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

Abstract

‎In this paper‎, ‎we obtain the usual stochastic order of series and parallel systems comprising heterogeneous discrete Weibull (DW) components‎. ‎Suppose X₁,...,X_n and Y₁,...,Y_n denote the independent component<span style="font-family: symbol;">¢</span>s lifetimes of two systems such that X_i <span style="font-family: symbol;">~</span> DW(<span style="font-family: symbol;">b</span>_i‎, ‎p_i) and Y_i <span style="font-family: symbol;">~</span> DW(<span style="font-family: symbol;">b</span>^*_i‎, ‎p^*_i), i=1,...,n. We obtain the usual stochastic order between series systems‎ ‎when the vector boldsymbol<span style="font-family: symbol;">b</span> is switched to the vector <span style="font-family: symbol;">b</span>^*with respect to the majorization order‎, ‎and when the vector log (1<span style="font-family: symbol;">-</span>p) is switched to the vector log (1<span style="font-family: symbol;">-</span>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<span style="font-family: symbol;">-</span>p) and log (1<span style="font-family: symbol;">-</span>p^*), and the p-majorization order between the parameters boldsymbol<span style="font-family: symbol;">b</span> and <span style="font-family: symbol;">b</span>^*. 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.