Energy Optimal Control for Time Varying Wireless Networks Michael J. Neely University of Southern California http://www-rcf.usc.edu/~mjneely
Part 1: A single wireless downlink ( L links) S={Totally Awesome} L 2 1 Slotted time t = 0, 1, 2, … t 0 1 2 3 … Power Vector: P(t) = (P 1 (t), P 2 (t), …, P L (t)) Channel States: S(t) = (S 1 (t), S 2 (t), …, S L (t)) (i.i.d. over slots) µ (P(t), S(t)) (where P(t) Π for all t) Rate-Power Function:
A 1 (t) A 2 (t) A L (t) S={Totally Awesome} µ L (P(t), S(t)) µ 1 (P(t), S(t)) µ 2 (P(t), S(t)) Random arrivals : A i (t) = arrivals to queue i on slot t (bits) Queue backlog : U i (t) = backlog in queue i at slot t (bits) Arrivals and channel states i.i.d. over slots (unknown statistics) Arrival rate: E [ A i (t) ] = λ i (bits/slot), i.i.d. over slots Rate vector: λ = ( λ 1 , λ 2 , … , λ L ) (potentially unknown) Allocate power in reaction to queue backlog + current channel state…
A 1 (t) A 2 (t) A L (t) S={Totally Awesome} µ L (P(t), S(t)) µ 1 (P(t), S(t)) µ 2 (P(t), S(t)) Random arrivals : A i (t) = arrivals to queue i on slot t (bits) Queue backlog : U i (t) = backlog in queue i at slot t (bits) P(t) Π ) Two formulations: (both have peak power constraint: 1. Maximize thruput w/ avg. power constraint: 2. Stabilize with minimum average power (will do this for multihop)
Some precedents: Energy optimal scheduling with known statistics: -Li, Goldsmith, IT 2001 [no queueing] -Fu, Modiano, Infocom 2003 [single queue] -Yeh, Cohen, ISIT 2003 [downlink] -Liu, Chong, Shroff, Comp. Nets. 2003 [no queueing, known stats or unknown stats approx] Stable queueing w/ Lyapunov Drift: MWM -- max µ i U i policy -Tassiulas, Ephremides, Aut. Contr. 1992 [multi-hop network] -Tassiulas, Ephremedes, IT 1993 [random connectivity] -Andrews et. Al. , Comm. Mag. 2001 [server selection] -Neely, Modiano, TON 2003, JSAC 2005 [power alloc. + routing] (these consider stability but not avg. energy optimality…)
A 1 (t) A 2 (t) µ 1 (t) µ 2 (t) Example: Can either be idle, or allocate 1 Watt to a single queue. S 1 (t), S 2 (t) { Good, Medium, Bad }
Capacity region Λ of the wireless downlink: (i) Peak power constraint: λ 1 P(t) Π λ 2 Λ = Region of all supportable input rate vectors λ Capacity region Λ assumes: -Infinite buffer storage -Full knowledge of future arrivals and channel states
Capacity region Λ of the wireless downlink: (i) Peak power constraint: λ 1 P(t) Π (ii) Avg. power constraint: λ 2 Λ = Region of all supportable input rate vectors λ Capacity region Λ assumes: -Infinite buffer storage -Full knowledge of future arrivals and channel states
Capacity region Λ of the wireless downlink: (i) Peak power constraint: λ 1 P(t) Π (ii) Avg. power constraint: λ 2 Λ = Region of all supportable input rate vectors λ Capacity region Λ assumes: -Infinite buffer storage -Full knowledge of future arrivals and channel states
To remove the average power constraint , we create a virtual power queue with backlog X(t). L Dynamics: X(t+1) = max[ X(t) - P av , 0] + P i (t) i =1 L A 1 (t) A 2 (t) A L (t) P i (t) i =1 µ L (P(t), S(t)) µ 1 (P(t), S(t)) µ 2 (P(t), S(t)) P av Observation: If we stabilize all original queues and the virtual power queue subject to only the peak power constraint , then P(t) Π the average power constraint will automatically be satisfied.
