PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Rate Control in Communication NetworksFrom Models to Algorithms
Yuedong Xu
Department of Computer Science & EngineeringThe Chinese University of Hong Kong
February 29, 2008
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Outline1 Preliminaries
Convex OptimizationTCP Congestion Control
2 Utility Maximization: Concepts and ModelMotivationBasic ModelOptimization Decomposition
3 Utility Maximization: AlgorithmsPrimal ProblemDual AlgorithmRelation to TCPStability and Convergence
4 Experiments5 Extensions
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Convex OptimizationTCP Congestion Control
Convex Optimization
Convex Set
Set C is a convex set if the line segment between any twopoints in C lies in C, i.e., if for any x1, x2 ∈ C and any θ ∈ [0, 1],we have
θx1 + (1 − θ)x2 ∈ C
Convex Hull
Convex hull of C is the set of all convex combinations of pointsin C:
{k
∑
i=1
θixi |xi ∈ C, θi ≥ 0, i = 1, 2, · · · , k ,k
∑
i=1
θi = 1}.
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Convex OptimizationTCP Congestion Control
Convex Optimization
Convex Function : Jensen’s inequality
f : Rn → R is a convex function if domf is a convex set and forall x , y ∈ dom f and t ∈ [0, 1], we have
f (tx + (1 − t)y) ≤ tf (x) + (1 − t)f (y)
f is strictly convex if above strict inequality holds for all x 6= yand 0 < t < 1.
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Convex OptimizationTCP Congestion Control
Convex Optimization
Standard FormA convex optimization problem with variables x :
minimize f0(x)
subject to fi(x) ≤ 0, i = 1, 2, · · · , m
hi(x) = 0, i = 1, 2, · · · , p.
where f0, f1, · · · , fm are convex functions; hi(x) are linear.
Objective Function: Minimize convex objective function (ormaximize concave objective function).Inequality Constraints: Upper bound inequality constraintson convex functions.Equality Constraints: Equality constraints must be affine.
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Convex OptimizationTCP Congestion Control
Convex Optimization
Lagrangian Function
Absorb the constraints as the penalties to the objective.Lagrangian function:
L(x , λ, ν) = f0(x) +
m∑
i=1
λi fi(x) +
p∑
i=1
νihi(x)
where Lagrange multipliers (dual variables): λ � 0, ν.
Dual Problem
Perform unconstrained maximization on L(x , λ, ν), thusobtaining Lagrangian dual function: g(λ, ν) = infx L(x , λ, ν).
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Convex OptimizationTCP Congestion Control
Convex Optimization
Example:
Primal problem: minimize f0(x) = −x1 log x1 − x2 log x2
subject to x1 + 2x2 ≤ 2, x1, x2 > 0;
Lag Function: L(x , λ) = −x1 log x1−x2 log x2+λ(x1+2x2−2); (λ ≥ 0)
Optimal x : x1 = eλ−1; x2 = e2λ−1;
Dual Function: maximize D(λ) = eλ−1 + e2λ−1 − 2λ;
over λ ≥ 0.
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Convex OptimizationTCP Congestion Control
Convex Optimization
Economics Interpretation
Primal objective (f0(x)): cost of operation
Primal constraints (fi(x)): can be violated
Dual variables (λ, v): price for violating the correspondingconstraint (dollar per unit violation). For the same price,can sell “unused violation” for revenue
Lagrangian (L(x , λ, v)): total cost
Lagrange dual problem (g(λ, v)): optimal cost as a functionof violation prices (Lagrangian multipliers)
Question
Optimal Solution of Primal Problem = Optimal Solution of DualProblem ?
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Convex OptimizationTCP Congestion Control
Convex Optimization
KKT Optimal Conditions
Karush-Kuhn-Tucker (KKT) conditions for a standard convexoptimization problem:
Primal constraints: fi(x) ≤ 0 and hi(x) = 0
Dual constraints: λ � 0
Complementary slackness: λi fi(x) = 0
Gradient of Lagrangian with respect to x vanishes:∇f0(x) +
∑mi=1 λi∇fi(x) +
∑pi=1 vi∇hi(x) = 0
If strong duality holds and x , λ, v are optimal, then they mustsatisfy the KKT conditions.
