Enhanced Markov Chain Model and Throughput Analysis of the Slotted CSMA/CA for IEEE Under Unsaturated Traffic Conditions

Transcription

1 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 1, JANUARY Enhanced Markov Chan Model and Throughput Analyss of the Slotted CSMA/CA for IEEE Under Unsaturated Traffc Condtons Chang Yong Jung, Student Member, IEEE, Ho Young Hwang, Member, IEEE, Dan Keun Sung, Senor Member, IEEE,and Gang Uk Hwang, Member, IEEE Abstract In ths paper, we propose an analytcal Markov chan model of the ted carrer-sense multple-access/collson-avodance (CSMA/CA) protocol for IEEE under unsaturated traffc condtons. Our proposed Markov chan model reflects the characterstcs of the IEEE medum-access control (MAC) protocol, such as a superframe structure, nowledgements, and retransmssons wth and wthout lmt. We evaluate the throughput performance of the ted CSMA/CA and verfy the analytcal model usng smulaton results. Index Terms Carrer sense multple access/collson avodance (CSMA/CA), IEEE , medum-access control (MAC), throughput. I. INTRODUCTION As wreless sensor networks (WSNs) have been wdely deployed n our lves, connectng sensor s n a smple and effcent manner wth low power and low cost has become an mportant ssue. The IEEE standard for low-rate wreless personal area networks (LR-WPANs) has been ntroduced to acheve these requrements [1. IEEE networks can operate n ether a beacon-enabled mode or a nonbeacon-enabled mode. In the nonbeacon-enabled mode, s n a personal area network (PAN) communcate wth each other accordng to an unted carrer-sense multple-access/collsonavodance (CSMA/CA) protocol. On the other hand, n the beaconenabled mode, s communcate wth each other accordng to a ted CSMA/CA protocol based on a superframe structure. Each superframe conssts of an actve perod and an nactve perod. The actve perod conssts of a beacon perod, a contenton access perod (), and a contenton-free perod (CFP). Durng the nactve perod, the coordnator and the s shall not nteract wth each other and may enter a low-power mode. To nvestgate the ted CSMA/CA protocol of IEEE , we consder the beacon-enabled mode n ths paper. Durng the n the beacon-enabled mode, IEEE has adopted a ted CSMA/CA protocol, whch s dfferent from that of the IEEE wreless local area network (WLAN) [2, [3. In the Manuscrpt receved July 24, 2007; revsed January 17, 2008 and March 20, Frst publshed Aprl 18, 2008; current verson publshed January 16, Ths work has been supported n part by a grant Next Generaton PC Project from the Insttute of Informaton Technology Assessment (IITA) and n part by the Korea Research Foundaton Grant funded by the Korean Government (KRF D00181). The revew of ths paper was coordnated by Prof.J.L. C. Y. Jung and D. K. Sung are wth the School of Electrcal Engneerng and Computer Scence, Korea Advanced Insttute of Scence and Technology, Daejeon , Korea (e-mal: cyjung@cnr.kast.ac.kr; dksung@ee.kast. ac.kr). H. Y. Hwang s wth the Department of Electrcal and Computer Engneerng, Unversty of Waterloo, Waterloo, ON N2L 3G1, Canada (e-mal: hyhwang@bbcr.uwaterloo.ca). G. U. Hwang s wth the Department of Mathematcal Scences, Korea Advanced Insttute of Scence and Technology, Daejeon , Korea (e-mal: guhwang@kast.edu). Color versons of one or more of the fgures n ths paper are avalable onlne at Dgtal Object Identfer /TVT case of the WLAN, the boff count value decreases by one only f the channel s dle; otherwse, t s frozen. On the other hand, n the case of IEEE , the boff count value decreases by one, regardless of whether the channel s dle or busy. Thus, we need to make a new analytcal model for the CSMA/CA algorthm that s used n IEEE There have been a few papers regardng ths techncal ssue. Park et al. [4, Tao et al. [5, Polln et al. [6, and Lee et al. [7 proposed analytcal Markov chan models for the ted CSMA/CA of the IEEE medum-access control (MAC) protocol. They assumed that the duraton