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Stochastic Network Optimization(tutorial)
M. J. NeelyUniversity of Southern California
See detailed derivations for these results in:
M. J. Neely, Stochastic Network Optimization with Application to Communication and Queueing Systems, Morgan & Claypool 2010.
http://www-bcf.usc.edu/~mjneely/
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Outline
• Queue Stability• Randomized Algs and Linear Programs for
Queue Stability (known statistics)• Dynamic Algs and Drift-Plus-Penalty (unknown statistics) • Backpressure Examples (if time)
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Queue Dynamics
Q(t+1) = max[Q(t) + a(t) – b(t), 0]
or
Q(t+1) = max[Q(t) + y(t), 0]
where y(t) = a(t) – b(t)
Q(t) b(t)a(t)
Note: Departures(t) = min[Q(t), b(t)]
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Rate Stability
Definition: Q(t) is rate stable if:
limt∞ Q(t)/t = 0 (with probability 1)
Q(t) b(t)a(t)
Why use this definition? Why use this definition?
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Rate Stability Theorem
Theorem: Recall y(t) = a(t) – b(t). Suppose:
limt∞ (1/t)∑τ=0 y(τ) = yav (w.p.1)
Then Q(t) is rate stable if and only if: yav ≤ 0
Q(t) b(t)a(t)
t-1
Proof: Necessary condition is straightforward. Sufficient condition requires a trick.
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Example Arrival Processes a(t) with Time Average Rate λ
• Deterministic: 1 packet periodically arrives every 10 slots.
limt∞ (1/t)∑τ=0 a(τ) = 1/10 (w.p.1)
• Random: {a(t)} i.i.d. over t in {0, 1, 2, …}.
limt∞ (1/t)∑τ=0 a(τ) = E{a(0)} (w.p.1)
t-1
t-1
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Outline
• Queue Stability• Randomized Algs and Linear Programs for
Queue Stability (known statistics)• Dynamic Algs and Drift-Plus-Penalty (unknown statistics) • Backpressure Examples (if time)
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Scheduling for Queue Stability
Q1(t) b1(t)Arrival rate λ1
QK(t) bK(t)Arrival rate λK
If we make service allocations every slot to ensurethat each bi(t) has time average bi,av , and:
then by the rate stability theorem, all queues are rate stable.
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Scheduling for Queue Stability
Q1(t) b1(t)Arrival rate λ1
QK(t) bK(t)Arrival rate λK
If we make service allocations every slot to ensurethat each bi(t) has time average bi,av , and:
then by the rate stability theorem, all queues are rate stable.
bi,av ≥ λi for all i in {1, …, K}
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A 3-Queue, 2-Server System:
Q1(t)λ1
Q2(t)λ2
Q3(t)
(b1(t), b2(t), b3(t)) in {(1, 1, 0), (1, 0, 1), (0, 1, 1)}
λ3
A server can serve 1 packet/slot. Every slot, choose:
How to schedule the servers to make all queues rate stable?
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Exercise (apply what we just learned):1λ1
2λ2
3λ3
Design a stabilizing server allocation algorithm for:(a) (λ1, λ2, λ3) = (1, 1/2, 1/2).(b) (λ1, λ2, λ3) = (2/3, 2/3, 2/3).(c) (λ1, λ2, λ3) = (1/2, 3/4, 3/4).(d) (λ1, λ2, λ3) = (0.4, 0.7, 0.8).(e) (λ1, λ2, λ3) = (0.46, 0.82, 0.70).
How many can you do in 5 minutes? (a) and (b) are easy.
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LP Approach to part (e):1λ1
2λ2
3λ3
Want to support (λ1, λ2, λ3) = (0.46, 0.82, 0.70).
(1) (1) (0) (0.46) p1 (1) + p2 (0) + p3 (1) ≥ (0.82) (0) (1) (1) (0.70)
p1 + p2 + p3 = 1
p1≥0, p2≥0, p3≥0
Choose i.i.d. with dist: p1 = Pr[choose (1,1,0)]p2 = Pr[choose (1,0,1)]p3 = Pr[choose (0,1,1)]
Linear Program
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General Problem (no channel state variation)
1λ1
λK K
b1(t)
bK(t)
Choose b(t)= (b1(t), …, bK(t)) in General set Γ Γ = {b(1), b(2) , b(3) , … , b(M-1) , b(M) }
Choose i.i.d. with distribution given by LP: ∑m=1 b(m)p(m) ≥ λ (where λ= (λ1, ..., λK) )p(m) ≥ 0 for all m in {1,…, M}∑m=1 p(m) = 1
M
M
Note: We cannot do better than this! This provably defines the capacity region!
