Moment Closures for Performance Models with Highly Non-linear Rates

Guenther, Marcel C.; Stefanek, Anton; Bradley, Jeremy T.

doi:10.1007/978-3-642-36781-6_3

Cited by 10 publications

(9 citation statements)

References 18 publications

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“…Experiments suggest that the following approximation results in accurate approximations: Figure 6 illustrates the improved accuracy for the discrete client-server model, where the approximation has been applied to the data transfer rate min(C 0 , S 0 )r data . This approximation has been recently evaluated on a large number of PCTMC models and been shown to be more accurate compared to the mean-field approximation (Guenther et al 2012). The result of Cain (1994) can be adapted to obtain an expression for the maximum of bivariate normal random variables, so that an analogous approximation can be applied to Figure 7 compares simulation estimates with the numerical solution to the mean-field equations obtained from Eq.…”

Section: Normal Approximationsmentioning

confidence: 97%

Mean-field analysis of hybrid Markov population models with time-inhomogeneous rates

2014

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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 timeinhomogeneous 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.

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Section: Normal Approximationsmentioning

confidence: 97%

Mean-field analysis of hybrid Markov population models with time-inhomogeneous rates

2014

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“…An alternative strategy for the analysis of the spatial PCTMC is to derive equations for the moments of the population variables [20,19], up to order k. Due to non-linearities in the rates, there is no exact closed form of these equations, and the differential equations for moments of order k depend on higherorder ones. Hence, equations are closed by relying to some heuristic [20,19]. Typically, moment closure equations give a better estimation of mean and variance than linear noise, as knowledge of higher-order moments introduce correction terms in the equations of lower order ones.…”

Section: Moment Closure Mean Field and Linear Noisementioning

confidence: 99%

“…where µ X@ and σ X@ denote the mean and variance of the population variable X@ at the steady state. To numerically compute the values of µ X@ and σ X@ , we resort to the normal momentclosure approximation of [19], which can be obtained at a much lower computational cost than by simulating the PCTMC, by integrating O(N 2 ) differential equations.…”

Section: Distance Between Locationsmentioning

confidence: 99%

“…Population processes are often modelled as Markovian stochastic processes, mostly as a subclass of continuous-time Markov chains (CTMCs) known as Population CTMCs (PCTMCs). These models have a very large or even infinite state space, a fact resulting in a lot of work in the computer science community to craft efficient algorithms for their analysis, ranging from specialised stochastic simulation to the use of mean-field [6] and moment closure techniques [20,19]. These methods, in particular, approximate the large or infinite set of linear Kolmogorov equations by a much smaller set of non-linear differential equations, capturing the mean, variance, and possibly other higher order moments.…”

Section: Introductionmentioning

confidence: 99%

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Location Aggregation of Spatial Population CTMC Models

Bortolussi

Feng

2016

Electron. Proc. Theor. Comput. Sci.

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In this paper we focus on spatial Markov population models, describing the stochastic evolution of populations of agents, explicitly modelling their spatial distribution, representing space as a discrete, finite graph. More specifically, we present a heuristic approach to aggregating spatial locations, which is designed to preserve the dynamical behaviour of the model whilst reducing the computational cost of analysis. Our approach combines stochastic approximation ideas (moment closure, linear noise), with computational statistics (spectral clustering) to obtain an efficient aggregation, which is experimentally shown to be reasonably accurate on two case studies: an instance of epidemic spreading and a London bike sharing scenario.

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“…One is through stochastic simulation, which is usually computationally expensive as performance metrics can only be derived after many simulation runs. The other is to build an analytical model in the form of initial value problems by fluidlimit theory [4] or moment closure techniques [5]. However, due to the intrinsic spatial-heterogeneity in collective systems (the same agent can exhibit different behaviour in different positions), the analytical model can be unresolvable due to the number of coupled ODEs in the model.…”

Section: Introductionmentioning

confidence: 99%

Speed-Up of Stochastic Simulation of PCTMC Models by Statistical Model Reduction

Feng

Hillston

2015

Computer Performance Engineering

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Abstract. We present a novel statistical model reduction method which can significantly boost the speed of stochastic simulation of a population continuous-time Markov chain (PCTMC) model. This is achieved by identifying and removing agent types and transitions from the simulation which have only minor impact on the evolution of population dynamics of target agent types specified by the modeller. The error induced on the target agent types can be measured by a normalized coupling coefficient, which is calculated by an error propagation method over a directed relation graph for the PCTMC, using a limited number of simulation runs of the full model. Those agent types and transitions with minor impact are safely removed without incurring a significant error on the simulation result. To demonstrate the approach, we show the usefulness of our statistical reduction method by applying it to 50 randomly generated PCTMC models corresponding to different city bike-sharing scenarios.

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Moment Closures for Performance Models with Highly Non-linear Rates

Cited by 10 publications

References 18 publications

Mean-field analysis of hybrid Markov population models with time-inhomogeneous rates

Mean-field analysis of hybrid Markov population models with time-inhomogeneous rates

Location Aggregation of Spatial Population CTMC Models

Speed-Up of Stochastic Simulation of PCTMC Models by Statistical Model Reduction

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