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

Abstract

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.

Appendix: Parameters used in the examples

Values of rate and initial population parameters used in the client–server example, Figs. 2, 3, 4, 5 and 6:

${r_{ data }}$	${r_{ task }}$	${r_{ reset }}$	${r_{ on }}$	${r_{ off }}$	${r_{ heat }}$	${r_{ cool }}$	${t_{ thresh }}$	${n_{{ C }}}$	${n_{{ S }}}$	${n_{{ A }}}$
0.6	0.2	0.1	0.2	0.2	0.2	0.4	30	40	30	20

Values of rate and initial population parameters used in the worked example, Figures 13, 14, 15:

${n_{{ C }}}$	${n_{{ S }}}$	${n_{{ A }}}$	${r_{q,1}}$	${r_{q,2}}$	${r_{s,1}}$	${r_{s,2}}$	${r_{ reset }}$	${r_{ wakeup }}$
20000	1000	100	0.2	0.5	0.2	0.2	0.2	0.3

${r_{ on }}$	${r_{ off }}$	${t_0}$	${t_{ thresh }}$	${t_{ sleep }}$	${p_{A,s}}$	${p_{A,{ sl }}}$	${p_{A,1}}$	${p_{A,2}}$	${r_{ cool }}$
0.2	0.2	25	20	23	10	1	30	37.5	0.026

The constants $p_{B,\cdot }$ are set as the respective $p_{A,\cdot }$ constants multiplied by $1.7$ and the heat constants $c_{\cdot , \cdot }$ are set as the corresponding $p_{\cdot ,\cdot }$ constants multiplied by a conversion factor $7.71\times 10^{-6}$.

Values of rate and initial population parameters used in the time-inhomogeneous worked example, Figures 17, 19, 20

${n_{{ S }}}$	${n_{{ A }}}$	${\mu }$	${r_{ up }}$	${r_{ down }}$	${r_{ fail }}$	${r_{ on }}$	${r_{ off }}$	${t_0}$	${t_{ thresh }}$	${t_{ fail }}$
150	50	50.0	10.0	5.0	1.0	2.0	0.5	25	25	35

${p_{S}}$	${p_{{ idle }}}$		${p_{A}}$		${h_{ cool }}$		${h_S}$		${h_{ idle }}$
1.0	0.2		0.2		5.0		1.0		0.1

We obtained the arrival rate $\lambda (t)$ from the World Cup 98 data (Arlitt 2000) available at http://ita.ee.lbl.gov/html/contrib/WorldCup.html. For illustration purposes, we have rescaled the arrival rate by a factor of $10^{-5}$.