Report

Stochastic optimization for power-aware distributed scheduling ω(t) t Michael J. Neely University of Southern California http://www-bcf.usc.edu/~mjneely Outline • Lyapunov optimization method • Power-aware wireless transmission – Basic problem – Cache-aware peering – Quality-aware video streaming • Distributed sensor reporting and correlated scheduling A single wireless device R(t) = r(P(t), ω(t)) chosen observed Timeslots t = {0, 1, 2, …} ω(t) = Random channel state on slot t P(t) = Power used on slot t R(t) = Transmission rate on slot t (function of P(t), ω(t)) Example R(t) = log(1 + P(t)ω(t)) chosen observed ω(t) t R(t) t Example R(t) = log(1 + P(t)ω(t)) chosen observed ω(t) t R(t) t Example R(t) = log(1 + P(t)ω(t)) chosen observed ω(t) t R(t) t Optimization problem Maximize: R Subject to: P ≤ c Given: • Pr[ω(t)=ω] = π(ω) , ω in {ω1, ω2, …, ω1000} • p(t) in P = {p1, p2, …, p5} • c = desired power constraint Consider randomized decisions • ω(t) in {ω1, ω2, …, ω1000} • P(t) in P = {p1, p2, …, p5} Pr[pk | ωi ] = Pr[P(t) = pk | ω(t)=ωi] 5 ∑Pr[pk | ωi ] = 1 k=1 ( for all ωi in {ω1, ω2, …, ω1000} ) Linear programming approach Max: R 1000 5 Max: ∑ ∑ π(ωi) Pr[pk|ωi] r(pk,ωi) i=1 k=1 S.t. : P ≤ c S.t.: 1000 5 ∑ ∑ π(ωi) Pr[pk|ωi] pk ≤ c i=1 k=1 Given parameters: π(ωi) (1000 probabilities) r(pk , ωi) (5*1000 coefficients) Optimization variables: Pr[pk|ωi] (5*1000 variables) Multi-dimensional problem R1(t) 1 R2(t) Access Point RN(t) 2 • Observe (ω1(t), …, ωN(t)) • Decisions: -- Choose which user to serve -- Choose which power to use N Goal and LP approach Maximize: R1 + R2 + … + RN Subject to: Pn ≤ c for all n in {1, …, N} LP has given parameters: π(ω1, …, ωN) (1000N probabilities) rn(pk , ωi) (N*5N*1000N coefficients) LP has optimization variables: Pr[pk|ωi] (5N*1000N variables) Advantages of LP approach • Solves the problem of interest • LPs have been around for a long time • Many people are comfortable with LPs Disadvantages of LP approach Disadvantages of LP approach • Need to estimate an exponential number of probabilities. • LP has exponential number of variables. • What if probabilities change? • Fairness? • Delay? • Channel errors? Lyapunov optimization approach Maximize: R1 + R2 + … + RN Subject to: Pn ≤ c for all n in {1, …, N} Lyapunov optimization approach Maximize: R1 + R2 + … + RN Subject to: Pn ≤ c for all n in {1, …, N} Virtual queue for each constraint: Pn(t) Qn(t) c Qn(t+1) = max[Qn(t) + Pn(t) – c, 0] Stabilizing virtual queue constraint satisfied! Lyapunov drift Q2 Q1 L(t) = ½ ∑ Qn(t)2 n Δ(t) = L(t+1) – L(t) Drift-plus-penalty algorithm Every slot t: • Observe (Q1(t), …., QN(t)), (ω1(t), …, ωN(t)) • Choose (P1(t), …, PN(t)) to greedily minimize: Δ(t) - (1/ε)(R1(t) + … + RN(t)) drift penalty • Update queues. Low complexity No knowledge of π(ω) probabilities is required Specific DPP implementation • Each user n observes ωn(t), Qn(t). • Each user n chooses Pn(t) in P to minimize: -(1/ε)rn(Pn(t), ωn(t)) + Qn(t)Pn(t) • Choose user n* with smallest such value. • User n* transmits with power level Pn*(t). Low complexity No knowledge of π(ω) probabilities is required Performance Theorem Assume it is possible to satisfy the constraints. Then under DPP with any ε>0: • All power constraints are satisfied. • Average thruput satisfies: R1 + … + RN ≥ throughputopt – O(ε) • Average queue size satisfies: ∑ Qn ≤ O(1/ε) General SNO problem ω(t) = Observed random event on slot t π(ω) = Pr[ω(t)=ω] (possibly unknown) α(t) = Control action on slot t Aω(t) = Abstract set of action options Minimize: y0(α(t), ω(t)) Subject to: yn(α(t), ω(t)) ≤ 0 for all n in {1, …, N} α(t) in Aω(t) for all t in {0, 1, 2, …} Such problems are solved by the DPP algorithm. Performance theorem: O(ε), O(1/ε) tradeoff. What we have done so far • Lyapunov optimization method • Power-aware wireless transmission – Basic problem – Cache-aware peering – Quality-aware video streaming • Distributed sensor reporting and correlated scheduling What we have done so far • Lyapunov