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General Problem (no channel state variation)
1λ1
λK K
b1(t)
bK(t)
Choose b(t)= (b1(t), …, bK(t)) in General set Γ Γ = {b(1), b(2) , b(3) , … , b(M-1) , b(M) }
Equivalently: Design a randomized alg thatchooses b(t) i.i.d. over slots so that:
E[b(t)] ≥ λ Note: We cannot do better than this! This provably defines the capacity region!
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General Problem (with channel state variation)
ω(t) = random states (i.i.d. every slot) with distribution π(ω) = Pr[ω(t)=ω]α(t) = control action (chosen in action space Aω(t) )
Penalties yi(t) = yi(α(t), ω(t)).
Minimize: y0
Subject to: (1) yi ≤ 0 for all i in {1, .., K}
(2) α(t) in Aω(t) for all slots t.
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LP Solution for known π(ω) = Pr[ω(t)=ω] Minimize: y0
Subject to: (1) yi ≤ 0 for all i in {1, .., K} (2) α(t) in Aω(t) for all slots t.
For simplicity of this slide, assume finite sets: ω(t) in {ω1, ω2, …, ωJ} , α(t) in {α1, α2, …, αΜ}Recall that yi(t) = yi(α(t), ω(t)). Let p(αm , ωj) = Pr[Choose α(t) = αm | ω(t) =ωj]
Exercise: Write an LP to solve this. Start with:
Minimize: ∑j=1∑m=1 π(ωj) p(αm, ωj) y0(αm, ωj) J M
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LP Solution for known π(ω) = Pr[ω(t)=ω] Minimize: y0
Subject to: (1) yi ≤ 0 for all i in {1, .., K} (2) α(t) in Aω(t) for all slots t.
For simplicity of this slide, assume finite sets: ω(t) in {ω1, ω2, …, ωJ} , α(t) in {α1, α2, …, αΜ}Recall that yi(t) = yi(α(t), ω(t)). Let p(αm , ωj) = Pr[Choose α(t) = αm | ω(t) =ωj]
Equivalently: Use randomized alg with:
Min: E{y0(t)} Subject to: E{yi(t)} ≤ 0 for all i in {1, …, K}
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Properties of the LP
Known Constants:• π(ω) for all ω in {ω1, ω2, …, ωJ}. (J is typically exponential in network size)• yi(αm, ωj) for all i, m, j in {1,…,K}x{1,….M}x{1,…,J}
LP Variables: • p(αm, ωj) for all m in {1, …, M}, j in {1, …, J}. (number of variables is typically exponential in network size)
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Disadvantages of this LP Approach?
• .• .• .• .
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Disadvantages of this LP Approach
• Problem is huge.- Non-polynomial number of variables.
• Need to know distribution π(ω). -Typically non-polynomial number of ω. -Good estimate would take “years”of samples. -Probabilities will certainly change by this time.• Not Adaptive to Changes.• What if ω(t) is non-i.i.d.? Non-ergodic? • For queueing networks: -Delay guarantees? -Flow control?
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Outline
• Queue Stability• Randomized Algs and Linear Programs for
Queue Stability (known statistics)• Dynamic Algs and Drift-Plus-Penalty (unknown statistics)• Backpressure Examples (if time)
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Dynamic ApproachMinimize: y0
Subject to: (1) yi ≤ 0 for all i in {1, .., K} (2) α(t) in Aω(t) for all slots t.
(1) Make “virtual queue” for each inequality constraint:
Qi(t+1) = max[Qi(t) + yi(t), 0] for i in {1, …, K}
Qi(t)/t 0 implies yi ≤ 0 Why? Because:
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Virtual Queue Stability Theorem:
Theorem: Qi(t)/t 0 implies yi ≤ 0.
Recall: Qi(t+1) = max[Qi(t) + yi(t), 0]
Proof: Qi(τ+1) = max[Qi(τ) + yi(τ), 0] ≥ Qi(τ) + yi(τ).
Thus: Qi(τ+1) - Qi(τ) ≥ yi(τ) for all τ.
Use telescoping sums over τ in {0, …, t-1}: Qi(t) – Qi(0) ≥ ∑τ=0 yi(τ).Divide by t and take limit as t ∞.
t-1
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Problem BecomesMinimize: y0
Subject to: (1) Qi(t) stable for all i in {1, .., K} (2) α(t) in Aω(t) for all slots t.
