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severely limit the gain offered by APR strategies. In an interesting application [66], multipath path transport (MPT) is combined with multiple description coding (MDC) in order to send video and image information in a multihop mobile radio network. In this section , we discuss a multipath scheme for mobile ad hoc networks based on diversity coding [45]. Data load is distributed over multiple paths in order to minimize the packet drop rate and achieve load balancing in a constantly changing environment. Suppose that n max paths are available for the transmission of data packets from a source to a destination. Any of the multipath schemes mentioned in the introduction can be employed in order to acquire these paths. No paths have nodes in common (mutually disjoint). Each path, indexed as i, i = 1, . . . n max , is either down at the time that the source attempts to transmit with probability of failure pi or the information is received correctly with probability 1 pi . Since there are no common nodes among the paths, they are considered independent in the sense that success or failure of one path cannot imply success or failure of another. It should be noted here that, in wireless ad hoc networks, nodes are sharing a single channel for transmission, so node disjointness does not guarantee the independence of the paths. Taking this into account, the paths are ideally considered independent as an approximation of a realistic ad hoc wireless network. For a more realistic modeling of the paths in a wireless network, one may refer to Tsirigos and Haas [67], where path correlation is included in the analysis. The failure probabilities of the available paths are organized in the probability vector p = { pi }, in such a way that pi pi+1 . The vector of success probabilities is de ned as q {qi } = 1 p = {1 pi }. Let us now suppose that we have to send a packet of D data bits utilizing the set of available independent paths in such a way as to maximize the probability that these bits are successfully communicated to the destination. This probability is denoted as P. In order to achieve this goal, we employ a coding scheme in which C extra bits are added as overhead. The resulting B = D + C bits are treated as one network-layer packet. The extra bits are calculated as a function of the information bits in such a way that, when splitting the B-bit packet into multiple equal-size nonoverlapping blocks, the initial D-bit packet can be reconstructed, given any subset of these blocks with a total size of D or more bits. First, we de ne the overhead factor r = B/D = b/d where b and d take integer values and the fraction b/d cannot be further simpli ed. One should note that 1/r would be equivalent to coding gain in channel coding theory. Next we de ne the vector v = {vi }, where vi is the number of equal-size blocks that is allocated to path i. Some of the paths may demonstrate such a poor performance that there is no point in using them at all. This means that we might require using only some of the available paths. If n is the number of the paths we have to use in order to maximize P, it would be preferable to de ne the block allocation vector v = {vi } as a vector of a variable size n, instead of xing its size to the number of available paths n max . Given the fact that the probability failure vector is ordered from the best path to the worst one, a decision to use n paths implies that these paths will be the rst n ones. Based on these observations, the allocation vector v = {vi } has the following form: v = {v1 , v2 , . . . , vn } , n n max . n If the block size is w, then w i=1 vi = B = r D. Therefore, the total number of n blocks that the B-bit packet is fragmented into is a = i=1 vi = r D/w. From pi pi+1 it follows that vi vi+1 , because a path with higher failure probability cannot be assigned fewer blocks than a path with a lower failure probability. The original D-bit packet is fragmented into N w-size blocks, d1 , d2 , d3 , . . . , d N , and the C-bit overhead packet
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