of a s nfnte wthout consderng the superframe structure under saturated traffc condtons. Snce most applcatons n wreless sensor networks (WSNs) are expected to operate under unsaturated traffc condtons, and they use superframes, ncludng nactve perods, to reduce power consumpton, we need an analytcal model consderng unsaturated traffc condtons and the superframe structure. From ths pont of vew, Ramachandran et al. [8 proposed an analytcal model of the ted CSMA/CA under unsaturated traffc condtons; however, they dd not consder the nactve perod n the superframe structure. M sć and M sć [9 also proposed a Markov chan model to analyze the ted CSMA/CA algorthm under unsaturated traffc condtons consderng the superframe structure, nowledgements, and retransmsson schemes of the IEEE MAC protocol. However, they assumed that there was no retransmsson lmt, and they dd not verfy ther analytcal model. Park et al. [4 showed that the model of M sćandm sć dd not match the smulaton results under saturated traffc condtons. In ths paper, we propose an enhanced Markov chan model to observe the throughput performance of the ted CSMA/CA algorthm under unsaturated traffc condtons, and the proposed Markov chan model reflects the characterstcs of the IEEE MAC protocol, such as a superframe structure, nowledgements, and retransmsson schemes wth and wthout a retry lmt. Through smulatons, we verfy that the analytc results from our model are well matched wth the smulaton results. In addton, we show that our model provdes more accurate results than the model of M sćandm sć[9. The rest of ths paper s organzed as follows. In Secton II, we brefly descrbe the IEEE MAC protocol. In Secton III, we propose an analytcal model of the ted CSMA/CA for the IEEE MAC protocol and evaluate the throughput performance of the ted CSMA/CA. Fnally, we present conclusons n Secton IV. II. OVERVIEW OF THE IEEE MAC PROTOCOL At the start of each superframe, the PAN coordnator transmts a beacon frame that carres system parameters, such as a beacon order (BO) that determnes the length of a beacon nterval (BI = BO symbols) and a superframe order (SO) that determnes the length of a superframe duraton (SD = SO symbols). In the perod, each communcates wth the PAN coordnator and other s usng the ted CSMA/CA. The duraton of one s auntboffp erod (default value =20symbols).When a has a new data frame to transmt, t ntalzes relevant parameters such as a boff exponent (BE) and the number of boffs or boff stages (NB), whch are set to macmnbe (default value =3) and 0, respectvely. In addton, t unformly selects a boff counter value from a wndow [0, 2 BE 1. The boff counter value s decremented by one for each tme, regardless of the channel state, and whenever the boff counter value s zero, the performs carrer sensng that requres two clear channel assessments (CCAs) at the physcal (PHY) layer before a transmsson. If the channel s assessed /$ IEEE

2 474 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 1, JANUARY 2009 to be dle at the two consecutve CCAs, then t transmts the data frame. If the channel s assessed to be busy, t ncreases the values of BE and NB by one and delays the transmsson for a random number of tme s that are unformly chosen from [0, 2 BE 1, wherebe s no more than amaxbe (default value =5). The above procedure s contnued untl the successful transmsson; however, f the NB value s greater than macmaxcsmaboffs (default value =4), then the CSMA/CA algorthm shall be termnated wth a channel access falure. If ether a channel access falure occurs or a frame transmsson falure occurs due to a collson, the retres the aforementoned procedure for retransmssons up to amaxf rameretres (default value =3) tmes. Snce the transmsson of an Ack frame shall commence at a boundary, the duraton t from the recepton of the last symbol of the data frame to the transmsson of the frst symbol of ts Ack frame s between at urnaroundt me (default