A MAC and Physical Layer Theoretical Model for IEEE a Networks Operating Simultaneously with Basic and RTS/CTS Access Schemes

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1 A MAC and Physical Layer Theoretical Model for IEEE 802.a Networks Operating Simultaneously with Basic and RTS/CTS Access Schemes Roger Pierre Faris Hoefel Department of Telecommunications Engineering - University La Salle Canoas - RS - Brazil roger@lasalle.tche.r Astract We have developed a theoretical cross-layer model that allows assessing the goodput and delay of the IEEE 802.a local area networks (WLANs) operating simultaneously under the distriuted coordination function (DCF) asic access (BA) and request-to-send/clear-to-send (RTS/CTS) medium access control (MAC) protocols under saturated traffic over correlated and uncorrelated fading channels. Resumo. Neste artigo é proposto um modelo teórico que permite analisar de maneira integrada o desempenho das camadas de enlace e física de redes locais sem fio IEEE 802.a operando simultaneamente no modo ásico e no modo RTS/CTS. As expressões analíticas propostas, validadas por meio de simulação da camada física IEEE 802.a, permitem verificar no desempenho do sistema tanto os efeitos dos protocolos de enlace (e.g. da carga no canal, do tamanho dos pacotes, do algoritmo de resolução da janela de contenção) em como os efeitos dos protocolos da camada física (e.g. esquemas de modulação, correção de erros, modelos de canal).. Introductions and Related Work The release of the first IEEE 802. standard (that specifies the MAC and the original slower frequency-hopping and direct sequence PHY layers) in 997 paved the way for a world wide development of a standardized cost-effective scalale technology for WLANs. Since then, intensive research activities have een carried out on analyze, design, implementation, and optimization issues of IEEE 802. networks. In 997, B. P. Crow et al pulished one of the first papers to explain the IEEE 802. protocol (with particular emphasis on the MAC layer) and to show simulation results for packetized data and a comination of packetized data and voice over WLANs [Crow 997]. In 2000, Bianchi proposed an analytical i-dimensional Markov model to estimate the performance of IEEE 802. networks [BIANCHI 2000] operating under saturated traffic conditions over ideal channels (i.e. only collisions were taken into account and the frames were not corrupted due to noise and interference). This Bianchi s model has een used as a framework to analyze others IEEE 802. technologies, as in [Roinson 2004] where it is proposed an analytical model to assess the performance of quality of service (QoS) schemes for IEEE 802. WLANs operating under saturated traffic conditions over ideal channels. In 2002, Qiao et al [Qiao 2002] derived an analytical model that takes the non-ideal channel into account on the performance of IEEE802.a WLANs. However, they assumed a very crude model for the MAC layer. Therefore, due to the properties of the radio channel and the use of advanced PHY layer techniques, there was a necessity to develop an accurate joint MAC and PHY theoretical

2 model to estimate the performance of IEEE802.ased networks. Hence, founded on the methodology proposed y Bianchi in [BIANCHI 2000], we have developed a crosslayer saturation goodput theoretical model that allows assessing the performance of IEEE 802.a WLANs over uncorrelated [HOEFEL 2005] and correlated fading channels [HOEFEL 2006]. We have also noticed a lack in the literature on theoretical analyzes of IEEE 802. WLANS when oth the BA and RTS/CTS access schemes are simultaneously operational, even when it is assumed an ideal PHY layer. Therefore, in the present contriution we have improved our previous theoretical results y: () developing a cross-layer model that allows to estimate the goodput and delay of IEEE 802.a WLANs operating simultaneously under the asic and RTS/CTS access schemes; (2) carrying out a unified comparison etween the performance of IEEE 802.a WLANs over uncorrelated and correlated fading channels when oth access modes are simultaneously operational. The aove contriutions are developed in Sections 4 to 0. We remark that this developed cross-layer methodology can e used, with minor modifications, to examine the performance of other 802. PHY layers (e.g and 802.g) as well as to e used as reference to analyze the effe of the PHY layer on MAC protocols for different systems (e.g. IEEE 802.6). In the next two sections we present meaningful aspe regarding to 802.a MAC (Section 2) and PHY (Section 3) layers. The contriutions of this work are developed in Sections 4 to IEEE 802. MAC The IEEE PHY standards 802., 802.a and 802. use the same MAC layer protocols. To accomplish it, a MAC service unit (MSDU) is segmented into a MAC protocol data unit (MPDU) that, on its turn, it is mapped to the physical layer using a standardized physical layer convergence procedure (PLCP). As the IEEE 802. MAC protocol is widely known, we only show at Figures a and the time diagram for the atomic transmission used y the DCF BA and RTS/CTS mechanisms [Gast 2002]. Notice that DIFS stands for DCF interframe spacing (IFS), PIFS for point coordination IFS, SIFS for short IFS and NAV for network allocation vector. Fig. a. The atomic BA scheme. Fig.. The atomic RTS/CTS scheme. Figure. The atomic cycle for the successful transmission using the atomic asic positive ACK of data and RTS/CTS schemes. 3. IEEE 802. Physical Layer In this paper we have followed the IEEE 802.a PHY layer [IEEE 802.a 999, pp. 6]. The IEEE 802.a is ased on orthogonal frequency division modulation (OFDM)

