1 Introduction
2 Literature review
2.1 Verifying signaling systems
2.2 Risk and reliability assessment of railway systems
3 Principles of DFT modeling
3.1 Fault trees
3.1.1 Static fault trees
3.1.2 Dynamic fault trees
3.2 Markov chains
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\({S}\) a finite set of states,
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\({R}:{S}\times {S}\rightarrow {\mathbb {R}_{\ge 0}}\) the transition rate matrix, and
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\({L}: {S}\rightarrow 2^{AP}\) a labeling function assigning a set of atomic propositions \({L}(s) \subseteq AP\) to each state \(s \in {S}\).
3.3 DFT analysis by model checking techniques
4 DFT model for reliability analysis of railway infrastructure
4.1 Modeling train runs
4.2 DFT model for railway infrastructure analysis
4.3 DFT models for infrastructure components
4.3.1 Switches
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Actuation (A): failures in the track switching process, e.g., blade movement, lock actuation,
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Control/Power (C): failures in control or power supply of switch subsystems,
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Detection (D): failure to detect/transmit the position of switch rails/locks,
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Locking (L): failure to lock the switch blades, and
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Permanent Way (P): mechanical failures of rails, stretcher bars, slide chairs, etc.
4.3.2 Slip switches
4.3.3 Crossings
4.3.4 Further components
4.4 Failure rates
Switches | Track segments | Signals | Axle counters | ||||||
---|---|---|---|---|---|---|---|---|---|
\(\lambda _P\) | \(\lambda _A\) | \(\lambda _C\) | \(\lambda _D\) | \(\lambda _L\) | \(\eta _{P,G}\) | Failure (per km) | Failure | Reset request | Failure |
1.46E–4 | 4.98E–4 | 2.26E–4 | 2.32E–4 | 1.28E–4 | 0.11 | 4.4E–4 | 2.9E–4 | 2.8E–4 | 1.1E–4 |
4.4.1 Switches and crossings
4.4.2 Further components
5 Quality metrics
5.1 Metrics for railway reliability modeling
Measure | Model-checking query |
---|---|
Unreliability | \({\mathsf {P}}^{s_0}\left( {\lozenge ^{\le t}\,{\text {Failed}}(\textsf {station})}\right) \) |
MTTF | \({\mathsf {ET}}^{s_0}\left( {\lozenge \,{\text {Failed}}(\textsf {station})}\right) \) |
Unreliability for route i | \({\mathsf {P}}^{s_0}\left( {\lozenge ^{\le t}\,{\text {Failed}}(\textsf {route i})}\right) \) |
Unreliability for train path i | \({\mathsf {P}}^{s_0}\left( {\lozenge ^{\le t}\,{\text {Failed}}(\textsf {tp i})}\right) \) |
Criticality of component v | \(\widetilde{I}_v(t)\) |
Unreliability after component v failed | \(\sum \limits _{s \in S, {\text {Failed}}(v) \in L(s)} {\mathsf {P}}^{s_0}\left( {\lnot {\text {Failed}}(v)\,\mathbf {\mathsf {U}}\,s}\right) \cdot {\mathsf {P}}^s\left( {\lozenge ^{\le t}\,{\text {Failed}}(\textsf {station})}\right) \) |
MTTF after component v failed | \(\sum \limits _{s \in S, {\text {Failed}}(v) \in L(s)} {\mathsf {P}}^{s_0}\left( {\lnot {\text {Failed}}(v)\,\mathbf {\mathsf {U}}\,s}\right) \cdot {\mathsf {ET}}^s\left( {\lozenge \,{\text {Failed}}(\textsf {station})}\right) \) |
Risk Achievement Worth for component v | \({\textsf {Unr}^{t}_{F[v \text { is always failed}]}({\text {Failed}}(\textsf {station}))} \,/\, {\textsf {Unr}^{t}_{F}({\text {Failed}}(\textsf {station}))}\) |
5.1.1 General metrics
5.1.2 Re-routing probability
5.1.3 Criticality of infrastructure elements
5.1.4 Further metrics
6 Evaluation
6.1 Input data
6.2 Infrastructure considered in the analysis
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Switches are the most interesting component from a routability perspective. They experience various modes of degradation where specific directions are unusable—by the blades being “stuck” in one direction—while other routing options are still available.
