Reliability Applications of Multivariate Exponential Distributions, Downtime Modeling, and Optimal Replacement of Deteriorating Parts.

摘要

In the area of application of probabilistic and stochastic modeling, sequences of random variables evolving over time are usually assumed to be sequences of independent random variables or Markov sequences. Here we introduce and apply a multivariate exponential distribution which may describe Markov or non-Markov sequences. The present work has examined one particular class of multivariate exponential distributions which preserve Markov sequence properties for both modeling of downtime distributions and modeling of stages of component deterioration. In downtime modeling, we study the distribution of the sum of several dependent random variables and compare the result with the distribution of a sum of independent variables as well as with the lognormal distribution. In deterioration modeling, we consider part replacement rules based on observation of the state of the part's quality and on specified reward structures. We identify the rate of deterioration by examining how long the component stays in each state and use dynamic programming to set up recursive optimization equations such that the expected reward per unit time is maximized. Sufficient conditions are given under which the optimum replacement rule has a very simple structure. (Author)