Some Studies On Reliability Based On The Residual Life And The Inactivity Time

As two important notions in theory of reliability, residual life and inactivity time have been paid much attentions in the past several decades. In this thesis, we mainly focus on the Renyi entropies for the residual life and the inactivity time, NBUmg (new better than used in moment generating function order) and IMIT (increasing mean inactivity time) life distributions, and a geometric process repair model for a repairable system with random lead time.The first part discusses the moment generating function (mgf) order and NBUmg life distribution. We prove the closure property of mgf order under the taking of independent but non-identical random sum. For a NBUmg unit, we carry out stochastic comparisons among the block replacement policy, the age replacement policy, the complete repair policy and the minimal repair policy, both the block replacement policy and the age replacement policy are proved to reduce the number of failures of the unit in the sense of mgf order.The second one deals with Renyi entropies for the residual life and the inactivity time. We prove that the decreasing residual Renyi entropy property of a stochastically smaller (in the sense of likelihood ratio order) random variable is preserved by a larger one, and based on this result, it is preserved by both the formation of parallel systems and the record value. Also, we prove that the increasing inactivity Renyi entropy property of a stochastically larger (in the sense of likelihood ratio order) random variable is preserved by a smaller one, and hence it is also preserved by the formation of series systems. Fur-thermore, we compare a random variable with its weighted version in terms of residual Renyi entropy.In the third part, we propose a new nonparametric testing method for IMIT prop-erty. Through evaluating the asymptotic Pitman efficiency, we compare the new testing method with two other related ones in literature. Meanwhile, Edge-worth expansion is employed to improve the new testing method. At last, we present some numerical results to demonstrate the new testing method. Finally, we study a simple repairable system with random lead time and analyze the effect of random lead time to the system. Assume that the successive operating times of the system form a decreasing geometric process, while the consecutive repair times of the system form an increasing geometric process. Under this assumption, we derive two important reliability indices of the system:the availability and the rate of occurrence of failures (ROCOF). Meanwhile, we investigate a repair replacement policy N based on the number of failures of the system and give the average cost rate of the system. Also, we present a numerical example for policy N.