(2) Define L(t) = sum of squares of queues (times ½): L(t) = (1/2)∑iQi(t)2
Why? Because if L(t) is “small,” then all queues are “small.”
(3) Define the “drift” Δ(t): Δ(t) = L(t+1) – L(t)
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Drift-Plus-Penalty AlgorithmMinimize: y0
Subject to: (1) Qi(t) stable for all i in {1, .., K} (2) α(t) in Aω(t) for all slots t.
(4) Every slot t, observe ω(t), Q(t). Then choose α(t) in Aω(t) to greedily minimize a boundon:
Drift-Plus-Penalty: Δ(t) + Vy0(α(t), ω(t))(5) Update virtual queues for i in {1, …, K}:
Qi(t+1) = max[Qi(t) + yi(α(t), ω(t)), 0]
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Drift-Plus-Penalty BoundRecall: Qi(t+1) = max[Qi(t) + yi(α(t), ω(t)), 0].
Thus: Qi(t+1)2 ≤ [Qi(t) + yi(α(t), ω(t))]2
= Qi(t)2 + yi(α(t), ω(t)) 2 + 2Qi(t)yi(α(t), ω(t))
Assuming squares of yi(α(t), ω(t)) 2 are bounded: Δ(t) ≤ B + ∑iQi(t)yi(α(t), ω(t))So: Δ(t) + Vy0(α(t), ω(t)) ≤ B + ∑iQi(t)yi(α(t), ω(t)) + Vy0(α(t), ω(t))
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Δ(t) + Vy0(α(t), ω(t)) ≤ B + ∑iQi(t)yi(α(t), ω(t)) + Vy0(α(t), ω(t))
Every slot t, observe ω(t), Q(t). Then choose α(t) by:
Minimize: ∑iQi(t)yi(α(t), ω(t)) + Vy0(α(t), ω(t))
Subject to: α(t) in Aω(t).
• No knowledge of π(ω) required.• No curse of dimensionality.• Provably adapts to changes, general sample paths.
The action is thus:
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Performance Tradeoff Theorem:
Theorem: For any V≥0. If the problem is feasible, then the drift-plus-penalty algorithm stabilizes allvirtual queues. Further: (i) yi ≤ 0 for all i in {1, .., K}.
(ii) y0 ≤ y0opt + B/V
(iii) If there is an “interior point” of the feasibility region, then Qi ≤ O(V) for all i in {1, .., K}.
Assume ω(t) i.i.d. for simplicity:
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Performance Tradeoff Theorem:
Proof of (ii) [parts (i) and (iii) just as easy]:
Assume ω(t) i.i.d. for simplicity:
Δ(t) + Vy0(α(t), ω(t)) ≤ B + ∑iQi(t)yi(α(t), ω(t)) + Vy0(α(t), ω(t)) ≤ B + ∑iQi(t)yi(α*(t), ω(t)) + Vy0(α*(t), ω(t))
For any other (possibly randomized) decision α*(t).
Let’s re-write the above to give more room to work…
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Performance Tradeoff Theorem:
Proof of (ii) [continued]:
Assume ω(t) i.i.d. for simplicity:
Δ(t) + Vy0(α(t), ω(t)) ≤ B + ∑iQi(t)yi(α*(t), ω(t))
+ Vy0(α*(t), ω(t))For any other (possibly randomized) decision α*(t).
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Δ(t) + Vy0(α(t), ω(t)) ≤ B + ∑iQi(t)yi(α*(t), ω(t))
+ Vy0(α*(t), ω(t))For any other (possibly randomized) decision α*(t).
Now use existence of randomized policy α*(t) that solves the problem: E[y0(α*(t), ω(t))] = y0
opt
E[yi(α*(t), ω(t))] ≤ 0 for all i in {1, …, K}
Performance Tradeoff Theorem:
Proof of (ii) [continued]:
Assume ω(t) i.i.d. for simplicity:
Now just take expectations…
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E[Δ(t) + Vy0(α(t), ω(t))] ≤ B + E[∑iQi(t)yi(α*(t), ω(t))]
+ VE[y0(α*(t), ω(t))]For any other (possibly randomized) decision α*(t).
Now use existence of randomized policy α*(t) that solves the problem: E[y0(α*(t), ω(t))] = y0
opt
E[yi(α*(t), ω(t))] ≤ 0 for all i in {1, …, K}
Performance Tradeoff Theorem:
Proof of (ii) [continued]:
Assume ω(t) i.i.d. for simplicity:
Now just take expectations…
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E[Δ(t) + Vy0(α(t), ω(t))] ≤ B + E[∑iQi(t)yi(α*(t), ω(t))]
+ VE[y0(α*(t), ω(t))]For any other (possibly randomized) decision α*(t).
