energy efficiency based resource allocation
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Energy efficiency based resource allocation
Philippe Ciblat
Joint work with C. Le Martret and X. Leturc
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Outline
Energy efficiency criterion
Application to Hybrid ARQ (HARQ) based system◦ Review on Fractional Programming◦ Short introduction to HARQ◦ Application to a practical scheme
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Generic resource allocation optimization problem
Energy-efficiency based problem
minθ
f ({Ek (θk )}k=1,...,K )
s.t. QoSk (θk ) ≥ QoS(0)k , ∀k ∈ {1, . . . ,K}
C({θk}k=1,...,K ) ≥ 0Ck (θk ) ≥ 0, ∀k ∈ {1, . . . ,K}
with the following energy efficiency
Ek =# total amount of data correctly delivered by link k
# total consumed energy on link k.
QoS constraints: FER, delay, data rate
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Why Energy Efficiency? example 1
O1: minimum power with data rate constraint (≥ 1Mbits/s)O2: maximum data rate with power constraint (≤ 35dBm)O3: maximum energy efficiency
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THALES GROUP INTERNAL
Thales Communications & Security
Optimal point O3
Optimal point O2
Optimal point O1
Remarks:we do not control the operating pointconsumed energy = transmit power + circuitry power (Pc)
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Why Energy Efficiency? example 2
Data rate: log2(1 + P) with transmit power PConsumed power: P + Pc
-30 -20 -10 0 10 20 30
P (dB)
0
0.5
1
1.5
EE
(b
its/J
ou
le)
EE vs P (for Pc=0)
EE vs P (for Pc = 0)
-10 -5 0 5 10
Pc (dB)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
EE
(b
its/J
ou
le)
-4
-2
0
2
4
6
8
10
P o
ptim
al (d
B)
Max EE and optimal P vs Pc
max EE
P optimal
Max EE (left) and best P (right) vs Pc
Remarks:EE makes sense iff Pc 6= 0EE operating point strongly depends on Pc
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Why Energy Efficiency? example 3
Qr remaining battery (%),Tt time to transmit the messages (s),Np number of transmitted messages,and Data rate (Mbits/s)
Qr Tt (s) Np Data rate
107 sent messagesEE 96 297 107 4.3
MTO 85 256 107 5MPO 89 1 280 107 1
Full battery useEE 0 8 327 2.8× 108 4.3
MTO 0 1 800 7× 107 5MPO 0 12 180 9.5× 107 1
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Capacity based Energy efficiency [Zappone2015]
K usersDownlink communications (EE computed at BTS)FDMA (with equal bandwidth for each user)
max{Pk}
∑Kk=1
Rk︷ ︸︸ ︷log2(1 + Gk Pk )∑Kk=1 Pk + Pc
s.t.K∑
k=1
Pk ≤ Pmax
Pk ≥ 0, ∀k
Remarks:Warning: the power constraint is not necessary saturated.Ratio between concave function and convex functionLinear constraints
⇒ Resorting to Fractional programming (FP)Philippe Ciblat Energy efficiency based resource allocation 7 / 21
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Review on Fractional Programming - 1
maxx∈C
f (x)g(x)
with concave function f , and positive convex function g.Let q∗ be the maximum value (assumed non-negative).Let x∗ be the argmax value.
Lemma 1x∗ is achieved iff
maxx∈C
f (x)− q∗.g(x) = 0
Consequence:If q∗ is known in advance, just solve
maxx∈C
f (x)− q∗.g(x)
As f concave and g convex, f − q∗.g is concaveConvex optimization
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Review on Fractional Programming - 2
Lemma 2Let
F (q) := maxx∈C
f (x)− q.g(x)
is a strictly decreasing function in q ∈ R+
Consequence:q 7→ F (q) is continuouslimq→0 F (q) = maxx∈C f (x) > 0 (as q∗ is non-negatve)limq→∞ F (q) = −∞ (as g is non-negative)q∗ is the unique root of FAny root-finding algorithm works!but each computation of F requires a convex optimization
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Practical algorithm
Lemma 3 : Dinkelbach algorithm [1967]
Start with q0 = 0, select an arbitrary small ε.Iterate over n
1. Given qn, findx∗n = argmax
x∈Cf (x)− qn.g(x)
2. Thenqn+1 =
f (x∗n)g(x∗n)
3. Stop when F (qn+1) < ε
Result: this algorithm converges to (q∗,x∗) up to ε.
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HARQ based energy efficiency
Only channel statistics known at the transmitter
− fast-varying Rayleigh/Rice fading channel− costly to report instantaneous channel realizations− cheaper to report statistics due to its coherence time
HARQ to handle unknown channel variation
k -th link characterizationOFDMASubcarriers are statistically equivalent
◦ γk : bandwidth proportion assigned to link k◦ Qk : energy used by link k in one OFDM symbol
− independent of subcarrier− Ek = Qk/γk : energy of link k in entire bandwidth
Rice fading channel
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Type-I HARQType-I HARQ: packet S is composed by coded symbols sn
.
S1
S1
S2
NACK
ACK
TX RX
T NO
YES
.
