Optimal Power-Down Strategies Chaitanya Swamy Caltech John Augustine Sandy Irani University of California, Irvine Dynamic Power Management Request i Idle period Request i+1 Machine/server serving jobs/requests in active state with high power consumption rate
Idle period between requests length is apriori unknown During idle period can transition to low power state incur power-down cost Idle power management: Determine when to transition so as to minimize total power consumed Active state s0 =1 Sleep state s1 =0 Transition cost down from s0 to s1 Idle period length : power consumption rate
: power consumption rate = d0,1 = cost to power- = t (not known in advance) A(t), OPT(t): total power Power Decide when to transition from active state to sleep consumed when idle period consume state. A length is t d 2d0,1 Competitive ratio (c.r.) of A Simply a continuous version of the ski-rental = maxt A(t)/OPT(t) = 2 problem.
OPT Suppose t is generated by a d0,1 probability distribution. Expected power ratio (e.p.r.) of t A d0,1 = Et [A(t)] / Et [OPT(t)] DPM with multiple sleep states Set of states S = (s0, s1,, sk) s0 : active state, rest are sleep states ri : power consumption rate of si r0 > r1 > > rk di,j : cost of transitioning from si to sj Power-down strategy is a tuple (S,T) S : sequence of states of S starting at s0
T : transition time sequence for S starting at t = 0 Power consume d s 0 s 1 s 2 d0,3 s 3
d0,2 d0,1 t = idle period length Power consume d s 0 Follow-OPT Strategy d2,3 s
d1,2 1 s 2 s d0,3 3 d0,1 OPT is lower envelop of lines d0,2
d0,1 t = idle period length Two Types of Bounds Global bound: what is the smallest c.r. (e.p.r.) * such that every DPM instance has a powerdown strategy of c.r. (or e.p.r.) at most * ? Instance-wise bound: Given a DPM instance I, what is the best c.r. (or e.p.r.) (I) for that instance? Clearly * = maxinstances I (I) Would like an algorithm that given instance I, computes strategy with c.r. (or e.p.r.) = (I). Related Work 2-state DPM ski-rental problem Karlin, Manasse, Rudolph & Slater: global bound of 2 for c.r.
Karlin, Manasse, McGeoch & Owicki: global bound of e/(e-1) for expected power ratio. easy to give instance-wise optimal strategies. Multi-state DPM Irani, Gupta & Shukla: global bounds for additive transition costs, di,k = di,j + dj,k for all i>j>k called DPM-A (additive). Show that Follow-OPT has c.r. = 2, give strategy with expected power ratio = e/(e-1). Other extensions capital investment problem (Azar et al.) Our Results Give the first bounds for (general) multi-state DPM. Global bounds: give a simple algorithm that
computes strategy with competitive ratio * 5.83. Instance-wise bounds: Given instance I find strategy with c.r. (I)+ in time O(k2log k.log(1/ )). Use this to show a lower bound of * 2.45. find strategy with optimal expected power ratio for the instance. Finding the Optimal Strategy DPM instance I is given. Want to find strategy with optimal competitive ratio for I. Decision procedure: given , find a strategy with c.r. or say that none exists. Need to determine a) state sequence, and b) transition times.
Claim: For any strategy A, c.r.(A) = maxt=transition time of A A(t)/OPT(t). Power consume d A OPT t = idle period length Suppose A=(S,T) has c.r. , and transitions to sS at time t1T s.t. A(t) < .OPT(t). Then, can find new transition times T' such that
Power a) A' = (S,T') has c.r. , b) A' transitions to s at time .OPT consume t' < t1. d A OPT t1 t = idle period length tA(s) = transition time of s in strategy A
Strat(s) = set of (partial) strategies A ending at s such that c.r.(A) in [0,tA(s)] E(s) = minA' Strat(s) tA' (s) = early transition time of s Power . OPT Let A = strategy attaining above minimum. Properties
of A: A a) A(E(s)) = .OPT(E(s)) OPT tA(s) = E(s) b) All transitions before s are at early transition times states q before s, tA(q) = E(q) t = idle period length Dynamic Programming Compute E(s) values using dynamic programming. Suppose we know E(s') for all states s' < s.
Then, E(s) = mins' before s (time when s' transitions to s). To calculate quantity in brackets, use that: Transition to s' was at t' = E(s') with A(t') = .OPT(t'), Transition to s must be at time t s.t. A(t) = .OPT(t). Finally, if E(s) is finite for state s with power consumption rate rS .rk, then we have a strategy ending at s with c.r. . Global Bound May assume that there are no power-up costs and di,j d0,j. s Scaling to ensure that d0,i / d0,i+1 Follow-OPT c where c < 1. Power
0 d1,2 d0,3 d2,3 s 1 Strategy s 2 s 3 OPT
d0,1 d0,2 d0,1 Theorem: Get a 5.83 competitive ratio. t = idle period Open Questions Randomized strategies: global or instance-wise bounds for randomized strategies. Better lower bounds. Thank You.
The aim is to make a material with the appearance and clarity of standard glass but with effective protection from small arms. usually made from a combination of two or more types of glass, one hard and one soft. ......
Density is defined as mass per unit volume. It is a measure of how tightly packed and how heavy the molecules are in an object. Density is the amount of matter within a certain volume. Proof that water and ice...
The Statement of Economic Interests, Form 700 is required by the California Fair Political Practices Commission also referred to as the FPPC. As a Technical Review Committee member who influences funding decisions of the Specialty Crop Block Grant Program, you...
ECG Signal processing (2) ECE, UA. ECG signal processing - Case  Diagnosis of Cardiovascular Abnormalities From Compressed ECG: A Data Mining-Based Approach IEEE TRANSACTIONS ON INFORMATION TECHNOLOGY IN BIOMEDICINE, VOL. 15, NO. 1, JANUARY 2011.
This understanding is advanced through a combination building selective factual knowledge and inculcating appropriate analytical skills. ... An AP course in English Language and Composition engages students in becoming skilled readers of prose written in a variety of rhetorical contexts,...
Definition of essential concepts . Part I - Policing in Democracies and the need for accountability, integrity, oversight . Conceptual origins and the evolution of the role of the police in democratic societies . Key police powers and implications for...
The Wildcare Trust was set up in 2005 with the aim of saving animals from extinction. It is a charity and relies on its members for financial support to continue its work. Membership fees are the main source of income,...
Ready to download the document? Go ahead and hit continue!