Michael J. Neely, University of Southern California CISS, Princeton University, March 2012 Asynchronous Scheduling for Energy Optimality in Systems with Multiple Servers t Server Server 2 Server 3 t0t0 t1t1 t2t2 t3t3 t4t4 t5t5 t6t6 t7t7 t8t8 t 10 t9t9

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

λ1λ1 λ2λ2 λΝλΝ Server 1 N classes of jobs, with Poisson arrivals. Jobs are queued according to class: Q 1 (t), …, Q N (t). S heterogeneous servers. Servers can have…  …different classes that are allowed for service.  …different processing mode options.  …different processing times and energy expenditures. Server 2 Server S

Consider one particular server: Active Idle Frame 0 Frame 1 D s [0] I s [0] D s [1] I s [1] D s [2] t Continuous Time operation. Each server s in {1, …, S} operates over frames. At frame r in {0, 1, 2, …}, server s decides:  m s [r] = processing mode for frame r. (chosen in some abstract set of options M s ).  I s [r] = Idle time for frame r (possibly 0). (chosen in the interval [0, I max ]).

Affect of Decisions on 1 Frame Active Idle Frame 0 Frame 1 D s [0] I s [0] D s [1] I s [1] D s [2] t Choosing m s [r], I s [r] determines: μ sn [r] = # jobs of type n that can be served by s = μ sn (m s [r]) (for each n in {1, …, N}) e s [r] = energy expended by server s = e s proc (m s [r]) + p s idle I s [r] T s [r] = frame duration for server s = D(m s [r]) + I s [r] ⌃ ⌃ ⌃

Now put all servers together: t Server Server 2 Server 3 t0t0 t1t1 t2t2 t3t3 t4t4 t5t5 t6t6 t7t7 t8t8 t 10 t9t9 t k = start of k th sub-frame. S (t k ) = {servers that start a frame at time t k }. What is the “state” at time t k ? State has size at least 2 S. (because each server is either ACTIVE or IDLE)

A “Simpler” Approach Look at the time averages that the network can achieve, and that are necessary. Define ē s, T s, μ sn as frame averages: The time average energy for server s is then:

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?

Special Case for Intuition Suppose set of mode options M s is finite for all s. Can show optimality of stationary, randomized policies with I s fixed, and with Pr[m s [r] = m s ] = q s (m s ):

Special Case for Intuition Suppose set of mode options M s is finite for all s. Can show optimality of stationary, randomized policies with I s fixed, and with Pr[m s [r] = m s ] = q s (m s ):

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.

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!

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 M s are infinite.  Adaptive when rates change. Extends Lyapunov optimization theory to treat coupled systems that operate asynchronously over their own timelines.

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).

Continuously Integrating the Energy Penalty time Accumulated energy expenditure for one server s Define: R s (t) = Current frame number of server s. Define: p s (t) = e s [R s (t)]/T s [R s (t)] = penalty rate function. If t = start of new frame for server s, then:

Continuously Integrating the Energy Penalty time Accumulated energy expenditure for one server s Define: R s (t) = Current frame number of server s. Define: p s (t) = e s [R s (t)]/T s [R s (t)] = penalty rate function. If t = start of new frame for server s, then:

A look at one sub-frame [t 4, t 5 ]: t Server Server 2 Server 3 t0t0 t1t1 t2t2 t3t3 t4t4 t5t5 t6t6 t7t7 t8t8 t 10 t9t9 We have effective penalty rates and effective service rates: p s (t) = e s [R s (t)]/T s [R s (t)] γ sn (t) = μ sn [R s (t)]/T s [R s (t)]

A look at one sub-frame [t 4, t 5 ]: t Server Server 2 Server 3 t0t0 t1t1 t2t2 t3t3 t4t4 t5t5 t6t6 t7t7 t8t8 t 10 t9t9 Define Lyapunov function on queue state (Q 1 (t), …, Q N (t)): L(t) = ∑ n Q n (t) 2 Define Lyapunov Drift Δ(t k ) for sub-frame k: Δ(t k ) = L(t k+1 ) – L(t k ) Define Drift-Plus-Integrated-Penalty:

Strategy t Server Server 2 Server 3 t0t0 t1t1 t2t2 t3t3 t4t4 t5t5 t6t6 t7t7 t8t8 t 10 t9t9 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.

Resulting Algorithm At each time t k, each server s that starts a new frame at this time does the following: Observe all queues Q 1 (t k ), …, Q N (t k ). Choose mode and idle time decisions m sk, I sk to solve: No knowledge of arrival rates λ 1, …, λ N. Decentralized. Generalizes the max-weight algorithm of Tassiulas & Ephremides 1992 to treat multiple asynchronous servers and joint stability and power optimization.

Resulting Algorithm At each time t k, each server s that starts a new frame at this time does the following: Observe all queues Q 1 (t k ), …, Q N (t k ). Choose mode and idle time decisions m sk, I sk to solve: *Note: Given m sk, it is easy to show that:

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

Conclusions 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 (λ 1, …, λ N ). t Server Server 2 Server 3 t0t0 t1t1 t2t2 t3t3 t4t4 t5t5 t6t6 t7t7 t8t8 t 10 t9t9