An Exposition of System Reliability Analysis with an
An Exposition of System Reliability Analysis with an Ecological Perspective Prof. Ali Muhammad Ali Rushdi Department of Electrical and Computer Engineering, King Abdulaziz University, Jeddah , Saudi Arabia. 1
Highlights of the presentation • Highlighting the redundancy-connectivity interplay between ecology and reliability. • Computing the survival reliability of species migration between habitat patches. • Covering the case when paths to destination habitat patches share common corridors. • Surveying sp-reduction, inclusion-exclusion, disjointness, and factoring techniques. • Stressing on Boolean-domain manipulations aided by the variable-entered Karnaugh map. 2
Introduction(1 of 5) • System reliability analysis might be described as an advanced application of probability theory. • System reliability analysis is a notable field of study within reliability engineering dealing with expressing the reliability of a system in terms of the reliabilities of its constituent components. • This field aims at utilizing redundancy to maximize system reliability and, in fact, seeks to obtain a more reliable system out of less reliable components. • The field encompasses several important issues more than its name suggests. • It pertains not only to system analysis as such, but also to system design and optimization, quantification of uncertainty, selection of most important components and optimal allocation of redundancy. 3
Introduction(2 of 5) • It is basically based on the algebra of events (a version of set algebra), which is isomorphic to the bivalent or 2 -valued Boolean algebra (switching algebra). • Instead of using the algebra of events, modern system reliability analysis utilizes switching algebra by employing the indicator variables for probabilistic events instead of the events themselves. • A switching or Boolean expression for the indicator variable of system success is sought as a function of the indicator variables of component (corridor) successes. • Switching from the Boolean domain to the probability domain is achieved via the probability or real transform which is a probability expression of system reliability as a function of component (corridor) reliabilities. • Most of the discussions in this paper pertain to various methods for converting a general switching expression into a PRE. 4
Introduction(3 of 5) 5
Introduction(4 of 5) • There are nice prospects of a potentially-fruitful interplay between ecology and system reliability. • This interplay might be centered at (or indicated by) a common thread and a shared issue of network connectivity and redundancy. • However, only few papers attempted to utilize reliability theory in ecology (Jordán, 2000; Rushdi and Hassan, 2015). • To set some foundation for future interaction between reliability and ecology, we offer herein a tutorial exposition of system reliability as applied to ecology. • We treat a problem that was earlier considered by Jordán (2000) and Rushdi and Hassan (2015) concerning survival reliability which is the probability of successful migration of a specific species from a critical habitat patch to one or more destination habitat patches via imperfect heterogeneous corridors. 6
Introduction(5 of 5) • We observe that connectivity and diversity issues are typically treated by ecologists in isolation of the methods followed by general reliability practitioners. • This paper reviews general reliability concepts and demonstrates how they can be effectively applied to a single important problem out of many potential problems pertaining to ecological networks. • Our exposition is not only a review of existing reliability techniques in an ecology setting, but it also introduces and evaluates a new reliability measure of connectivity when the ecological network has several destination habitat patches that share some edges (corridors) in common. 7
Preliminaries-Assumptions • The analysis concerns one particular species, henceforth called the pertinent or concerned species. The analysis does not take into account any characteristic of the species. • The pertinent species is in danger of local extinction in a certain habitat patch called the critical habitat patch. It escapes such extinction by migrating to a new habitat patch (one out of a few destination habitat patches) through imperfect heterogeneous corridors and perfect stepping stones. • Each of the corridors is in one of two states, either good (permeable) or failed (deleted or destroyed). • The migration system is also in one of two states, either successful or unsuccessful. • Destination habitat patches and stepping stones are not susceptible to failure. • Corridor states are statistically independent. 8
Preliminaries-Ecology Nomenclature • Habitat patch: a place where the local population of the pertinent species may reproduce and survive for a long term. • Stepping stone: a relatively small place that helps the migration of the local population of the pertinent species, but is not suitable for its long-term survival. • Ecological corridor: a physical area which connects patches (habitat patches and stepping stones) and makes migration possible for a given species between habitat patches. However, a corridor is not expected to support long-term survival for the species. 9
Preliminaries-Reliability Nomenclature(1 of 2) 10
Preliminaries-Reliability Nomenclature(2 of 2) • • • Reliability-Ready Expression (RRE): An expression in the switching (Boolean) domain, in which logically multiplied (ANDed) entities are statistically independent and logically added (ORed) entities are disjoint. Such an expression can be directly transformed, on a one-to-one basis, to the algebraic or probability domain by replacing switching (Boolean) indicators by their statistical expectations, and also replacing logical multiplication and addition (ANDing and ORing) by their arithmetic counterparts. sp-complex: a coherent system is a series-parallel (s-p) complex iff it has no components (corridors) in series or in parallel, and hence such a system cannot be further simplified via series-parallel reductions. Path (tie-set): an implicant of system success; a set of components (corridors) whose functioning ensures that the system functions. Minimal path: a prime implicant of system success; a path for which all components must function for the system to function. A minimal path in reliability theory can be visually drawn on the network graph as it corresponds to the graph-theoretic concept of a "path". Cut (cut-set): an implicant of system failure; a set of components (corridors) whose failure ensures that the system fails. Minimal cut: a prime implicant of system failure; a cut-set for which all components must fail for the system to fail. A minimal cut-set in reliability theory can be visually drawn on the network graph as it corresponds to the graph-theoretic concept of a "cut-set ". 11
Networks with a single destination habitat patch • This Section reviews techniques for the evaluation of system reliability-unreliability of an ecological network with a single destination habitat patch. • Focus herein is to (a) make this work self-contained, (b) set the base for new techniques of the following sections, (c) stress the viewpoint of starting in the Boolean domain before going to the probability domain, and (d) give priority to attaining reliability expressions that are as compact or minimal as possible. This involves utilization of duality, use of factored expressions, and preservation of statistical independence 12
Networks with a single destination habitat patch (1 of 3) Fig. 2. A small ecological network with a single destination habitat patch D. This network allows further series-parallel reductions, and hence is not an sp-complex network. 13
Sp-reduction, Minimum paths/cut-sets Fig. 3. (a) The small ecological network in Fig. 2 reduced to an sp-complex network, (b) Minimal paths for the network in Fig. 3(a), (c) Minimal cut-sets for the 14 network in Fig. 3(a).