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Convex OptimizationTCP Congestion Control
TCP Congestion Control
Problem Description
Congestion is the conflict between demand and capacity.
Congestion control is a problem of resource management.
Congestion leads to buffer overflow, large delay, bandwidthunderutilization.
Current Solutions
Rate Adaption in the source (e.g. TCP)
Controllers in the buffer (e.g. AQM)
Question
Is current TCP merely a heuristic algorithm?
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Convex OptimizationTCP Congestion Control
TCP Congestion Control
TCP Versions
Tahoe (Jacobson 1988)Slow Start, Congestion Avoidance, Fast Retransmit
Reno (Jacobson 1990)Further Adding Fast Recovery
Vegas (Brakmo & Peterson 1994)Delay(RTT)-based Congestion Avoidance
Active Queue Management
Random Early Detection (Floyd & Jacobson 1993)
Proportional Integral (Hollot,Misra & Towsley 2001)
Random Exponential Marking (Athuraliya & Low 2000)
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Convex OptimizationTCP Congestion Control
TCP Congestion Control
TCP Reno/RED Dynamics:
time
window
host
1
routerB Avg
marking/dropping
routerqueue
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
MotivationBasic ModelOptimization Decomposition
Utility Maximization: Concepts and Model
Motivating Example
Given a network with
Two Links that have the capacities c1 and c2.
Three end-to-end flows x1, x2 and x3.
Question: How to allocate bandwidth for the end-to-end flows?
c1 c2c1 c2
x1
x2
x3
x1
x2
x1
x2
x3
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
MotivationBasic ModelOptimization Decomposition
Utility Maximization: Concepts and Model
Potential Solutionsx1 = 0, x2 = c1, x3 = c2.
x1 = c1/2, x2 = c1/2, x3 = c2 − c1/2.
· · · · · ·Which solution is the “BEST”? It depends on our objective!
Definition
In economics, utility is a measure of the relative happiness orsatisfaction (gratification) gained by consuming differentbundles of goods and services. — Wikipedia
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
MotivationBasic ModelOptimization Decomposition
Utility Maximization: Concepts and Model
Notation
Cj , (j ∈ J): the finite capacity of link j ;
r , (r ∈ R): a router that has non-empty link set;
xs, (s ∈ S): the flow rate allocated to user s;
A, (Ajs, j ∈ J, s ∈ S): defines a 0-1 matrix that depicts therouting; set Ajs = 1 if s uses the resource j .
U, (Us(·), s ∈ S): the utility of user s with the rate xs.
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
MotivationBasic ModelOptimization Decomposition
Utility Maximization: Concepts and Model
Assumption
The utility function Us(xs) is an increasing, strictly concave andcontinuously differentiable function of xs. (The traffic that leadsto such a utility function is called elastic traffic by S. Shenker.)
System Model
SYSTEM(U,A,C):max
∑
s∈S
Us(xs)
subject toAx ≤ C
overx ≥ 0.
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
MotivationBasic ModelOptimization Decomposition
Utility Maximization: Concepts and Model
Example 1
Let Us(xs) = log xs, what is the optimal solution?
Example 2
Let Us(xs) = −x−1, what is the optimal solution?
2Mbps
x1
x2
x3
x1
x2
x1
x2
x3
2Mbps
=
101
011A
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
MotivationBasic ModelOptimization Decomposition
Utility Maximization: Concepts and Model
Solving SYSTEM(U, A, C)
Lagrangian form:
L(x ;µ) =∑
s∈S
Us(xs) + µT (C − Ax)
=∑
s∈S
(Us(xs) − xs
∑
j∈s
µj) +∑
j∈J
µjCj ,
where µ = (µj , j ∈ J) are Lagrangian multipliers. Then,
∂L∂xs
= U′
s(xs) −∑
j∈s
µj .
