Switching diffusion approximations for optimal power management in parallel processing systems

Abstract

In this paper, we investigate optimal power management in parallel processing systems composed of one queue and several identical processing stations. Power consumption is controlled by setting some of the stations into an inactive state, where they consume less power but are unable to provide service. This way, we are faced with the conflicting objective of minimizing power consumption while maintaining a desired quality of service. A distinguishing feature here, regarding most previous literature on this subject, is that we consider systems operating a policy that may turn the reserve machines on or off with setup times and under general inter-arrival or service time distributions, subject to some conditions. When these condition fail, we also provide a model with general inter-arrival times and exponentially distributed service times. To some extent, a controlled switching diffusion obtained in this paper via heavy traffic analysis and stochastic optimal control theory are the technical underpinning of the paper. We also propose a numerical approach to the solutions of the optimal control problems based on the Markov chain approximation method. Finally, we consider some numerical experiments that illustrate the efficiency of the proposed approach.

Keywords:

• Heavy traffic analysis
• switching diffusion
• stochastic optimal control
• parallel processing queuing systems

Notes

1 Note that the definition of randomized Markovian control used here relates to the definition given in [15] by ${\pi }_t(C,i)=v(z^n(t),i)(C)$ for a state $i\in E$ and $C\in \mathcal{B}(\mathcal{U}).$ In [15], the control policy is not dependent on another process as it is the case here.

Additional information

Funding

This research was partially supported by the Minas Gerais Research Foundation (FAPEMIG) under the grant APQ 00945/14 and by the Brazilian National Council for Scientific and Technological Development (CNPq) under the grants CNPq-304801/2015-1 and CNPq-421486/2016-3.