value =12symbols) and at urnaroundt me + auntboffp erod (32 symbols), that s, one tme can be ncluded n the duraton t at most. After transmttng a data frame, the wats for ts Ack frame durng the duraton that s specfed by the parameter macackw atduraton (default value =54symbols). III. ANALYSIS OF THE SLOTTED CSMA/CA IN THE IEEE MAC PROTOCOL A. Markov Chan Models In ths paper, we consder a sngle-hop wreless network consstng of a PAN coordnator and n sensor s. We assume that all s are wthn the transmsson range of each other and tme-synchronzed by the PAN coordnator s beacon. We also assume that there are no transmsson errors and no channel sensng errors. We consder a star topology and an uplnk data transmsson scenaro so that transmtted frames can be lost only due to collsons. Note that WSNs mght have a dfferent topology other than a star topology; however, consderng a general topology s beyond the scope of ths paper. We assume that the data frame arrves at each accordng to a Posson process wth rate λ and that each can store a sngle data frame. Thus, when a has a data frame to transmt, t cannot accept any more new data frames from upper layers. We further assume that all n s are homogeneous, and accordngly, the performance of all s s dentcal. For ths reason, we tag an arbtrary and call t the tagged. For the analyss, we construct a dscrete-tme Markov chan, whch models the operaton of the CSMA/CA algorthm n the tagged and captures the key characterstcs of the IEEE MAC protocol such as a superframe structure, nowledgements, and retransmsson schemes wth and wthout a retry lmt. For convenence, when we construct the Markov chan, we consder only the tme epochs where the states of the Markov chan defned below are changed. The state transton dagram of the Markov chan s gven n Fg. 1. The states of the Markov chan at tme t are classfed nto three types. The frst type s of the form {s(t),b(t),w(t),r(t)}. Here, s(t) [0,m represents the value of NB at tme t, where m = macmaxcsmaboff. b(t) represents the value of the boff counter at tme t.whens(t) =, b(t) s n [0,W 1,where W 0 =2 macmnbe,andw = W 0 2 mn(,amaxbe macmnbe), 1 m. w(t) {1, 2} represents the remanng number of CCAs to be done for a transmsson at tme t. Thus, f b(t) =0and w(t) =2, then the tagged performs the frst CCA at tme t. Smlarly, f b(t) =0and w(t) =1, the tagged performs the second CCA at tme t. P I and P II n Fg. 1 denote the probabltes that the channel s dle when performng the frst CCA and the second CCA, respectvely. Fnally, r(t) [0,R represents the value of the retransmsson plane at tme t, whch s shown by a rectangular box n Fg. 1, where R = amaxf rameretres. Note that our model can smply be appled to the retransmsson scheme wthout lmt f R tends to. The second type s of the form {T nd s(t),r(t)} and {T d s(t),r(t)}. These states represent the nondeferred and deferred transmssons n the s(t)th boff stage on the r(t)th retransmsson plane at tme t. Snce we consder the superframe structure wth a fnte and an nactve perod, note that the tagged needs to defer a transmsson untl the start of the next when the transmsson cannot be completed wthn the current. c and P d c n Fg. 1 denote the collson probabltes for nondeferred transmssons and deferred transmssons, respectvely. The thrd type s of the form {D s(t),r(t)}. These states represent the watng state n the s(t)th boff stage on the r(t)th retransmsson plane at tme t n order for the tagged to defer the transmsson untl the next due to the l of the remanng s n the current. P d s the probablty that a deferred transmsson occurs. In Fg. 1, we have, n fact, one more state { 1, 1, 1, 1} that represents the state where the tagged has no data frame to transmt. The transton probablty α s the probablty that a new data frame occurs between state transton tmes at the tagged. Now, to complete the constructon of the Markov chan n Fg. 1, we need to determne the probabltes P I,P II,Pc