Optimal Power-Down Strategies

Chaitanya SwamyCaltech

John Augustine Sandy IraniUniversity of California, Irvine

Dynamic Power Management

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

Request i

Request i+1

Idle period

Active state s0 : power consumption rate = 1Sleep state s1 : power consumption rate = 0Transition 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.

d0,1

OPT

t

Power consume

d

d0,1

A(t), OPT(t): total power consumed when idle period length is tA

2d0,1

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

t = idle period length

Power consume

d

d0,1

s0

s1

s2

s3

d0,2

d0,3

OPT is lower envelop of lines

t = idle period length

Power consume

d

d0,1

s0

s1

s2

s3

d0,2

d0,3

Follow-OPT Strategy

d0,1

d1,2

d2,3

Two Types of Bounds

• Global bound: what is the smallest c.r. (e.p.r.) * such that every DPM instance has a power-down 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.)

– 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 * ≤ 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).

t = idle period length

Power consume

d

OPT

A

t = idle period length

Power consume

d

OPT

A

Suppose A=(S,T) has c.r. ≤ , andtransitions to sS at time t1T s.t. A(t) <

.OPT(t).

Then, can find new transition times T' such that a) A' = (S,T') has c.r. ≤ , b) A' transitions to s at time t' < t1.

.OPT

t1

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 sLet A = strategy attaining above minimum.Properties of A:

a) A(E(s)) = .OPT(E(s))

b) All transitions before s are at early transition times – states q before s,

tA(q) = E(q)

t = idle period length

Power

OPT

.OPT

tA(s) = E(s)

A

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

OPT

t = idle period length

Power

d0,1

s0 s

1s2

s3

d0,2

d0,3

Follow-OPT Strategy

d0,1

d1,2

d2,3

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.

Theorem: Get a 5.83 competitive ratio.

Open Questions

•Randomized strategies: global or instance-wise bounds for randomized strategies.

•Better lower bounds.

Thank You.