second kind of collision s, we will assume in the following discussion o f IPCS that no two tran smitters will tr ansmit simultaneou s ‡ . Con ceptually , we cou ld imagin e the rand om variable associate d with backoff coun tdown to be continuou s rather tha n discr ete, which means that th e starting /ending of one link’ s transmission will coincide with the starting/endin g of a nother lin k’ s transmission with zero pro bability . The key idea of IPCS is to utilize the whole carr ier- sensing power history , not just the carrier-sensing po wer at one particular time. I n CSMA networks, each tr ansmitter T i carrier senses th e channel excep t during the time when it transmits †† This corresponds to the e xposed-node phen omenon. ‡ Collisi ons due to s imultan eous countdo wn-to-zero can be tackled by an expo nential backof f mechani sm in which the transmission probability of each node is adjusted in a dynamic way based on the busyness of the network. In Wi Fi, for example, the countdo wn window is doubled af ter each collision. The probabilit y of this kind of collisions can be made small with a proper design of the back off m echani sm D A T A or receives A CK. The power b eing sensed increases if a link starts to tran smit, and decreases if a link finishes transmission. As a result, the power sensed by tran smitter T i , denoted by P C S i ( t ) , is a contin uous fu nction of time t . In I PCS, instead of ch ecking the absolute power sensed at time t , the transmitter checks increments of power in the past up to time t . If the p acket du ration t packet (includin g both D A T A a nd A CK frames an d the SIFS in between) is a c onstant for all lin ks, then it suffices to check the power increments dur- ing the tim e wind ow [ t − t packet , t ] § . Let { t 1 , t 2 , · · · , t k , · · · } denote the time instan ces whe n th e power being sensed changes, and { ∆ P C S i ( t 1 ) , ∆ P C S i ( t 2 ) , · · · , ∆ P C S i ( t k ) , · · · } denote the correspo nding increme nts, r espectively . In IPCS, transmitter T i will d ecide the ch annel to be id le at time t if the f ollowing condition s are met: ∆ P C S i ( t k ) ≤ P th , ∀ t k such th at t − t packet ≤ t k ≤ t, (9 ) where P th is th e carrier-sensing power th reshold determin ed accordin g to C S R ; othe rwise, the channel is d eemed to be busy . Since ∆ P C S i ( t k ) is negativ e if a lin k stops transmission at some time t k , we only need to check the instances where the p ower incremen ts are positive. By checkin g ev ery incr ement in the detected power , T i can separate the powers fro m all conc urrent tra nsmitters, a nd c an map the power profile to the re quired distance informa tion. In this way , IPCS can ensure the separation s between all transmitters are tig ht in accordan ce with Theorem 1. Theor em 2: If the car rier-sensing power thresho ld P th in the I PCS mechanism is set as: P th = P t ( Safe- C S R cumulative ) − α , (10) where Safe- C S R cumulative is the safe c arrier-sensing ra nge in (6), then it is suf ficient to prevent hidden -node collisions under the cu mulative interf erence mode l. Pr oof: Th e pr oof is given in Append ix C. t 1 t 2 t 3 1 ( ) CS P t 3 2 ( ) CS P t 3 ( ) CS P t Fig. 4. The power sensed by transmitter T ′ 3 as a function of time Let us use Fig. 3 again to show how IPCS can implement the saf e carrier-sensing range successfully . W e set the carr ier- sensing power threshold P th as in (1 0). W e will show that the location requ irement of th e third link u nder IPCS is the same as in dicated by the safe carrier-sensing range (lo cation l ′ 3 in Fig. 3) . Th e tran smitter of the th ird link will only initiate its § This assumption is used to simplify explanati on only . In general, we could check a time windo w suffici ently lar ge to cover the maximum packet size among al l links. 