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Design of Fault-tolerant Structures for Underwater Sensor Networks Based on Markov Chains

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31 mar 2025
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Figure 1.

Generalized functional structure of the hierarchical underwater sensor network.
Generalized functional structure of the hierarchical underwater sensor network.

Figure 2.

Schematic hierarchical structure of the USN with X0 = 13 and imax = 3.
Schematic hierarchical structure of the USN with X0 = 13 and imax = 3.

Figure 3.

A homogeneous Markov chain for calculating the probability of failure of the i-th level of the USN.
A homogeneous Markov chain for calculating the probability of failure of the i-th level of the USN.

Figure 4.

A homogeneous Markov chain for calculating the probability of failure of the last (highest) level of the USN.
A homogeneous Markov chain for calculating the probability of failure of the last (highest) level of the USN.

Figure 5.

A block diagram of the designing method of fault-tolerant structures for USNs.
A block diagram of the designing method of fault-tolerant structures for USNs.

Figure 6.

Changing of the real value NΣ during the implementation of the method.
Changing of the real value NΣ during the implementation of the method.

Main parameters of the USN’s structure at the 1st iteration

i Type of hierarchy level pmi Xi Xic Ni max
0 Initial sensors 0.025 16 8 781
1 UUVs 0.0125 8 4 1045
2 Buoys 0.01 4 2 924
3 Terrestrial communication nodes 0.001 2 1 6955
4 Main terrestrial communication node 0.0005 1 1 13914

Changing of the vector X during the implementation of the method

Number of iteration Vector X NΣmaxR
1 {16,8,4,2,1} 439
2 {32,8,4,2,1} 535
3 {32,8,8,2,1} 635
4 {32, 16, 8, 2, 1} 761

Main parameters of the USN’s structure at the 4th iteration

i Type of hierarchy level pmi Xi Xic Ni max
0 Initial sensors 0.025 32 16 1245
1 UUVs 0.0125 16 8 1569
2 Buoys 0.01 8 4 1307
3 Terrestrial communication nodes 0.001 2 1 6955
4 Main terrestrial communication node 0.0005 1 1 13914

Main parameters of the initial USN’s structure

i Type of hierarchy level pMi Xi Xic
0 Initial sensors 0.025 16 8
1 UUVs 0.0125 8 4
2 Buoys 0.01 4 2
3 Terrestrial communication nodes 0.001 2 1
4 Main terrestrial communication node 0.0005 1 1

Main parameters of the USN’s structure at the 2nd iteration

i Type of hierarchy level pmi Xi Xic Ni max
0 Initial sensors 0.025 32 16 1245
1 UUVs 0.0125 8 4 1045
2 Buoys 0.01 4 2 924
3 Terrestrial communication nodes 0.001 2 1 6955
4 Main terrestrial communication node 0.0005 1 1 13914

Main parameters of the USN’s structure at the 3rd iteration

i Type of hierarchy level pmi Xi Xic Ni max
0 Initial sensors 0.025 32 16 1245
1 UUVs 0.0125 8 4 1045
2 Buoys 0.01 8 4 1307
3 Terrestrial communication nodes 0.001 2 1 6955
4 Main terrestrial communication node 0.0005 1 1 13914