Decoding Collided Acknowledgement Transmission in IEEE 802.11 ...

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which shows that c is a column of MN. Since Ω+(G1, c) = Ω−(G1, c), by (3) we thus have F (G1, c) = 0. Proposition 1: Consider a certain system with N stations where N ≥ 1. For any G1, G2 ⊂ G and G1, G2 = ∅, −→ G1 = −→ G2 iif G1 = G2. Proof: By definition given in (4), we can easily see that if G1 = G2, then −→ G1 = −→ G2. In the following, we shall prove that if G1 = G2, then −→ G1 = −→ G2. We will examine two cases: (i) |G1| is an odd integer; (ii) |G1| is an even integer. Without loss of generality, we assume that |G1| ≥ |G2|. Case (i): |G1| is an odd integer. Let g = |G1|. Based on lemma 1, there exists a column c in MN such that and hence according to (4), we also have −→ G1(c) = 1. F (G1, c) = 1, (6) This column will have the following property Ω+(G1, c) = g+1 2 Ω−(G1, c) = Ω+(G1, c) − 1 = g−1 2 Ω+(G ′ 1, c) = Ω−(G ′ 1, c) = N−g 2 with any permutation. If G1 = G2, there exists K1 = G1 \ G2 = ∅ and K2 = G2 \ G1 where |K1| = k1 ≥ 1 and |K2| = k2 ≥ 0. It is clear that K1 ⊆ G1, K2 ⊆ G ′ 1 imply 1 ≤ k1 ≤ g, k2 ≤ N − g. where g is an odd integer and N − g is an even integer. With the above condition, there exists a permutation within c such that Ω+(K1, c) = ⌊ k1 g+1 2 ⌋ + 1 ≤ 2 Ω−(K1, c) = ⌊ k1−1 g−1 2 ⌋ ≤ 2 Ω+(K2, c) = ⌊ k2 N−g 2 ⌋ ≤ 2 Ω−(K2, c) = ⌊ k2+1 N−g 2 ⌋ ≤ 2 that confirms K1 ⊆ G1, K2 ⊆ G ′ 1 given (7). The immediate result gives and which yield F (K1, c) = F (K2, c) = 1, k1 = 1, 3, ... 2, k1 = 2, 4, ... −1, k2 = 1, 3, ... 0, k2 = 0, 2, 4, ... (7) (8) (9) F (K1, c) − F (K2, c) ≥ 1. (10) Since G2 = (G1 \ K1) ∪ K2, with (6) and (10), we obtain F (G2, c) = F (G1, c) − F (K1, c) + F (K2, c) ≤ 0 and hence −→ G2(c) = 0. Since −→ G1(c) = 1, there exists a column in MN such that −→ G2(c) = −→ G2(c), which is sufficient to show that −→ G1 = −→ G2. Case (ii): |G1| is an even integer. Let g = |G1|. Based on lemma 2, there exists a column c in MN such that and hence according to (4), we also have −→ G1(c) = 0. F (G1, c) = 0 (11) This column will have the following property Ω+(G1, c) = Ω−(G1, c) = g 2 Ω+(G ′ 1, c) = N−g+1 2 Ω−(G ′ 1, c) = N−g−1 2 (12) with any permutation. If G1 = G2, there exist K1 = G1 \ G2 = ∅ and K2 = G2 \ G1 where |K1| = k1 ≥ 1 and |K2| = k2 ≥ 0. It is clear that K1 ⊆ G1, K2 ⊆ G ′ 1 imply 1 ≤ k1 ≤ g, k2 ≤ N − g. where g is an nonzero even integer and N −g is an odd integer. We first exclude the condition where k2 = 0. With the above condition, there exists a permutation within c such that Ω+(K1, c) = ⌊ k1 g 2 ⌋ ≤ 2 Ω−(K1, c) = ⌊ k1+1 g 2 ⌋ ≤ 2 Ω+(K2, c) = ⌊ k2 N−g+1 2 ⌋ + 1 ≤ 2 Ω−(K2, c) = ⌊ k2−1 N−g−1 2 ⌋ ≤ 2 that confirms K1 ⊆ G1, K2 ⊆ G ′ 1 given (12). The immediate result gives and which yield F (K1, c) = F (K2, c) = −1, k1 = 1, 3, ... 0, k1 = 2, 4, ... 1, k2 = 1, 3, ... 2, k2 = 2, 4, ... (13) (14) F (K2, c) − F (K1, c) ≥ 1. (15) For k2 = 0 which implies G2 ⊂ G1, since G2 cannot be empty, k1 = g − |G2| must be ≤ g, there exists another permutation in c such that Ω+(K1, c) < Ω−(K1, c) ≤ Ω−(G1, c) giving F (K1, c) = −1, k1 = 1, 3, ... −2, k1 = 2, 4, ... (16) and F (K2, c) − F (K1, c) ≥ 1 as in (15) since F (K2, c) = 0. Given that G2 = (G1 \ K1) ∪ K2, with (11) and (15), we obtain F (G2, c) = F (G1, c) − F (K1, c) + F (K2, c) ≥ 1 and hence −→ G2(c) = 1. Since −→ G1(c) = 0, there exists a column in MN such that −→ G2(c) = −→ G2(c) which is sufficient to show that −→ G1 = −→ G2. With the above two cases, we have proven that if G1 = G2, then −→ G1 = −→ G2. Together with the earlier establishment that if G1 = G2, then −→ G1 = −→ G2, we conclude that for any G1, G2 ⊂ G and G1, G2 = ∅, −→ G1 = −→ G2 iif G1 = G2.
