Increasing demand for computational power has led cloud providers to employ multi-NUMA servers and offer multi-NUMA virtual machines to their customers. However, multi-NUMA VMs introduce additional complexity to scheduling algorithms. Beyond merely selecting a host for a VM, the scheduler has to map virtual NUMA topology onto the physical NUMA topology of the server to ensure optimal VM performance and minimize interference with co-located VMs. Under these constraints, maximizing the number of allocated multi-NUMA VMs on a host becomes a combinatorial optimization problem. In this paper, we derive closed-form expressions to compute the maximum number of VMs for a given flavor that can be additionally allocated onto a physical server. We consider nontrivial scenarios of mapping 2- and 4-NUMA symmetric VMs to 4- and 8-NUMA physical topologies. Our results have broad applicability, ranging from real-time dashboards (displaying available cluster capacity per VM flavor) to optimization tools for large-scale cloud resource reorganization.

翻译：随着计算能力需求的增长，云服务提供商开始采用多NUMA服务器，并向客户提供多NUMA虚拟机。然而，多NUMA虚拟机给调度算法带来了额外的复杂性。调度器不仅需要为虚拟机选择宿主机，还必须将虚拟NUMA拓扑映射到服务器的物理NUMA拓扑上，以确保虚拟机的最佳性能并最小化与共置虚拟机的干扰。在这些约束条件下，最大化宿主机上可分配的多NUMA虚拟机数量成为一个组合优化问题。本文推导出闭式表达式，用于计算在给定规格下可额外分配到一台物理服务器上的最大虚拟机数量。我们考虑了将2-NUMA和4-NUMA对称虚拟机映射到4-NUMA和8-NUMA物理拓扑的非平凡场景。研究结果具有广泛适用性，涵盖从实时仪表盘（按虚拟机规格显示可用集群容量）到大规模云资源重组的优化工具。