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

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.