An exposition of system reliability analysis with an ecological perspective

Highlights

• We emphasize the redundancy-connectivity interplay between ecology and reliability.

• We compute the survival reliability of species migration between habitat patches.

• We cover the case when paths to destination habitat patches share common corridors.

• We survey reduction, inclusion–exclusion, disjointness, and factoring techniques.

• We stress Boolean-domain manipulations aided by the variable-entered Karnaugh map.

Abstract

This paper attempts to set the stage for a prospective interplay between ecology and reliability theory concerning the common issues of network connectivity and redundancy. The paper treats the problem of survival reliability which is the probability of successful migration of a specific species from a critical habitat patch to destination habitat patches via heterogeneous imperfect corridors. The paper surveys techniques of system reliability in an ecological setting and contributes methods for computing a new measure of reliability that arises when paths to destination habitat patches share common corridors. Care is taken to ensure that the reliability expressions obtained are as compact as possible and to check them for correctness.

Introduction

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 (Von Neumann, 1956, Moore and Shannon, 1956). 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.

System reliability analysis might be described as an advanced application of probability theory. 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 (see Fig. 1) is achieved via the probability or real transform (Papaioannou et al., 1975, Kumar and Breuer, 1981, Rushdi, 1983b, Heidtmann, 1991, Jain, 1996, Rushdi and Ghaleb, 2015), which is a probability expression of system reliability as a function of component (corridor) reliabilities. Table 1 cites well-known results from elementary probability theory showing how probability formulas are simplified when events in union are mutually exclusive or intersected events are statistically independent (i.e., when their indicators are disjoint or independent, respectively). Based on these ideas, many algorithms have emerged for converting the switching (Boolean) expression for the indicator variable of system success into a probability-ready expression (PRE), i.e., into an expression that is directly convertible, on a one-to-one basis, to the corresponding probability transform. Note that in a PRE

• all ORed terms (products) are disjoint, and

• all ANDed terms (sums) are statistically independent.

The conversion from a PRE to a probability expression is trivially achieved by replacing Boolean variables by their expectations, AND operations by multiplications, and OR operations by additions (Rushdi and Goda, 1985, Rushdi, 1987b, Rushdi, 1988b, Rushdi, 1990, Rushdi and AbdulGhani, 1993, Rushdi and Ba-Rukab, 2005a, Rushdi and Ba-Rukab, 2005b). Most of the discussions in this paper pertain to various methods for converting a general switching expression into a PRE.

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 (see, e.g., Fahrig and Paloheimo, 1988, Goodwin and Fahrig, 2002, Jordán, 2003, Jordán et al., 2003, Schooley and Wiens, 2003, Kindlmann and Burel, 2008, Baranyi et al., 2011, Gallardo et al., 2011, Orsi et al., 2011, Reza and Abdullah, 2011, Saura et al., 2011, Foltête and Giraudoux, 2012, Blazquez-Cabrera et al., 2014, Hoque et al., 2014, Pinto et al., 2014). However, only few papers attempted to utilize reliability theory in ecology (Jordán, 2000, Rushdi and Hassan, 2015), and the authors are unaware of any attempts to make use of ecological paradigms and concepts in reliability theory. 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. 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.

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. The paper will be of a significant impact if it succeeds in triggering some ecologists to reformulate some of their connectivity problems in a network-reliability context, and then to challenge reliability experts to apply their advanced tools to these problems. Beneficial mutual interaction would arise between ecology and reliability fields, and an interdisciplinary subfield might emerge.

The organization of the rest of this paper is as follows. Section 2 lists our assumptions, notation and certain useful nomenclature in the ecology and reliability domains. Section 3 reviews techniques for the evaluation of system reliability–unreliability of an ecological network with a single destination habitat patch. These are the old techniques used to handle two-terminal (source-to-terminal) reliability in classical reliability theory (Rushdi and Al-Khateeb, 1983). Sections 4 Networks with several destination habitat patches having no edges in common, 5 Network with several destination habitat patches having edges in common extend the work in Section 3 to treat networks with several destination habitat patches. Section 4 demonstrates that the reliability analysis of an ecological network that has several destination habitat patches sharing no edges in common reduces to the reliability analysis of several ecological networks each having a single destination habitat patch (i.e., to problems of the kind discussed in Section 3). Section 5 deals with the problem of networks with several destination habitat patches that share some edges in common. This problem is similar to (albeit with a subtle difference from) the problem of broadcast reliability in classical reliability theory (Bienstock, 1988, Kulkarni and Trahan, 1991, Ball et al., 1992, Bao et al., 1998, Ramachandran et al., 2007), which considers connectivity from one node to all the nodes in a set of other nodes. The problem in Section 5 considers connectivity from one node to at least one node among the nodes in a set of other nodes. The problem is solved via enumeration of cut-sets, path-sets, or factored groups of paths. This enumeration reveals certain particular features that are specific to ecology networks. The symbolic reliability–unreliability expressions obtained herein are all checked via the exhaustive tests set by Rushdi (1983b). Section 5 concludes the paper and points out new directions for further research.