A Unified Approximation of Optimal Shutdown Schedules Based on a Brownian Motion Process (Mathematical Modeling and Optimization under Uncertainty)

Since the Energy Star Project by $\mathrm{U}. \mathrm{S}$ . government, the power management for computer systems has received considerable attention all over the world. As a computer consists of a number of electric components and devices, the problems of power management to reduce the energy consumption have to be discussed in terms of each component unit such as IC chip [1], microprocessor [2], CPU, disk drive, display and so on. Also, since the measurement technique to estimate the electrical power consumed in each component has been developed recently $[3, 4]$ , some interesting attempts have been made to reduce the electrical power in the real computer operation $[5, 6]$ . In general, the power management should be carried out at each level of the hierarchical computer design process; circuit level, layout level, logic level, behavioral level, architectural level, etc. In particular, the system level power management techniques have emerged as one of the most applicable design methodologies in practice, because they do not assume the development of new low-power devices. For the details on the system level power management techniques, see $[7, 8]$ . The dynamic power management, as it is generically known, can provide a control scheme that dynamically reconfigures an electric system to provide the requested services and performance levels with a minimum number of active components or a minimum load on such components $[7, 8]$ . The design methods will be useful for the operating system and the control system of peripheral devices. Especially, the dynamic power management plays an important role to achieve energy efficiency in the operating system, since the application software programs are monitored and controlled by it. It is, however, known that typical operating systems like UNIX, $\mathrm{W}\mathrm{i}\mathrm{n}\mathrm{d}_{0}\mathrm{w}\mathrm{s}\mathrm{o}\mathrm{s}$ and $\mathrm{M}\mathrm{a}\cos$ were not designed originally with energy efficiency in mind. The most simple way to establish the power reduction in the current operating systems is to add the ability to selectively shutdown the peripherals which are not currently being used. In fact, this method called the shutdown approach or the shutdown policy has been applied to the power saving in the hard disk [9] as well as VLSI circuits system [7, 10, 11]. The typical example for the dynamic power management is the mobile computing with limited capacity of battery [12, 13, 14]. Unlike a desktop personal computer, the electrical power in a mobile computer must be carefully rationed among all of the components and peripherals. For such systems, the shutdown approach will be useful to reduce the electrical power consumed in the operating period. In this paper, we present a stochastic model for computer systems $\mathrm{w}\iota_{1}\mathrm{i}_{\mathrm{C}\mathrm{h}}$ employ the shutdown approach. We consider two models proposed in Okamura et al. $[15, 16]$ and develop a unified approach to integrate these models theoretically. Based on the general arrival assumption on tasks, we formulate the expected electrical power consumption per unit time in steady state and propose a unified approximate method by applying the so-called diffusion approximation.