1. Jennifer Campbell November 30, 2010

2.  Problem Statement and Motivation  Analysis of previous work  Simple - competitive strategy  Near optimal deterministic algorithm  Analysis  Other Results

3.  Power-Down strategies for systems with multiple sleep states.  System pays per time unit to reside in high- cost state OR transition to low cost state for a one time fixed cost.  In a single sleep state is similar to ski rental problem.

4.  Shared memory in multi-processor machines  Networks, whether to keep a connection open between bursts of packets.  Critical to maximizing battery usable in mobile systems.  In most computers BIOS support 5 power states ◦ Hibernate ◦ 3 levels of Sleep

5.  Problem Statement and Motivation  Analysis of previous work  Simple - competitive strategy  Near optimal deterministic algorithm  Analysis  Other Results

6. Best deterministic online algorithm ▫ Stay at high power until the total energy spend is equal to the cost to power up from a low power state ▫ Optimal competitive ratio of 2 Best Randomized Algorithms ▫ Competitive ratios of If idle periods are generated by a known probability distribution ▫ Competitive ratios of

7.  Pervious work assumed additive transition costs. This is not so in general. ◦ Additional energy is spent in transitioning to lower power states. ◦ Could be overhead in stopping at intermediate states.

8.  Problem Statement and Motivation  Analysis of previous work  Simple - competitive strategy  Near optimal deterministic algorithm  Analysis  Other Results

9.  Given OPT is optimal offline algorithm  Consider a strategy A, which “follows” OPT. ◦ Making each transition to a new state as the idle period gets longer ◦ Same strategy as 2- competitive ratios for 2- state case  Theorem: There exists a competitive strategy for ANY system.

10.  Problem Statement and Motivation  Analysis of previous work  Simple - competitive strategy  Near optimal deterministic algorithm  Analysis  Other Results

11.  Given an input of a system described by (K, d) with state sequence S, for which the optimal online schedule has a competitive ration of ◦ k is the number of sleep states. ◦ K is a vector for power-consumption rates ◦ S is a set of states of the system ◦ d is the cost to move from state s i to s j.

12.  Goal: Provide an algorithm which returns completive schedule in  The algorithm uses a decision procedure to decide if a - competitive ratio schedule exists given

14.  Slide t to be t’, which makes t’ eager ◦ This means ◦ Do this for every t  The new set of t’ is still - competitive if the system still ends at the final state s, where

15.  This method is exponential in k as it enumerates all subsequences of all states.  It can be modified to use dynamic programming ◦ Take a system and constant and output YES if a - competitive strategy exists and NO o.w.

16.  Problem Statement and Motivation  Analysis of previous work  Simple - competitive strategy  Near optimal deterministic algorithm  Analysis  Other Results

17.  Computing every eager transition takes  Finding all eager transitions for all states using dynamic programming takes:  Once all transitions are found, it must be decided if this is a -competitive strategy, and find it.  Total time:  This provides a lower bound of 2.45 on the competitive ratio for deterministic algorithms.

18.  The Dynamic programming in the pervious algorithm can be adapted to have a bound on m ◦ m is the number of states that can be used by the online algorithm.  This change adds a factor of m to the running time ◦ New running time is:

19.  Problem Statement and Motivation  Analysis of previous work  Simple - competitive strategy  Near optimal deterministic algorithm  Analysis  Other Results

20.  Algorithm which takes system description and probability distribution as input and produced a power-down strategy ◦ The probability distribution is generating the idle period length. ◦ Running time is based on the representation of the distribution. It is often an histogram ◦ Running time:  Where B is the number of bins in the histogram

21.  J. Augustine, S. Irani, and C. Swamy, ``Optimal Power-Down Strategies'', SIAM Journal on Computing, Vol. 37, pages 1499- 1516, 2008.