The current document is directed to methods and systems for efficiently executing OSL-virtualization containers within the execution environments provided by virtual machines that execute above traditional virtualization layers within large, virtualized, distributed computing systems. The currently disclosed methods and systems anticipate the need for additional virtual machines in order to meet anticipated demands for one or more computational resources by the containers. In addition, the methods and systems provision and launch virtual machines with computational-resource allocations that minimize overhead and computational-resource wastage. In one implementation, computational-resource utilization of ATMs and containers within the virtualized, distributed computer system are periodically monitored in order to estimate future demand for the computational resource and, when necessary, to launch additional virtual machines to meet the estimated future demand for the computational resource.