3 using a total of 52 sucarriers, of which 48 sucarriers carry actual data and four sucarriers are pilots used to facilitate coherent detection. The channel symol rate R s is of 2 Msymols/sec since the OFDM symol interval (ts) is set to 4µs. Ta. shows the OFDM PHY modes, where BpS means Bytes per Symol. For instance, the PHY mode carries 3 ytes per symol, i.e. 6 Mps* tsymol/8.0=3 BpS. Mode Modulation Code Data BpS p Rate R c Rate BPSK /2 6 Mps 3 2 BPSK 3/4 9 Mps QPSK /2 2 Mps 6 4 QPSK 3/4 8 Mps 9 Ta.. The IEEE 802.a PHY modes. Mode Modulation Code Data BpS p Rate R c Rate 5 6-QAM /2 24 Mps QAM 3/4 36 Mps QAM 2/3 48 Mps QAM 3/4 54 Mps 27 Considering the IEEE 802.a convolutional codes generator polynomials, g 0 =(33) 8 and g =(7) 8, of rate r=/2 and constrain length K=7, then the union ound on the proaility of decoding error is given y [Conan 984] Pe ( γ, p ) < P0( γ, p ) + 38 P2( γ, p ) + 93 P4( γ, p ) + () where the used notation emphasizes the dependence of P d with the received signal-tointerference-plus-noise (SINR) per it γ, and the PHY mode p. The union ound on the proaility of decoding error for the higher code rates of 2/3 and 3/4 (which are otained y puncturing the original rate-/2 code), are given y (2) and (3), respectively [Haccoun 989]. Pe ( γ, p) < P6 ( γ, p) + 6 P7 ( γ, p) + 48 P8 ( γ, p) + (2) Pe ( γ, p) < 8P5 ( γ, p) + 3 P6 ( γ, p) + 60 P7 ( γ, p) + (3) Assuming that the convolutional forward error correcting code (FEC) is decoded using hard-decision Viteri decoding, then (4-5) model the proaility of selecting incorrectly a path when the Hamming distance d is even and odd, respectively. The average it error rate (BER) for the PHY mode p modulation scheme is denoted y ρ p. P d d d d k d k, ρ p ( ρp) 2 d 2 k= d + k 2 ( ) d/2 d 2 γ p = p p ( ρp) + d d k d k, ρ p ( ρ p) k = d + k ( ) 2 ( γ p) = (4) (5) The numer of octets of the PHY layer protocol data unit (PDU) for the RTS frame is given y 6 6 N rts = N pre + Nsrv + lrts + Ntail = (6) 8 8 where N pre, N srv and N tail denote the numer of octets of the preamle, service and tail fields, respectively. The numer of octets used to carry the logical control information sent y the RTS, CTS and ACK control frames is laelled y l rts, l and l ack, respectively. The numer of octets of the PPDU that transports the CTS and ACK control frames is given y (7), where l=l =l ack =4 octets. The MAC PCLP PDU (i.e. a PPDU) length is given y (8), where l pl denotes the payload length in octets. The MPDU header and the cyclic redundant checking (CRC) fields have together a size of 34 ytes [GAST 2002] and 6 its are used to flush out the convolutional coder to the zero state N = Nack= Npre+ Nsrv+ l+ Ntail= (7) N mp= Npre+ Nsrv+ Nmh+ Npl + Ntail= lpl + (8) P d

4 The time period spent to transmit a MPDU with a payload of l pl octets over the IEEE 802.a using the PHY mode p mp is given y (9). The length of RTS, CTS and ACK control frames are given y (0-2), respectively. l ( 6+ 6 )/8 l + ( 6+ 6) / 8 pl Tmp( pmp) = tpclp_pre+ tpclpsig _ + ts BpS( pmp) (9) ( 6 6) l + + /8 T( p) = tpclp_pre+ tpclp_ SIG+ ts. BpS ( p) () 4. Analytical Results for the Saturation Goodput T ( p ) tpclp_pre tpclp_ SIG rts rts rts = + + ts BpS ( prts) (0) ( 6 6) l T p tpclp e tpclp SIG ack + + /8 ack( ack) = _Pr + _ + ts BpS ( pack) (2) It is assumed a fixed numer of n STAs operating in saturation conditions, i.e. each STA has a packet to transmit after the completion of each successful transmission. It is also postulated that each packet collides with a constant and independent conditional collision proaility p. The capture effect is neglected in such way that the lost of frames due to collisions is independent of the lost of frames due to noise and interference. The post ackoff procedure has not een taken into account [Roison 2004]. The proaility that a transmitted MPDU is successful depends upon the following events: () no collision and successful transmission using the BA scheme; (2) no collision and successful transmission using the RTS/CTS access scheme. Thus, { P ( S S ) + P ( S S S S )} Ps = ( p) a mp ack rts rts mp ack (3) where S, S rts and S ack denote, respectively, the proaility that the RTS, CTS and ACK control frames e transmitted with success. S mp denotes the proaility the transmission of a MPDU frame is successful. Eq. (4) and (5) model the proaility that a MPDU is transmitted using the asic access and RTS/CTS schemes, respectively. Notice that these proailities depend upon the MAC payload length l pl and the RTSThreshold defined at the Management Information Base (MIB). = Pr{ l RTSThreshold} (4) Prts Pr{ lpl RTSTheshold} Pa pl < = (5) Eq. (6) details the events that cause an unsuccessful MPDU transmission: () collision; (2) no collision, ut a corrupted frame on the asis access scheme (see 7); (3) no collision, ut a corrupted frame on the RTS/CTS access scheme (see 8). ( p) ( P P + P P ) Pf = p + a f,a rts f,rts (6) = - Smp Sack Pf, a = (- Smp ) + Smp (- Sack ) [(- S ) + S ( S ) + S S (- S ) + S S S (- S )] Pf,rts = rts rts = - Srts S Smp Sack rts mp Fig. 2 shows a discrete i-dimensional Markov chain (s(t),(t)) for the ackoff window size. It is assumed that: () a MPDU is transmitted using the asic access scheme with proaility P a. ; (2) a MPDU is transmitted using the RTS/CTS scheme with proaility P rts ; (3) s(t) is the stochastic process of the ackoff stage (0,,m) of the STA at time t, where m denotes the numer of ackoff stages; (4) (t) is the random rts mp ack (7) (8)