nd,pc d,p d, and α. As shown below, we can drectly determne the probabltes P d and α, but t seems to be dffcult to drectly determne P I,P II,Pc nd,andpc d. To solve ths problem, assumng that P I,P II,Pc nd, and Pc d are ndependent of the boff stages and retransmsson planes, we obtan expressons of P I,P II,Pc nd,and Pc d n terms of the steady-state probabltes of the Markov chan and numercally solve them. The detals are summarzed n the followng sectons. B. Probabltes P d and α To compute the probablty P d, note that the total number of tme s that are needed for a sngle transmsson s 2+Ndata + N t +. Here, two s are ncluded due to the number of tme s N for performng two CCAs, Ndata s the number of tme s for the transmsson of a data frame, Nt s the number of tme s for the perod t,andn s the number of tme s for the transmsson of an Ack frame. Snce the transmsson s deferred due to the l of the remanng tmes n a, P d s approxmated by P d N def 2+N = N data + N t + N N where N s the total number of tme s n a. Next, α s the probablty that a new data frame occurs between state transton tmes at the tagged. Snce the data frame arrves accordng to a Posson process wth rate λ, α s expressed as α = N 1 N T 0 λe λτ dτ + 1 N T o 0 λe λτ dτ (1) where T o = T + T CFP + T Inactve + T BEP. Here, T, T CFP, T Inactve, and T BEP are the duratons of one tme, a CFP, an nactve perod, and a beacon perod, respectvely. Note that (N 1)/N s the probablty that an arbtrary tme s not the last tme of a.

3 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 1, JANUARY Fg. 1. Markov chan model. C. Probabltes Pc nd and Pc d Let b,j,k,l = lm t P { s(t) =, b(t) =j, w(t) =k, r(t) =l}, b T nd,l = lm t P {T s(t) nd = T nd,r(t) =l}, b T d,l = lm t P {T d s(t) = T d,r(t) =l}, andb D,l = lm t P {D s(t) = D,r(t) =l}, where [0,m, j [0,W 1, k [1, 2, and l [0,R, be the statonary probabltes of the Markov chan. We assume that the probabltes P I, P II, Pc nd,andpc d are ndependent of the boff stages and the retransmsson planes. Then, some transton probabltes n Fg. 1 can be expressed as b T d,l = P db,0,2,l, [0,m, l [0,R b T nd,l = C 1b,0,2,l = P I P II (1 P d )b,0,2,l [0,m,l [0,R b +1,0,2,l = C 2 b,0,2,l =(1 P I P II )(1 P d )b,0,2,l [0,m 1, l [0,R b 0,0,2,l+1 = C 3 b 0,0,2,l [ (P ) m nd = c C 1 + Pc d P d C2 + C m+1 2 l [0,R 1 =0 b 0,0,2,l where C 1 represents the transton probablty from the state where the tagged performs the frst CCA to the state where the tagged performs a nondeferred transmsson, and C 2 and C 3 represent the transton probabltes between the boff stages and the retransmsson planes, respectvely. Let τ nd and τ d be the statonary probabltes that the tagged carres out nondeferred and deferred transmssons, respectvely. Then, they satsfy τ nd = τ d = R m l=0 =0 R l=0 =0 b T nd,l I m b T d,l. Let τ be the statonary probablty that the tagged transmts a data frame. Then, t follows that τ = τ nd + τ d. The collson probabltes c and P d c for nondeferred and deferred transmssons of the tagged, respectvely, are expressed as c P d c =1 (1 τ nd ) (n 1) C 1 (2) =1 (1 P d_tx )(n 1) (3)

4 476 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 1, JANUARY 2009 where τ nd /C 1 n (2) s the condtonal probablty that the tagged performs the frst CCA, gven that a nondeferred transmsson occurs. P d_tx n (3) s the probablty that the tagged defers the transmsson n a. To compute P d_tx, note that τ dn /Nstate represents the average number of deferred transmssons n a at the tagged, where Nstate s the average number of tme s stayng n an arbtrary state of the Markov chan n Fg. 1. Nstate s gven by N state =(1 τ) 1+τ(1 P c )(N data + N t + N ) + τp c (N data + N tmeout) where Ntmeout s the number of tme s untl the Ack tmer expres due to no Ack. Snce there s at most one deferred transmsson n a for the tagged, we have P d_tx Nstate = τ dn. (4) From (2) and (3), the overall collson probablty P c for the tagged s expressed as P