SIFS SIFS SIFS DIFS Data Poll ACK Poll ACK … Td SIFS DIFS Data ACKs Td Ta Tp (a) (b) Fig. 1. Illustration of protocol handshake procedure for (a) the IEEE 802.11b PCF, (b) our proposed mechanism. With Proposition 1, we have shown that based on our coding scheme, there is a one-to-one relationship between the decoded bitstreams and the combinations of the receivers. This one-toone relationship allows a decoder to identify the presence of each individual station in a collided transmission. In the case that no station replies acknowledgement, an idle period will occur on the channel. This idle channel will trigger the MAC layer retransmission of the failed multicast transmission. IV. PROTOCOL PERFORMANCE In this section, we show the performance advantage of our proposed approach. We consider the popular IEEE 802.11b standard [1] and summarize values of protocol parameters needed for our performance evaluation in Table IV. The full data rate of 11 Mbps transmission for data is assumed. While the IEEE 802.11 standard does not enforce explicit acknowledgement for a one-to-many transmission, for the performance comparison, we consider the use of the standardized but optional point coordination function (PCF) to achieve acknowledgement. PCF uses power-save poll (PS-Poll) frame to perform individual polling. Its protocol handshake procedure is shown in Fig. 1(a), and the protocol handshake procedure for our proposed method is also depicted in Fig. 1(b). Based on Fig. 1 and Table IV, consider a d-byte payload to multicast, we first have Td = TP HY + 272+8·d 11 Tp = SIF S + δ + TP OLL + SIF S + δ + TACK Ta = SIF S + δ + TCCF (17) where Td, Tp, and Ta is the duration of a data frame transmission at 11 Mbps, the duration to poll a station, and the duration for all stations to simultaneously return an acknowledgement, respectively. With these timings, consider a multicasting to n stations, we can determine the duration of the protocol handshake procedure by TST D = Td + nTp + DIF S + δ TCC = Td + ⌈ n 9 ⌉Ta + DIF S + δ (18) where TST D and TCC are the durations using the standard PCF mechanism and our proposed collision code mechanism respectively. For our proposed mechanism, when the number of stations exceed 9, we partition them into several groups for acknowledgement collisions. Fig. 2 plots and compares the two protocol handshake durations. As can be seen, our Protocol handshake duration (second) 0.05 0.04 0.03 0.02 0.01 IEEE 802.11b PCF Our proposed scheme 0 0 10 20 30 40 50 Number of stations Fig. 2. Protocol handshake duration comparison between the IEEE 802.11b PCF and our proposed mechanism with payload size of 8184 bits. TABLE IV TIMING FOR PROTOCOL PARAMETERS. Protocol parameter Duration (µs) Signal propagation delay, δ 1 SIFS 20 DIFS 50 PHY overhead, TP HY 192 PS-Poll frame, TP OLL ACK frame, TACK MAC data frame (d-byte payload), TDAT A TP HY + 160 TP HY + 112 TP HY + 272+8·d 11 Collision code frame ∗ , TCCF TP HY + 272 + 128 ∗ Note: This collision code design supports up to 9 stations which requires 126 bits or 16 bytes (with byte alignment). mechanism using collision codes requires very short period to complete the acknowledgement whereas the duration using the IEEE 802.11b PCF grows as the number of stations increases. V. CONCLUSION Motivated by the concept of PNC, we proposed a collision decoding scheme that improves the reliability of wireless multicasting for the IEEE 802.11 MAC protocol. We first established the majority principle between the input and the output bitstreams based on BPSK modulation. Using the majority principle, we developed a coding scheme achieving the identification of individual acknowledgement transmission from a collided transmission. We applied our design to increase the reliability and reduce the overhead of wireless multicasting. We believe that this paper pioneers in the MAC protocol enhancement using PNC. There are other related issues that require further research to address. In our scheme, although acknowledgement transmissions appear simultaneously, they are not necessarily synchronous in the physical layer. As a future work, we will investigate the penalty of the time synchronization and the impact on our coding scheme. REFERENCES [1] Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY), IEEE 802.11 Std., 1999. [2] E. Pagani and G. P. Rossi, “Reliable broadcast in mobile multihop packet networks,” in Proceedings of MobiCom, 1997. [3] K. Tang and M. Gerla, “MAC reliable broadcast in ad hoc networks.” [4] K. H. Lee and D. H. Cho, “A multiple access collision avoidance protocol for multicast service in mobile ad hoc networks.” [5] P. Rogers and N. Abu-Ghazaleh, “Directed broadcast: a MAC level primitive for robust network broadcast.” [6] S. Zhang, S. C. Liew, and P. P. Lam, “Hot topic: physical-layer network coding,” in Proc. of MobiCom’06, 2006, pp. 358–365.
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