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Michael J. Neely, University of Southern California www- bcfc /~ mjneely /

Asynchronous Scheduling for Energy Optimality in Systems with Multiple Servers. Server 3. 0. 1. 2. 3. 4. Server 2. 0. 1. 2. Server 1. 0. 1. 2. 3. t. t 0. t 1. t 2. t 3. t 4. t 5. t 6. t 7. t 8. t 9. t 10. Michael J. Neely, University of Southern California

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Michael J. Neely, University of Southern California www- bcfc /~ mjneely /

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  1. Asynchronous Scheduling for Energy Optimality in Systems with Multiple Servers Server 3 0 1 2 3 4 Server 2 0 1 2 Server 1 0 1 2 3 t t0 t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 Michael J. Neely, University of Southern California http://www-bcf.usc.edu/~mjneely/ CISS, Princeton University, March 2012

  2. Motivation • The following things are important: • Multi-core processing architectures. • Distributed data center processing. • Energy-efficiency.

  3. Server 1 λ1 λ2 Server 2 λΝ Server S • N classes of jobs, with Poisson arrivals. • Jobs are queued according to class: Q1(t), …, QN(t). • S heterogeneous servers. Servers can have… • …different classes that are allowed for service. • …different processing mode options. • …different processing times and energy expenditures.

  4. Consider one particular server: Frame 1 Frame 0 Active Active Active Idle Idle t Ds[0] Ds[2] Ds[1] Is[0] Is[1] • Continuous Time operation. • Each server s in {1, …, S} operates over frames. • At frame r in {0, 1, 2, …}, server s decides: • ms[r] = processing mode for frame r. • (chosen in some abstract set of options Ms). • Is[r] = Idle time for frame r (possibly 0). • (chosen in the interval [0, Imax]).

  5. Affect of Decisions on 1 Frame Frame 1 Frame 0 Active Active Active Idle Idle t Ds[0] Ds[2] Ds[1] Is[0] Is[1] • Choosing ms[r], Is[r] determines: • μsn[r] = # jobs of type n that can be served by s • = μsn(ms[r]) (for each n in {1, …, N}) • es[r] = energy expended by server s • = esproc(ms[r]) + psidleIs[r] • Ts[r] = frame duration for server s • = D(ms[r]) + Is[r] ⌃ ⌃ ⌃

  6. Now put all servers together: Server 3 0 1 2 3 4 Server 2 0 1 2 Server 1 0 1 2 3 t t0 t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 • tk = start of kthsub-frame. • S(tk) = {servers that start a frame at time tk}. • What is the “state” at time tk? • State has size at least 2S. • (because each server is either ACTIVE or IDLE)

  7. A “Simpler” Approach • Look at the time averages that the network can achieve, • and that are necessary. • Define ēs, Ts, μsnas frame averages: • The time average energy for server s is then:

  8. The Optimization Problem • Goal: Minimize total time average power subject to supporting all data (i.e., stabilizing all queues). • Minimum power is given by the following problem: Is this really “simpler”? How to solve this?

  9. Special Case for Intuition • Suppose set of mode options Msis finite for all s. • Can show optimality of stationary, randomized policies • with Is fixed, and with Pr[ms[r] = ms] = qs(ms):

  10. Special Case for Intuition • Suppose set of mode options Msis finite for all s. • Can show optimality of stationary, randomized policies • with Is fixed, and with Pr[ms[r] = ms] = qs(ms):

  11. Reminiscent of a “Linear Fractional (LF) Program” • LF Problems are non-convex, but can be solved efficiently: • Boyd, Vandenberghe book (non-linear change of variables). • Neely 2011 presents online method (for 1-server problem). • Our problem is not LF. It has: • Fractional terms in the constraints. • Fractional terms with different denominators. • Such problems are generally intractable.

  12. Reminiscent of a “Linear Fractional (LF) Program” • LF Problems are non-convex, but can be solved efficiently: • Boyd, Vandenberghe book (non-linear change of variables). • Neely 2011 presents online method (for 1-server problem). • Our problem is not LF. It has: • Fractional terms in the constraints. • Fractional terms with different denominators. • Such problems are generally intractable. • BUT: Our problem has special physical structure!

  13. Main Result • We exploit the physical structure to solve the problem. • Our solution is an online algorithm and: • Does not require knowledge of arrival rates. • Works even if the option sets Ms are infinite. • Adaptive when rates change. • Extends Lyapunov optimization theory to treat coupled systems that operate asynchronously over their own timelines.

  14. We use 2 Ideas from Restless Bandit Theory • Drift-Plus-Penalty Ratio from our prior work • [Li, Neely 2010, 2011]. • View discrete rewards as being continuously integrated over time (similar to Tsitsiklis 94 • “short proof” of Gitten’s index theorem).

  15. Continuously Integrating the Energy Penalty Accumulated energy expenditure for one server s time • Define: Rs(t) = Current frame number of server s. • Define: ps(t) = es[Rs(t)]/Ts[Rs(t)] = penalty rate function. • If t = start of new frame for server s, then:

  16. Continuously Integrating the Energy Penalty Accumulated energy expenditure for one server s time • Define: Rs(t) = Current frame number of server s. • Define: ps(t) = es[Rs(t)]/Ts[Rs(t)] = penalty rate function. • If t = start of new frame for server s, then:

  17. A look at one sub-frame [t4, t5]: Server 3 0 1 2 3 4 Server 2 0 1 2 Server 1 0 1 2 3 t t0 t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 • We have effective penalty rates and effective service rates: • ps(t) = es[Rs(t)]/Ts[Rs(t)] • γsn(t) = μsn[Rs(t)]/Ts[Rs(t)]

  18. A look at one sub-frame [t4, t5]: Server 3 0 1 2 3 4 Server 2 0 1 2 Server 1 0 1 2 3 t t0 t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 • Define Lyapunov function on queue state (Q1(t), …, QN(t)): • L(t) = ∑nQn(t)2 • Define Lyapunov Drift Δ(tk) for sub-frame k: • Δ(tk) = L(tk+1) – L(tk) • Define Drift-Plus-Integrated-Penalty:

  19. Strategy Server 3 0 1 2 3 4 Server 2 0 1 2 Server 1 0 1 2 3 t t0 t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 When it is time for server s to make a decision, it chooses a processing mode and idle time to minimize its contribution to the drift-plus-integrated-penalty expression.

  20. Resulting Algorithm • At each time tk, each server s that starts a new frame at this time does the following: • Observe all queues Q1(tk), …, QN(tk). • Choose mode and idle time decisions msk, Isk to solve: • No knowledge of arrival rates λ1, …, λN. • Decentralized. • Generalizes the max-weight algorithm of Tassiulas& Ephremides1992 to treat multiple asynchronous servers and joint stability and power optimization.

  21. Resulting Algorithm • At each time tk, each server s that starts a new frame at this time does the following: • Observe all queues Q1(tk), …, QN(tk). • Choose mode and idle time decisions msk, Isk to solve: *Note: Given msk, it is easy to show that:

  22. Performance Theorem If the problem is feasible, then: All queues are stable. Average power satisfies for all k in {1, 2, 3, …}: Average queue size is O(V).

  23. Conclusions Server 3 0 1 2 3 4 Server 2 0 1 2 Server 1 0 1 2 3 t t0 t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 • Lyapunov optimization method extended to treat multi-server systems operating over asynchronous timelines. • Non-Convex Fractional Problem is solved. • Algorithm is decentralized at each server. • Algorithm does not require knowledge of • rates (λ1, …, λN).

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