7 transmission wh en it senses the cha nnel to be idle. Its carrier- sensed power is sho wn in Fig. 4. W ithout loss of gen erality , suppose that link l 1 starts transmission bef ore l 2 . T he third transmitter detects two increme nts in its carrier-sensed power at time instances t 1 and t 2 which ar e due to the tr ansmissions of T 1 and T 2 , respectively . I n th e IPCS mecha nism, the thir d transmitter will believe that the ch annel is idle (i.e ., it can start a new transmission) if the following is true: ( ∆ P C S 3 ( t 1 ) = P t d ( T ′ 3 , T 1 ) − α ≤ P th , ∆ P C S 3 ( t 2 ) = P t d ( T ′ 3 , T 2 ) − α ≤ P th . (11) Substituting P th in (10) to (11), we f ind that the require ments in (1 1) are e quiv alent to the following distan ce req uirements: ( d ( T ′ 3 , T 1 ) ≥ Safe- C S R cumulative , d ( T ′ 3 , T 2 ) ≥ Safe- C S R cumulative . So the third link can be located at the position of l ′ 3 , as shown in Fig. 3, in stead of far away at the location o f l 3 as in the conv entional carrier-sensing mechan ism. Compared with the conv entional carrier-sensing mechanism, the advantages of IPCS are 1) IPCS is a p airwise car rier-sensing mechan ism. In the IPCS m echanism, the power fro m each and every con- current link is checked individually . This is equ iv alen t to checking the sepa ration between every p air o f concurrent transmission links. W ith IPCS, all the analyses ba sed on the co ncept of a carrier-sensing ran ge rema in valid. 2) IPCS imp roves sp atial reuse and network throug hput. In th e conventional carrier-sensing mech anism, the link separation r equireme nt in creases as the nu mber of co n- current links in creases. In IPCS, howe ver, th e link separation r equiremen t remains the same. Furth ermore, because I PCS is a pairw ise mechanism, the order o f the transmissions of links will not affect the spatial r euse. V . S I M U L AT I O N S R E S U LT S W e perfo rm simulations to evaluate the r elativ e perfo r- mance of IPCS an d con ventional Carrier Sensing (CS). In our sim ulations, the nod es are located within in a squ are area o f 300 m × 30 0 m . The locations of the transmitters are generated accor ding to a Poisson p oint pr ocess. T he leng th of a link is unifor mly d istributed between 10 and 20 meters. More spe cifically , th e r eceiv er associated with a tran smitter is rando mly located between the two concentric circles of radii 10 m and 20 m ce ntered o n the tra nsmitter . W e study th e system pe rforman ce u nder different link densities by varying the nu mber o f links in the squar e f rom 1 to 200 in o ur simulations. The simulatio ns are car ried ou t based o n the 802.1 1b protoco l. T he com mon phy sical lay er lin k rate is 11 M bps . The p acket size is 1 460 Bytes. The minimum and maximu m backoff window C W min and C W max are 31 a nd 102 3, respec- ti vely . The slot time is 20 µs . The SIFS and DIFS are 10 µs and 50 µs , resp ectiv ely . The transmit power P t is set as 100 m W . The path-loss expon ent α is 4 , the SIR requirement γ 0 is 0 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 average number of links per unit area spatial reuse IPCS Traditional CS theorectical result (optimal) 0 1 2 3 4 5 6 7 throughput per unit area (Mbps) Fig. 5. Spatial reuse and networ k throughput under IPCS and the con ventio nal CS m echani sms 20 , and the correspon ding Safe- C S R cumulative equals 1 17 . 6 m based on (6). That is, th e c arrier-sensing power thr eshold P th = P t ( Safe- C S R cumulative ) − α = 5 . 2 3 × 10 − 7 mW . In Fig. 5, we plot spatial reuse an d network throughpu t under IPCS and the conventional CS me chanisms. Simulation results show that network throug hput is pro portion al to spatial reuse. So we plot these two results in the same figu re. W e define a “u nit ar ea” as the “consum ed area” of each “activ e” transm itter und er th e tig htest p acking. Giv en Sa fe- C S R cumulative = 117 . 6 m , accor ding to the ca rrier-sensing range analy sis, the “un it area” is √ 3 2 Safe- C S R 2 cumulative = 1 . 197 × 10 4 m 2 . Th e x-axis is the a verage nu mber of link s (i.e., all lin ks, includ ing active and inactive links) per un it area as we vary th e total num ber of links in the whole square. That is, th e x-axis co rrespond s to the link den sity of the network. The le ft y-axis is the spatial re use, o r the average “active” link density in the network. The optimal value o f th e spatial reuse is 1 , wh ich is shown as a dashed line in Fig. 5. The right y-axis is th e throu ghput per unit area. It is clear fr om Fig. 5 th at IPCS outperfo rms the co n ven- tional CS. T he improvement beco mes mo re sign ificant wh en the network becomes denser . At th e densest point in the figu re, spatial reu ses u nder I PCS an d co n ventional CS are 0 . 9424 an d 0 . 5834 , respectively . The network thr oughp uts per unit ar ea are 6 . 66 M bp s an d 4 . 