Abstract: With the growing popularity of cloud-based data center networks (DCNs), task resource allocation has become more and more important to the efficient use of resource in DCNs. This paper considers provisioning the maximum admissible load (MAL) of virtual machines (VMs) in physical machines (PMs) with underlying tree-structured DCNs using the hose model for communication. The limitation of static load distribution is that it assigns tasks to nodes in a once-and-for-all manner, and thus requires a priori knowledge of program behavior. To avoid load redistribution during runtime when the load grows, we introduce maximum elasticity scheduling, which has the maximum growth potential subject to the node and link capacities. This paper aims to find the schedule with the maximum elasticity across nodes and links. We first propose a distributed linear solution based on message passing, and we discuss several properties and extensions of the model. Based on the assumptions and conclusions, we extend it to the multiple paths case with a fat tree DCN, and discuss the optimal solution for computing the MAL with both computation and communication constraints. After that, we present the provision scheme with the maximum elasticity for the VMs, which comes with provable optimality guarantee for a fixed flow scheduling strategy in a fat tree DCN. We conduct the evaluations on our testbed and present various simulation results by comparing the proposed maximum elastic scheduling schemes with other methods. Extensive simulations validate the effectiveness of the proposed policies, and the results are shown from different perspectives to provide solutions based on our research.