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Switch failures have been shown to be one of the most important factors in delay build-up [6] and have been the continuous focus of research on design and asset monitoring improvements in reliability engineering.
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Compared to rails, switches are more complex and vulnerable and, hence, fail significantly more often than track segments. In addition, design specifications of railway line segments are fixed, such that reliability can mainly be improved by shortening inspection intervals.
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Failures of signals and axle counters tend to yield milder disruptions compared to switch failures.
6.3 Set-up
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sched: each route set only contains the scheduled route. The unreliability then corresponds to the re-routing probability.
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alt 5: each route set contains the five most feasible routes according to the priorities in the input data.
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single: each component is modeled by a single basic event. The resulting model is a static fault tree, because no MUTEX is present.
7 Results
7.1 Model characteristics
Scenario | Railway | DFT nodes | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
Id | Station | Routing | Detail | Route sets | Routes | Train paths | Comp. | BE | Static | Dynamic |
1 | Aachen | Sched | Single | 59 | 59 | 44 | 54 | 54 | 325 | 0 |
2 | Refined | 545 | 438 | 54 | ||||||
3 | Alt 5 | Single | 14 | 66 | 29 | 46 | 46 | 248 | 0 | |
4 | Refined | 464 | 344 | 46 | ||||||
5 | Herzogenrath | Sched | Single | 11 | 11 | 13 | 22 | 22 | 96 | 0 |
6 | Refined | 194 | 135 | 19 | ||||||
7 | Alt 5 | Single | 10 | 36 | 25 | 25 | 25 | 141 | 0 | |
8 | Refined | 224 | 186 | 22 | ||||||
9 | Mönchengladbach | Sched | Single | 30 | 30 | 31 | 41 | 41 | 229 | 0 |
10 | Refined | 481 | 333 | 48 | ||||||
11 | Alt 5 | Single | 11 | 55 | 41 | 47 | 47 | 259 | 0 | |
12 | Refined | 523 | 371 | 52 | ||||||
13 | Wuppertal | Sched | Single | 26 | 26 | 23 | 27 | 27 | 163 | 0 |
14 | Refined | 300 | 226 | 30 | ||||||
15 | alt 5 | Single | 14 | 49 | 28 | 27 | 27 | 179 | 0 | |
16 | Refined | 300 | 242 | 30 |
7.2 Results for station failure
CTMC construction | Model checking queries | ||||||
---|---|---|---|---|---|---|---|
Id | States | Transitions | Time (s) | Unreliability | MTTF (d) | Time (s) | |
Aachen | 1 | 2 | 2 | 0.00 | 0.997 | 15.04 | 0.00 |
2 | 2049 | 13,313 | 0.43 | 0.996 | 16.38 | 0.01 | |
3 | 769 | 5329 | 0.15 | 0.913 | 36.94 | 0.15 | |
4 | – | – | MO | – | – | – | |
