Functional survivability analysis of structurally complex technical systems

Abstract

Aim. The paper analyzes the functional survivability of structurally complex technical systems. This approach is the evolution of the structural survivability paradigm, when the system/element failure criterion is binary. The paper shows that given a wide variety of probabilistic scenarios of adverse effects (AE) on a system, an invariant model kernel is identified that is responsible for the interpretation of functional redundancy. The aim is to identify the proportion of retained operable states within the acceptable computational time, when the fixed number u of elements is disabled as the result of AE. In this case the analysis of survival law is conducted at the confluence of functional redundancy analysis and probabilistic AE models of arbitrarily wide variety. Methods . A technical system is considered a controllable cybernetic system equipped with specialized survival facilities (SF). System survivability analysis uses logic and probabilistic methods, as well as the results of the combinatorial theory of random allocation. It is assumed that: a) AE are localized and single (one effect affects exactly one element); b) each of the system’s elements has a binary logic (operability – failure) and zero resilience, i.e. destruction after one effect is guaranteed. Subsequently this assumption is generalized for the case of r-fold AE and L-resilient element. Results. The paper reconstructs a number of variants of the destruction law and survivability functions of technical systems. It is identified that those distributions are based on prime and generalized Morgan numbers, as well as Stirling numbers of the second kind that can be recovered using the simplest recurrence formulas. While the assumptions of the mathematical model are generalized for the case of nr-fold AE and L-resilient elements, the generalized Morgan numbers involved in the estimation of the destruction law are identified using the random allocation theory by means of n-fold differentiation of the generating polynomial. In this case it does not appear to be possible to establish a recursive relation between the generalized Morgan numbers. It is shown that under homogeneous assumptions regarding the survivability model (equally resilient system elements, equally probable AEs) in the correlation kernel for the system survivability function, regardless of the destruction law, is the functional redundancy vector F(u, e ), where u is the number of affected elements, e is the system’s limiting efficiency criterion, below which its functional failure is diagnosed, F(u, e ) is the number of system states operable in terms of e under u failures (destructions) of its elements. Conclusions. Point models of survivability are an excellent tool of express analysis of structurally complex systems and tentative estimation of survivability functions. The most simple assumptions of structural survivability can be generalized in cases when the system’s operability logic is not binary, yet is associated with the level of system operation efficiency. In this case we must speak of functional survivability. The PNP computational complexity of the survivability evaluation problem does not allow solving it by means of a simple enumeration of the system states and AE variants. Ways must be found of avoiding simple enumeration, e.g. by using conversion of the system operability function and its decomposition by means of generalized logical and probabilistic methods.