We consider the auto-scaling problem for application hosting in a cloud, where applications are elastic and the number of requests changes over time. The application requests are serviced by Virtual Machines (VMs), which reside on Physical Machines (PMs) in a cloud. We aim to minimize the number of hosting PMs by intelligently packing VMs into PMs, while the VMs are auto-scaled, i.e., dynamically acquired and released, to accommodate varying application needs. We consider a shadow routing based approach for this problem. The proposed shadow algorithm employs a specially constructed virtual queueing system to dynamically produce an optimal solution that guides the VM auto-scaling and the VM-to-PM packing. The proposed algorithm runs continuously without the need to re-solve the underlying optimization problem “from scratch”, and adapts automatically to the changes in the application demands. We prove the asymptotic optimality of the shadow algorithm. The simulation experiments further demonstrate the algorithm's good performance and high adaptivity.

中文翻译：

---

用于在云中托管应用程序的在线 VM 自动缩放算法

我们考虑在云中托管应用程序的自动扩展问题，其中应用程序是有弹性的，并且请求的数量会随着时间的推移而变化。应用程序请求由驻留在云中物理机 (PM) 上的虚拟机 (VM) 提供服务。我们的目标是通过智能地将 VM 打包到 PM 中来最小化托管 PM 的数量，同时 VM 是自动缩放的，即动态获取和释放，以适应不同的应用程序需求。针对这个问题，我们考虑了一种基于影子路由的方法。所提出的影子算法采用特殊构造的虚拟排队系统来动态生成指导VM 自动缩放和VM 到PM 打包的最佳解决方案。所提出的算法连续运行，无需“从头”重新解决底层优化问题，并自动适应应用需求的变化。我们证明了影子算法的渐近最优性。仿真实验进一步证明了该算法的良好性能和高适应性。