4* | 1,174,596 | 5,891,462 | 103.61 | 0.784 | 57.09 | 2.22 | |
Herzogenr. | 5 | 2 | 2 | 0.00 | 0.879 | 42.69 | 0.00 |
6 | 257 | 1281 | 0.01 | 0.826 | 51.54 | 0.00 | |
7 | 232 | 1489 | 0.02 | 0.704 | 73.86 | 0.00 | |
8 | 13,801 | 153,049 | 2.75 | 0.495 | 127.70 | 0.12 | |
8* | 8636 | 61,743 | 1.06 | 0.496 | 124.62 | 0.04 | |
M’gladbach | 9 | 2 | 2 | 0.00 | 0.995 | 16.94 | 0.00 |
10 | 8193 | 61,441 | 1.25 | 0.991 | 19.01 | 0.05 | |
11 | 22,658 | 228,251 | 3.45 | 0.867 | 47.81 | 0.22 | |
12 | – | – | MO | – | – | – | |
12* | 5,912,302 | 32,950,979 | 480.08 | 0.692 | 72.37 | 15.07 | |
Wuppertal | 13 | 2 | 2 | 0.00 | 0.964 | 27.11 | 0.00 |
14 | 65 | 257 | 0.01 | 0.953 | 29.50 | 0.00 | |
15 | 312 | 1637 | 0.03 | 0.855 | 47.04 | 0.00 | |
16 | 145,925 | 1,631,261 | 36.28 | 0.612 | 89.64 | 1.55 | |
16* | 44,219 | 273,656 | 5.11 | 0.617 | 86.22 | 0.15 |
7.3 Criticality analysis
BDD-based | Model checking-based | Criticality | ||||||
---|---|---|---|---|---|---|---|---|
Id | Elem. | Nodes | Tot. time [s] | States | Time [s] | Tot. time [s] | Results | |
Aachen | 1 | 54 | 55 | 0.98 | [2, 3] | [0.03, 0.03] | 1.49 | [ 0.0025, 0.0031] |
2 | 113 | – | n.a. | [2048, 6144] | [2.73, 5.07] | 551.57 | [– 0.0021, 0.0047] | |
3 | 46 | 22 | 1.00 | [769, 1282] | [1.14, 1.81] | 63.26 | [ 0.0000, 0.1092] | |
4 | 96 | – | n.a. | – | – | MO | – | |
4* | 96 | – | n.a. | [1,109,775, 1,432,106] | [643.96, 789.21] | 68,203.81 | [ 0.0535, 0.2512] | |
Herzogenr. | 5 | 22 | 23 | 0.66 | [2, 3] | [0.02, 0.02] | 0.37 | [ 0.1216, 0.1515] |
6 | 42 | – | n.a. | [257, 768] | [0.24, 0.28] | 10.52 | [– 0.0079, 0.1903] | |
7 | 25 | 25 | 0.84 | [232, 365] | [0.23, 0.31] | 6.50 | [ 0.0000, 0.3690] | |
8 | 48 | – | n.a. | [13,801, 22,880] | [18.17, 30.45] | 1098.22 | [– 0.1185, 0.5631] | |
8* | 48 | – | n.a. | [8636, 12,184] | [6.30, 12,184] | 351.56 | [– 0.1070, 0.5628] | |
M’gladbach | 9 | 41 | 42 | 0.91 | [2, 3] | [0.02, 0.04] | 0.96 | [ 0.0049, 0.0061] |
10 | 97 | – | n.a. | [8192, 24,576] | [6.84, 13.63] | 1260.52 | [– 0.0039, 0.0101] | |
11 | 47 | 1,033 | 1.09 | [22,658, 34,970] | [29.05, 46.82] | 1645.05 | [ 0.0000, 0.1665] | |
12 | 107 | – | n.a. | – | – | MO | – | |
12* | 107 | – | n.a. | [5,486,213, 6,956,578] | [2523.45, 2991.12] | 297,626.65 | [0.2236, 0.3926] | |
Wuppertal | 13 | 27 | 28 | 0.76 | [2, 3] | [0.02, 0.03] | 0.52 | [ 0.0404, 0.0451] |
14 | 60 | – | n.a. | [65, 192] | [0.33, 0.37] | 20.52 | [– 0.0120, 0.0525] | |
15 | 27 | 52 | 1.32 | [302, 408] | [0.37, 0.46] | 10.36 | [ 0.0000, 0.1815] | |
16 | 60 | – | n.a. | [145,925, 259,200] | [312.31, 614.00] | 25,762.23 | [– 0.1021, 0.4307] | |
16* | 60 | – | n.a. | [44,219, 62,996] | [40.48, 53.39] | 2764.00 | [– 0.0661, 0.4283] |