Adaptive Disk Spindown via Optimal Rent-to-Buy in Probabilistic Environments

Abstract

In the {single rent-to-buy decision} problem, without a priori knowledge of the amount of time a resource will be used we need to decide when to buy the resource, given that we can rent the resource for \$1 per unit time or buy it once and for all for \$$c$. In this paper we study algorithms that make a sequence of single rent-to-buy decisions, using the assumption that the resource use times are independently drawn from an unknown probability distribution. Our study of this rent-to-buy problem is motivated by important systems applications, specifically, problems arising from deciding when to spindown disks to conserve energy in mobile computers~[DKM, LKH, MDK], thread blocking decisions during lock acquisition in multiprocessor applications~[KLM], and virtual circuit holding times in IP-over-ATM networks~[KLP, SaK]. We develop a provably optimal and computationally efficient algorithm for the rent-to-buy problem and evaluate its practical merit for the disk spindown scenario via simulation studies. Our algorithm uses $O(\sqrt{t})$ time and space, and its expected cost for the $t$th resource use converges to optimal as $O(\sqrt{\log t/t})$, for any bounded probability distribution on the resource use times. Alternatively, using $O(1)$ time and space, the algorithm almost converges to optimal. We describe the results of simulating our algorithm for the disk spindown problem using disk access traces obtained from an HP workstation environment. We introduce the natural notion of {\em effective cost\/} which merges the effects of energy conservation and response time performance into one metric based on a user specified parameter~$a$, the relative importance of response time to energy conservation. (The buy cost~$c$ varies linearly with~$a$.) We observe that by varying~$a$, we can model the tradeoff between power and response time well. We also show that our algorithm is best in terms of effective cost for almost all values of~$a$, saving effective cost by 6--25\% over the optimal online algorithm in the competitive model~(i.e., the 2-competitive algorithm that spins down the disk after waiting~$c$ seconds). In addition, for small values of~$a$ (corresponding to when saving energy is critical), our algorithm when compared against the optimal online algorithm in the competitive model reduces excess energy by 17--60\%, and when compared against the 5~second threshold reduces excess energy by 6--42\%.