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50 Chapter 4: Unequal Erasure Protection (UEP) in Network Coding
The temporary expected utility E t [U i ] of the i th sink at the t th iteration is given by
E t [U i ] =
ω∑
φ t,i,j · ∆ j , (4.14)
j=1
where φ t,i,j represents the probability that the prefix P j can be recovered at the sink i
after the GEK assignment at the iteration t. φ t,i,j equals ρ i,j in Eq. (4.8) if P j can be
recovered by the GEKs assigned so far up to the iteration t. Otherwise, it equals zero.
∆ j denotes the incremental utility if P j is recovered [5].
We suggest that the number n(t 1 ) of GEKs assigned at the iteration t 1 should be
greater than or equal to the number n(t 2 ) at the iteration t 2 if t 1 < t 2 , i.e., it is better
to take larger search spaces into consideration during some first iterations, after which
things do not improve much.
In our network example, we assume that every edge-disjoint path has an erasure
probability of 0.1 and the utility vector ∆U k = [∆ 1 , ∆ 2 , ∆ 3 ] = [1, 0.5, 0.25] for every sink
node R k . If we let n(1) = 3 and n(2) = n(3) = 1, then the minimum temporary expected
data quality in the first iteration is maximized by allocating the GEKs [1 0 0] T to T 3 ,
[0 1 0] T to T 1 , and [1 1 0] T to T 5 . The sink R 1 now has the temporary expected utility
of E 1 [U 1 ] = (0.9)(1) + (0.81)(0.5) = 1.305, since, by receiving [1 0 0] T
from T 3 with the
probability of 0.9, R 1 can recover the first prefix, and by receiving both [1 0 0] T from T 3
and [0 1 0] T from T 1 with the probability of 0.81, it can recover the second prefix.
In the same manner, the temporary expected data qualities E 1 [U 2 ], E 1 [U 3 ], and E 1 [U 4 ]
of R 2 , R 3 , and R 4 become 1.305, 1.305, and 1.215, respectively. This gives the minimum
temporary expected data quality, min E 1 [U i ], of 1.215 at the sink R 4 , which is the maximum
that one can find at this iteration.
Table 4.2 shows the resulting expected data qualities at all sink nodes and the minimum
one, after the third iteration finishes and all subtrees have obtained their GEKs.
Table 4.3 shows that, in our network example, the suggested strategy achieves the
global optimum. In addition, on average, a random GEK assignment results in the minimum
expected quality that is closer to that of the worst case than the best one. Thus,