Stochastic Bounds and Dependence Properties of Survival Times in a Multicomponent Shock Model

AbstractConsider a system that consists of several components. Shocks arrive according to a counting process (which may be non-homogeneous and with correlated interarrival times) and each shock may simultaneously destroy a subset of the components. Shock models of this type arise naturally in reliability modeling in dynamic environments. Due to correlated shock arrivals, individual component lifetimes are statistically dependent, which makes the explicit evaluation of the joint distribution intractable. To facilitate the development of easily computable tight bounds and good approximations, an analytic analysis of the dependence structure of the system is needed. The purpose of this paper is to provide a general framework for studying the correlation structure of shock models in the setup of a multivariate, correlated counting process and to systematically develop upper and lower bounds for its joint component lifetime distribution and survival functions. The thrust of the approach is the interplay between a newly developed notion, majorization with respect to weighted trees, and various stochastic dependence orders, especially orthant dependence orders of random vectors and orthant dependence orders of stochastic processes. It is shown that the dependence nature of the joint lifetime is inherited from spatial dependence and temporal dependence; that is, dependence among various components due to simultaneous arrivals and dependence over different time instants introduced by the shock arrival process. The two types of dependency are investigated separately and their joint impact on the performance of the system is analyzed. The results are used to develop computable bounds for the statistics of the joint component lifetimes, which are tighter than the product-form bounds under certain conditions. The shock model with a non-homogeneous Poisson arrival process is studied as an illustrative example. The result is also applicable to the cumulative damage model with multivariate shock arrival processes

A discrete MMAP for analysing the behaviour of a multi-state complex dynamic system subject to multiple events.

A complex multi-state system subject to different types of failures, repairable and/or nonrepairable, external shocks and preventive maintenance is modelled by considering a discrete Markovian arrival process with marked arrivals (D-MMAP). The internal performance of the system is composed of several degradation states partitioned into minor and major damage states according to the risk of failure. Random external events can produce failures throughout the system. If an external shock occurs, there may be an aggravation of the internal degradation, cumulative external damage or extreme external failure. The internal performance and the cumulative external damage are observed by random inspection. If major degradation is observed, the unit goes to the repair facility for preventive maintenance. If a repairable failure occurs then the system goes to corrective repair with different time distributions depending on the failure state. Time distributions for corrective repair and preventive maintenance depend on the failure state. Rewards and costs depending on the state at which the device failed or was inspected are introduced. The system is modelled and several measures of interest are built into transient and stationary regimes. A preventive maintenance policy is shown to determine the effectiveness of preventive maintenance and the optimum state of internal and cumulative external damage at which preventive maintenance should be taken into account. A numerical example is presented, revealing the efficacy of the model. Correlations between the numbers of different events over time and in non-overlapping intervals are calculated. The results are expressed in algorithmic-matrix form and are implemented computationally with Matlab.Junta de Andalucía, Spain, under the grant FQM307Ministerio de Economía y Competitividad, España, MTM2017-88708-PEuropean Regional Development Fund (ERDF

Government spending shocks and the multiplier: New evidence from the U.S. based on natural disasters

The literature on estimating macroeconomic effects of fiscal policy requires suitable instruments to identify exogenous and unanticipated spending shocks. So far, the instrument of choice has been military build-ups. This instrument, however, largely limits the analysis to the US as few other countries have been involved in mainly extraterritorial conflicts. Moreover, the expenditure associated with military build-ups affects primarily the defense sector so that the resulting multiplier does not necessarily approximate the effects of changes to general government spending. We propose an alternative instrument: government relief expenditure in the wake of natural disasters which is more similar in its scope to general government spending. We construct a rich data set of natural disasters and the corresponding government responses at the US state level. We apply this methodology both at the state as well as national levels and show that natural disasters serve as a powerful instrument for identifying government spending shocks. Furthermore, we show that the multiplier pertaining to non-defense government spending is higher than the defense-spending multiplier estimated in the literature using military build-ups

A complex multi-state k-out-of-n: G system with preventive maintenance and loss of units

In this study, a multi-state k-out-of-n: G system subject to multiple events is modeled through a Markovian Arrival Process with marked arrivals. The system is composed initially of n units and is active when at least k units are operational. Each unit is multi-state, each of which is classified as minor or major according to the level of degradation presented. Each operational unit may undergo internal repairable or non-repairable failures, external shocks and/or random inspections. An external shock can provoke extreme failure, while cumulative external damage can deteriorate internal performance. This situation can produce repairable and non-repairable failures. When a repairable failure occurs the unit is sent to a repair facility for corrective repair. If the failure is non-repairable, the unit is removed. When the system has insufficient units with which to operate, it is restarted. Preventive maintenance is employed in response to random inspection. The system is modeled in an algorithmic and computational form. Several interesting measures of performance are considered. Costs and rewards are included in the system. All measures are obtained for transient and stationary regimes. A numerical example is analyzed to determine whether preventive maintenance is profitable, financially and in terms of performance.Junta de Andalucía (Spain) FQM-307Ministerio de Economía y Competitividad (España) MTM2017-88708-PEuropean Regional Development Fund (ERDF

On engineering reliability concepts and biological aging

Some stochastic approaches to biological aging modeling are studied. We assume that an organism acquires a random resource at birth. Death occurs when the accumulated dam-age (wear) exceeds this initial value, modeled by the discrete or continuous random vari-ables. Another source of death of an organism is also taken into account, when it occurs as a consequence of a shock or of a demand for energy, which is a generalization of the Strehler-Mildwan’s model (1960). Biological age based on the observed degradation is also defined. Finally, aging properties of repairable systems are discussed. We show that even in the case of imperfect repair, which is certainly the case for organisms, aging slows down with age and eventually can even fade out. This presents another possible explanation for the human mortality rate plateaus.mortality