On the information properties of working used systems using dynamic signature

Abstract

Shannon entropy is a useful criterion for measuring the uncertainty (predictability) of lifetimes of engineering systems. In this work, we provide an explicit expression for the entropy of the residual lifetime of a working used system with exactly i failed components at time t, using dynamic signature. We also present additional results on bounds and ordering properties for the proposed entropy. We find an expression for the Jensen‐Shannon (JS) divergence of the residual lifetime of a working used system, and show that the JS divergence of the system is equal to that of its dual. An improved bound for the JS divergence is also obtained. Finally, based on the proposed entropy, we introduce a criterion using which we can prefer a system. This criterion, a distribution‐free measure that only depends on the dynamic signature, ranks systems based on their closeness to extreme systems.

Tsallis Entropy of a Used Reliability System at the System Level

“…To see some recent research on bounds on the uncertainty of the lifetime of coherent systems, we refer the reader, for example, to Refs. [ 15 , 16 , 23 ] and the references there. In the following theorem, we provide bounds on the residual Tsallis entropy of the lifetime of the coherent system in terms of the residual Tsallis entropy of the parent distribution .…”

TRényi’s Entropy for the Past Life Distribution With Application in Inactive Coherent Systems

“…One promising approach is to use previous Rényi entropy bounds, which have been shown to accurately approximate the lifetime of coherent systems under such circumstances. Toomaj and Doostparast [28,29] pioneered the development of such barriers for a new system, while more recently Toomaj et al [30] has extended this work by deriving bounds on the entropy of a coherent system when all its components are working; see also Mesfioui et al [15]. In the following theorem, we introduce new bounds on the past Rényi entropy of the coherent system's lifetime, expressed in terms of the past Rényi entropy of the higher-order distribution H α (X t ).…”

Cumulative Residual Entropy of the Residual Lifetime of a Mixed System at the System Level

“…This is a common practical issue that arises in many applications. To address this challenge, researchers have recently developed bounds for the uncertainty of the lifetimes of mixed systems, as discussed in studies such as [14], and their related references. In the following theorem, we provide bounds for the residual cumulative residual entropy of a mixed system's lifetime, in terms of the residual entropy of the parent distribution E (X t ).…”

“…Most recently, Toomaj [13] and Toomaj et al [8] delved into stochastic comparisons of R'enyi entropy and cumulative residual entropy of mixed systems, respectively, demonstrating that both systems yield similar signatures. Exciting recent research has delved into the study of coherent systems comprising n components, where all components are alive at time t. Toomaj et al [14] investigated the Shannon differential entropy of the system's lifetime, while [15] explored the Tsallis entropy of the same. Mesfioui et al [16] also investigated the Tsallis entropy of coherent systems with identical properties, making this a fascinating area of current research.…”