Comparisons for series and parallel systems with discrete Weibull components via separate comparisons of parameters

‎In this paper‎, ‎we obtain the usual stochastic order of series and parallel systems comprising heterogeneous discrete Weibull (DW) components‎. ‎Suppose X۱,...,Xn and Y۱,...,Yn denote the independent component¢s lifetimes of two systems such that Xi ~ DW(bi‎, ‎p i) and Yi ~ DW(b*i‎, ‎p *i), i=۱,...,n. We obtain the usual stochastic order between series systems‎ ‎when the vector \boldsymbolb is switched to the vector b*with respect to the majorization order‎, ‎and when the vector log (۱-p) is switched to the vector log (۱-p *) in the sense of the weak supermajorization order‎. ‎We also discuss the usual stochastic order between series systems by using the unordered majorization between the vectors log(۱-p) and log (۱-p *), and the p-majorization order between the parameters \boldsymbolb and b*. It is also shown that the usual stochastic order between parallel systems comprising heterogeneous discrete Weibull components when the vector log p is switched to the vector log p *in the sense of the weak supermajorization order‎. ‎These results enable us to find some lower bounds for the survival functions of a series and parallel systems consisting of independent heterogeneous discrete Weibull components.

کلیدواژه ها:

Discrete Weibull distribution ، P-majorization order ، Unordered majorization order ، Weak submajorization order ، Weak supermajorization order