5 process that models the ackoff time counter for a given STA; (5) the contention window (CW) size at ackoff stage i is laeled as W i = 2 i W, where i (0,m) is the ackoff stage and W is the MAC CW size parameter CW min The maximum CW size is denoted as W m =2 m W =CW max -. (-p) {P a (S mp S ack ) + P rts (S rts S S mp S ack ) } 0, 0 0, 0, W 0 -, , ', W - [p+(-p) (P a P f,a + P rts P f,rts ) ]/ W m- m cw, 0 m cw,,... m cw, W mcw - [p+(-p) (P a P f,a + P rts P f,rts ) ]/ W [p+(-p) (P a P f,a + P rts P f,rts ) ]/ W m - Figure 2. Bi-dimensional Markov chain (s(t),(t)) model for the ackoff window size and a non-ideal channel. 4.. Packet Transmission Proaility Using the notation P{i,k /i 0,k o }= P{s(t+)=i,(t+)=k s(t)=i o,(t)=k o }, equations (9) to (22) model the null one-step transition proailities of the i-dimensional Markov chain depicted at Fig. 2. The decreasing of the ackoff timer at the eginning at each slot time of size σ is modeled as { i,k i,k + } = for k ( 0,W 2) and i ( 0,m) P i (9) Eq. (20) takes into account that a new PPDU starts at the ackoff stage 0 and that the ackoff is uniformly distriuted into the range (0, W 0 -) after a successful PPDU transmission. ( Smp Sack ) + ( S S S S ) Pa P{ 0,k i, 0} = [ p] / W0 for k ( 0,W0 ) and i ( 0,m) (20) Prts rts mp ack The property that a new ackoff value is uniformly chosen in the range (0,W i ) after an unsuccessful transmission at the ackoff stage i- can e modeled as P Pa ( Smp Sack ) + { }/ Wi for k ( 0,Wi ) and i (,m) (2) Prts ( Srts S Smp Sack ) { i,k i, 0} = p + [ p] Eq. (22) models the fact that the ackoff is not increased in susequent frame transmissions once the ackoff stage has reached the value m. P { m,k m, } = p + [ p] { [ P ( S S ) + P ( S S S S )] }/ W fork ( 0,W ) 0 a mp ack rts rts mp ack m mi (22) The transmission occurs when the ackoff timer counter is equal to zero. Therefore, using an algeraic procedure similar to the one developed in [Hoefel 2005], then we can show that the proaility that a STA transmits in a randomly chosen slot time is given y (23), where, the stationary proaility of a given STA e in the time

6 slot 0 for first contention window is given y (24). The average frame success transmission proaility is denoted y (25). m 0, 0 τ = i, = (23) ( p) P S S + P S S S S 0,0 i= 0 0 = m + 2 W [ ( ) ( )] a mp ack rts rts mp ack 2 ( p) S [ + 2 S ( p ) ] m m [ + ( + p) S] + ( + p) S 2+ W+ 2 W [ + ( + p) S] ( S S ) + P ( S S S S ) { m } S = Pa mp ack rts rts mp ack (25) 0,0 Notice that for an ideal channel (i.e. S rts =S = S mp = S ack =), Eq. (24) resumes to 2( 2 p) ( p) m ( 2 p) ( W + ) + pw ( 2 p) [ ] = (26) which is in agreement with (6) of [Bianchi 2000]. Each STA transmits with proaility τ. Therefore, the conditional proaility that a transmitted PPDU encounters a collision in a given slot time can e stated as n p = ( τ ). (27) The nonlinear system represented y (23) and (27) can e solved using numerical techniques. 4.2 Goodput The goodput (net throughput) in its per second (ps) can e modeled as the ratio of the payload its transmitted with success to the average cycle time, i.e. P N P N G a a + rts rts ps = (28) Pa Ta + Prts Trts The average numer of payload its transmitted with success for the BA and RTS/CTS schemes are given y (29) and (30), respectively. The numer of payload octets when is used the BA and RTS/CTS schemes are given y l pl,a and l pl,rts, respectively. Na =8 lpl,a Ps Ptr Smp Sack (29) Nrts =8 lpl,rts Ps Ptr Srts S Smp Sack (30) The proaility that there is no collision on the channel conditioned to the fact that at least one STA transmits is given y n n τ ( τ ) n n n τ ( τ ) n τ ( τ ) P s = = = P n tr Ptr ( τ ) where P tr is the proaility that there is at least one transmission in the considered slot time. The average cycle time for the asic access and RTS/CTS scheme is given y (32) and (33), respectively. T a = Bs,a + B f,a + B f 2,a + B f 3,a + I (32) Trts = Bs,rts + Bf,rts + Bf 2,rts, + Bf 3,rts + Bf 4,rts + Bf 5, rts + I (33) For the asic access scheme, the average usy time when the atomic positive ACK asic access transmission is successful is given y ( DIFS + T ( p ) + a + SIFS + T ( p ) a) Bs,a = Ps Ptr Smp Smac mp mp ack ack + (34) (24) (3)

