Mean-field analysis of hybrid Markov population models with time-inhomogeneous rates
Mean-field analysis of hybrid Markov population models with time-inhomogeneous rates
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DOI：
10.1007/s10479-014-1664-9
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发表时间：
2014-08
影响因子：
4.8
通讯作者：
Anton Stefanek;R. A. Hayden;J. Bradley
中科院分区：
文献类型：
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作者：
Anton Stefanek;R. A. Hayden;J. Bradley
我们考虑种群连续时间马尔可夫链（PCTMC）的混合扩展-一类马尔可夫过程捕获大群行为相同的代理之间的相互作用。我们用连续变量扩展PCTMC的离散状态空间，这些连续变量在总体向量上作为积分演变，并且可以同时为PCTMC中的转换速率提供反馈。此外，我们还包含了时间非均匀速率参数，可用于将实际测量数据纳入模型。我们扩展了PCTMC的平均场技术，并展示了如何推导一个积分方程组，该方程组近似混合PCTMC中种群和连续变量的均值和高阶矩的演化。我们证明了一阶和二阶收敛的结果，证明了近似的正确性。我们使用基于正态分布的矩闭包，这提高了在比例控制情况下矩近似的准确性，其中过渡率取决于连续变量高于或低于固定阈值的数量。我们演示了该框架如何适用于模拟大型系统中全球累积量的反馈，例如数据中心的能源消耗、总成本或温度。我们提出了一个具有温度管理和随时间变化的外部工作负载的多服务器系统模型。我们将展示如何使用真实数据来表示框架内的工作负载。我们使用随机模拟来验证这个例子和一个假设的异构计算集群的早期例子。
We consider a hybrid extension of population continuous time Markov chains (PCTMC)—a class of Markov processes capturing interactions between large groups of identically behaved agents. We augment the discrete state space of a PCTMC with continuous variables that evolve as integrals over the population vector and that can simultaneously provide feedback to the rates of transitions in the PCTMC. Additionally, we include time-inhomogeneous rate parameters, which can be used to incorporate real measurement data into the models. We extend mean-field techniques for PCTMCs and show how to derive a system of integral equations that approximate the evolution of means and higher-order moments of populations and continuous variables in a hybrid PCTMC. We prove first- and second-order convergence results that justify the approximations. We use a moment closure based on the normal distribution which improves the accuracy of the moment approximation in case of proportional control where transition rates depend on the amount a continuous variable is above or below a fixed threshold. We demonstrate how this framework is suitable for modelling feedback from globally-accumulated quantities in a large scale system, such as energy consumption, total cost or temperature in a data centre. We present a model of a many server system with temperature management and external workload that varies with time. We show how to use real data to represent the workload within the framework. We use stochastic simulation to validate the example and an earlier example of a hypothetical heterogeneous computing cluster.