Control policy: In this slide we show special case when Π restricts power options to full power to one queue, or idle (general case in paper). A 1 (t) A 2 (t) A L (t) µ 1 (t) µ 2 (t) µ L (t) U i (t) µ i (t) - X(t)P tot Choose queue i that maximizes: Whenever this maximum is positive. Else, allocate no power at all. Then iterate the X(t) virtual power queue equation: L X(t+1) = max[ X(t) - P av , 0] + P i (t) i =1
Performance of Energy Constrained Control Alg. (ECCA): Theorem: Finite buffer size B , input rate λ Λ or λ Λ (r 1 *,…, r L *) = optimal vector (r 1 , …, r L ) = achieved thruput vec. L L r i * - C/(B - A max ) r i (a) Thruput: i =1 i =1 (b) Total power expended over any interval (t 1 , t 2 ) P av (t 2 -t 1 ) + X max where C, X max are constants independent of rate vector and channel statistics. 2 + P peak 2 + P av 2 )/2 C = (A max
Part 2: Minimizing Energy in Multi-hop Networks N node ad-hoc network ( λ ic ) = input rate matrix = (rate from source i to destination node j ) (Assume ( λ ic ) Λ ) S ij (t) = Current channel state between nodes i,j Goal: Develop joint routing, scheduling, power allocation to minimize N E [ g i ( P ij ) ] j n =1 (where g i ( ) are arbitrary convex functions)
Part 2: Minimizing Energy in Multi-hop Networks N node ad-hoc network ( λ ic ) = input rate matrix = (rate from source i to destination node j ) (Assume ( λ ic ) Λ ) S ij (t) = Current channel state between nodes i,j Goal: Develop joint routing, scheduling, power allocation to minimize N E [ g i ( P ij ) ] j n =1 To facilitate distributed implementation, use a cell-partitioned model…
Part 2: Minimizing Energy in Multi-hop Networks N node ad-hoc network ( λ ic ) = input rate matrix = (rate from source i to destination node j ) (Assume ( λ ic ) Λ ) S ij (t) = Current channel state between nodes i,j Goal: Develop joint routing, scheduling, power allocation to minimize N E [ g i ( P ij ) ] j n =1 To facilitate distributed implementation, use a cell-partitioned model…
Analytical technique: Lyapunov Drift L ( U(t) ) = U n2 (t) Lyapunov function: n Lyapunov drift: Δ (t) = E [ L(U(t+1) - L(U(t)) | U(t) ] Theorem: (Lyapunov drift with Cost Minimization) n U n (t) + Vg ( P (t) ) - Vg ( P * ) Δ (t) If for all t : C - ε C + VGmax Then: (a) (stability and bounded delay) n E [ U n ] ε g( P *) + C/V (b) E [ g( P ) ] (resulting cost)
Joint routing, scheduling, power allocation: link l c l *(t) = ( (similar to the original Tassiulas differential backlog routing policy [92])
l i * l j * (2) Each node computes its optimal power level P i * for link l from (1): P i * maximizes: µ l (P, S l (t))W l * - Vg i (P) (over 0 < P < P peak ) Q i * (3) Each node broadcasts Q i * to all other nodes in cell. Node with largest Q i * transmits: Transmit commodity c l * over link l *, power level P i *
Performance: ε = “distance” to capacity ε region boundary. ε Theorem: If ε >0, we have…
Example Simulation: Two-queue downlink with { G, M, B } channels A 1 (t) A 2 (t) µ 1 (t) µ 2 (t)
Conclusions: 1. Virtual power queue to ensure average power constraints. 2. Channel independent algorithms (adapts to any channel). 3. Minimize average power over multihop networks over all joint power allocation, routing, scheduling strategies. 4. Stochastic network optimization theory
http://www-rcf.usc.edu/~mjneely/
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