Inclusion-Exclusion (IE) • 15
Disjointness • 16
Boole-Shannon Expansion • There is a technique for replacing the success or failure of the network by a combination of its restrictions or subfunctions referenced to an orthonormal basis. • The success or failure attains a PRE form provided these restrictions or subfunctions are themselves in PRE form. • This technique utilizes the Boole-Shannon Expansion in the Boolean domain. It is equivalent to the Total Probability Theorem in the probability domain and to the Factoring Theorem in the “Graph Domain. ” 17
Boole-Shannon Expansion Fig. 4. A variable-entered Karnaugh map whose entries depict system success for sub-networks obtained for the network in Fig. 3(a) with corridors 2 and 5 as pivoting elements. 18
Networks with several destination habitat patches having no edges in common (a) (b) 19
Networks with several destination habitat patches having no edges in common Table 2. Unreliability U and subnetwork unreliabilities U(A)-U(H ) versus corridor unreliability p U p 0. 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1. 0 UA 0. 0 0. 27100 0. 48800 0. 65700 0. 78400 0. 87500 0. 93600 0. 97300 0. 99200 0. 99900 1. 0 UB 0. 02152 0. 08864 0. 19840 0. 34048 0. 45000 0. 65952 0. 80164 0. 91136 0. 97848 1. 0 UC 0. 00361 0. 12960 0. 26010 0. 40960 0. 56250 0. 70560 0. 82810 0. 92160 0. 98010 1. 0 UD 0. 0 0. 10000 0. 20000 0. 30000 0. 40000 0. 50000 0. 60000 0. 70000 0. 80000 0. 90000 1. 0 UE 0. 0 0. 34390 0. 59040 0. 75990 0. 87040 0. 93750 0. 97440 0. 99190 0. 99840 0. 99990 1. 0 UF 0. 05150 0. 17568 0. 33507 0. 50176 0. 65625 0. 78624 0. 88543 0. 95232 0. 98901 1. 0 UG 0. 00686 0. 04666 0. 13265 0. 26214 0. 42188 0. 59270 0. 75357 0. 88470 0. 97030 1. 0 UH 0. 01000 0. 04000 0. 09000 0. 16000 0. 25000 0. 36000 0. 49000 0. 64000 0. 81000 1. 0 20
Networks with several destination habitat patches having no edges in common Fig. 6. Representation of monotonic Unreliability curves 21
Comparison of numerical results obtained by the purely-additive formula (19) and the all-q formula (20) [Rushdi and Hassan, 2015] Table 3. Comparisons q 0. 8 0. 9999 0. 99999 Fig. 7. Log-scale comparisons 22
Network with several destination habitat patches having edges in common • When the network has several destination habitat patches sharing some edges in common, then it can be analyzed via a “Boole-Shannon Expansion” of the network success with respect to the successes of the common edges. • Alternatively, we can adapt the enumeration techniques to fit the problem of connectivity from one habitat patch to any-out-of-many destination habitat patches. • For this purpose, we utilize the ecological network in Fig. 4(a) of Rushdi and Hassan (2015) (originally taken from Jordán (2000)) as an example. 23
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Fig. 9. A variable-entered Karnaugh map whose entries depict system failure for sub 25 networks obtained for the network in Fig. 8(a) with corridors 2, 4 and 9 as pivoting elements.
Fig. 10. (a) A small ecological network with one-to-any-out-of-many reliability, (b)-(e) Enumeration of minimal cut-sets for the network in (a), (f) A cut-set that is not minimal 26
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Network with several destination habitat patches having edges in common Fig. 12. Enumeration of groups of minimal paths for the network in Fig. 8(a): (a) Direct paths, (b) Paths via stepping stone 1, (c) Paths via stepping stone 2. 28
Conclusion • Bidirectional interplay between ecology and reliability. • A tutorial exposition of system reliability techniques applied to the survival reliability problem in ecology. • A survey existing well-known techniques such as seriesparallel reductions, enumeration of path sets and cut-sets, inclusion-exclusion, disjointness, and Boole-Shannon expansion. • Priority is given to attaining a reliability expression that is as compact or minimal as possible. • Dealing with a more involved situation that arises when there are definitely several destination habitat patches with the paths to them from the critical habitat patch sharing some edges (corridors). 29
References This presentation is loosely based on the following two references and interested audience/readers can extract the elaborated details of references from thereof. • Rushdi, A. M. A. , & Hassan, A. K. (2015). Reliability of migration between habitat patches with heterogeneous ecological corridors. Ecological Modelling, 304, 1 -10. • Rushdi, A. M. A. , & Hassan, A. K. (2016). An Exposition of System Reliability Analysis with an Ecological Perspective. Ecological Indicators, Accepted for Publication. 30
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