08 M bps , resp ectiv ely . Using con ventio nal CS as the base line, the IPCS improves spatial reuse and network thro ughp ut b y mo re than 60 % . Under the conventional CS, in order to make sure the cu- mulative d etected power is no larger than the p ower threshold P th , the packing of concurrent tr ansmission links will become more and more sparse as add itional n umber of links attempt to tran smit. Un der IPCS, this do es not occur . As a result, the improvement in spatial reuse is more significant as the network becomes d enser . 8 W e also find that wh en the ne twork becomes den ser and denser, spatial reuse und er IPCS b ecomes very close to the theoretical result. The small gap is likely due to th e fact that a link w hich could b e activ e concu rrently und er IPCS does not exist in the g iv en topolog y . The pr obability of th is happen ing decreases as the network b ecomes denser . V I . C O N C L U S I O N In this paper , we derive a threshold on the safe carrier- sensing ran ge that is sufficient to p revent hidde n-nod e colli- sions under the cum ulativ e interfer ence mod el. W e show that the safe carrier-sensing range required under the cumulative interferen ce mod el is larger than th at requir ed und er th e pairwise interferen ce m odel by a constant multiplicati ve factor . W e propo se a novel carr ier-sensing mechan ism called Incremen tal-Power Carrier -Sensing (I PCS) th at can re alize the safe carrier-sensing r ange co ncept in a simple way . The IPCS checks e very incremen t in the detected p ower so that it can separate the d etected power of every concurrent transmitter , and then maps the power pro file to the required distance informa tion. Our simulation results show that IPCS can boost spatial re use an d network through put by more tha n 6 0% relativ e to the conventional carr ier-sensing m echanism in the current 802.1 1 proto col. One future research dir ection is to further tighten the safe carrier-sensing r ange accor ding to the topology informatio n. In this p aper, we have assumed a commo n safe carrier- sensing r ange for all transmitters. Allowing th e carrier-sensing range to vary fr om transmitter to transmitter accordin g to the local network to pologic al s tructures m ay impr ove spatial reuse further . In this p aper, we h av e no t con sidered vir tual carrier sensing (i.e., th e R TS/CTS mod e in 80 2.11). En suring hidden- node free oper ation u nder v irtual carrier sensing is rather complicated even u nder the pairwise interfere nce mode l (see [11] f or d etails.) The study of interfe rence-safe transmission s for v irtual car rier sensing und er the cumulative interference model is a subject for furth er study . A P P E N D I X A T H E N E E D F O R R S ( R E - S TA RT ) M O D E It is shown in [8] that althou gh the c arrier-sensing range is suf ficiently large for th e SINR requ irements of all nodes, transmission failur es can still occu r due to th e “Receiver - Capture effect”. 1 T 1 R max d 2 T 2 R max d cannot carrier sense each other can carrier sense each other Fig. 6. Collisi on due to “Recei ver-Capt ure ef fect” T ake a two-link ca se shown in Fig. 6 as an example. In Fig. 6, d ( T 1 , T 2 ) > C S R and d ( T 1 , R 2 ) < C S R . So the transmitters T 1 and T 2 can n ot carrier-sense each othe r , b ut R 2 can sen se the sign al transmitted from T 1 . Suppo se that C S R is set large enoug h to guaran tee the SINR re quiremen ts on l 1 and l 2 (both the D A T A frames and the A CK fra mes). If T 1 transmits first, then R 2 will h av e sensed the signal of T 1 and the default operatio n in m ost 802. 11 p roduc ts is that R 2 will not attempt to receive the later signal from T 2 , even if th e signal from T 2 is stronger . This will cause the transmission on link l 2 to fail. It is further shown in [8 ] that no m atter how large the car rier-sensing range is, we can always come up with an example that giv es rise to tran smission failures, if the “Receiver - Capture effect” is no t dealt with properly . Th is kind o f collision s c an b e solved with a receiv er “RS (Re-Start) mode”. W ith RS mode, a recei ver will switch to recei ve th e stronger pa cket as long as the SINR th reshold γ 0 for th e later link c an b e satisfied. A P P E N D I X B P R O O F O F T H E O R E M 1 Pr oof: W ith the rece i ver’ s RS mod e, in order to p rev ent hidden- node collisions in 802.1 1 network s, we only need to show that con dition (6) is sufficient to gu arantee the satisfaction of b oth the SIR requir ements (1) and ( 2) o f all the co ncurren t tra nsmission lin ks. Let S C S denote a subset of li nks that are allo wed to transmit concur rently under the S afe- C S R cumulative setting. Con sider any two lin ks l i and l j in S C S , we have d ( T j , T i ) ≥ Safe- C S R cumulative = ( K + 2) d max . Because b oth th e lengths of links l i and l j satisfy d ( T i , R i ) ≤ d max , d ( T j , R j ) ≤ d max , we have th e fo llowing b ased o n th e triangular ine quality d ( T j , R i ) ≥ d ( T j , T i ) − d ( T i , R i ) ≥ ( K + 1) d max , d ( R j , T i ) ≥ d ( T i , T j ) − d ( T j , R j ) ≥ ( K + 1) d max , d ( R j , R i ) ≥ d ( R i , T j ) − d ( T j , R j ) ≥ K d max . W e take th e most conservati ve distance K d max in our interferen ce analysis (i.e., we will p ack the interfer ence links in a tightest manne r g i ven th e S afe- C S R cumulative in ( 6)). Consider any two links l i and l j in S C S . The fo llowing four inequalities are satisfied: d ( T i , T j ) ≥ K d max , d ( T i , R j ) ≥ K d max , d ( T j , R i ) ≥ K d max , d ( R i , R j ) ≥ K d max . Consider any link l i in S C S . W e will show that the SIR requirem ents fo r both the D A T A f rame and the ACK fr ame can be satisfied. W e first consider the SIR r equireme nt of the D A T A fram e. Th e SIR at R i is: S I R = P t d − α ( T i , R i ) P l j ∈S C S ,j 6 = i P t d − α ( S j , R i ) For the recei ved signal power we c onsider the worst case that d ( T i , R i ) = d max . So we have P t d − α ( T i , R i ) ≥ P t · d − α max . (12) 9 T o calculate the cumu lativ e interfere nce power , we consider the worst case th at all the other concu rrent transmission links have th e de nsest pack ing, in which the link lengths of all the othe r concu rrent tran smission links ar e equal to zero. In this case, the links degenerate to nod es. Th e minimum distance between any two links in S C S is K d max . Th e den sest packing of nodes with th e min imum distance req uirement is the hexagon p acking (a s shown in Fig. 7) . If link l j is the fir st layer neighbo r link of link l i , we h av e d ( S j , R i ) ≥ K d max . Thus we have P t d − α ( S j , R i ) ≤ P t ( K d max ) − α = 1 K α · P t d − α max , and there are at most 6 neighb or links in the first layer . If link l j is the second layer n eighbo r link of link l i , we have d ( S j , R i ) ≥ √ 3 K d max . Thus we have P t d − α ( S j , R i ) ≤ P t  √ 3 K d max  − α = 1  √ 3 K  α P t d − α max , and there are at most 12 neigh bor links in the secon d layer . If link l j is the n th layer neighbo r link of link l i with n ≥ 2 , we have d ( S j , R i ) ≥ √ 3 2 n · K d max . Thus we have P t d − α ( S j , R i ) ≤ P t √ 3 2 nK d max ! − α = 1  √ 3 2 nK  α P t d − α max , and there are at most 6 n neigh bor link s in the n th lay er . So the cumulative inte rference power satisfies: X l j ∈S C S ,j 6 = i P t d − α ( S j , R i ) ≤ 6 ·  1 K  α + ∞ X n =2 6 n  2 √ 3 nK  α ! · P t d − α max =6 ·  1 K  α 1 + ∞ X n =2 n  2 √ 3 n  α ! · P t d − α max =6 ·  1 K  α 1 +  2 √ 3  α ∞ X n =2 n  1 n  α ! · P t d − α max =6 ·  1 K  α 1 +  2 √ 3  α ∞ X n =2 1 n α − 1 ! · P t d − α max ≤ 6 ·  1 K  α  1 +  2 √ 3  α 1 α − 2  · P t d − α max (13) = P t d − α max γ 0 , (14) where (1 3) follows from a boun d on Riemann’ s ze ta function and (14) follows from the definitio n of K in (7). According to (12) and (14), we find that th e SIR of the D A T A fram e of link l i at the r eceiv er R i satisfies: S I R = P t d − α ( T i , R i ) P l j ∈S C S ,j 6 = i P t d − α ( S j , R i ) ≥ P t · d − α max P t d − α max γ 0 = γ 0 . This m eans that th e SI R r equireme nt o f the successful transmission of the DA T A fr ame o n link l i can be satisfied. First layer link max Kd i T i R m ax d Second layer link Third layer link Fig. 7. The packing of the interfering links in the worst case The proof that th e SIR requir ement of the A CK frame on link l i can b e satisfied follows a similar pr ocedur e as above. So for any lin k l i in the concur rent tr ansmission link set S C S , condition (6) is sufficient to satisfy the SIR requiremen ts of the successful transmissions of both its D A T A and A CK frames. This mean s that, together with the receiver’ s RS mod e, condi- tion (6) is su fficient for preventing h idden-n ode co llisions in CSMA networks u nder the cu mulative interferen ce mode l. A P P E N D I X C P R O O F O F T H E O R E M 2 Pr oof: Consider any link l i in th e link set L . T ransmitter T i will always do carr ier sen sing except wh en it transmits D A T A fra me or rec ei ves ACK frame . W e show that co ndi- tion (10) is sufficient to prevent hidd en-nod e c ollisions in the following two situations, which cover all the po ssible transmission scenarios: 1) Link l i has m onitored the channe l for at least t packet before its backoff counter reaches zero and it transmits. 