Building around the idea of a large scale server infrastructure with a potentially large number of tailored resources, which are capable of interacting to facilitate the deployment, adaptation, and support of services, cloud computing needs to frequently reschedule and manage various application tasks in order to accommodate the requests of a wide range and number of users. One of the challenges of cloud computing is to support and manage Quality-of-Service (QoS) by designing efficient techniques for the allocation of tasks between users and the cloud virtual resources, as well as assigning virtual resources to the cloud physical resources. The migration of virtual resources across physical resources is another challenge that requires considerable attention; especially in federated cloud computing environments wherein, providers might be willing to offer their unused resources as a service to the federation (cooperative allocation) and pull back these resources for their own use when they are needed (competitive allocation). This paper revisits the issue of QoS in cloud computing by formulating and presenting i) a multi-QoS task allocation model for the assignment of tasks to virtual machines and ii) a virtual machine migration model for a federated cloud computing environment by considering cases where resource providers are operating in cooperative or competitive mode. A new differential evolution (DE) based binding policy for task allocation and a novel virtual machine model are proposed as solutions for the problem of QoS support in federated cloud environments. The experimental results show that the proposed solutions improved the quality of service in the cloud computing environment and reveal the relative advantages of operating a mixed cooperation and competition model in a federated cloud environment.