Now use existence of randomized policy α*(t) that solves the problem: E[y0(α*(t), ω(t))] = y0
opt
E[yi(α*(t), ω(t))] ≤ 0 for all i in {1, …, K}
Performance Tradeoff Theorem:
Proof of (ii) [continued]:
Assume ω(t) i.i.d. for simplicity:
Now just take expectations…
y0opt
0
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E[Δ(t) + Vy0(α(t), ω(t))] ≤ B + Vy0opt
Performance Tradeoff Theorem:
Proof of (ii) [continued]:
Assume ω(t) i.i.d. for simplicity:
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E[Δ(t) + Vy0(α(t), ω(t))] ≤ B + Vy0opt
Performance Tradeoff Theorem:
Proof of (ii) [continued]:
Assume ω(t) i.i.d. for simplicity:
Sum over t in {0, …, T-1}, divide by T:
(1/T){ E[L(T)]–E[L(0)] + V∑t=0E[y0(α(t), ω(t))] } ≤ B+V y0opt
Thus (since L(T)≥0, L(0) =0): (1/T) ∑t=0E[y0(α(t), ω(t))] ≤ B/V + y0
opt
T-1
T-1
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Outline
• Queue Stability• Randomized Algs and Linear Programs for
Queue Stability (known statistics)• Dynamic Algs and Drift-Plus-Penalty (unknown statistics)• Backpressure Examples (if time)
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Backpressure and Power Optimizationfor Multi-Hop Networks
(An Application of “Drift-Plus-Penalty”)
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(a) (b)
(c) (d)
μ15=2μ45=1 μ23=1
μ32=1μ54=1μ45=1μ21=1
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Diversity Backpressure Routing (DIVBAR)From: M. J. Neely and R. Urgaonkar, “Optimal Backpressure Routing in Wireless Networks with Multi-Receiver Diversity,” Ad Hoc Networks (Elsevier), vol. 7, no. 5, pp. 862-881, July 2009. (conference version in CISS 2006).
• Multi-Hop Wireless Network with Probabilistic Channel Errors.
• Multiple Neighbors can overhear same transmission, increasing the prob. of at least one successful reception (“multi-receiver diversity”).
• Prior use of multi-receiver diversity: Routing heuristics: (SDF Larsson 2001)(GeRaf Zorzi, Rao 2003) (ExOR Biswas, Morris 2005). Sending one packet (DP): (Lott, Teneketzis 2006)(Javidi, Teneketzis 2004) (Laufer, Dubois-Ferriere, Kleinrock 2009).
• Our backpressure approach (DIVBAR) gets max throughput, and can also minimize average power subject to network stability.
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Cut capacity = 0.9 packets/slot
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Cut capacity = 0.9 packets/slot
• Network capacity: λΑ = λΒ = 0.9/2 = 0.45• Max throughput of ExOR: λA = λΒ = (.3 + .7*.3)/2 = 0.255*Note: The network capacity 0.45 for this example was mistakenly stated as 0.455 in [Neely, Urgaonkar 2009].
