Load balancing of complex stochastic tasks using stochastic majorization

The authors consider the static load balancing problem of assigning several large tasks to a (smaller) system of homogeneous processors, where a task´s structure is modeled as a branching process, and all tasks are assumed to have stochastically identical behavior. They show how the theory of majorization can be used to obtain a partial order among possible task assignment. The power of this approach may be summarized as follows: a simple comparison between assignments creates an ordering between them that holds for a variety of objective functions as well as for several statistics such as the mean and variance. This partial ordering is particularly useful when heterogeneous constraints are placed on the numbers of tasks that one may assign to the processors. The results show that if the vector of numbers of tasks assigned to each processor under one mapping is majorized by that of another mapping, then the former mapping is better than the latter with respect to a large number of objective functions. In particular, it is shown how measurements of finishing time, resource utilization, and reliability are all captured by the theory