Elasticity is a fundamental feature of cloud computing and can be considered as a great advantage and a key benefit of cloud computing. One key challenge in cloud elasticity is lack of consensus on a quantifiable, measurable, observable, and calculable definition of elasticity and systematic approaches to modeling, quantifying, analyzing, and predicting elasticity. Another key challenge in cloud computing is lack of effective ways for prediction and optimization of performance and cost in an elastic cloud platform. The present paper makes the following significant contributions. First, we present a new, quantitative, and formal definition of elasticity in cloud computing, i.e., the probability that the computing resources provided by a cloud platform match the current workload. Our definition is applicable to any cloud platform and can be easily measured and monitored. Furthermore, we develop an analytical model to study elasticity by treating a cloud platform as a queueing system, and use a continuous-time Markov chain (CTMC) model to precisely calculate the elasticity value of a cloud platform by using an analytical and numerical method based on just a few parameters, namely, the task arrival rate, the service rate, the virtual machine start-up and shut-down rates. In addition, we formally define auto-scaling schemes and point out that our model and method can be easily extended to handle arbitrarily sophisticated scaling schemes. Second, we apply our model and method to predict many other important properties of an elastic cloud computing system, such as average task response time, throughput, quality of service, average number of VMs, average number of busy VMs, utilization, cost, cost-performance ratio, productivity, and scalability. In fact, from a cloud consumer's point of view, these performance and cost metrics are even more important than the elasticity metric. Our study in this paper has two significance. On one hand, a cloud service provider can predict its performance and cost guarantee using the results developed in this paper. On the other hand, a cloud service provider can optimize its elastic scaling scheme to deliver the best cost-performance ratio. To the best of our knowledge, this is the first paper that analytically and comprehensively studies elasticity, performance, and cost in cloud computing. Our model and method significantly contribute to the understanding of cloud elasticity and management of elastic cloud computing systems.

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云计算弹性的定量建模与分析计算

弹性是云计算的基本特征，可以说是云计算的一大优势和关键优势。云弹性的一个关键挑战是对弹性的可量化、可测量、可观察和可计算的定义以及建模、量化、分析和预测弹性的系统方法缺乏共识。云计算的另一个关键挑战是缺乏在弹性云平台中预测和优化性能和成本的有效方法。本文做出了以下重要贡献。首先，我们提出了云计算弹性的新的、定量的、正式的定义，即云平台提供的计算资源与当前工作负载匹配的概率。我们的定义适用于任何云平台，并且可以轻松测量和监控。此外，我们通过将云平台视为排队系统开发了一个分析模型来研究弹性，并使用连续时间马尔可夫链（CTMC）模型通过基于解析和数值的方法精确计算云平台的弹性值。只是几个参数，即任务到达率、服务率、虚拟机启动和关闭率。此外，我们正式定义了自动缩放方案，并指出我们的模型和方法可以轻松扩展以处理任意复杂的缩放方案。其次，我们应用我们的模型和方法来预测弹性云计算系统的许多其他重要属性，例如平均任务响应时间、吞吐量、服务质量、平均虚拟机数量、平均繁忙虚拟机数量、利用率、成本、性价比、生产力和可扩展性。事实上，从云消费者的角度来看，这些性能和成本指标甚至比弹性指标更重要。我们在本文中的研究有两个意义。一方面，云服务提供商可以使用本文开发的结果预测其性能和成本保证。另一方面，云服务提供商可以优化其弹性扩展方案，以提供最佳的性价比。据我们所知，这是第一篇对云计算的弹性、性能和成本进行分析和综合研究的论文。我们的模型和方法显着有助于理解云弹性和弹性云计算系统的管理。