RELIABILITY AND IMPORTANCE MEASURES OF WEIGHTED-k-OUT-OF-n SYSTEM

Recently many popular reliability systems are being extended to weighted systems where weight associated with each component refers to load/capacity of that component. The simplest k-out-of-n system extended under this concept is known as weighted-k-out-of-n system. A weighted-k-out-of-n: G(F) system consists of n components each one having a positive integer weight wi, i = 1, 2, …, n such that the total system weight is w and the system works (fails) if and only if the accumulated weight of working (failed) components is at least k. In this paper, we introduce the concept of Weighted Markov Bernoulli Trial (WMBT) {Xi, i ≥ 0} and study the distribution of [Formula: see text], the total weight of successes in the sequence of n WMBT and conditional distribution of [Formula: see text] given that uth trial results in failure/success. We refer the distribution of [Formula: see text] as Weighted Markov Binomial Distribution (WMBD). Further we discuss application of WMBD and conditional WMBD in evaluation of reliability, Birnbaum reliability importance (B-importance) and improvement potential importance of weighted-k-out-of-n: G/F system. The numerical work is included to demonstrate the computational simplicity of the developed results. Further we compare our study with the existing results in terms of efficiency and find that our results are efficient for large values of k.

A Simple Stochastic Kinetic Transport Model

We introduce a discrete-time microscopic single-particle model for kinetic transport. The kinetics are modeled by a two-state Markov chain, and the transport is modeled by deterministic advection plus a random space step. The position of the particle after n time steps is given by a random sum of space steps, where the size of the sum is given by a Markov binomial distribution (MBD). We prove that by letting the length of the time steps and the intensity of the switching between states tend to 0 linearly, we obtain a random variable S(t), which is closely connected to a well-known (deterministic) partial differential equation (PDE), reactive transport model from the civil engineering literature. Our model explains (via bimodality of the MBD) the double peaking behavior of the concentration of the free part of solutes in the PDE model. Moreover, we show for instantaneous injection of the solute that the partial densities of the free and adsorbed parts of the solute at time t do exist, and satisfy the PDEs.

A Simple Stochastic Kinetic Transport Model

We introduce a discrete-time microscopic single-particle model for kinetic transport. The kinetics are modeled by a two-state Markov chain, and the transport is modeled by deterministic advection plus a random space step. The position of the particle after n time steps is given by a random sum of space steps, where the size of the sum is given by a Markov binomial distribution (MBD). We prove that by letting the length of the time steps and the intensity of the switching between states tend to 0 linearly, we obtain a random variable S(t), which is closely connected to a well-known (deterministic) partial differential equation (PDE), reactive transport model from the civil engineering literature. Our model explains (via bimodality of the MBD) the double peaking behavior of the concentration of the free part of solutes in the PDE model. Moreover, we show for instantaneous injection of the solute that the partial densities of the free and adsorbed parts of the solute at time t do exist, and satisfy the PDEs.

Multimodality of the Markov Binomial Distribution

We study the shape of the probability mass function of the Markov binomial distribution, and give necessary and sufficient conditions for the probability mass function to be unimodal, bimodal, or trimodal. These are useful to analyze the double-peaking results of a reactive transport model from the engineering literature. Moreover, we give a closed-form expression for the variance of the Markov binomial distribution, and expressions for the mean and the variance conditioned on the state at time n.

Multimodality of the Markov Binomial Distribution

We study the shape of the probability mass function of the Markov binomial distribution, and give necessary and sufficient conditions for the probability mass function to be unimodal, bimodal, or trimodal. These are useful to analyze the double-peaking results of a reactive transport model from the engineering literature. Moreover, we give a closed-form expression for the variance of the Markov binomial distribution, and expressions for the mean and the variance conditioned on the state at time n.