• first packet is more protected• there is less retransmission• transmission delay is reduced• Efficiency is upper-bounded by the code rate
DrawbacksEach received packet is treated independentlyMis-decoded packet is thrown in the trash
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Type-II HARQMemory at RX side is considered⇒ Type-II HARQ
.
NACK
ACK
TX RX
S1(1)
S1(2)
S2(1)
NO
YES
.
Main examples:Chase Combining (CC)Incremental Redundancy (IR)
Y1 = S1(1) + N1Y2 = S1(2) + N2
then joint detection on
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Performance metrics
Packet Error Rate (PER):
PER = Prob(message is not decoded)
Efficiency (Throughput/Goodput/etc):
η =information bits received without error
transmitted bits(Mean) delay:
d = # transmitted packets when message is correctly received
Jitter:σd = delay standard deviation
Quality of Service (QoS)
Data: PER and efficiencyVoice on IP: delayVideo Streaming: efficiency and jitter
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Our practical optimization problem
Type-I HARQ with Rice channel and minimum efficiency constraints
maxγ,E
K∑k=1
mk Rkγk (1− πk (Gk Ek ))
κ1,kγk Ek + κ2,k
s.t. mk Rkγk (1− πk (Gk Ek )) ≥ η(0)k ,∑K
k=1 γk ≤ 1, γk ≥ 0, Ek ≥ 0with
Gk = |Ak |2 + ς2k
πk probability that one frame in error
πk (Gk Ek ) ≈ ak
bk
4∑`=1
c`e− |Ak |2Gk Ekθ`dk
1+ς2k Gk Ekθ`dk
1 + ς2k Gk Ekθ`dk
δk
Remark: More difficult than just information-theoretic metric
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How to solve it?
fk : x 7→ 1− πk (Gk x) concavechange of variables (γk ,Ek ) 7→ (γk ,Qk ), then
(γk ,Qk ) 7→ γk (1− πk (Gk Qk/γk )) = γk fk (Qk/γk )
is concave as perspective of fkConsequently, in (γk ,Qk )
◦ Numerator: concave◦ Denominator: convex (as linear)◦ Constraints set: convex set
Results (fractional programming tool)
Jong’s algorithm: solve at iteration i (with the above constraints)
maxγ,Q
K∑k=1
u(i)k mk Rkγk (1− πk (Gk Qk/γk ))− v (i)
k κ1,k Qk
and update u(i)k and v (i)
k according to well-defined equationsKKT can be written in closed-form
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Numerical results
K = 10 links, Bandwidth W = 5 MHzQPSK, convolutional code of rate 1/2Rician factor: 10 (dashed line), 0 (solid line)
1 2 3 4 5 6 7 8
x 105
0.5
1
1.5
2
2.5
3
3.5
4
4.5x 10
7
η(1 )`
(bits/s)
SE
E (
bits
/joul
e)
MSEEMPEEMMEEMGEEMPO [Litt.]
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Future works
Model for Pc :◦ How to take into account the manufacturing◦ How to take into account the decoding power consumption
= a − b × log(C − R)
with C the Shannon capacity and R the current data rate◦ Extension to massive MIMO done but without two previous items
and also depends on the hybrid RF.
More philosophical thoughts: what do we want to do with anetwork?
◦ If Pc is large (not efficient), then EE leads to high P, and moreGreen House Gas (GHG)
◦ If Pc is low (very efficient), then EE leads to low P, and to lowtech(low rate, for instance)
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Sketch of proof for Lemma 1
Let x∗ be the optimal solution of RHS of Lemma 1. It means
f (x∗)− q∗g(x∗) = 0⇒ f (x∗)g(x∗)
= q∗
Moreover ∀x 6= x∗,
f (x)− q∗g(x) ≤ 0⇒ f (x)g(x)
≤ q∗
which proves that x∗ is the optimal solution of FP.Let x∗ be the optimal solution of FP. As q∗ = f (x∗)/g(x∗), we get
f (x)g(x)
< q∗ ⇒ f (x)− q∗g(x) ≤ 0
for any x 6= x∗.
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Sketch of proof for Lemma 2
Assume q1 > q2
x∗1 the argmax with q1, and x∗2 the argmax with q2
F (q1) = f (x∗1)− q1g(x∗1)(a)< f (x∗1)− q2g(x∗1)(b)< f (x∗2)− q2g(x∗2) = F (q2)
(a) q1 > q2 and g is a positive function. Strict inequality.(b) x∗2 is the argmax for q2
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Sketch of proof for Lemma 3
Step 1: sequence {qn}n is strictly increasing.Assuming F (qn) = f (x∗n)− qng(x∗n) > 0 (True for F (q0))f (x∗n)− qn+1g(x∗n) = 0
qn+1 − qn =F (qn)
g(x∗n)>
F (qn)
gmax> 0 (1)
with gmax = maxx∈C g(x) (it exists if C compact)Step 2: convergence to q∗
Due to stopping criterion, bounded increasing sequence, and solimn→∞ qn = qAssuming that limn→∞ F (qn) = F (q) > ε, i.e, q < q∗ − δ withF (q∗ − δ) = ε.but as qn converges, (qn+1 − qn) converges to 0, and Eq. (1)implies
F (qn)→ 0⇒ qn → q∗
which leads to a contradiction.
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