7 where a is the propagation delay. T mp (p mp ) is the time period necessary to transmit a MPDU when it is used the PHY mode p mp (see 9), and T ack (p ack ) is the time period spent to transmit a positive ACK control frame using the PHY mode p ack (see 2). The average usy time when the transmission is successful using the RTS/CTS scheme is given y B = P P S S S S [ DIFS + T ( p ) + a + SIFS + s,rts s tr T ( p rts mp mac ) + a + SIFS + T mp ( p mp rts rts ) + a + SIFS + Tack ( p ack where T rts (p rts ), T (p ) and denote the time necessary to transmit the RTS and CTS control frames when it is used the PHY mode p rts and p, respectively (see 0 and ). B f, a models the average amount of time that the channel is usy due to collisions of a MPDU frames when it used the BA scheme (see 36). However, for the RTS/CTS MAC scheme the waste time occurs due to RTS control frame collisions, as modeled y B f, rts (see 37). B = P ( P ) ( DIFS + T ( p ) a) ( P ) ( DIFS + T ( p ) a) B f,a = Ptr s mp mp + (36) ) + a] (35) f,rts tr s rts rts + (37) B f 2, a and B f 3, a model the average lost time due to an unsuccessful transmission of data and ACK control frames, respectively, when it is used the BA scheme. ( S ) ( DIFS + T ( p a) B f 2, a = Ptr Ps mp mp mp ) + (38) DIFS + T ( ) 3, ( ) ( ) mp pmp + a + B f a = Ptr Ps Smp Sack SIFS + Tack pack + a (39) B f 2, rts, B f 3, rts, B f 4, rts and B f 5, rts model the average time that the channel is usy with unsuccessful transmissions due to noise and interference, of RTS, CTS, data and ACK frames, respectively, when it is used the RTS/CTS access scheme. Finally, the average time that a slot time is idle is given y (39), where s is the slot time length. B ( S rts ) [ DIFS + Trts ( prts ) ]. S ( S )[ DIFS + T ( p ) + a + SIFS + T ( p ) a]. f 2, rts Ptr Ps + a = (40) Bf 3,rts = Ptr Ps rts rts rts + (4) B f 4,rts = Ptr Ps S rts S ( S ) [ mp Trcs ( p ) + a + SIFS + Tmp ( p mp ) + ( S )[ DIFS + Trts ( prts ) + a + SIFS + a] B f 5,rts = Ptr Ps S rts S mp ack DIFS + Trts ( prts ) + a + SIFS + (43) Trcs ( p ) + a + SIFS + Tmp ( pmp ) + a + SIFS + Tack ( pack ) + a] I = tr σ. (44) ( P ) 5. Analytical Results: Average Delay The average delay etween the time that a MPDU arrived at the queue until the time that an ACK for this MPDU is received can e modeled as the average numer of cycle times necessary to accomplish a successful MPDU transmission, as modeled y (45). The average cycle time for the BA and RTS/CTS schemes is given y (32) and (33), respectively. The average numer of time slots spent for a successful transmission is given y (46), where the average proaility that an atomic transmission is not successful, P f, is given y (6). (42)