Optimize the dual function over the feasibility region.Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
MotivationBasic ModelOptimization Decomposition
Utility Maximization: Concepts and Model
Solving SYSTEM(U, A, C)
Using KKT conditions, we can express these conditions morecompactly: (x) solves SYSTEM(U, A, C) if and only if theseexists multipliers (µ) such that:
Ax ≤ C, x ≥ 0;
µ ≥ 0;
µT (C − Ax) = 0, (∑
j∈s
µj − U′
(x))T x = 0;
The first row is primal feasibility; the second row is dualfeasibility; and the third row comprises complementaryslackness.
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
MotivationBasic ModelOptimization Decomposition
Utility Maximization: Concepts and Model
Discussion
THE GOOD: Mathematically tractable due to the convexity;
THE BAD: Utilities are unlikely to be known by the network;
THE UGLY: You derive the flow rates in a centralizedmanner.
Question
How can we obtain a distributed algorithm to allocate rates?
Road Map
Solve “THE BAD” first, then the “THE UGLY”.
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
MotivationBasic ModelOptimization Decomposition
Utility Maximization: Concepts and Model
Definition
Shadow price is the change in the objective value of theoptimal solution of an optimization problem obtained by relaxingthe constraint by one unit. In a business application, a shadowprice is the maximum price that management is willing to payfor an extra unit of a given limited resource. · · · · · · Shadowprice is the value of the Lagrange multiplier at the optimalsolution. — Wikipedia
Decomposition
From the perspective of economic theory, the original problemis replaced by two simpler problems for users and network.
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
MotivationBasic ModelOptimization Decomposition
Utility Maximization: Concepts and Model
User’s Angle
USERs(Us;λs):
max Us(ws
λs) − ws
overws ≥ 0.
where ws is the amount to pay per-unit time and λs is regardedas a charge per unit flow for user s. Hence, the flow rate xs isexactly ws
λs.
The Important Idea:User s wants to maximize profit by choosing optimal ws to pay.
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
MotivationBasic ModelOptimization Decomposition
Utility Maximization: Concepts and Model
Network’s Angle
Network(A, C; w):max
∑
s∈S
ws log xs
subject toAx ≤ C,
overx ≥ 0.
The Important Idea:Network knows payments ws from all users s and chooses rateallocation to maximize the revenue.
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
MotivationBasic ModelOptimization Decomposition
Utility Maximization: Concepts and Model
Question
Is the optimization decomposition true?
Theorem 1
There always exist vectors w ,λ, and x , satisfying ws = λsxs,such that ws solves USERs(Us;λs) and x solvesNETWORK (A, C; w); further, the vector x is then the uniquesolution to SYSTEM(U, A, C).
Proof
The combinations of KKT conditions of the USER andNETWORK problems are identical to the SYSTEM problem.
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
MotivationBasic ModelOptimization Decomposition
Utility Maximization: Concepts and Model
Physical Meaning
The users and the network optimize their individual benefit, andthe social welfare is automatically achieved.
Recap
SYSTEM is decomposed into many local USER problems andone global NETWORK problem where local utility functions arenot needed.
Fairness?
What are the relationships between optimal rate allocation andfairness ?
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
MotivationBasic ModelOptimization Decomposition
Utility Maximization: Concepts and Model
Definition
Max-Min Fairness : A vector of rates x is max-min fair if it isfeasible, and if for each s, xs cannot be increased withoutdecreasing xs∗ for some s∗ for which xs∗ ≤ xs.Proportional Fairness : Feasible x is proportionally fair (perunit charge) if for any other feasible x
′
,
∑
s∈S
x′
s − xs
xs≤ 0.
Theorem 2
x solves NETWORK (A, c; w) if and only if it is proportionallyfair (per unit charge).
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
MotivationBasic ModelOptimization Decomposition
Utility Maximization: Concepts and Model
Physical Meaning
The network cannot achieve a better social revenue bychanging the rate vector x .
A tradeoff between maximum capacity and max-minthroughput. Generally, their relationship can be depicted by
Max-min throughput
Maximum Capacity
Proportional Fairness
Fair Unfair
Road Map
“THE BAD” is solved, and the remaining problem is “THEUGLY”.