2) Link l i finishes a tran smission; th en m onitors the ch an- nel for less than t packet when its backoff counter reaches zero; then it transmits its next p acket. Let us f irst c onsider case 1 ) : W e show that for the links that are allowed to trans- mit simultaneou sly , the separation b etween any pair of transmitters is no less than the safe carrier-sensing range Safe- C S R cumulative . W e use in ductive pr oof method . Suppo se that befo re l i starts to tr ansmit, there are a lready M links transmitting and they ar e collectively denoted b y the link set S C S . Without loss of gener ality , suppose that these M links begin to tra nsmit o ne by o ne, accordin g to th e orde r l 1 , l 2 , · · · , l M . For any link l j ∈ S C S , let t j and t ′ j denote the times when link l j starts to transmit the D A T A frame and the A CK fr ame, respectively . 10 In o ur inductive p roof, by assump tion we ha ve d ( T j , T k ) ≥ Safe- C S R cumulative , ∀ j, k ∈ { 1 , · · · , M } , j 6 = k . (15) W e now sh ow that con dition (15) will still h old after link l i starts its transmission. Before link l i starts its tra nsmission, tran smitter T i monitors the c hannel f or a time period of t packet . So T i at least senses M incremen ts in the car rier-sensing power P C S i ( t ) that happen at time t 1 , t 2 , · · · , t M when the links in S C S start to tr ansmit th eir D A T A frames. Ther e may also be som e increments in the P C S i ( t ) th at hap pen at t ′ 1 , t ′ 2 , · · · , t ′ M if the links in S C S start to transmit the ACK frames bef ore link l i starting it transmission. In the IPCS mechan ism, at least the following M in equalities must be satisfied if T i can start its transmission: ∆ P C S i ( t j ) ≤ P th , for j = 1 , · · · , M . Because ∆ P C S i ( t j ) = P t d ( T i , T j ) − α , P th = P t ( Safe- C S R cumulative ) − α , we have d ( T i , T j ) ≥ Safe- C S R cumulative for j = 1 , · · · , M . Thus, we have shown that the separation be tween any pair of transmitters in the link set S C S ∪ l i is no less than Safe- C S R cumulative after link l i starting transmission. Now let u s consider case 2 ) : Before starting the tran smission of the ( m + 1) th packet, link l i first finishes the transmission of the m th pa cket (from time t i ( m ) to t i ( m ) + t packet ), and waits f or a DIFS plus a backoff time (from time t i ( m ) + t packet to t i ( m + 1) ). L et S C S denote th e set of links th at are tran smitting when l i starts the ( m + 1) th packet at time t i ( m + 1) . Consider any link l j in set S C S . Because the tra nsmission time of every p acket in the network is t packet . W e kn ow th at the start time t j of the concur rent tr ansmission o n lin k l j must range fr om t i ( m ) to t i ( m + 1) , i.e ., t i ( m ) < t j < t i ( m + 1) . If t i ( m ) + t packet < t j < t i ( m + 1) , this means t j is in the DIFS or th e backoff time o f link l i . During this p eriod, transmitter T i will d o carrier sen sing. T he I PCS mechanism will make sure that th e distance between T i and T j satisfies d ( T i , T j ) ≥ Safe- C S R cumulative . If t i ( m ) < t j < t i ( m ) + t packet , this means t j falls into the transmission time of the m th packet of link l i . Du ring the tr ansmission time, T i is n ot able to do carrier sensing because it is in the process of transmitting the D A T A fram e or receiving the ACK frame. Howe ver, the transmitter T j will do carrier sensing before it starts to transmit at time t j . The carrier sensing don e by T j can make sure tha t the d istance between T i and T j satisfies d ( T i , T j ) ≥ Safe- C S R cumulative . So for any link l j in S C S , w e h av e d ( T i , T j ) ≥ Safe- C S R cumulative . R E F E R E N C E S [1] G. Brar , D. M. Blough, and P . Santi, “Computati onally ef ficien t schedul- ing with the physical interfer ence m odel for throughput improv ement in wireless mesh networks” , in Pr oc. ACM Mobic om , 2006. [2] S. A. Borbash and A. 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