Bivariate characterization of some stochastic order relations

Bivariate (and multivariate) functional representations for the following stochastic order relations are established: the likelihood ratio ordering, hazard rate ordering, and the usual stochastic ordering. The motivation of the study is (i) to provide a general approach that supports the ‘pairwise interchange' arguments widely used in various settings, and (ii) to develop new notions of stochastic order relations so that dependent random variables can be meaningfully compared. Applications are illustrated through problems in stochastic scheduling, closed queueing network and reliability.