c = c D. Probabltes P I and P II τ nd τ d + τ nd + P d c τ d τ d + τ nd. To obtan the probabltes P I and P II that the channel s dle at the frst and second CCAs, respectvely, we need to compute the average number of busy and dle tme s n a from the vewpont of the network or the channel. The average number of busy s n a s N b_ = [ ( N s_tx net N data + N ) + N c_tx net N data where N s_tx net and N c_tx net are the average numbers of successful transmssons and collded transmssons n the network except for the tagged n a, respectvely, and can be expressed as N s_tx net N c_tx =(n 1) [ N nd_tx d_tx c n 2,0 +N P d c n 2,0 net =(n 1) [ N nd_tx n 2 =1 c n 2, +1 +N d_tx n 2 =1 P d c n 2, +1 where N nd_tx s the average number of nondeferred transmssons for a nontagged n a and s gven by N nd_tx = τ nd N /Nstate, and N d_tx s the average number of deferred transmssons for a nontagged n a and s gven by N d_tx = τ d N /Nstate, whch s the same as P d_tx n (4). In addton, c n 2, s the condtonal collson probablty that, under the gven condton that one among n 1 nontagged s carres out a nondeferred transmsson, s among n 2 other nontagged s carry out nondeferred transmssons and +1s collde accordngly and s expressed as c n 2, = ( n 2 (5) (6) (7) ) ( τnd C 1 ) ( 1 τ nd C 1 ) n 2. In the same manner, P d c n 2, for deferred transmssons s expressed as ( ) n 2 P d c n 2, = P d_tx (1 P d_tx )n 2. Snce c n 2,0 n (6) s the probablty that a nondeferred transmsson of one among n 1 nontagged s s successful, c n 2,0 n (6) s the average number of successful nondeferred transmssons for the network except the tagged n a. Regardng (7), we consder the case that +1 nontagged s collde. From the vewpont of the network or the channel, t s consdered as a sngle collded transmsson. Therefore, we can say that each of the +1nontagged s that s nvolved n the collson contrbutes the amount of the 1/( +1)porton to the sngle collded (n 1)N nd_tx n 2 transmsson. (n 1)N nd_tx nd (P =1 c n 2, /( +1))n (7) s the average number of nondeferred collded transmssons for the network except the tagged n a. In the same way, (n 1)N d_tx N d_tx n 2 (P d =1 c n 2, P d c n 2,0 n (6) and (n 1) /( +1)) n (7) are the average numbers of successful deferred transmssons and collded deferred transmssons, respectvely, for the network except the tagged n a. Note that the frst and second CCAs for nondeferred transmssons cannot be performed at any tme s of a but can be performed n a lmted range of a. The length of the lmted range s n (5) can nclude the number of busy s n the range where the tagged does not perform the frst CCA due to deferred transmssons. Snce the number of those busy s s very small compared wth that of s that the tagged can perform N N def. N b_ the frst CCA,.e., (N N def ), we assume that ths effect can be neglgble. Therefore, the probablty P I s approxmated as P I ( N N ) def N b_ N N. (8) def Next, we observe that the second CCA can occur when the channel s dle at the frst CCA. Note that N b _ s the average number of s where the channel s dle, but t s busy at the next. Snce those s can just occur before data or Ack frame transmssons, N b _ s expressed as N b _ = [ ( ) N s_tx net 1+N t + N c_tx net 1. Thus, the probablty P II s approxmated as ) N b_ ( P II 1 N N def P I N N def N b _. (9) Fnally, each parameter of P I, P II, Pc nd,andpc d can be numercally solved from (2), (3), (8), (9), and the normalzaton condton of the Markov chan, as mentoned before. E. Throughput Analyss Let S be the system throughput. Then, S s computed as S = nd R τ(1 P c )E[T Payload (1 τ)e [T +τ(1 P c)e[t suc +τp c E[T col where D R s the data rate (n bts per second), E[T Payload s the average duraton of a data frame payload, and E[T suc and E[T col are the average duratons of a successful transmsson and a collded transmsson, respectvely, and are expressed as E[T suc =T N suc E[T col =T N col ( ) = T N data + Nt + N = T ( N data + N tmeout).

