Majorization of Weighted Trees: A New Tool to Study Correlated Stochastic Systems

Stochastic models with simultaneous arrivals arise naturally in the areas such as synchronized communication networks, flexible manufacturing systems, production/inventory systems, reliability modeling in random environment, etc., with a wide variety of interpretations. Simultaneous arrivals introduce dependence among various components of the system and make the explicit solutions of joint system performance measures either computationally intensive or intractable. This calls for structural analysis of dependence natures of such systems so that the insight gained from the analysis can suggest plausible approaches to develop tight bounds and efficient approximations to key system performance measures.The usual majorization order on a totally ordered index set and Schur convexity are powerful tools for establishing inequalities, in particular, for stochastic systems. However, the investigation on stochastic systems with simultaneous arrivals leads naturally to the comparison of distributions of parameter values on a tree-structured index set. In this paper, we introduce the notion ofmajorization with respect to weighted trees. The fundamental strength of tree majorization lies in its ability to make versatile comparisons over two parameter sets defined on a partially ordered index set. We identify several classes of transformations that provide simple characterizations of tree majorization orders. Finally, we apply the new notion to an assemble-to-order system and to a shock model with simultaneous subsystem failures and study the dependence structure of these systems. Our results show that tree majorization and its interplay with various notions of stochastic orders are useful tools to study the dependency of stochastic systems with multivariate, synchronized input processes.