04
A(t) Q(t) μ(t) Q(t+1) = max[Q(t) + A(t) – μ(t), 0] A Single Wireless Link Uncontrolled: A(t) = random arrivals, λ Controlled: μ(t) = bits served
[depends on power use & channel state]<br>
05
Random Channel States ω(t) Observe ω(t) on slot t
ω(t) in {0, ω1, ω2, … , ωM}
ω(t) ~ i.i.d. over slots
π(ωk) = Pr[ω(t) = ωk]
Probabilities are unknown<br>
06
Opportunistic Power Allocation p(t) = power decision on slot t
[based on observation of ω(t)]
Assume:
p(t) in {0, 1} (“on” or “off”)
μ(t) = p(t)ω(t)
Time average expectations:
p(t) = (1/t) ∑ E[ p(τ) ] τ=0 t-1<br>
07
Stochastic Optimization Problem Minimize : lim p(t)
Subject to: lim μ(t) ≥ λ
p(t) in {0, 1} for all slots t p* = ergodic optimal average power Define: Fix ε>0. ε-approximation on slot t if:
p(t) ≤ p* + ε
μ(t) ≥ λ - ε Challenge: Unknown probabilities!<br>
08
Prior algorithms and analysis E[Q] Tε
Neely 03, 06 (DPP) O(1/ε) O(1/ε2)
Georgiadis et al. 06
Neely, Modiano, Li 05, 08: O(1/ε) O(1/ε2)
Neely 07: O(log(1/ε)) O(1/ε2)
Huang et. al. ‘13 (DPP-LIFO): O(log2(1/ε)) O(1/ε2)
Li, Li, Eryilmaz ‘13, ’15: O(1/ε) O(1/ε2)
(additional sample path results)<br>
09
Prior algorithms and analysis E[Q] Tε
Neely 03, 06 (DPP) O(1/ε) O(1/ε2)
Georgiadis et al. 06
Neely, Modiano, Li 05, 08: O(1/ε) O(1/ε2)
Neely 07: O(log(1/ε)) O(1/ε2)
Huang et. al. ‘13 (DPP-LIFO): O(log2(1/ε)) O(1/ε2)
Li, Li, Eryilmaz ‘13, ’15: O(1/ε) O(1/ε2)
(additional sample path results)
Huang et al. ’14: O(1/ε2/3) O(1/ε1+2/3)<br>
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Main Results Lower Bound: No algorithm can do better than O(1/ε) convergence time.
Upper Bound: Provide tighter analysis to show that Drift-Plus-Penalty (DPP) algorithm achieves:
Convergence Time: Tε = O( log(1/ε) / ε)
Average queue size:
E[Q] ≤ O( log(1/ε) )<br>
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Part 1: Ω(1/ε) Lower Bound for all Algorithms Example system:
ω(t) in {1, 2, 3}
Pr[ω(t) = 3], Pr[ω(t) = 2], Pr[ω(t) = 1] unknown.
Proof methodology:
Case 1: Pr[ transmit | ω(0) = 2 ] > ½.
Assume Pr[ω(t) = 3] = Pr[ω(t) = 2] = ½.
Optimally compensate for mistake on slot 0.
Case 2: Pr[ transmit | ω(0) = 2 ] ≤ ½.
Assume different probabilities.
Optimally compensate for mistake on slot 0.<br>
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Case 1: Fix λ=1, ε > 0 Rate E[μ(t)] Power E[p(t)] X 1 0 0 1 h(μ) curve<br>
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Case 1: Fix λ=1, ε > 0 Rate E[μ(t)] Power E[p(t)] A X 1 0 0 1 (E[μ(0)], E[p(0)])
is in this region.<br>
14
Case 1: Fix λ=1, ε > 0 Rate E[μ(t)] Power E[p(t)] A X 1 0 0 1 (E[μ(0)], E[p(0)])
is in this region.<br>
15
Case 1: Fix λ=1, ε > 0 Rate E[μ(t)] Power E[p(t)] A X 1 0 0 1 (E[μ(0)], E[p(0)])
is in this region.<br>
16
Case 1: Fix λ=1, ε > 0 Rate E[μ(t)] Power E[p(t)] A 1 0 0 1 (E[μ(0)], E[p(0)])
is in this region. X Optimal compensation
Requires time Ω(1/ε).<br>
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Part 2: Upper Bound Channel states 0 < ω1 < ω2 < … < ωM
General h(μ) curve (piecewise linear) Power E[p(t)] Rate E[μ(t)] λ h(μ) curve p*<br>
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Part 2: Upper Bound Channel states 0 < ω1 < ω2 < … < ωM
General h(μ) curve (piecewise linear) Rate E[μ(t)] λ h(μ) curve Transmit iff
ω(t) ≥ ωκ-1 Transmit iff
ω(t) ≥ ωκ Power E[p(t)]<br>
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Drift-Plus-Penalty Alg (DPP) Δ(t) = Q(t+1)2 – Q(t)2
Observe ω(t), choose p(t) to minimize:
Δ(t) + V p(t) Weighted penalty Drift<br>
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Drift-Plus-Penalty Alg (DPP) Δ(t) = Q(t+1)2 – Q(t)2
Observe ω(t), choose p(t) to minimize:
Δ(t) + V p(t) Weighted penalty Drift Algorithm becomes:
P(t) = 1 if Q(t)ω(t) ≥ V
P(t) = 0 else Q(t) ω(t)<br>
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Drift Analysis of DPP V/ωk V/ωk+1 V/ωk-1 0 Q(t) Positive drift Negative drift 0 < ω1 < ω2 < … < ωM<br>
22
Useful Drift Lemma (with transients) Z(t) Negative drift: -β 0 Lemma: E[erZ(t)] ≤ D + (erZ(0) – D)ρt “steady state” “transient” Apply 1: Z(t) = Q(t)
Apply 2: Z(t) = V/ωk – Q(t)<br>
23
After transient time O(V) we get: V/ωk V/ωk+1 V/ωk-1 0 Q(t) Positive drift Negative drift Pr[ Red intervals ] = O(e-cV) Choose V = log(1/ε) Pr[ Red ] = O(ε)<br>
24
After transient time O(V) we get: V/ωk V/ωk+1 V/ωk-1 0 Q(t) Positive drift Negative drift Pr[ Red intervals ] = O(e-cV) λ<br>
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Analytical Result λ But queue is stable, so E[μ] = λ + O(ε).
So we timeshare appropriately and:
E[Q(t)] ≤ O( log(1/ε) )
Tε ≤ O( log(1/ε) / ε ) p*<br>
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Simulation: E[p] versus queue size<br>
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Simulation: E[p] versus time<br>
28
Non-ergodic simulation
(adaptive to changes)<br>
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Conclusions Fundamental lower bound on convergence time
Unknown probabilities
“Cramer-Rao” like bound for controlled queues
Tighter drift analysis for DPP algorithm:
ε-approximation to optimal power
Queue size O( log(1/ε) ) [optimal]
Convergence time O( log(1/ε)/ε ) [near optimal]<br>