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Primal ProblemDual AlgorithmRelation to TCPStability and Convergence
Utility Maximization: Algorithms
Global PictureDesign distributed algorithms using the gradient-basedmethod;
The congestion indications (link prices) can be generatedby considering different performance goals (e.g. loss rate,delay, robustness etc.);
The congestion indications (link prices) can be feedback tothe source in several ways;
Rate control system is globally stable without regard totime delay.
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Primal ProblemDual AlgorithmRelation to TCPStability and Convergence
Utility Maximization: Algorithms
NETWORK (A, C; w) Problem with Variable x
max∑
r∈R
ws log xs
subject toAx ≤ C, x ≥ 0.
Lagrangian Function
L(x ; w) =∑
r∈R
ws log xs + µT (C − Ax)
Unique Optimum : xs =ws
∑
j∈r µj.
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Primal ProblemDual AlgorithmRelation to TCPStability and Convergence
Utility Maximization: Algorithms
Primal Algorithm
dxs(t)dt
= κ(ws − xs
∑
j∈r
µj(t))
µj(t) = pj(∑
s:j∈s
xs(t))
where κ is a small constant.
Interpretation
Link j charges pj(y) per unit flow, when total flow on link j is y .Each source tries to equalize the total cost with target value ws.
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Primal ProblemDual AlgorithmRelation to TCPStability and Convergence
Utility Maximization: Algorithms
Another Interpretation
Link j generates feedback signal pj(y), when total flow on link jis y . Each source linearly increase its rate (proportional to ws)and multiplicatively decrease its rate (proportional to totalfeedback).
Implementation
xs(t + 1) = xs(t) + κ(ws − xs
∑
j∈r
µj(t))
µj(t + 1) = pj(∑
s:j∈s
xs(t + 1))
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Primal ProblemDual AlgorithmRelation to TCPStability and Convergence
Dual Algorithm
dµj(t)dt
= κ(
∑
s:j∈s
xs(t) − qj(µj(t)))
xs(t) =ws
∑
j∈r µj(t).
where qj(µj(t)) is the amount of flow on link j that wouldgenerate price µj(t).
Implementation
Link algorithm: µj(t + 1) = µj(t)+κ(∑
s:j∈s xs(t)−qj(µj(t)))
Source algorithm: xs(t + 1) = U−1s (ps(t))
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Primal ProblemDual AlgorithmRelation to TCPStability and Convergence
Utility Maximization: Algorithms
Primal Algorithm VS Dual Algorithm
Primal algorithm: a system where rates vary gradually, andshadow prices are given as functions of the rates.Dual algorithm: a system where shadow prices vary gradually,with rates given as functions of the shadow prices.
In the primal algorithm, sender adjusts rate according to thefeedback of the congestion signals. In the dual algorithm, thenetwork computes the shadow prices directly and send themback to the sender.
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Primal ProblemDual AlgorithmRelation to TCPStability and Convergence
Utility Maximization: Algorithms
Mapping to TCP/AQM
A TCP scheme may be mapped into a specific utilityfunction.
Major TCP schemes approximately carrying out primal ordual algorithm.
Congestion Measures
Price
Queueing delay
Queue Length
Packet Loss
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Primal ProblemDual AlgorithmRelation to TCPStability and Convergence
TCP Utility Functions
TCP Reno:
U renos (xs) =
√2
Dstan−1(xsDs
2
)
TCP Vegas:Uvegas
s (xs) = αsds log xs
Queue Management
FIFO: pl =1cl
[(yl (t) − cl)]+
RED: bl = (yl(t) − cl)+; rl = −αlcl(rl(t) − bl(t)); pl = ml(rl);
REM: pl(t + 1) = [pl(t) + γ(yl(t) − cl)]+
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Primal ProblemDual AlgorithmRelation to TCPStability and Convergence
Utility Maximization: Algorithms
System Block Diagram
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Primal ProblemDual AlgorithmRelation to TCPStability and Convergence
Utility Maximization: Algorithms
Question
What is the most important property for a rate controller?Stability!
Basic Notions
An “insensitivity” to small perturbations, where perturbationsare modeling errors of system, environment, noise etc.
