Optimal Power-Down Strategies Chaitanya Swamy Caltech John Augustine Sandy Irani University of California, Irvine

Dynamic Power Management Idle period Request i 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 : power consumption rate = 1 Sleep state s1 : power consumption rate = 0 Transition cost = d0,1 = cost to power-down from s0 to s1 Idle period length = t (not known in advance) Decide when to transition from active state to sleep state. Simply a continuous version of the ski-rental problem. Want to try skiing but unsure of the number of ski trips Rent at cost $10/trip OR Buy paying $100 A(t), OPT(t): total power consumed when idle period length is t A 2d0,1 d0,1 OPT t Power consumed Suppose t is generated by a probability distribution. Expected power ratio (e.p.r.) of A = Et [A(t)] / Et [OPT(t)] Competitive ratio (c.r.) of A = maxt A(t)/OPT(t) = 2

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 consumed s0 s1 s2 s3 d0,3 d0,2 d0,1 t = idle period length

Follow-OPT Strategy d0,1 d1,2 d2,3 Power consumed s0 s1 s2 s3 OPT is lower envelop of lines d0,3 d0,2 d0,1 t = idle period length

Two Types of Bounds Global bound: what is the smallest c.r. (e.p.r.) r* such that every DPM instance has a power-down strategy of c.r. (or e.p.r.) at most r* ? Instance-wise bound: Given a DPM instance I, what is the best c.r. (or e.p.r.) r(I) for that instance? Clearly r* = maxinstances I r(I) Would like an algorithm that given instance I, computes strategy with c.r. (or e.p.r.) = r(I).

Related Work 2-state DPM – ski-rental problem Multi-state DPM 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.) can view as DPM where states “arrive” over time, but with more restrictive transition costs.

Our Results Give the first bounds for (general) multi-state DPM. Global bounds: give a simple algorithm that computes strategy with competitive ratio r* ≤ 5.83. Instance-wise bounds: Given instance I find strategy with c.r. r(I)+e in time O(k2log k.log(1/e)). Use this to show a lower bound of r* ≥ 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 r, find a strategy with c.r. ≤ 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 consumed A OPT t = idle period length

Suppose A=(S,T) has c.r. ≤ r, and transitions to sÎS at time t1ÎT s.t. A(t) < r.OPT(t). Then, can find new transition times T' such that a) A' = (S,T') has c.r. ≤ r, b) A' transitions to s at time t' < t1. r.OPT t1 Power consumed A OPT 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) ≤ r in [0,tA(s)] E(s) = minA' ÎStrat(s) tA' (s) = early transition time of s Let A = strategy attaining above minimum. Power r.OPT tA(s) = E(s) A Properties of A: a) A(E(s)) = r.OPT(E(s)) b) All transitions before s are at early transition times – "states q before s, tA(q) = E(q) OPT 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') = r.OPT(t'), – Transition to s must be at time t s.t. A(t) = r.OPT(t). Finally, if E(s) is finite for state s with power consumption rate rS ≤ r.rk, then we have a strategy ending at s with c.r. ≤ r.

Global Bound May assume that there are no power-up costs and di,j ≤ d0,j. Scaling to ensure that d0,i / d0,i+1 ≤ c where c < 1. s0 Follow-OPT Strategy Power d2,3 s1 s2 d1,2 s3 d0,3 OPT d0,1 d0,2 Theorem: Get a 5.83 competitive ratio. d0,1 t = idle period length

Open Questions Randomized strategies: global or instance-wise bounds for randomized strategies. Better lower bounds.

Thank You.