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

2. 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

3. 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

4. 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

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

6. 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

7. 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).

8. 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.

9. 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.

10. 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.

11. 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

12. 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

13. 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

14. 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.

15. 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

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

17. Thank You.