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.
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