Stochastic Comparisons of Lifetimes of Series and Parallel Systems with Dependent Heterogeneous MOTL-G Components under Random Shocks

To measure the magnitude among random variables, we can apply a partial order connection defined on a distribution class, which contains the symmetry. In this paper, based on majorization order and symmetry or asymmetry functions, we carry out stochastic comparisons of lifetimes of two series (parallel) systems with dependent or independent heterogeneous Marshall–Olkin Topp Leone G (MOTL-G) components under random shocks. Further, the effect of heterogeneity of the shape parameters of MOTL-G components and surviving probabilities from random shocks on the reliability of series and parallel systems in the sense of the usual stochastic and hazard rate orderings is investigated. First, we establish the usual stochastic and hazard rate orderings for the lifetimes of series and parallel systems when components are statistically dependent. Second, we also adopt the usual stochastic ordering to compare the lifetimes of the parallel systems under the assumption that components are statistically independent. The theoretical findings show that the weaker heterogeneity of shape parameters in terms of the weak majorization order results in the larger reliability of series and parallel systems and indicate that the more heterogeneity among the transformations of surviving probabilities from random shocks according to the weak majorization order leads to larger lifetimes of the parallel system. Finally, several numerical examples are provided to illustrate the main results, and the reliability of series system is analyzed by the real-data and proposed methods.

Reliability optimization in series-parallel and parallel-series systems subject to random shocks

Consider a series-parallel (parallel-series) system composed of [Formula: see text] subsystems subject to [Formula: see text] independent stochastically identical nonhomogeneous Poisson processes, its reliability optimization problems are analysed in this paper. The reliability function of the series-parallel (parallel-series) system is presented. We study the optimal subsystems grouping policy to maximize the system reliability. It is shown that the series-parallel (parallel-series) system is more reliable (unreliable) when more subsystems share a common combined shock process. Different allocation policies of components are compared in terms of majorization order. Some examples are given to illustrate the results.

h-Type indices, partial sums and the majorization order

We study the array of partial sums, P X, of a given array X in terms of its h-type indices. Concretely, we show that h( P X) can be described in terms of the Lorenz curve of the array X and obtain a relation between the sum of the components of P X and the Gini index of X. Moreover, we obtain sharp lower and upper bounds for h-type indices of P X.

SOME NEW RESULTS ON THE LARGEST ORDER STATISTICS FROM MULTIPLE-OUTLIER GAMMA MODELS

In this paper, we carry out stochastic comparisons of the largest order statistics arising from multiple-outlier gamma models with different both shape and scale parameters in the sense of various stochastic orderings including the likelihood ratio order, star order and dispersive order. It is proved, among others, that the weak majorization order between the scale parameter vectors along with the majorization order between the shape parameter vectors imply the likelihood ratio order between the largest order statistics. A quite general sufficient condition for the star order is presented. The new results established here strengthen and generalize some of the results known in the literature. Numerical examples and applications are also provided to explicate the theoretical results.

COMMENTS ON “ORDERING PROPERTIES OF ORDER STATISTICS FROM HETEROGENEOUS POPULATIONS”

Balakrishnan and Zhao does an excellent job in this issue at reviewing the recent advances on stochastic comparison between order statistics from independent and heterogeneous observations with proportional hazard rates, gamma distribution, geometric distribution, and negative binomial distributions, the relation between various stochastic order and majorization order of concerned heterogeneous parameters is highlighted. Some examples are presented to illustrate main results while pointing out the potential direction for further discussion.

Discrete Dynamics by Different Concepts of Majorization

For the description of complex dynamics of open systems, an approach is given by different concepts of majorization (order structure). Discrete diffusion processes with both invariant object number and sink or source can be represented by the development of Young diagrams on lattices. As an experimental example, we investigated foam decay, dominated by sinks. The relevance of order structures for the characterization of certain processes is discussed.

AN AGING CONCEPT BASED ON MAJORIZATION

In this article, we introduce a new dispersion order weaker than the classic dispersion order discussed by Lewis and Thompson (1981). We study the equivalence of this order with the majorization order under the assumption of unimodality. Finally, we use this equivalence to characterize the IFR aging notion for unimodal distributions by means of the notion of decreasing in randomness.