4. Analysis
4.1. Reliability Analysis
4.1.6. The Special Model
The previous analysis of the general network model is based on the assumption that node failures can be ignored. However, in a realistic network, the situation is usually more complicated with the existence of node compromise. In order to realize the effect of node failures, we illustrate a special example of network environment to explain the influence of node failure over R(m, n). Figure 4-8 shows the special network model, where N disjoint paths with identical number of hops and success probability are considered. All wireless links and intermediate nodes have the same success probability respectively, which are denoted as Pl and Pn. That is, the influence ratio of the two factors is fixed under special network model.
Figure 4-8 The special network case
Based on the assumptions of the network environment, the original path success probability is the same as equation (5). When (m, n) FAMIDS is applied, the path success probability of transmitting a sub-packet of L/m bits can be expressed as equation (6). Using (m, n) FAMIDS, the base station can reconstruct the original information if more than m sub-packets are correctly received. As a result, the
communication reliability R(m, n) can be modified as equation (27):
On account of the definition of cumulative node and link success probability, equation (27) can be rewritten as
( ) ∑ ( ) ( )
Figure 4-9 illustrates an example of the relationship between the original Ps and the communication reliability after (m, 6) FAMIDS is applied. In this example, the number of hops is 3 and the node success probability Pn is 0.9. As shown in Figure 4-9, we can observe that the communication reliability is greatly improved when the original path success probability is relative small because of the increase in the link success probability. In order to determine the optimal (m, n) set with the highest communication reliability under different Ps, some fundamental theorems regarding the relationship between R(m, n)’s are derived.
Figure 4-9 Relationship between Ps and (m,6) MIDS
• Fundamental Theorems
Here are some fundamental theorems that the results help us to determine the optimal value of (m, n) set with highest communication reliability.
Theorem 4-5: R(m, n) is strictly increasing for 0 < Ps < 1.
Theorem 4-5 implies that the communication reliability achieved by a specific (m, n) FAMIDS is proportional to Ps. When the successful transmission probability of each path increases, the communication reliability is also improved.
Theorem 4-6: if 0 < Ps < 1, then R(m, n)< R(m, n+1).
Theorem 4-6 suggests the communication reliability can be improved if more pieces of sub-packets are sent and the same number of sub-packets is needed to reconstruct the original data packet. Figure 4-10 shows the distribution of the reliability curves of (1, n) MIDS, where1≤ n≤6, and each path has 3 hops with Pn=0.9. The results implied in Figure 4-10 are consistent with Theorem 4-6.
Figure 4-10 Relationship between the Ps and (1,n) MIDS
Theorem 4-7: R(m, n) > R(m+1, n) when PsÆ1-
From Theorem 4-7, we can observe that the fewer the number of sub-packets needed to reconstruct the original data packet, the higher the communication reliability when the message dispersal degree n is fixed and Ps is approaching to 1.
The only likelihood that Ps is approaching to 1 is both Pl and Pn are approaching to 1.
As a result, we can infer Corollary 2 from Theorem 4-7.
Corollary 2: R(1, n) is the optimal of R(i, n) when PsÆ1-, for 1≤i≤n.
Under the condition that Ps is approaching to 1, it means that both the node and link failure rate are relative small and thus the effect of packet splitting is so trivial that can be neglected. As a consequence, the maximum information expansion ratio achieves the optimal reliability under such network condition. The result is also consistent with the simple model. Compared with the case when PsÆ1-, the case when PsÆ0+ is more complicated. Theorem 4-8 &
Theorem 4-9 give the explanation of such case.
Theorem 4-8: R(η,n) is the optimal of R(i, n) when Pl Æ 0+, for 1≤i≤n,
where ⎥
⎥
⎢ ⎤
⎢
⎡ +
−
= ⋅
1 1
N N
P n η P
Theηvalue derived by Theorem 4-8 is a critical number of (m, n) FAMIDS.
When link success probability is approaching to zero, i.e. SNR is extremely small, the optimal (m, n) set is mainly determined by node success probability. From Theorem 4-8, we can guarantee the correctness of Corollary 3 & Corollary 4.
Corollary 3: R(1, n) is the optimal of R(i, n) when PlÆ0+ and PnÆ0+, for n
i≤
≤
1 .
Corollary 3 conforms to our intuition that the source node should maximize the information expansion ratio when the network is in an exceptionally poor condition.
This case happens when the sensor network is deployed in an extremely noisy environment and full of compromised or out-of-battery nodes.
Corollary 4: R(η2,n) is the optimal of R(i, n) when PlÆ0+ and PnÆ1-, for n
i≤
≤
1 , where η2 =
⎣ ⎦
n 2 .The result of Corollary 4 is similar with the analysis in Tsai’s paper [10] where the sender and receiver share n parallel wireless communication channels with no intermediate nodes. Because of the high node success probability, the communication reliability is mainly constrained by the unreliable links.
Figure 4-11 shows the relationship between PN and the optimal (m, n) FAMIDS when PlÆ0+ based on Theorem 4-8, for n = 1~10. It is noted that the only optimal solution is (1, n) when number of paths n≤3. With the decrease in PN, the effect of packet splitting also decreases.
