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 X1,...,Xn and Y1,...,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=1,...,n. We obtain the usual stochastic order between series systems when the vector oldsymbolb is switched to the vector b*with respect to the majorization order, and when the vector log (1-p) is switched to the vector log (1-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(1-p) and log (1-p *), and the p-majorization order between the parameters oldsymbolb 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.