Orderings for series and parallel systems comprising heterogeneous exponentiated Weibull-geometric components

Abstract

ABSTRACTLet X1, …, Xn be independent random variables with Xi ∼ EWG(α, β, λi, pi), i = 1, …, n, and Y1, …, Yn be another set of independent random variables with Yi ∼ EWG(α, β, γi, qi), i = 1, …, n. The results established here are developed in two directions. First, under conditions p1 = ⋅⋅⋅ = pn = q1 = ⋅⋅⋅ = qn = p, and based on the majorization and p-larger orders between the vectors of scale parameters, we establish the usual stochastic and reversed hazard rate orders between the series and parallel systems. Next, for the case λ1 = ⋅⋅⋅ = λn = γ1 = ⋅⋅⋅ = γn = λ, we obtain some results concerning the reversed hazard rate and hazard rate orders between series and parallel systems based on the weak submajorization between the vectors of (p1, …, pn) and (q1, …, qn). The results established here can be used to find various bounds for some important aging characteristics of these systems, and moreover extend some well-known results in the literature.