Stochastic comparisons of residual lifetimes and inactivity times of coherent systems with dependent identically distributed components

In this paper we compare the residual lifetime of a used coherent system of age t>0$t>0$ with the lifetime of the similar coherent system made up of used components of age t. Here ‘similar’ means that the system has the same structure and the component lifetimes have the same dependence (joint reliability copula). Some comparison results are obtained for the likelihood ratio order, failure rate order, reversed failure rate order and the usual stochastic order. Similar results are reported for comparing inactivity time of a coherent system with lifetime of similar coherent system having component lifetimes same as inactivity times of failed components.     

Component and system active redundancies for coherent systems with dependent components

The prevailing engineering principle that redundancy at the component level is superior to redundancy at the system level is generalized to coherent systems with dependent components. Sufficient (and necessary) conditions are presented to compare component and system redundancies by means of the usual stochastic, hazard rate, reversed hazard rate, and likelihood ratio orderings. Explicit numerical examples are provided to illustrate the theoretical findings. Some related results in the literature are generalized and extended. Copyright © 2017 John Wiley & Sons, Ltd.     

Redundancy allocation at component level versus system level

In this paper, we treat the problem of stochastic comparison of standby [active] redundancy at component level versus system level. In the case of standby redundancy, we present some interesting comparison results of both series systems and parallel systems in the sense of various stochastic orderings for both the matching spares case and non-matching spares case, respectively. In the case of active redundancy, a likelihood ratio ordering result of series systems is presented for the matching spares case; and for the non-matching spares case, a counterexample is provided to show that there does not exist similar result even for the hazard rate ordering. The results established here strengthen and generalize some of those known in the literature. Some numerical examples are also provided to illustrate the theoretical results.     

On the optimal allocation of two active redundancies in a two-component series system

Components C₁ and C₂ form a series system. Suppose we can allocate the spare R₁ in parallel with C₁ and the spare R₂ in parallel with C₂, or otherwise, allocate R₁ with C₂ and R₂ with C₁. In this paper, we compare these two options using hazard rate ordering.     

Stochastic comparisons of lifetimes of series and parallel systems with dependent and heterogeneous components

This work considers stochastic comparisons of lifetimes of series and parallel systems with dependent and heterogeneous components having lifetimes following the proportional odds (PO) model. The joint distribution of component lifetimes is modeled by Archimedean survival copula. We discuss some potential applications of our findings in system reliability and actuarial science.     

On the hazard rate and reversed hazard rate orderings in two-component series systems with active redundancies

The lifetimes of two-component series systems with two active redundancies are compared using the hazard rate and the reversed hazard rate orders. We study the problem of where to allocate the spares in a system to obtain the best configuration. We compare redundancy at component level vs. system level using the likelihood ratio order. For this problem we find conditions under which there is no hazard rate ordering between the lifetimes of the systems.     

Orderings of coherent systems with randomized dependent components

Consider a general coherent system with independent or dependent components, and assume that the components are randomly chosen from two different stocks, with the components of the first stock having better reliability than the others. Then here we provide sufficient conditions on the component’s lifetimes and on the random numbers of components chosen from the two stocks in order to improve the reliability of the whole system according to different stochastic orders. We also discuss several examples in which such conditions are satisfied and an application to the study of the optimal random allocation of components in series and parallel systems. As a novelty, our study includes the case of coherent systems with dependent components by using basic mathematical tools (and copula theory).     

Stochastic comparisons of series and parallel systems with randomized independent components

Consider a series or parallel system of independent components and assume that the components are randomly chosen from two different batches, with the components of the first batch being more reliable than those of the second. In this note it is shown that the reliability of the system increases, in usual stochastic order sense, as the random number of components chosen from the first batch increases in increasing convex order. As a consequence, we establish a result analogous to the Parrondo’s paradox, which shows that randomness in the number of components extracted from the two batches improves the reliability of the series system.     

On stochastic comparisons of series systems with heterogeneous dependent and independent location-scale family distributed components

This paper carries out comparisons of heterogeneous series systems with location-scale family distributed components It is shown that the systems with dependent components in series sharing Archimedean copula with more dispersion in the location or scale parameters result in better performance in the sense of the usual stochastic order. Moreover, if the components are independently distributed, it is possible to obtain more generalized results as compared to the dependent set-up.     

On allocation of redundancies in two-component series systems

Allocation of active [standby] redundancies in a system is a topic of great interest in reliability engineering and system safety because optimal configurations can significantly increase the reliability of a system. In this paper, we study the problem of allocating two exponentially distributed active [standby] redundancies in a two-component series system using the tools of stochastic ordering. We establish two interesting results on likelihood ratio ordering which have no restriction on the parameters.     

On redundant weighted voting systems with components having stochastic arrangement increasing lifetimes

As one generalization of the k-out-of-n structure, the weighted voting system has been paid much attention during the past two decades. This paper has a further study on active redundancies allocation to weighted voting reliability systems of components having LWSAI lifetimes. For redundancies with SAI lifetimes, allocating a more reliable redundancy to a weaker and more heavily weighted component is found to produce a more reliable system in the sense of having higher reliability. Also, in the context of redundancies with identically distributed lifetimes, we show that allocating more redundancies to a weaker and more heavily weighted component produces a more reliable system. Some numerical examples are presented to illustrate the main results as well.     