5 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 1, JANUARY Fg. 2. Normalzed system throughput for varyng SO and BO values and retransmssons wthout lmt. E[T s the average duraton of a nontransmsson. Note that the transmsson s nclude s of data and Ack frame transmssons and t for successful transmssons, as well as s of data frame transmssons and watng untl the Ack tmer expres for collded transmssons. To compute E[T, we frst consder the average number of transmsson s of the tagged n a, whch s denoted by N tx_. Snce the average number of transmssons of the tagged n a s gven by t follows that = N nd_tx + N d_tx N tx N tx_ = N tx [ (1 Pc )N suc = τn Nstate + P c Ncol. Observng that, among the nontransmsson s, the last n a s followed by a CFP perod, an nactve perod, and a beacon perod, E[T s gven by ( N E [T = N tx_ 1 ) T +1 T o N N tx_ where T o s gven n (1). F. Performance Evaluaton To evaluate the throughput performance of the IEEE MAC protocol, we consder a 2.4-GHz PHY layer and D R = 250 kbps. A data frame conssts of PHY and MAC headers and a MAC payload. Let the sze of data frames L data be 32 bytes, the sze of PHY and MAC headers of data frames be 15 bytes, and the sze of Ack frames L be 11 bytes. The number of s n a network s 10. Each perod s set as T BEP = SO symbols, T = SO symbols, T CFP = SO symbols, and T Inactve = BO SO symbols, respectvely. Fg. 2 shows the normalzed system throughput S/D R when we vary the normalzed offered load per λ n (= λl data /D R ) and the values of SO and BO. In Fg. 2, we consder the case of retransmssons wthout lmt to compare our results wth those of M sć and M sć. Note that we can obtan the results for retransmssons wthout lmt by lettng the maxmum number of retransmssons R go to. Fg. 3. Normalzed system throughput for varyng data szes and retransmssons wth lmt. As shown n Fg. 2, the model of M sć andm sć fals to match the smulaton results as the normalzed offered load λ n ncreases. There are some reasons for the msmatch of the model of M sć and M sć. Frst, they dd not separately obtan the collson probabltes for nondeferred and deferred transmssons. Second, when calculatng the average number of busy s n a to obtan the probablty that the channel s dle at the frst CCA, they dd not consder the fact that there are no nowledgements for collded transmssons, and they dd not properly obtan the average number of transmssons n a n the vewpont of the network. Thrd, when obtanng the probablty that the channel s dle at the second CCA, the condton that the channel s dle at the frst CCA was not consdered n the model of M sć and M sć. These factors affect the naccurate results of the model of M sćandm sć. On the other hand, our model s well matched wth the smulaton results. For the normalzed offered load λ n smaller than 10 2, snce most of the transmssons are successful due to low traffc, the model of M sć andm sć seems to be well matched wth the smulaton results. As the value of BO ncreases wth a fxed SO value, the normalzed system throughput decreases for the value of λ n larger than Ths s because the proporton of a s gettng smaller due to a longer nactve perod under these traffc condtons. On the other hand, for the value of λ n that s smaller than 10 3, there s almost no dfference n the system throughput, regardless of the duraton of a beacon nterval BI due to low traffc. Fg. 3 shows the normalzed system throughput S/D R when we vary the normalzed offered load per λ n and the data