Figure 4-11 Relationship between optimal R(m, n) and PN
Theorem 4-9: R(1, n) is the optimal of R(i, n) when Pn Æ 0+, for 1≤i≤n.
From Theorem 4-9, we can conclude that in the circumstance of numerous invalid intermediate nodes, the source node should split the original data packet with maximum information expansion ratio. The result of Theorem 4-9 coincides with the result of Theorem 4-8.
Theorem 4-10: If m≥η, then R(m, n) > R(m+1, n).
We can conclude from Theorem 4-10 that the curves of R( η ,n), R( η +1,n),…,R(n,n) do not have intersections. R(η,n) would be the optimal solution if
η
≥
m . The next theorem discusses the relationship between R(m, n) if m≤η.
Theorem 4-11: If 2≤ m≤η, then there exists exactly one critical probability
*
As we can see in Theorem 4-11, in the intervalm≤η, a particular FAMIDS does not always achieve better communication reliability than another. A FAMIDS can give better reliability in a range of Ps, but worse reliability in the other range. Besides, it is noted that the value ofηdepends on the cumulative node success probability of each path, PN. With the decrease in PN, the value of η also reduces, i.e. fewer pairs of FAMIDS have intersections.
Figure 4-12 shows the relationship between the R(m, n) under different PN, for n
= 8, m = 1 ~ 4. In this example, each path has three hops as Figure 4-12. Figure 4-12(a) shows the curves of R(m, n) when Pn=0.9, i.e. η= 4. As we can see, the curves of R(1,8) ~ R(4, 8) have intersections. On the other hand, Figure 4-12(b) represents the curves of R(m, n) when Pn=0.75, i.e.η= 3, such that only R(1,8) ~ R(3,
8) have intersections. When Pn decreases to 0.65, i.e. η= 2, only the first two curves, R(1,8) and R(2,8), have intersections, as shown in Figure 4-12(c). Finally, we can obviously see that the curves of R(m, n) do not have intersections if η= 1, as shown in Figure 4-12(d).
Figure 4-12 R(m, n) under different Pn
Theorem 4-12: For fixed n≥6 &
1 2
> −
PN n , if 2≤ m≤η , then
( ) ( )
[
1, , ,]
*[ ( )
, ,( 1, )]
* m n m n P m n m n
Ps − > s + , where ⎥
⎥
⎢ ⎤
⎢
⎡ +
−
= ⋅
1 1
N N
P n η P
Theorem 4-12 proves that the intersections of Ps*
[ (
m+i−1,n) (
, m+i,n) ]
have increasing order as i increases for fixed n. The following Figure 4-13 shows the relationship between Ps*[ ]
i,n ≡Ps*[ ( ) (
i,n, i+1,n) ]
and PN for n = 4 ~ 9. For example, if PN is equal to 0.9 and the number of available paths is 9, the following conclusioncan be drawn:
1) R(1,9) > R(2,9) if Ps> Ps*[1,9] = 0.68 2) R(2,9) > R(3,9) if Ps> Ps*[2,9]= 0.2 3) R(3,9) > R(4,9) if Ps> Ps*[3,9]= 0.0028
We provide the approximation function of each critical probability for different number of available paths in the appendix I to K. The approximation functions can assist the source node with the determination of the optimal (m, n) FAMIDS with highest communication reliability.
Figure 4-13 Relationship between Ps* and PN
• Optimal solution discussion
Based on the fundamental theorems derived in this section, we can determine the optimal set of (m, n) FAMIDS with highest communication reliability under special network condition. First of all, according to Theorem 4-6, we can infer that one achieves the maximum communication reliability when total available paths are used for packet transmission. This conclusion holds for each kind of network. Second, about the relationship between all (m, N) FAMIDS, we have found that any particular FAMIDS does not always achieves better reliability than another. According to the knowledge on Ps and PN, one can determine the optimal value of m. Algorithm 4-4 finds the optimal value of m under special network model that achieves the optimal reliability.
Algorithm 4-4: Finding Optimal m for (m, N) FAMIDS under Special Model Input N: the number of available paths
Input Ps: the path success probability
Input PN: the cumulative node success probability
Output: m: the optimal value of m, such that (m, N) FAMIDS achieves the optimal communication reliability.
1. Calculate the critical number ⎥
⎥
⎢ ⎤
⎢
⎡ +
−
= ⋅
1 1
N N
P N
η P .
2. Determine the critical probabilities Ps*[(m, N), (m+1, N)] for m=1..η-1 by PN
and the approximation functions.
3. Search the optimal m as follows.
• If Ps*
[ ( ) (
1,N , 2,N) ]
<Ps <1, then the optimal value of m is 1.• If Ps*
[ (
i+1,N) (
, i+2,N) ]
<Ps <Ps*[ ( ) (
i,N , i+1,N) ]
, then the optimal value of m is i+1.• If 0<Ps <Ps*
[ (
η−1,N) (
,η,N) ]
, then the optimal value of m is η.4. If the optimal m cannot satisfy the information expansion ratio requirement, then
choose m=m+1, m=m+2, …, m=N respectively, until the information expansion ratio is satisfied.
5. Output m.