# Comparison of Network Coding and Non-Network Coding Schemes for Multi-hop Wireless Networks

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4 Fig. 7. Reliability-throughput tradeoff for the multiple path network of Fig 6. Here (1) is for coding, () is for repetition with combining and (3) is for repetition without combining. l =1,k=.5, for (a): d =5.5, = 800; for (b) d =8, = 800; for (c): d =8, =40. because when d goes up, both the effects of physical layer combining and interference decrease, making coding more advantageous. When goes up, the reliability of both coding and repetition increases, shifting all the curves upwards. As d increases to, the transmission schemes reduce to the multiple path wireless network with interference only within a path. For this case, we consider the coding scheme and the repetition scheme with combining under different values of p, and l 1, showing our results in Figures 8 and 9. As before, we observe that coding is useful in high-throughput region, and can show the following asymptotic result: Proposition 1: For the multiple path network with d =, when throughput 0, Rel coding, Rel repetition 1 and Rel repetition >Rel coding ; when throughput, Rel coding, Rel repetition 0 and Rel coding >Rel repetition. Proof: From Equations 5, we have: 1. For repetition scheme: Pr succ (Tp)=exp ( α ( β Tp 1 )) γ lk i lk i=1 i + li k (β Tp 1) Rel =Pr 3 succ + 3Pr succ (1 Pr succ ) + 3Pr succ (1 Pr succ ) Fig. 8. Reliability-throughput tradeoff for multiple path wireless network. For coding scheme: where d =. Herel 1 =64,p=8,l=1,k=3, = 800. Pr succ (Tp)=exp ( α ( β Tp 1 )) γ lk i lk i + li k (β Tp 1) Rel = Pr 3 succ +3 Pr succ i=1 (1 Pr succ ) where Tp denotes throughput, α, β are certain positive coefficients. Rel, Rel are the network reliability, Pr succ, Prsucc are the success probabilities for each path. When Tp 0, using exp (x) x +1; x 1 (ln) x as x 0, weget: Pr succ (Tp) 1 λ Tp; Prsucc (Tp) 1 λ Tp Rel 1 (λ Tp) 3 ; Rel 1 3(λ Tp) where λ is a certain coefficient. So Rel, Rel 1, and since 3(λ Tp) > (λ Tp) 3 for sufficiently small Tp, Rel > Rel. When Tp, it is easy to note that Rel, Rel 0, and we find: Rel Rel µ exp ( α β Tp) 1 ( ) α β Tp ( exp µexp ( α β Tp) 0 β Tp ) 1 β Tp where µ is a certain positive coefficient, so Rel < Rel. Also, the range of reliability over which coding is advantageous increases as p and increase (see Fig 9(a) (b)), or as l 1 decreases (see Fig 9(c)), because reliability is generally improved. C. Generalized Multiple path Wireless Network Finally, we consider a wireless network with multiple groups of paths, where paths within a group are physically close together while paths in different groups are far apart. The results of the previous section show that it is advantageous to send the same information on the closely-spaced paths within a group to take advantage of physical layer combining. Here we investigate the benefit of coding across different groups of paths. Analogously to the previous section, we consider two transmission schemes for a simple network consisting of three widely-spaced groups of paths, illustrated in Fig 10, with negligible interference across groups: a coding scheme where independent information is transmitted on groups and the binary sum of corresponding data bits is transmitted on the third 00

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