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Primal ProblemDual AlgorithmRelation to TCPStability and Convergence
Utility Maximization: Algorithms
Lyapunov Stability
Consider an autonomous nonlinear dynamical system
x = f (x(t)), x(0) = x0,
where x(t) ∈ D ⊆ Rn denotes the system state vector, D an
open set containing the origin, and f : D → Rn continuous on
D. Without loss of generality, we may assume that the origin isan equilibrium. The origin of the above system is said to beLyapunov stable, if, for every ǫ > 0, there exists a δ = δ(ǫ) > 0such that, if ‖x(0)‖ < δ, then ‖x(t)‖ < ǫ, for every t ≥ 0.— from Wikipedia
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Primal ProblemDual AlgorithmRelation to TCPStability and Convergence
Utility Maximization: Algorithms
Lyapunov second theorem on stability
Consider a function V (x) : Rn → R such that
V (x) ≥ 0 with equality if and only if x = 0 (positivedefinite).
V (x(t)) < 0 (negative definite).
Then V (x) is called a Lyapunov function candidate and thesystem is asymptotically stable in the sense of Lyapunov(i.s.L.).
Interpretation
The kinesthetic energy of an autonomous dynamic system willvanish eventually.
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Primal ProblemDual AlgorithmRelation to TCPStability and Convergence
Utility Maximization: Algorithms
Stability of Primal Algorithms
Establish a Lyapunov function under mild regularity conditions:
U(x) =∑
s∈S
ws log xs −∑
j∈J
∫
P
s:j∈s xs
0pj(y)dy
Stability of Dual Algorithms
Establish a Lyapunov function under mild regularity conditions:
V(x) =∑
s∈S
ws log(∑
j∈s
µj) −∑
j∈J
∫ µj
0qj(η)dη
Prove the global stability under the Lyapunov second theorem.Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Primal ProblemDual AlgorithmRelation to TCPStability and Convergence
Utility Maximization: Algorithms
Related Issues
Rate of convergence
Stochastic perturbation
Time-delay systems
Routing
Other possible decomposition
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Experiments
Logical Network:
S1 Switch Switch Switch
S2 S3 D1
D2
D3
Link 1 Link 2
Note
Note: We use an alternative dualalgorithm (REM) in the simulationsince F.P.Kelly’s work presents atheoretic framework instead of animplementable algorithm in the realnetwork.
We evaluate the price update in REM algorithm. Each PC wasequipped with 64 MB of RAM and 100-MB/s PCI ethernetcards. The packets are 500B long, containing 489B datapayload. Utility function: Ui(xi) = αi log xi . γ = 1.5 × 10−2.
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Homogeneous Case:Each source transmitted data for a totalof 120 s, with their starting times staggered by intervals of 40 s:source 1 started transmitting at time 0, source 2 at time 40 s,and source 3 at time 80 s. α1 = α2 = α3 = 1 × 104.
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Heterogeneous Case:The setup in this experiment is thesame as in Experiment 1, except that the utility function ofsource 3 has α3 = 2 × 104, double that of sources 1 and 2.
Yuedong Xu Rate Control in Communication Networks
PreliminariesUtility Maximization: Concepts and Model
Utility Maximization: AlgorithmsExperiments
Extensions
Extensions
Examples
The assumption of above network is that C is a fixed vector,which is not true in wireless networks, e.g. the link capacity is afunction of scheduler or transmission power.
The original maximization is then decomposed into a flow ratecontrol subproblem and a lower-layer subproblem
maximize∑
j∈J
µjcj
subject to MAC or PHY layer constraints
Yuedong Xu Rate Control in Communication Networks
Appendix Key Reference
Key Reference I
F.P. Kelly, A.K. Maulloo, et.al, “Rate control forcommunication networks: shadow prices, proportionalfairness and stability”, Journal of the Operational ResearchSociety, 1998
S.H. Low, D.E. Lapsley, “Optimization Flow Control¡aI:Basic Algorithm and Convergence”, IEEE/ACM Trans.Networking, 1999.
M. Chiang, S.H. Low, et.al,“Layering as optimizationdecomposition”, Proceeding of IEEE, 2007.
Yuedong Xu Rate Control in Communication Networks