Stochastic comparisons for bulk queues

We consider two important classes of single-server bulk queueing models: M^(X)/G^(Y)/1 with Poisson arrivals of customer groups, and G^(X)/m^(Y)1 with batch service times having exponential density. In each class we compare two systems and prove that one is more congested than the other if their basic random variables are stochastically ordered in an appropriate manner. However, it must be recognized that a system that appears congested to customers might be working efficiently from the system manager's point of view. We apply the results of this comparison to (i) the family {M/G^(s)/1,s 1} of systems with Poisson input of customers and batch service times with varying service capacity; (ii) the family {G^(s)/1,s 1} of systems with exponential customer service time density and group arrivals with varying group size; and (iii) the family {M/D/s,s 1} of systems with Poisson arrivals, constant service time and varying number of servers. Within each family, we find the system that is the best for customers, but this turns out to be the worst for the manager (or vice versa). We also establish upper (or lower) bounds for the expected queue length in steady state and the expected number of batches (or groups) served during a busy period. The approach of the paper is based on the stochastic comparison of random walks underlying the models.This research was partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University.     

Stochastic approximation with dependent noise

In this work we derive the usual limit laws (weak and strong convergence, central limit theorem, invariance principle) for stochastic approximation with stationary noise. The idea is to introduce an artificial sequence, related to the SA scheme, but which clearly obeys the desired limit law. This sequence is subtracted from the SA scheme and the remainder, which behaves more or less deterministically, is shown to vanish using simple limit arguments.     

On the joint signature of several coherent systems with some shared components

The concept of joint signature (JS), introduced by Navarro, Samaniego, and Balakrishnan (2010), is a useful tool for investigating the joint reliability of two coherent systems with shared components. In this article, by considering several coherent systems which share some components, with independent and identically distributed lifetimes, we obtain a pseudo-mixture representation for the joint distribution of the systems lifetimes based on a general notion of joint signature which is referred to as generalized joint signature (GJS). It is shown how the GJS is separated from the effect of the components’ lifetime distribution and this relationship helps us to represent the GJS as a two-dimensional matrix instead of a high-dimensional one. Based on the GJS, some ordering results are obtained for comparing two clusters of coherent systems with some shared components. Several examples are provided to illustrate the results established here.     

Scheduling starting times for an active redundant system with non-identical lifetimes

Here we examine an active redundant system with scheduled starting times of the units. We assume availability of n non-identical, non-repairable units for replacement or support. The original unit starts its operation at time s₁ = 0 and each one of the (n − 1) standbys starts its operation at scheduled time s_i (i = 2, …, n) and works in parallel with those already introduced and not failed before s_i. The system is up at times s_i (i = 2, …, n), if and only if, there is at least one unit in operation. Thus, the system has the possibility to work with up to n units, in parallel structure. Unit-lifetimes T_i (i = 1, …, n) are independent with cdf F_i, respectively. The system has to operate without inspection for a fixed period of time c and it stops functioning when all available units fail before c. The probability that the system is functioning for the required period of time c depends on the distribution of the unit-lifetimes and on the scheduling of the starting times s_i. The reliability of the system is evaluated via a recursive relation as a function of the starting times s_i (i = 2, …, n). Maximizing with respect to the starting times we get the optimal ones. Analytical results are presented for some special distributions and moderate values of n.     

On a problem of optimal harvesting from a stochastic system with a jump component

In this paper, we study a problem of optimal harvesting from a stochastic system modeled by a geometric Lévy process. A verification theorem of the variational inequality type is also given and proved. The paper has been motivated by I. Elsanosi et al. [Stochastics Stochastics Rep. (2000)], where the authors considered an optimal harvesting problem with price dynamics following a stochastic differential delay equation.     

Active redundancy allocation for a k-out-of-n:F system of dependent components

We characterize active redundancy through compensator transform and use the reverse rule of order 2 (RR₂) property between compensator processes to investigate the problem of where to allocate a spare in a k-out-of-n:F system of dependent components through active redundancy.     

The behavior of warm standby components with respect to a coherent system

This paper is concerned with a coherent system consisting of active components and equipped with warm standby components. In particular, we study the random quantity which denotes the number of surviving warm standby components at the time of system failure. We represent the distribution of the corresponding random variable in terms of system signature and discuss its potential utilization with a certain optimization problem.     

Stochastic Kolmogorov-type system with infinite delay

The paper considers a stochastic functional Kolmogorov-type population system with infinite delay under the general probability measures. Main aim is to show that the environment noise will not only suppress a potential population explosion but also make the solution be stochastically ultimately bounded and asymptotic stable. Moreover, two stochastic functional Lotka-Volterra equations as examples are provided to illustrate the main results.     

Stochastic ordering of folded normal random variables

Horn (1988) proved a necessary and sufficient condition for the stochastic ordering of an arbitrary folded normal variable and a standard folded normal variable. This paper studies the more general case and presents the necessary and sufficient conditions for the stochastic ordering of two arbitrary folded normal variables.