On cumulative residual extropy of coherent and mixed systems

In system reliability, coherent systems play a central role since without them any system is considered flawed. A system is coherent if all the components are relevant to that system i.e. functioning of every component has an effect in the functioning of the system and the structure of the system is monotone. Monotone structure means that improving any component will not deteriorate the system. A mixed system is a stochastic mixture of coherent systems and any coherent system is a special case of a mixed system. In reliability engineering, one major problem is to compare various coherent and mixed systems so that the better system can be used to increase the overall reliability. Another important problem is to measure the complexity of systems. A highly complex system will naturally have a higher running and maintenance cost associated with it and it is desirable for a reliability engineer to have an understanding regarding the complexity of systems beforehand. In this paper, we address these two problems from an information theoretic approach. Extropy is a measure of information which is the dual of the famous Shannon entropy measure. Recently, a new measure related to extropy, called cumulative residual extropy (CREx), was introduced in the literature by Jahanshahi et al. (Probab Eng Inf Sci 1–21, 2019). This measure is based on the survival function of the underlying random variable and it has some advantages over extropy measure. In this work, we analyze the CREx measure for coherent and mixed systems and develop some comparison results among systems. We also obtain some bounds of CREx of coherent and mixed systems consisting of independent and identically distributed (iid) and dependent and identically distributed (d.i.d.) components. We propose a new divergence measure to calculate the complexity of systems having iid components. Also, we introduced a new discrimination measure to compare various coherent and mixed systems when pairwise comparisons by usual stochastic order is not possible. Finally, we discuss analysis of the CREx measure of coherent systems having heterogeneous components. We also provide applications involving redundancy allocation. Various numerical examples are considered for illustrative purposes.