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ExOR[Biswas, Morris 05]
DIVBAR, E-DIVBAR[Neely, Urgaonkar 2009]
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ExOR[Biswas, Morris 05]
DIVBAR, E-DIVBAR[Neely, Urgaonkar 2009]
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ExOR[Biswas, Morris 05]
DIVBAR, E-DIVBAR[Neely, Urgaonkar 2009]
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ExOR[Biswas, Morris 05]
DIVBAR, E-DIVBAR[Neely, Urgaonkar 2009]
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DIVBAR for Mobile Networks
= Stationary Node
= Locally Mobile Node
= Fully Mobile Node
= Sink
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DIVBAR for Mobile Networks
= Stationary Node
= Locally Mobile Node
= Fully Mobile Node
= Sink
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DIVBAR for Mobile Networks
= Stationary Node
= Locally Mobile Node
= Fully Mobile Node
= Sink
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DIVBAR for Mobile Networks
= Stationary Node
= Locally Mobile Node
= Fully Mobile Node
= Sink
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DIVBAR for Mobile Networks
= Stationary Node
= Locally Mobile Node
= Fully Mobile Node
= Sink
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DIVBAR for Mobile Networks
= Stationary Node
= Locally Mobile Node
= Fully Mobile Node
= Sink
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DIVBAR for Mobile Networks
= Stationary Node
= Locally Mobile Node
= Fully Mobile Node
= Sink
![Page 54: Stochastic Network Optimization (tutorial) M. J. Neely University of Southern California](https://reader035.vdocuments.us/reader035/viewer/2022081520/56816293550346895dd306e9/html5/thumbnails/54.jpg)
DIVBAR for Mobile Networks
= Stationary Node
= Locally Mobile Node
= Fully Mobile Node
= Sink
![Page 55: Stochastic Network Optimization (tutorial) M. J. Neely University of Southern California](https://reader035.vdocuments.us/reader035/viewer/2022081520/56816293550346895dd306e9/html5/thumbnails/55.jpg)
DIVBAR for Mobile Networks
= Stationary Node
= Locally Mobile Node
= Fully Mobile Node
= Sink
![Page 56: Stochastic Network Optimization (tutorial) M. J. Neely University of Southern California](https://reader035.vdocuments.us/reader035/viewer/2022081520/56816293550346895dd306e9/html5/thumbnails/56.jpg)
DIVBAR for Mobile Networks
= Stationary Node
= Locally Mobile Node
= Fully Mobile Node
= Sink
![Page 57: Stochastic Network Optimization (tutorial) M. J. Neely University of Southern California](https://reader035.vdocuments.us/reader035/viewer/2022081520/56816293550346895dd306e9/html5/thumbnails/57.jpg)
DIVBAR for Mobile Networks
= Stationary Node
= Locally Mobile Node
= Fully Mobile Node
= Sink
![Page 58: Stochastic Network Optimization (tutorial) M. J. Neely University of Southern California](https://reader035.vdocuments.us/reader035/viewer/2022081520/56816293550346895dd306e9/html5/thumbnails/58.jpg)
Average Total Backlog Versus Input rate for Mobile DIVBAR
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Average Power Versus Delay(Fix a set of transmission rates for each node)
Avg. Power Avg. Delay
Performance-Delay Tradeoff: [O(1/V), O(V)]
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Avg. Power Avg. Delay
Average Power Versus Delay(Fix a set of transmission rates for each node)
Performance-Delay Tradeoff: [O(1/V), O(V)]
![Page 61: Stochastic Network Optimization (tutorial) M. J. Neely University of Southern California](https://reader035.vdocuments.us/reader035/viewer/2022081520/56816293550346895dd306e9/html5/thumbnails/61.jpg)
Avg. Power Avg. Delay
Average Power Versus Delay(Fix a set of transmission rates for each node)
Performance-Delay Tradeoff: [O(1/V), O(V)]
![Page 62: Stochastic Network Optimization (tutorial) M. J. Neely University of Southern California](https://reader035.vdocuments.us/reader035/viewer/2022081520/56816293550346895dd306e9/html5/thumbnails/62.jpg)
Avg. Power Avg. Delay
Average Power Versus Delay(Fix a set of transmission rates for each node)
Performance-Delay Tradeoff: [O(1/V), O(V)]
![Page 63: Stochastic Network Optimization (tutorial) M. J. Neely University of Southern California](https://reader035.vdocuments.us/reader035/viewer/2022081520/56816293550346895dd306e9/html5/thumbnails/63.jpg)
Avg. Power Avg. Delay
Average Power Versus Delay(Fix a set of transmission rates for each node)
Performance-Delay Tradeoff: [O(1/V), O(V)]
![Page 64: Stochastic Network Optimization (tutorial) M. J. Neely University of Southern California](https://reader035.vdocuments.us/reader035/viewer/2022081520/56816293550346895dd306e9/html5/thumbnails/64.jpg)
Avg. Power Avg. Delay
Average Power Versus Delay(Fix a set of transmission rates for each node)
Performance-Delay Tradeoff: [O(1/V), O(V)]
![Page 65: Stochastic Network Optimization (tutorial) M. J. Neely University of Southern California](https://reader035.vdocuments.us/reader035/viewer/2022081520/56816293550346895dd306e9/html5/thumbnails/65.jpg)
Avg. Power Avg. Delay
Average Power Versus Delay(Fix a set of transmission rates for each node)
Performance-Delay Tradeoff: [O(1/V), O(V)]
![Page 66: Stochastic Network Optimization (tutorial) M. J. Neely University of Southern California](https://reader035.vdocuments.us/reader035/viewer/2022081520/56816293550346895dd306e9/html5/thumbnails/66.jpg)
Avg. Power Avg. Delay
Average Power Versus Delay(Fix a set of transmission rates for each node)
Performance-Delay Tradeoff: [O(1/V), O(V)]