Standby Allocation Of Reliability Systems And Stochastic Comparisons Of Coherent Structures

Allocation of a standby and stochastic comparisons have been proven to be very useful in applied probability,statistics,reliability engineering,operations research,military technique and other fields.This paper includes two parts.The first part conducts some stochastic comparisons of residual life or inactivity time of general coherent systems,respectively. The second one consists of three aspects:(i)Some stochastic comparisons are conducted between general standby systems with one active unit and one standby one;(ii)Allocation of a single standby in a series system with two independent components is investigated.(iii)Allocation of one of two general standbys in a series system or parallel system with two independent components is researched.At first,coherent systems with a set of independent and identical components are researched.In terms of the concept of signatures,stochastic comparisons on the residual life or inactivity time of two systems with stochastically ordered signatures are conducted.Secondly,general standby systems are developed with one active unit and one standby one.The hazard rate order between two different standby units, is proven to be a sufficient condition for the usual stochastic order between two corresponding standby systems with a common active unit,to hold;On the other hand,the usual stochastic order(likelihood ratio order)between two different active units,is proven to be a sufficient condition for the usual stochastic (reversed hazard rate)order between two corresponding standby systems with a common standby unit,to hold.Thirdly,we investigate the allocation of a single standby in a series system with two independent components.It is proven to be optimal that the standby is assigned to the weaker active unit,in the sense of stochastic precedence order and usual stochastic order,respectively.At last,we investigate the allocation of one of two general standbys in a series system or parallel one with two independent components.On one hand, that the stronger standby is assigned to the weaker active unit of the series system,is proven to be better than,that the weaker standby is assigned to the stronger active one,in the sense of stochastic precedence order and usual stochastic order,respectively;On the other hand,that the stronger standby is assigned to the stronger active unit of the parallel system,is proven to be better than,that the weaker standby is assigned to the weaker active one,in the sense of usual stochastic order.In the last four chapters,some examples are given to illustrate the main results.