TANG AND WANG:MAC FOR EFFICIENT COEXISTENCE BETWEEN FULL AND HALF-DUPLEX COMMUNICATIONS 5879 law [20].n is the path loss index (usually with a range 2-4). 0.22 P.are the same for all nodes.To guarantee success of capture 0.18 effect,the SIR needs to exceed the capture threshold z.In our 016 50.14 ng Chent 1.The 50.12 Poa(y>)=1+z(ri/ru)-m (20) 0.069 0.04 15 30 35 wherer is the distance between the APand the receiving client and ri is the distance between the interfering client iand Fig.7.The probability of collision s are unilormly (PDF)of is given by h()=2ra,0<≤ (21 The PDF of ri is calculated by [22] 034 full duplex g-802.11DcF =5B,25(-).0<n≤2.(2m 1 3 10 15 3035 Fig-8.The ility of successful transmission Pea (23) 0.1+z() VII.PERFORMANCE EVALUATION From the above equation we know that P is determined by rate.As expl ；seting up a du is set larger.then the threshold is hisher.which means it demands a higher capture link rate.When the link rate and data A.Performance Results Based on the Protocol Model lthcult to denve a cl an iven a fixe d link for the AD and1024for 8 as follows.First.the allowed additional time is determined. (Eg.(1)of our protocol and 802.11 DCF are shown in Fig.7 where the result of 802.11 DCF follows the analysis in [19] ength the t on time of The pro sful rans e given by where T is the transmission time for data frame in half probability of A-Duplex is lower than 821 DCE and the duplex link.Since the link rate and the data frame length are successful transmission probability is higher than 802.11 DCF TI is also xed value th /2 and the which leads to higher roughput in A-Duplex eve en if th capture effect is Substituting Eqs.(14)(18)and (23)into Eq.(13)and let RTS/CTS is small. TT/B.we can derive the throughput.Note the abo To evaluate the saturation thr roughp ut.the sy the mi s Mb 22C0 2.the link rate considering capture effect needs TANG AND WANG: MAC FOR EFFICIENT COEXISTENCE BETWEEN FULL- AND HALF-DUPLEX COMMUNICATIONS 5879 law [20], n is the path loss index (usually with a range 2 − 4), Ptx is the transmit power at the transmitter, and r is the distance between the transmitter and the receiver. We assume A, n, and Ptx are the same for all nodes. To guarantee success of capture effect, the SIR needs to exceed the capture threshold z. In our design, the SIR (γ = Pu/Pi) at a client u is the ratio of signal power from the AP over that from an interfering Client i. The conditional capture probability is given by [21] Pca(γ > z|i) = 1 1 + z(ri/ru)−n , (20) where ru is the distance between the AP and the receiving client u, and ri is the distance between the interfering client i and the receiving client u. Considering the clients are uniformly distributed around the AP, the probability density function (PDF) of ru is given by h(ru) = 2ru, 0 < ru ≤ 1. (21) The PDF of ri is calculated by [22] h(ri) = 1 2 1 B(2, 2.5) ri 2  1 − ri 2 3 2 , 0 < ri ≤ 2, (22) where B(·,·)is the Beta function. The approximate average con￾ditional capture probability can be computed as follows [22]: Pca = 1 0 2 0 h(ri)h(ru) 1 + z  ri ru −n dridru. (23) From the above equation we know that Pca is determined by the capture threshold z, and z is determined by the capture link rate. As explained in Section IV-C, setting up a dual link needs to check an additional condition, i.e., Tadd ≤ TAP/β. If the β is set larger, then the threshold z is higher, which means it demands a higher capture link rate. When the link rate and data frame length are variable, it is difficult to derive a closed-form relationship between β and z. However, given a fixed link rate and data frame length, the value of z can be determined from β as follows. First, the allowed additional time is determined, i.e., Tadd = TAP/β, where TAP = TS1 since link rate and data frame length are fixed. Following that, the transmission time of the capture link T2 is determined as T2 = Tadd + T1 + TSIFS − TACK, where T1 is the transmission time for data frame in half duplex link. Since the link rate and the data frame length are fixed, T1 is also a fixed value. With T2 and the data frame length, we can get the transmission rate of downlink. Finally, the capture threshold z is determined by the transmission rate. Substituting Eqs. (14)–(18) and (23) into Eq. (13) and let Tadd = Ts1/β, we can derive the throughput. Note the above derivation assumes the maximum additional time that satis- fies Tadd ≤ TAP/β, so the derived throughput is the minimum throughput that can be achieved by our protocol. Fig. 7. The probability of collision. Fig. 8. The probability of successful transmission. VII. PERFORMANCE EVALUATION In this section the performance of A-Duplex is first evaluated based on a protocol model, and then a physical model is used to evaluate A-Duplex in a more practical setup. A. Performance Results Based on the Protocol Model We set the minimum contention window to 32 for both the AP and clients and the maximum contention window to 128 for the AP and 1024 for clients. The collision probabilities (Eq. (18)) of our protocol and 802.11 DCF are shown in Fig. 7, where the result of 802.11 DCF follows the analysis in [19]. The probability of successful transmission (PA + Pc given by Eqs. (16) and (17)) are shown in Fig. 8. Thus, the collision probability of A-Duplex is lower than 802.11 DCF and the successful transmission probability is higher than 802.11 DCF, which leads to higher throughput in A-Duplex even if the capture effect is not considered as shown in Fig. 9. However, without capture effect, the improvement over 802.11 DCF with RTS/CTS is small. To evaluate the saturation throughput, the system parameters are selected according to Table I. We set the link rate to 18 Mbps. We set β = 2.2. Considering the requirement of Tadd = TAP/2.2, the link rate considering capture effect needs