5s78 LEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS,VOL 14.NO.10.OCTOBER 2015 can become negative after it is deducted by the downlink as the AP transmits successfully when all clients keep silent transmiss Following [9.the throughput S is defined as the fraction of sending payload bits.It can be be negative.When the AP in case a)finds a client has ne ative deficit.it just skips this client to the next one.Actually we can (P+P.+PPEUL) negative value of the deficit.In othe (13 ca where the parameters are defined as follows.Pr is the prob- case b).This limit improves fairess,but results in loss of dual ability of transmission from at least one node.PA and Peare Ias the proba ion at the Al e for orobability and Pis the collision and T several reasons:D Although the deficit is updated in the same are the average time of successful transmission for the AP and way as that in the deficit round robin algorithm,the deficit in case bin way:2)The deficit in case not strictly followed (due to dual links via The above quation reflects that the throughput is contributed effect).perfect faimness like deficit round robin cannot always by three components,i.e.,data transmission from the AP to be ensured. a VI PERFORMANCE ANALYSIS re effect.but it d from RTS. CTS.ACK.etc.Such overhead is captured by Ta and T as In this follows nt ITs=E(TAP)=E(Tdaal TsIEs TAck TDIES T2=E(TA]=TRTS+3TsIFs+Tcrs+E(+TAcK+TDIrs el.The nalysis can be adopted her (14) Since there are N clients and one AP.P is given by A-Duplex Por=1-(1-P)(1-P )W (15) s of the Ap and a p n be e for are different.We set the minimum contention window Wo and maximum 2Wo for the AP and the minimum contention for clients.F llowing simila PA Po(1-P) (16) can get th fully when 1+W+pW∑m(2p (9) active.Thus,Pis given by: In Pe=NP:(I-Pr)(N-D) 17) one of the clients gets the channel no matter whether the AP The collision probability is given by: gets the channel or not.Thus,p is given by P=P-P-P. (18) p=1-1-P)N- (10) where n is the nur agation mode for the aP in a slot as with deterministic power and Rayleigh fading.the (11)instantaneous powerPis exponentially distributed as: 1+W+pWo∑(2po e的y he AP and D)=元元，P>0, (19 m=1-(1-P,)5878 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 10, OCTOBER 2015 can become negative after it is deducted by the downlink transmission time. Even after it is updated according to Eq. (8), it may still be negative. As a result, even if the procedure in case a) is similar to the deficit round robin algorithm, the deficit can be negative. When the AP in case a) finds a client has negative deficit, it just skips this client to the next one. Actually we can set a limit to the negative value of the deficit. In other words, if the deficit of a client is not enough for a packet transmission (by considering the negative limit), the client cannot be selected in case b). This limit improves fairness, but results in loss of dual link chances. Thus, fine-tuning such a limit provides a tradeoff between fairness and throughput. We call our scheme a virtual deficit round robin scheme for several reasons: 1) Although the deficit is updated in the same way as that in the deficit round robin algorithm, the deficit in case b) is not used in a round robin way; 2) The deficit in cases a) and b) can be negative; 3) Since the round robin order is not strictly followed (due to exploring dual links via capture effect), perfect fairness like deficit round robin cannot always be ensured. VI. PERFORMANCE ANALYSIS In this section we analyze saturation throughput of A-Duplex to study its throughput improvement over 802.11 DCF. “Satu￾ration” means the AP and clients always have packets for trans￾mission. In [19], the performance of 802.11 DCF is analyzed via a Markov model. The same analysis can be adopted here, but two major differences in A-Duplex need to be considered. First, when a client and the AP collide, the client can still get the channel successfully due to full duplex capability at the AP. Second, capture effect exists in A-Duplex. In a practical system, the Markov chains of the AP and clients can be different since the contention windows for them are different. We set the minimum contention window W0 and maximum 2m0W0 for the AP and the minimum contention window W and maximum 2mW for clients. Following similar Markov chain analysis in [19], we can get the transmission probability Pt for a client in a slot as Pt = 2 1 + W + pW m−1 i=0 (2p)i , (9) where p is the conditional collision probability for a client. In A-Duplex, a successful transmission from a client occurs when one of the clients gets the channel no matter whether the AP gets the channel or not. Thus, p is given by p = 1 − (1 − Pt) N−1, (10) where N is the number of clients. In the same way, we can get the transmission probability Pt0 for the AP in a slot as Pt0 = 2 1 + W0 + p0W0 m0−1 i=0 (2p0)i , (11) where p0 is the conditional collision probability for the AP and is given by p0 = 1 − (1 − Pt) N, (12) as the AP transmits successfully when all clients keep silent. Following [19], the throughput S is defined as the fraction of time for successfully sending payload bits. It can be calculated as follows S = (PA + Pc + PcPca)E{Lp} (1 − Ptr)Tslot + PATs1 + PcTs2 + PcPcaTadd + PcolTc , (13) where the parameters are defined as follows. Ptr is the prob￾ability of transmission from at least one node. PA and Pc are defined as the probability of successful transmission at the AP and a client, respectively. Pca is the average conditional capture probability, and Pcol is the collision probability. Ts1 and Ts2 are the average time of successful transmission for the AP and for a client, respectively. Tadd is the additional time for capture effect, as defined in Section IV-C. Moreover, Tc is the collision time and E{Lp} is the average payload size in a data frame. The above equation reflects that the throughput is contributed by three components, i.e., data transmission from the AP to clients without utilizing capture effect, data transmission from clients to the AP, and data transmission from the AP to clients with capture effect, but it must exclude overhead from RTS, CTS, ACK, etc. Such overhead is captured by Ts1 and Ts2 as follows ⎧ ⎪⎨ ⎪⎩ Ts1 = E{TAP} = E{Tdata} + TSIFS + TACK + TDIFS Ts2=E{TA}=TRTS+3TSIFS+TCTS+E{Tdata}+TACK+TDIFS Tc = TRTS + TDIFS. (14) Since there are N clients and one AP, Ptr is given by: Ptr = 1 − (1 − Pt0)(1 − Pt) N. (15) When all clients keep silent and only the AP sends packet, the AP sends packet successfully. Thus, PA is given by: PA = Pt0(1 − Pt) N. (16) The client transmits packet successfully when only one of the clients transmits no matter whether the AP keeps silent or active. Thus, Pc is given by: Pc = NPt(1 − Pt) (N−1) . (17) The collision probability is given by: Pcol = Ptr − PA − Pc. (18) The average conditional capture probability Pca is asso￾ciated with the capture threshold, i.e., Pca decreases as the capture threshold increases. Considering a propagation model with deterministic power attenuation and Rayleigh fading, the instantaneous power P is exponentially distributed as: fp(P) = 1 P0 e − P P0 ,P > 0, (19) where P0 is the received local-mean power and is determined by P0 = Ar−nPtx, where Ar−n is the deterministic path-loss