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 b 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 b 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 -logp* 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.