on hazard rate ordering of $p$-spacings between failures in sequential (n-r+1)-out-of-n systems

‎the hazard rate is an important measures to study the lifetime random variables in reliability theory‎, ‎survival analysis and stochastic modeling‎. ‎in this article‎, ‎we study the hazard rate ordering of $p$-spacings in sequential $(n-r+1)$-out-of-$n$ systems with different structures‎, ‎including resolving an open problem‎. ‎since we do not impose restrictions as previous studies did‎, ‎our findings yield new results for various useful models of ordered random variables‎, ‎too‎. ‎finally‎, ‎an illustrative example is also given‎.