8 D = P Pa T a+ Prts T rts m W i i (45) P ( P ) i = ( P ) i= 0 f 2 + i= m f Wm 2 (46) 6. Analytical Results for Frame Success Proaility over Uncorrelated Fading Channels In order to have a unified analytical framework upon the modeling of fading effe on the MAC protocols we have summarized in this Section the theoretical results developed in [Hoefel 2005]. We elieve that this approach is time saving to the reader. Postulating that the errors inside of the decoder are interdependent, then the upper ound for a successful transmission of a frame with l octets is given y (47) [Puersley 987]. Notice that the union ound on the proaility of decoding error is given y: () Eq. () for the PHY modes, 3 and 5; (2) Eq. (2) for the PHY mode 7; (3) Eq. (3) for the PHY modes 2, 4, 6 and 8. S( l, γ 8 [ P ( γ, p ] l, p ) < e ) (47) The PCLP header is always transmitted using PHY mode. Then, for an uncorrelated multipath fading channel, the successful MPDU transmission when it is used the PHY mode p mp is given y (48). Correspondingly, the RTS, CTS and ACK control frames success proailities are given y (49), (50) and (5), respectively. Smp(l mp, γ,pmp) = S( 24 / 8, γ, )S( 34+ ( 6+ 6)/ 8+ lmp, γ,pmp) (48) Srts( lrts, γ, ρrts ) = S( 24 / 8, γ, ) S( 20 + ( ) / 8, γ, ρrts ) (49) S (l, γ, ρ ) = S( 24 / 8, γ, ) S( 4 + ( 6 + 6)/ 8, γ, ρ ) (50) Sack(lack, γ, ρack ) = S( 24 / 8, γ, ) S( 4 + ( )/ 8, γ, ρack) (5) In this susection we have assumed a flat fading Nakagami-m channel temporally independent at symol level and independent across of the OFDM carriers. Therefore, the average BER for the PHY mode p can e stated as ρ p = Pp ( γ, p) p( γ ) dγ (52) 0 where the P p (γ ) is a function that depends on the signal-to-noise-plus-interference ratio (SINR)γ, the modulation/coding scheme p, the receiver structure and the channel. Considering a maximum ratio comining (MRC) receiver matched with the channel diversity and that the same average power Ω is received at each diversity ranch, then the proaility distriution function (pdf) of the SINR per it at the Viteri decoder input is of gamma kind [Hoefel 2005], i.e. L m m p( γ ) = Γ( L mn ) γ n m γ γ n L m ( ) n n γ exp if γ > 0, m 0.5 (53) where m n is the fading figure, γ is the average SINR per it at the detector output (or at Viteri decoder input) and L is the numer of diversity ranches. Notice that m n = models a Rayleigh channel. n

9 For inary phase-shift keying (BPSK used in PHY modes and 2) and quaternary PSK (QPSK used in PHY modes 3 and 4) signaling schemes, then the BER is given y [Proakis 200, pp. 269] ( 2 R ) for p =, 2, 3,, Pp ( γ, p ) = Q γ c 4 (54) where R c is the code rate and Q(x) is the complementary Gaussian cumulative distriution function. For rectangular M-ary quadrature amplitude modulation (M-QAM signaling used in PHY modes 5, 6, 7 and 8), P p (γ ) is given y y (55), where M is the cardinality of the modulation and erfc(z) is the complementary error function and p=5, 6, 7 and 8 [Yang 2000]. M 2log M R M 2 c 3log 2 M Rc Pp (, p ) erfc erfc. M log M ( M ) M log M ( M ) γ 2 γ γ = (55) 7. Analytical Results for Frame Success Proaility over Block Correlated Fading Channels Here, we have assumed that the fading is correlated at each atomic cycle (see Fig. and 2) and uncorrelated among distinct atomic cycles. As stated for uncorrelated fading channels, the union ound on the proaility of decoding error is given y: () Eq. () for the PHY modes, 3 and 5; (2) Eq. (2) for the PHY mode 7; (3) Eq. (3) for the PHY modes 2, 4, 6 and 8. Assuming that the errors inside of the hard decision Viteri decoder are interdependent, then Pursley and Taipale have shown that the upper ound for a successful transmission of a frame with l octets is given y [Puersley 987] 8 [ P ( γ,m ] l S( l, γ,m ) < e ) (56) For a lock-fading channel, this upper ound may e modified to (57a), where the p(g ) is the pdf of the SINR per it at the demodulator output (e.g. p(g ) is given y (53) for MRC receiver over a Nakagami-m fading channel) and g inf is chosen to satisfy the inequality (57). 8l S( l, γ, p ) < [ Pe( γ, p )] p( γ )dγ Pe ( γ inf, p). (57a,) γ inf 7. PHY Mode (BPSK@6Mps) For the PHY mode, the union ound on the decoding error can e estimated using () and (4), where ρ m is given y (58) with R c =/2. ρ 2 ( γ R ) = Pp ( γ ) = Q c. (58) Since all frames are transmitted using the PHY mode, then S rts, S, S mp and S ack can e estimated using (59-65) with p=. Srts( prts ) = S( Nrts, γ, prts ); (59) S ( p ) = P{ CTS is ack / RTS was ack}; (60) S ( p ) = S( N rts + N, γ, p ) ; Srts (6)