sze. In ths case, we consder retransmssons wth lmt (R =3). Let the values of SO and BO be set to 1. As the data sze ncreases, the normalzed system throughput ncreases. Snce most of the transmssons are successful n a low traffc load such as λ n smaller than 10 2,large data frame szes yeld hgh throughput performance. In a hgh traffc load such as λ n larger than 10 2, f a transmts a large data frame, t occupes the channel for a long tme. Ths fact causes a low transmsson probablty and a low collson probablty, whch results n hgh throughput performance. Fg. 4 shows the normalzed system throughput S/D R when we vary the number of s n and the value of the retransmsson lmt R. In ths case, the normalzed offered load per λ n s set to 0.03, the values of SO and BO are set to 1, and the maxmum value of boff stages m s set to 0. As the value of R ncreases, the normalzed system throughput slghtly ncreases for a low traffc load

6 478 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 1, JANUARY 2009 Effcent Sum Rate Maxmzaton and Resource Allocaton n Block-Dagonalzed Space-Dvson Multplexng Boon Chn Lm, Wtold A. Krzymeń, Senor Member, IEEE, and Chrstan Schlegel, Senor Member, IEEE Fg. 4. Normalzed system throughput for varyng the value of the retransmsson lmt R. such as n smaller than 6. Retransmssons decrease the probablty of frame drops. However, for a hgh traffc load such as n larger than 12, the system throughput decreases as the value of R ncreases. A smaller value of R causes earler frame drops. Snce these frame drops reduce the number of contendng s n the network, the collson probablty decreases. Ths yelds hgher system throughput. Abstract For space-dvson multplexng (SDM) va block dagonalzaton on multuser multple-nput multple-output (MIMO) wreless downlnk, t s shown that receve antenna selecton (RAS) s necessary for maxmzng the achevable sum rate. Ths s true even when all receve antennas are equpped wth rado frequency (RF) chans and RAS reduces the upper bound on the broadcast sum capacty, and when the orthogonalzed channels use optmal processng. Smlarly, spatal-mode selecton (SMS) s necessary for sum rate maxmzaton when receve-weght matrces are used for spatal-mode allocaton. RAS/SMS may release transmsson resources that can fully be utlzed va addtonal user schedulng to yeld further sum rate gans. Optmal user selecton for sum rate maxmzaton s subsumed wthn an exhaustve RAS/SMS process for multantenna termnals, and both selecton processes become dentcal for sngle-antenna termnals. RAS/SMS thus helps reduce the performance gap from the optmal sum capacty even for small user pool szes. A block antenna/mode selecton approach s ntroduced to help overcome the drawbs of exstng algorthms. Snce RAS/SMS nvolves antenna/mode rankng, a systematc method for resource allocaton wth sum rate loss mnmzaton s nherently provded. Ths way, a streamlned process that combnes user selecton, RAS/SMS, and resource allocaton s developed for sum rate maxmzaton of block-dagonalzed SDM. Index Terms Block dagonalzaton, MIMO spatal multplexng, multuser MIMO downlnk, receve-antenna selecton, spatal-mode selecton, sum rate maxmzaton. IV. CONCLUSION In ths paper, we have proposed an analytcal Markov chan model of the ted CSMA/CA n the IEEE MAC protocol consderng a superframe structure, nowledgements, and retransmssons wth and wthout lmt under unsaturated traffc condtons. Wth the proposed model, we have evaluated the throughput performance of the ted