Anton Stefanek

Fluid computation of passage-time distributions in large Markov models

Recent developments in the analysis of large Markov models facilitate the fast approximation of transient characteristics of the underlying stochastic process. Fluid analysis makes it possible to consider previously intractable models whose underlying discrete state space grows exponentially as model components are added. In this work, we show how fluid-approximation techniques may be used to extract passage-time measures from performance models. We focus on two types of passage measure: passage times involving individual components, as well as passage times which capture the time taken for a population of components to evolve. Specifically, we show that for models of sufficient scale, global passage-time distributions can be well approximated by a deterministic fluid-derived passage-time measure. Where models are not of sufficient scale, we are able to generate upper and lower approximations for the entire cumulative distribution function of these passage-time random variables, using moment-based techniques. Additionally, we show that, for passage-time measures involving individual components, the cumulative distribution function can be directly approximated by fluid techniques. Finally, using the GPA tool, we take advantage of the rapid fluid computation of passage times to show how a multi-class client-server system can be optimised to satisfy multiple service level agreements.

A new tool for the performance analysis of massively parallel computer systems

We present a new tool, GPA, that can generate key performance measures for very large systems. Based on solving systems of ordinary differential equations (ODEs), this method of performance analysis is far more scalable than stochastic simulation. The GPA tool is the first to produce higher moment analysis from differential equation approximation, which is essential, in many cases, to obtain an accurate performance prediction. We identify so-called switch points as the source of error in the ODE approximation. We investigate the switch point behaviour in several large models and observe that as the scale of the model is increased, in general the ODE performance prediction improves in accuracy. In the case of the variance measure, we are able to justify theoretically that in the limit of model scale, the ODE approximation can be expected to tend to the actual variance of the model.

Fluid analysis of energy consumption using rewards in massively parallel markov models

Capturing energy consumption directly from a stochastic behavioural model is a computationally expensive process. Using a so-called fluid analysis technique we are able to access accumulated reward measures in much larger scale stochastic systems than has been previously possible.These accumulated rewards are ideal for deriving energy and power consumption from stochastic process models. In previous work, it has been shown how to derive a set of ordinary differential equations (ODEs) whose solutions approximate the moments of component counts in a continuous-time Markov chain(CTMC) described in a stochastic process algebra. In this paper, we show how to extend the method to provide rapid access to moments of accumulated rewards in CTMCs. In addition to measuring the amount of energy used by a system, we are also interested in the time taken to reach a particular level of energy consumption. In reward terms, this is a so-called completion time. In this paper, we are able to use higher moments of rewards to give us access to completion time distributions. We demonstrate the technique on a model of energy consumption in a client-server system with server failure and hibernation. Moreover, we are able to use these new and rapid techniques to capture the trade-off between energy consumption and service level agreement (SLA) compliance. We use a standard optimisation approach to find the precise configuration of the system which minimises the energy consumption while satisfying an operational response-time quantile.

Moment closures for performance models with highly non-linear rates

Fluid analysis of Population CTMCs with non-linear evolution rates requires moment closures to transform a linear system with infinitely many ordinary differential equations (ODEs) into a non-linear one with a finite number of ODEs. Due to the ubiquity of kinetics with quadratic rates in physical processes, various closure techniques have been discussed in the context of systems biology and performance analysis. However, little research effort has been put into moment closures for higher-order moments of models with piecewise linear and higher-order polynomial evolution rates. In this paper, we investigate moment closure techniques applied to such models. In particular we look at moment closures based on normal and log-normal distributions. We compare the accuracy of the moment approximating ODEs with the exact results obtained from simulations. We confirm that by incorporating higher-order moment ODEs, the moment closure techniques give accurate approximations to the standard deviation of populations. Moreover, they often improve the accuracy of mean approximations over the traditional mean-field techniques.

GPA - A Tool for Fluid Scalability Analysis of Massively Parallel Systems

Recent ordinary differential equation (ODE) based techniques allow efficient analysis of Markovian population models with extremely large state spaces. In most cases of realistic scale, they provide the only alternative to stochastic simulation. Moreover, numerical solution of the ODEs is cheaper computationally than simulation by orders of magnitude. We present the Grouped PEPA Analyser (GPA) tool with new functionality to exploit computationally inexpensive fluid analysis techniques to allow the exploration of large numbers of system configurations in models with large state spaces. GPA provides an efficient implementation of the fluid analysis techniques for models described in a stochastic process algebra. It implements recently developed extensions allowing specifications of complex reward measures using combinations of state based, rate accumulated and impulse rewards. Combined with the ability to efficiently capture various passage time metrics, GPA can be used to solve optimisation problems with a reward objective function under different service level agreement type constraints.