10 S S ( p ) P{ MPDU mp mp = is correct / RTS, CTS were correct}; S ( N rts + N + N mp,γ, pmp ) Smp ( pmp ) = ; (63) (62) Srts S ( pack) P{ ACKiscorrect/ RTS, CTS,MPDUwerecorrect} S( Nrts + N + Nmp + Nack, γ, pack) Sack( pack) =. (65) (64) SrtsSSmp ack = 7.2 PHY Mode 2 (BPSK@9Mps) When the MPDU is transmitted using the PHY mode 2, then all control frames are transmitted using the PHY mode. Thus, the S rts and S are still given y (59) and (6) with p=. The union ound on the decoding error can e estimated using (3) and (5), where ρ m is given y (58) with R c =3/4. The S md can e stated as S( N mp, γ, 2 ) S( N mp, γ, 2 ) Smp ( 2 ) = (66) Srts ( )S ( ) S( N rts + N, γ, ) Notice that the 3 octets of the preamle were not taken into account in (66) since they are transmitted using the PHY mode (i.e. a more roust modulation scheme). We also have used the following approximation: P{ MPDU P{ MPDU,RTS,CTS } P{ MPDU } is ack / RTS and CTS were ack} = (67) P{ RTS, CTS } P{ RTS, CTS} since N mp >>(N rts +N ), and the MPDU is transmitted using the PHY mode 2 (i.e. a signaling scheme with lesser immunity to noise and interference than the signaling scheme used to transmit the control frames). The ACK control frame is transmitted using the PHY mode, while the MPDU is transmitted using the PHY mode 2 (i.e. a signaling scheme more suitale to the decoding errors). Thus, the ACK control frame success proaility can e approximated y (68) for lock fading channels. S ack ( m ) = P{ ACK is ack / RTS,CTS and MPDU were ack) (68) 7.3 PHY Mode 3(QPSK@2Mps) In this case all control and data frames are transmitted using the PHY mode 3. Therefore, S rts and S are given y (59) and (6), respectively, with p=3. Correspondingly, S md and S ack are given y (63) and (65) with p=3. The union ound on the decoding error can e estimated using () and (4), where ρ p given y (58) with R c =/2 for coherent demodulation [Proakis 2000]. 7.4 PHY Mode 4 (QPSK@8Mps) Here, all the control frames are transmitted using the PHY mode 3 and the MPDU is transmitted using the PHY mode 4. Consequently, S rts and S are given y (59) and (6), respectively, with p=3. Using similar reasoning developed for PHY mode 2, then S md is given y (69) and S ack is given y (68). The union ound on the decoding error can e estimated using (3) and (5), where ρ p given y (58) with R c =3/4 for coherent demodulation [Proakis 2000]. S( Nmpd, γ, 4 ) S( Nmpd, γ, 4 ) Smpd ( 4 ) =. (69) Srts( 3 )S( 3 ) S( Nrts + N, γ, 3 ) 7.5 PHY Mode 5 (6QAM@24Mps)

11 In this case all control and data frames are transmitted using the PHY mode 5. Therefore, S rts and S are given y (59) and (6), respectively, with p=5. Correspondingly, S md and S ack are given y (63) and (65) with p=5. The union ound on the decoding error can e estimated using () and (4), where ρ p for QAM signaling is given y (55) with M=6 and R c =/ PHY Mode 6 (6QAM@36Mps), PHY Mode 7 (64QAM@48Mps) and PHY Mode 8(64QAM@64Mps) For all these PHY modes the control frames are transmitted using the PHY mode 5 and the MPDU is transmitted using the PHY modes 6, 7, or 8. Thus, the S rts and S are still given y (59) and (6) with p=5. The S md is given y S( N mpd, γ, p ) S( N mpd, γ, p ) S mpd ( p ) = (70) S rts ( 5 )S ( 5 ) S( N rts + N, γ, 5 ) where p=6, 7 and 8. The S ack can e estimated y (68) since the control frames are transmitted using 6QAM with R c =/2 while the MPDU is transmitted using 6QAM with R c =3/4 (PHY 6), 64QAM with R c =2/3 (PHY 7) or 64QAM with R c =3/4 (PHY 8). 8. Joint Link and Physical Layer IEEE 802. Simulator The C a joint MAC and PHY layer has the following main characteristics: It implements the MAC state machine that fulfills the DCF BA and RTS/CTS MAC schemes (Fig. ). The OFDM PHY layer is implemented assuming perfect synchronism. The PHY layer signal processing algorithms implements the maximum-likelihood hard decision detection for the PHY mode to PHY mode 8. The convolutional hard-decision decoding is implemented using a semi-analytic approach as follows. The average BER is estimated at a frame asis using on-line statistics collected at the demodulator output. Then the average BER is used in (4-5) to estimate the proaility of the successful MPDU transmission. It is assumed the following parameters: slot time σ=9µs, SIFS=6 µs, DIFS=EIFS=34 µs, CW min =6, CW max =023, m=6, a=µ.s, N pl =023 octets. It is used a confidence interval of 98% [Hoefel 2005] 9. Analytical and Simulation Results: a Comparative Approach In this section, assuming correlated and uncorrelated flat fading Rayleigh channels, we shall show results for the following system configurations: () IEEE 802.a networks operating under the RTS/CTS access scheme (P a =0 and P rts = in (3) and susequent equations); (2) IEEE 802.a networks operating under the asic access mode (P a = and P rts =0 in (3) and in the following); (3) joint operation of asic and RTS/CTS schemes with P a = P rts =0.5 in (3) and in the following. Unless otherwise noticed, l pl is set to 023 octets. 9. Single Operation of Basic Mode Scheme and RTS/CTS Access Schemes Over Uncorrelated Fading Channel Some results on the joint link and physical layer performance of IEEE 802.a networks over uncorrelated mulitpath fading channel can e found in [Hoefel 2005], where we investigated the system performance assuming that only the RTS/CTS mechanism is