optimization method • Power-aware wireless transmission – Basic problem – Cache-aware peering – Quality-aware video streaming • Distributed sensor reporting and correlated scheduling What we have done so far • Lyapunov optimization method • Power-aware wireless transmission – Basic problem – Cache-aware peering – Quality-aware video streaming • Distributed sensor reporting and correlated scheduling What we have done so far • Lyapunov optimization method • Power-aware wireless transmission – Basic problem – Cache-aware peering – Quality-aware video streaming • Distributed sensor reporting and correlated scheduling Mobile P2P video downloads Mobile P2P video downloads Mobile P2P video downloads Access Point Mobile P2P video downloads Access Point Mobile P2P video downloads Access Point Mobile P2P video downloads Access Point Access Point Mobile P2P video downloads Access Point Access Point Access Point Mobile P2P video downloads Access Point Access Point Access Point Mobile P2P video downloads Access Point Access Point Access Point Mobile P2P video downloads Access Point Access Point Access Point Mobile P2P video downloads Access Point Access Point Access Point Mobile P2P video downloads Access Point Access Point Access Point Cache-aware scheduling • Access points (including “femto” nodes) • Typically stationary • Typically have many files cached • Users • Typically mobile • Typically have fewer files cached • Assume each user wants one “long” file • Can opportunistically grab packets from any nearby user or access point that has the file. Quality-aware video delivery Quality Bits: 40968 Bits: 277256 Layer L D: 0 Bits: 419640 Bits: 299216 D: 0 D: 0 D: 0 Quality Layer 2 Quality Layer 1 Bits: 7370 D: 10.777 Bits: 41304 Bits: 72800 D: 6.716 D: 6.261 Bits: 59984 D: 6.129 Bits: 8176 D: 11.045 Bits: 58152 D: 7.363 Bits: 97864 D: 6.971 Bits: 120776 D: 7.108 Video chunks as time progresses • D = Distortion. • Results hold for any matrices Bits(layer, chunk), D(layer, chunk). • Bits are queued for wireless transmission. Fair video quality delivery Minimize: f( D1 ) + f( D2 ) + … + f( DN ) Subject to: Pn ≤ c for all n in {1, …, N} Video playback rate constraints Fair video quality delivery Minimize: f( D1 ) + f( D2 ) + … + f( DN ) Subject to: Pn ≤ c for all n in {1, …, N} Video playback rate constraints Recall the general form: Min: y0 S.t. : yn ≤ 0 for all n α(t) in Aω(t) for all t Fair video quality delivery Minimize: f( D1 ) + f( D2 ) + … + f( DN ) Subject to: Pn ≤ c for all n in {1, …, N} Video playback rate constraints Recall the general form: Min: y0 S.t. : yn ≤ 0 for all n α(t) in Aω(t) for all t Define Yn(t) = Pn(t) - c Fair video quality delivery Minimize: f( D1 ) + f( D2 ) + … + f( DN ) Subject to: Pn ≤ c for all n in {1, …, N} Video playback rate constraints Recall the general form: Min: y0 S.t. : yn ≤ 0 for all n α(t) in Aω(t) for all t Define auxiliary variable γ(t) in [0, Dmax] Equivalence via Jensen’s inequality Minimize: f( D1 ) + f( D2 ) + … + f( DN ) Subject to: Pn ≤ c for all n in {1, …, N} Video playback rate constraints Minimize: f( γ1(t)) + f( γ2(t)) + … + f( γN(t)) Subject to: Pn ≤ c for all n in {1, …, N} γn = Dn for all n in {1, …, N} Video playback rate constraints Example simulation BS • Region divided into 20 x 20 subcells (only a portion shown here). • 1250 mobile devices, 1 base station • 3.125 mobiles/subcell • • • • Phases 1, 2, 3: File availability prob = 5%, 10%, 7% Basestation Average Traffic: 2.0 packets/slot Peer-to-Peer Average Traffic: 153.7 packets/slot Factor of 77.8 gain compared to BS alone! What we have done so far • Lyapunov optimization method • Power-aware wireless transmission – Basic problem – Cache-aware peering – Quality-aware video streaming • Distributed sensor reporting and correlated scheduling Distributed sensor reports ω1(t) 1 ω2(t) 2 Fusion Center • ωi(t) = 0/1 if sensor i observes the event on slot t • Pi(t) = 0/1 if sensor i reports on slot t • Utility: U(t) = min[P1(t)ω1(t) + (1/2)P2(t)ω2(t),1] Maximize: U Subject to: P1 ≤ c P2 ≤ c What is optimal? Agree on plan 0 1 2 3 4 t What is optimal? Agree on plan 0 1 2 3 4 Example plan: User 1: • t=even Do not report. • t=odd Report if ω1(t)=1. User 2: • t=even Report if ω2(t)=1 • t=odd: Report with prob ½ if ω2(t)=1 t Common source of randomness Day 1 Day 2 Example: 1 slot = 1 day Each user looks at Boston Globe every day: • If first letter is a “T” Plan 1 • If first letter is an “S” Plan 2 • Etc. Specific example Assume: • Pr[ω1(t)=1] = ¾, Pr[ω2(t)=1] = ½ • ω1(t), ω2(t) independent • Power constraint c = 1/3 Approach 1: Independent reporting • If ω1(t)=1, user 1 reports with probability θ1 • If ω2(t)=1, user 2 reports with probability θ2 Optimizing θ1, θ2 gives u = 4/9 ≈ 0.44444 Approach 2: Correlated reporting Pure strategy 1: • User 1 reports if and only if ω1(t)=1. • User 2 does not report. Pure strategy 2: • User 1 does not report. • User 2 reports if and only if ω2(t)=1. Pure strategy 3: • User 1 reports if and only if ω1(t)=1. • User 2 reports if and only if ω2(t)=1. Approach 2: Correlated reporting X(t) = iid random variable (commonly known): • Pr[X(t)=1] = θ1 • Pr[X(t)=2] = θ2 • Pr[X(t)=3] = θ3 On slot t: • Users observe X(t) • If X(t)=k, users use pure strategy k. Optimizing θ1, θ2, θ3 gives u = 23/48 ≈ 0.47917 Summary of approaches Strategy u Independent reporting 0.44444 Correlated reporting 0.47917 Centralized reporting 0.5 Summary of approaches Strategy u Independent reporting 0.44444 Correlated reporting 0.47917 Centralized reporting 0.5 It can be shown that this is optimal over all distributed strategies! General distributed optimization Maximize: U Subject to: Pk ≤ 0 for k in {1, …, K} ω(t) = (ω1(t), …, ωΝ(t)) π(ω) = Pr[ω(t) = (ω1, …, ωΝ)] α(t) = (α1(t), …, αΝ(t)) U(t) = u(α(t), ω(t)) Pk(t) = pk(α(t), ω(t)) Pure strategies A pure strategy is a deterministic vectorvalued function: g(ω) = (g1(ω1), g2(ω2), …, gΝ (ωΝ)) Let M = # pure strategies: M = |A1||Ω1| x |A2||Ω2| x ... x |AN||ΩN| Optimality Theorem There exist: • K+1 pure strategies g(m)(ω) • Probabilities θ1, θ2, …, θK+1 such that the following distributed algorithm is optimal: X(t) = iid, Pr[X(t)=m] = θm • Each user observes X(t) • If X(t)=m use strategy g(m)(ω). LP and complexity reduction • The probabilities can be found by an LP • Unfortunately, the LP has M variables • If (ω1(t), …, ωΝ(t)) are mutually independent and the utility function satisfies a preferred action property, complexity can be reduced • Example N=2 users, |A1|=|A2|=2 --Old complexity = 2|Ω1|+|Ω2| --New complexity = (|Ω1|+1)(|Ω2|+1) Lyapunov optimization approach • Define K virtual queues Q1(t), …, QK(t). • Every slot t, observe queues and choose strategy m in {1, …, M} to maximize a weighted sum of queues. • Update queues with delayed feedback: Qk(t+1) = max[Qk(t) + Pk(t-D), 0] Separable problems If the utility and penalty functions are a separable sum of functions of individual variables (αn(t), ωn(t)), then: • There is no optimality gap between centralized and distributed algorithms • Problem complexity reduces from exponential to linear. Simulation (non-separable problem) • • • • • • 3-user problem αn(t) in {0, 1} for n ={1, 2, 3}. ωn(t) in {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} V=1/ε Get O(ε) guarantee to optimality Convergence time depends on 1/ε Utility Utility versus V parameter (V=1/ε) V (recall V = 1/ε) Average power up to time t Average power versus time V=100 V=50 V=10 power constraint 1/3 Time t Adaptation to non-ergodic changes Conclusions • Drift-plus-penalty is a strong technique for general stochastic network optimization • Power-aware scheduling • Cache-aware scheduling • Quality-aware video streaming • Correlated scheduling for distributed stochastic optimization Conclusions • Drift-plus-penalty is a strong technique for general stochastic network optimization • Power-aware scheduling • Cache-aware scheduling • Quality-aware video streaming • Correlated scheduling for distributed stochastic optimization