CSMA/CA. We have valdated our proposed analytcal model by smulaton. REFERENCES [1 Wreless Medum Access Control (MAC) and Physcal Layer (PHY) Specfcatons for Low Rate Wreless Personal Area Networks (LR-WPANs), IEEE Std Specfcaton, Oct. 1, [2 Wreless LAN Medum Access Control (MAC) and Physcal Layer (PHY) Specfcatons, IEEE Std Specfcaton, Jun. 26, [3 G. Banch, IEEE saturaton throughput analyss, IEEE Commun. Lett., vol. 2, no. 12, pp , Dec [4 T. R. Park, T. H. Km, J. Y. Cho, S. Cho, and W. H. Kwon, Throughput and energy consumpton analyss of IEEE ted CSMA/CA, Electron. Lett., vol. 41, no. 18, pp , Sep [5 Z. Tao, S. Panwar, D. Gu, and J. Zhang, Performance analyss and a proposed mprovement for the IEEE contenton access perod, n Proc. IEEE WCNC, Las Vegas, NV, Apr. 3 6, 2006, pp [6 S. Polln, M. Ergen, S. C. Ergen, B. Bougard, L. V. Perre, F. Catthoor, I. Moerman, A. Baha, and P. Varaya, Performance analyss of ted carrer sense IEEE medum access layer, n Proc. IEEE GLOBECOM, San Francsco, CA, Nov. 27 Dec. 1, 2006, pp [7 T. Lee, H. R. Lee, and M. Y. Chung, MAC throughput lmt analyss of ted CSMA/CA n IEEE WPAN, IEEE Commun. Lett., vol. 10, no. 7, pp , Jul [8 I. Ramachandran, A. K. Das, and S. Roy, Analyss of Contenton Access Perod of IEEE , Unv. Washngton, Seattle, WA, UWEE Tech. Rep. UWEETR , Feb [9 J. M sć andv. B. M sć, Access delay for s wth fnte buffers n IEEE beacon enabled PAN wth uplnk transmssons, Comput. Commun., vol. 28, no. 10, pp , Jun I. INTRODUCTION For a wreless base staton (BS) equpped wth M antennas where coordnaton s feasble among the transmt chans but not among the K moble user termnals, smultaneous downlnk transmssons to multple users are possble when channel state nformaton s avalable at the transmtter. The optmal approach for sum rate maxmzaton s drty paper codng (DPC) [1, whch s very complex, and beamformng s a reduced-complexty alternatve. For a system wth N receve antennas per user, the optmal beamformng sum rate scales as for DPC,.e., wth M log log KN [2, when K s very large and schedulng s appled. However, optmal beamformng nvolves sgnal-to-nterference-plus-nose rato balancng and s also qute complex. A suboptmal beamformng method s zero-forcng beamformng (ZFBF), whch enforces zero nterference among spatal layers (streams). For multantenna termnals, t s better to mpose orthogonalty between users only (cancel only multuser nterference), because antennas located at the same termnal can effectvely cooperate. Ths s commonly referred to as block dagonalzaton (BD) [3, [4. Although nferor when compared wth optmal beamformng, the Manuscrpt receved June 11, 2007; revsed December 6, 2007, February 15, 2008, and March 20, Frst publshed May 7, 2008; current verson publshed January 16, Ths work was supported n part by DSO Natonal Laboratores (Sngapore), by TRLabs, by the Roht Sharma Professorshp, and by the Natural Scences and Engneerng Research Councl of Canada. The revew of ths paper was coordnated by Dr. C. Yuen. B. C. Lm and C. Schlegel are wth the Department of Electrcal and Computer Engneerng, Unversty of Alberta, Edmonton, AB T6G 2R3, Canada (e-mal: bclm@ece.ualberta.ca; schlegel@ece.ualberta.ca). W. A. Krzymeń s wth the Department of Electrcal and Computer Engneerng, Unversty of Alberta, Edmonton, AB T6G 2R3, Canada, and also wth TRLabs, Edmonton, AB T6G 2W3, Canada (e-mal: wak@ece.ualberta.ca). Color versons of one or more of the fgures n ths paper are avalable onlne at Dgtal Object Identfer /TVT /$ IEEE