Jennifer Campbell November 30, 2010. Problem Statement and Motivation Analysis of previous work Simple - competitive strategy Near optimal deterministic.

Jennifer CampbellNovember 30, 2010

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

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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.

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

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

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• 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

ee 1

ee 1

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.

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

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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.

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

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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 si to sj.

*

kKKK ...0

kSSS ...0

Goal: Provide an algorithm which returns completive schedule in

The algorithm uses a decision procedure to decide if a - competitive ratio schedule exists given

)( * )1log(log2

kkO

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

)'()'( tOPTtA

ss kk

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.

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

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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.

)(log kO

)log( 2 kkO

O k 2logklog(1 )

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: )1log(log2

kkmO

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

)223(

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

)(log2 BkkO

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