12 active. In this susection, we complement the results shown in the [Hoefel 2005] comparing the goodput and delay performance of oth access methods in relation to the PHY mode and payload length. Ta. 2 shows the maximum goodput and the minimum delay for the asic access and RTS/CTS schemes. For a payload of 023 octets, we have noticed that when a modulation with low cardinality is used to transmit a MPDU, then a superior performance is attainale with the MAC RTS/CTS scheme since the minor time spent with collisions counteralances the greater overhead due to the control frames. However, for a payload of 255 octets the maximum system performance is always otained with the asic access scheme due to the lesser MAC overhead. Ta. 2. Maximum goodput and minimum delay for the asic access and RTS/CTS schemes: N pl =023 and 255 octets. Mode p Max. Goodput ( Mps) Min. Delay (ms) 255 octe 023 octe 255 octe 023 octe BA RTS/CTS BA RTS/CTS BA RTS/CTS BA RTS/CTS ) Comparison of the Performance of RTS/CTS Scheme Over Correlated Fading and Uncorrelated Fading Channel Fig 3 shows that for temporally uncorrelated fading channel there is a well-defined short range of SINR per it where the system performance is acceptale. On the other hand, when the fading is strongly correlated there is a wide and smooth variation of the goodput with the average SINR per it. Fig. 3 also shows that the spatial diversity (assumed uncorrelated) provides greater gain in the required γ on environments where the fading is uncorrelated. However, the diversity gain is also sustantial on temporally correlated fading environments. 24.0M 20.0M Flat Rayleigh Channel RTS+CTS+Data+ACK 0 STS Saturation Goodput in ps 6.0M 2.0M 8.0M 4.0M 0.0 PHY 3 Uncorrelated Fading L= L=3 Correlated Fading L= L=3 PHY 8 Uncorrelated Fading L= L=3 Correlated Fading L= L= Average γ per Antenna in db Figure 3. Comparison etween analytical (straight lines) and simulation (marks) results for the goodput in ps over a Jakes correlated fading channel and a Rayleigh temporally uncorrelated fading channel. RTS/CTS Scheme.

13 9.3 Comparison of the Performance of Basic Access Scheme Over Correlated Fading and Uncorrelated Fading Channel Fig. 4 shows results similar to those ones depicted in Fig. 3, except that now it is assumed the asic access scheme instead of the RTS/CTS mechanism. We again point out a good agreement etween numerical and simulation results. Notice that the minimum attainale average delay does not depend on the fading e correlated or uncorrelated. The minimum delay for the PHY mode 3 is otained with the RTS/CTS scheme and with the asic access scheme for the PHY mode 8, consistently with the remarks carried out on the results shown in Ta. 2. For uncorrelated fading, we can determine a threshold for the average SINR where the delay increases significantly when the average SINR is elow of this threshold. For correlated fading, consistent with the goodput performance, this threshold is also oserved, ut for minor values of the average SINR. 28.0M 24.0M Flat Rayleigh Channel Data+ACK 0 STS Saturation Goodput in ps 20.0M 6.0M 2.0M 8.0M 4.0M 0.0 PHY 3 Uncorrelated Fading L= L=3 Correlated Fading L= L=3 PHY 8 Uncorrelated Fading L= L=3 Correlated Fading L= L= Average γ per Antenna in db Figure 4a. Goodput in ps versus the average SINR per it. Average Delay [s] 00m 0m Flat Rayleigh Channel 0 STS; L= (no spatial diversity) PHY 3 Correlated Fading RTS/CTS Data+ACK Uncorrelated Fading RTS/CTS Data+Ack PHY 8 Correlated Fading RTS/CTS Data+Ack Uncorrelated Fading RTS/CTS Data+Ack Average γ per Antenna in db Figure 4. Average delay in seconds versus the average SINR per it. Figure 4. Comparison etween analytical (straight lines) and simulation (marks) results for the goodput in ps over a Jakes correlated fading channel and a Rayleigh temporally uncorrelated fading channel. Basic Mode Scheme.

14 Hereafter, we shall present results for IEEE 802.a networks operating simultaneously under the asic access and RTS/CTS access schemes. It is generated MAC data payloads of 255 and 023 octets with equal proaility (i.e. P a =P rts =0.5). The RTSThreshold is set to 256 octets. 9.4 Joint Operation of Basic Mode Scheme and RTS/CTS Access Scheme Over Correlated Fading Channel In despite of the complexity of the MAC and PHY layers, we can verify a good agreement etween analytical and simulation results for the goodput (Fig. 5a) and delay (Fig. 5) for the joint operation of BA and RTS/CTS schemes. We can also draw the following remarks. First, we can verify that the PHY mode 3 (QPSK with R c =/2) allows a superior performance in relation to that one otained with the PHY mode (BPSK with R c =/2) since the QPSK signalling has a etter spectral efficiency when it is implemented coherent demodulation. Notice that this also explains the etter performance of PHY mode 4 (QPSK with R c =3/4) in relation to the PHY mode 2 (BPSK with R c =3/4) signalling scheme. Second, PHY mode 5 (6QAM with R c =/2) has a etter performance than the PHY mode 2 (BPSK with R c =3/4) and PHY mode M Saturation Goodput in ps 8.0M 6.0M 4.0M 2.0M 0.0M 8.0M 6.0M 4.0M PHY PHY 2 PHY 3 PHY 4 PHY 5 PHY 6 PHY 7 PHY 8 Flat Rayleigh Channel 0 STS; L= (no spatial diversity) 2.0M Average γ per Antenna in db Figure 5a. Goodput in ps versus the average SINR per it. Flat Rayleigh Channel 0 STS; L= (no spatial diversity) Average Delay [s] 00m 0m PHY PHY 2 PHY 3 PHY 4 PHY 5 PHY 6 PHY 7 PHY 8 m Average γ per Antenna in db Figure 5. Average delay in seconds versus the average SINR per it. Figure 5. Comparison etween analytical (straight lines) and simulation (marks) results for the joint operation of asic and RTS/CTS access schemes. Correlated Rayleigh Channel.

