rate control in communication networkscslui/csc6480/rate_control.pdf · preliminaries utility...

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



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




Extensions


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}.




Extensions


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.




Extensions


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.




Extensions


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).




Extensions


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.




Extensions


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 ?




Extensions


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.




Extensions


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?




Extensions



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)




Extensions



TCP Reno/RED Dynamics:

time

window

host

1

routerB Avg

marking/dropping

routerqueue




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




Extensions



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




Extensions



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.




Extensions



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.




Extensions



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




Extensions



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



Extensions



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.




Extensions



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”.




Extensions



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.




Extensions



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.




Extensions



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.




Extensions



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.




Extensions



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 ?




Extensions



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).




Extensions



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”.




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.




Extensions



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.




Extensions



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.




Extensions



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))




Extensions


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))




Extensions



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.




Extensions



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




Extensions


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)]+




Extensions



System Block Diagram




Extensions



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.




Extensions



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




Extensions



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.




Extensions



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



Extensions



Related Issues

Rate of convergence

Stochastic perturbation

Time-delay systems

Routing

Other possible decomposition




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.




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.




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.




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


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.


rate control in communication networkscslui/csc6480/rate_control.pdf · preliminaries utility...

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