Mean-Field analysis of markov models with reward feedback

We extend the population continuous time Markov chain formalism so that the state space is augmented with continuous variables accumulated over time as functions of component populations. System feedback can be expressed using accumulations that in turn can influence the Markov chain behaviour via functional transition rates. We show how to obtain mean-field differential equations capturing means and higher-order moments of the discrete populations and continuous accumulation variables. We also provide first- and second-order convergence results and suggest a novel normal moment closure that can greatly improve the accuracy of means and higher moments. We demonstrate how such a framework is suitable for modelling feedback from globally-accumulated quantities such as energy consumption, cost or temperature. Finally, we present a worked example modelling a hypothetical heterogeneous computing cluster and its interaction with air conditioning units.

Fluid computation of the performance: energy tradeoff in large scale Markov models

Recent fluid analysis techniques allow fast and effcient calculation of complex reward metrics and passage time probabilities in systems with very large state space. We demonstrate how to incorporate these to look at the trade-off between service level agreement (SLA) satisfaction and complex reward optimisation. We show how the uid analysis naturally leads to a constrained global optimisation problem with embedded differential equations. We illustrate this problem on an abstract model of a virtualised execution environment that accurately captures resource allocations.

Specification and Efficient Computation of Passage-Time Distributions in GPA

We present a significant extension to the Grouped PEPA Analyser (GPA) tool. We have augmented the tool with the ability to specify complex passage-time distributions with the Unified Stochastic Probes formalism and implemented efficient fluid analysis techniques to compute the distributions. The extension incorporates immediate signalling and weighted passive rates and permits two classes of passage time, namely global and individual passage times, to be computed. We summarise how the different classes of passage-time query can be expressed using the Unified Stochastic Probe formalism and present some results from probed GPA models.

Time-inhomogeneous Population Models of a Cycle-Stealing Distributed System

Organisations such as research institutions and universities often increase utilisation of their office workstations by deploying a high-throughput cycle-stealing distributed system. Such systems allow users to submit a large number of computing tasks into a central pool. The system observes activity of workstations and continually assigns tasks to idle machines. When a user becomes active on the machine, the scheduler interrupts the task execution. This approach can significantly increase utilisation of the resources. However, it can also lead to wastage of computing cycles if tasks get interrupted too often.In this paper, we develop a detailed Population Continuous Time Markov Chain (PCTMC) model of the whole system that accurately captures the contention between the interactive users and high-throughput tasks. The PCTMC framework is well suited to the inherently time-inhomogeneous nature of the user behaviour and allows to capture a large number of performance and energy consumption metrics. We fit the PCTMC model to real data and propose a methodology to forecast cluster availability in the near future. We show how to use historically collected and live data to parametrise the PCTMC model and use efficient fluid analysis techniques to predict the desired metrics. Additionally, the fast analysis enables exploration of various what-if scenarios. We demonstrate a working implementation of the method using the existing GPA tool for analysis of PCTMC models. We argue that this methodology could allow the system maintainers to optimise the energy and performance parameters of the system. Moreover, it would benefit the users who could use the model forecasts to better distribute and plan their large scale computations.

Energy consumption in the office

In this paper we present measurements of energy usage of standard office computing equipment. Using a data trace lasting for all of March 2012 we analyse the energy use of office equipment such as desktop computers, a printer and a fridge. The interest in a more detailed knowledge of the energy usage patterns of these appliances is driven by the desire to manage, and if possible reduce, the energy consumption of computing equipment in a university department. The reason behind this can be financial to reduce electricity costs and/or environmental to reduce the carbon foot print of an office environment. We analyse the data and show simple autoregressive time series models to predict the energy usage of appliances. We also show that its feasible to accurately approximate the power consumption of a desktop computer using the CPU utilisation information. We describe a future set-up where we plan to monitor the energy usage of a student lab.