15 Fig. 6 shows a comparison etween analytical (straight lines) and simulation (marks) results for the following system configurations: () asic access (BA) scheme; (2) RTS/CTS scheme; (3) joint operation of BA and RTS/CTS access schemes. When it is assumed the PHY mode 3, then the etter performance is otained with the joint operation of the BA and RTS/CTS protocols. However, for the PHY mode 8, the maximum goodput is otained with the BM scheme since the overhead of the RTS, CTS and ACK control frames is sustantial in relation to the MPDU payload. Notice that in this case the greater waste of time due to collisions of MPDU frames in the asic access scheme does not counteralance the control frames overhead. Saturation Goodput in ps 22.0M 20.0M 8.0M 6.0M 4.0M 2.0M 0.0M 8.0M 6.0M 4.0M 2.0M M Flat Rayleigh Channel 0 STS; L= (no spatial diversity) PHY 3 PHY 8 BM RTS/CTS BM+RTS/CTS BM RTS/CTS BM+RTS/CTS Average γ per Antenna in db Figure 6. Comparison etween analytical (straight lines) and simulation (marks) results for the goodput. Correlated fading channel. 9.5 Joint Operation of Basic Mode Scheme and RTS/CTS Access Scheme Over Uncorrelated Fading Channel Finally, Fig. 7 is similar to Fig.5a, except that it is assumed an uncorrelated Rayleigh fading channel. Notice that the interrelations etween the system performance and the different PHY modes are conceptually similar to those ones pointed out at Fig. 5a. 0. Conclusions In this paper we have derived and validated a joint MAC and PHY cross-layer goodput saturation model that can e confidentially used to estimate first order results for the goodput and the delay of IEEE 802.a ad hoc networks operating simultaneously under the asic access and RTS/CTS operational modes. We have considered the following: () a flat fading Rayleigh channel that is uncorrelated at symol level and independent across the OFDM carriers; (2) a flat fading Rayleigh channel that is correlated at symol level and dependent across the OFDM carriers. To sum up we strongly elieve that the unified methodological approach developed here can e helpful for a widely audience of researchers, from Computer Science to Telecommunications Engineering arenas, since the results that we have developed permit a thorough understanding of a multitude of aspe involved in the challenging interrelations etween complex MAC protocols and advanced PHY techniques.

16 Saturation Goodput in ps 20.0M Flat Rayleigh Channel Uncorrelated Fading 0 STS; L= (no spatial diversity) 5.0M 0.0M 5.0M 0.0 PHY PHY 2 PHY 3 PHY 4 PHY 5 PHY 6 PHY 7 PHY Average γ per Antenna in db Figure 7. Comparison etween analytical (straight lines) and simulation (marks) results for the joint operation of asic and RTS/CTS access schemes. Uncorrelated Rayleigh Channel.. References Bianchi, G. (2000) Performance Analysis of the IEEE 802. Distriuted coordination function, In: IEEE Journal on Selected Areas of Communication, v.8, no. 3, pp , March Conan, J. (984) The weight spectra of some short low-rate convolutional codes, In: IEEE Trans. Communications, vol. 32, pp , Sept B. P. Crow et al (997) IEEE 802. wireless local area networks, In: IEEE Communications Magazine, vol. 35, no. 9, pp. 6-26, Sept Gast, M. S Wireless Networks, New York: O Reilly, Haccoun, D. and Bégin, G (989) High-rate punctured convolutional codes for Viteri and Sequential decoding, In: In: IEEE Trans. Communications, vol. 37, n., p. 3-20, Nov Hoefel, R. P. F. (2005) A Joint MAC and Physical Layer Analytical Model for IEEE 802.a Networks Operating under RTS/CTS Access Scheme, In: 22 0 Simpósio Brasileiro de Redes de Computadores, May Hoefel, R. P. F. (2006) Goodput and Delay Cross-Layer Analysis of IEEE 802.a Networks over Block Fading Channels, In: IEEE Consumer Communication and Network Conference, Jan IEEE 802.a (999) Part : Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specification Amendment : High-speed Physical Layer in the 5 GHz and, supplemented to IEEE 802. standard, Sept Proakis, J. G. (200) Digital Communications, New York, 200. Puersley, M. B and Taipale, D. J. (987) Error Proailities for Spread-Spectrum Packet Radio with Convolutional Codes and Viteri Decoding, In: IEEE Trans. Communications, vol. 35, n., p. -2, Jan Roinson, J. W. and Randhawa, T. S. (2004) Saturation throughput analysis of IEEE 802.e enhanced distriuted coordination function, In: IEEE J. on Select. Areas on Comm., vol. 22, no. 5, p , June Qiao, S. D., S. Choi and Shin, K. G. (2002) Goodput analyzes and link adaptation for the IEEE 802.a wireless LANs, In: IEEE Trans. Moile Comp., pp , Yang, L. and Hanzo, L. (2000) A recursive algorithm for the error proaility evaluation of M- QAM, In: IEEE Comm. Letters, vol. 4, pp , Oct