Flow Control
KAIST CS644
Advanced Topics in Networking
Jeonghoon Mo
<[email protected]>
School of Engineering
Information and Communications University
1
Acknowledgements

Part of slides is from
- tutorial of R. Gibbens and P. Key at
SIGCOMM 2000
- S. Low’s OFC presentation
Jeonghoon Mo
October 2004
2
Overview
Problem
 Objectives
 Kelly’s Framework - Wired Data Networks
 Extensions
- Quality of Service
- Wireless Network
- High Speed Network: Aggregated Flow
Control

Jeonghoon Mo
October 2004
3
Problem
Flows share links:
How to share the links bandwidth?
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October 2004
4
Problem

How to control the network to share the
bandwidth efficiently and fairly?
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October 2004
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Link Model
Set of resources, J; set of routes, R
 A route r is a subset r  J.
 Let

A

jr

1
jr
0 otherwise
Capacity of resource j is Cj.
A’x  c
x0
Jeonghoon Mo
October 2004
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A Few System Objectives
Max Throughput
 Max-min Fairness (Most Common)
 Proportional Fairness (Kelly)
 -Fairness (Mo, Walrand)

Jeonghoon Mo
October 2004
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Max System Throughput
6
6
x2

Maximize:
x(1) + x(2) + x(3)

x*= (0,6,6)
maximizes the total
system throughput.

However, user 1 does
not get anything. =>
unfair
x1 x
3
Two links with capacity 6
 Three users: 1,2,3
 x(i) : bandwidth to user i
 x(1)+x(2) <= 6
 x(1)+x(3) <= 6

Jeonghoon Mo
October 2004
8
Max-Min Fairness
6
6
x2
x1 x
3

Most commonly used definition of fairness.
Maximize Minimum of the x(i), recursively.
x*= (3,3,3) is the max-min allocation.

However, user 1 uses more resources.


Jeonghoon Mo
October 2004
9
Proportional Fairness
6
6
x2





x1 x
3
Proposed by Frank Kelly
Social Welfare: Sum of Utilities of Users  log( xi )
i
Maximize the Social Welfare
x*= (2,4,4) is the Proportional Fair Allocation.
Can be generalized into “Utility Fairness”.
Jeonghoon Mo
October 2004
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-Fairness
Generalized Fairness Definition
(1-)
x
 System Objective:
Max i pi

1-

includes proportional-fair, max-min-fair, max
throughput
-  = 0 : Maximum allocation (p=1)
-   1 : Proportional fair allocation
-  = 2 : TCP-fair allocation
-   : Max-min fair allocation (p=1)
Jeonghoon Mo
October 2004
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-Fairness

Trade-off between Fairness and Efficiency
- Bigger  favors Fairness
- Smaller  favors Efficiency
(source: Is Fair Allocation Inefficient, INFOCOM 04)
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October 2004
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Fairness and Efficiency (Infocom 04)

Counter-Example
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October 2004
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Algorithms
How to achieve those system objectives?
14
Players
source
- controls its rate or window based on
(implicit or explicit) network feedback
 router (link)
- Generate (implicit) feedback or controls
packets

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October 2004
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Source Algorithm
TCP Vegas, RENO, ECN
 XCP

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October 2004
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Active Queue Management (AQM)





Priority Queue
WFQ
RED
REM
XCP Router
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October 2004
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Kelly’s Model and Algorithm
18
User: rate and utility
Each route has a user: if xr is the rate on
route r, then the utility to user r is Ur(xr).
 Ur() --- increasing, strictly concave,
continuously differentiable on xr  [0 , ) --elastic traffic
 Let C=(Cj, j J), x=(xr, r  R) then Ax  C.

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October 2004
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System problem

Maximize aggregate utility, subject to capacity
constraints
max
 U x 
r
rR
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October 2004
subject to
Ax  C
over
x0
r
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User problem

User r chooses an amount to pay per unit time
wr, and receives in return a flow xr = wr/r
 wr 
max U r    wr
 r 
over wr  0
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October 2004
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Network problem

As if the network maximizes a logarithmic
utility function, but with constants (wr, rR)
chosen by the users
max
 w log x
rR
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October 2004
r
subject to
Ax  C
over
x0
r
22
Three optimization problems

SYSTEM(U,A,C)

USERr(Ur;r)

NETWORK(A,C;w)
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October 2004
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Decomposition theorem

There exist vectors  , w and x such that
1. wr = rxr for r  R
2. wr solves USERr(Ur; r)
3. x solves NETWORK(A, C; w)
The vector x then also solves SYSTEM(U, A, C).
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October 2004
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Thus the system problem may be solved by
solving simultaneously the network and user
problems
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October 2004
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Result

A vector x solves NETWORK(A, C; w) if and
only if it is proportionally fair per unit charge
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October 2004
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Solution of network problem
Strategy: design algorithms to implement
proportional fairness
 Several algorithms possible: try to mimic
design choices made in existing standards

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October 2004
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Primal algorithm
d


xr t     wr  xr t   j t  
dt


rR


 j t   p j   xs t 
 s: js

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October 2004
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Interpretation of primal algorithm
Resource j generates feedback signals at rate j(t)
 signals sent to each user r whose route passes
through resource j
 multiplicative decrease in flow xr at rate proportional
to stream of feedback signals received
 linear increase in flow xr at rate proportional to wr

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October 2004
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Related Work
Optimization Flow Control (S. Low)
 Window based Model (Mo, Walrand)

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October 2004
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Optimization Flow Control



Distributed algorithm to share network resources
Link algorithm: what to feed back
- RED
Source algorithm: how to react
- TCP Tahoe, TCP Reno, TCP Vegas
Source alg
Link alg
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October 2004
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Welfare maximization
Primal problem:
max
ms  x s  M s
subject to
U ( x )
s
s
s
x
sS ( l )
s
 cl , l  L
Capacity cl can be less than real link
capacity
 Primal problem hard to solve & does not adapt

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October 2004
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Model
Network: Links l each of capacity cl
 Sources s:
(L(s), Us(xs), ms, Ms)
L(s) - links used by source s
Us(xs) - utility if source rate = xs

x1
ms  xs  M s
x1  x2  c1
x1  x3  c2
c1
c2
x2
x3
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October 2004
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Distributed Solution
D(p)
Dual problem: min
p 0
D( p)   Bs ( p s )   pl cl
s
l
Bs ( p )  max U s ( xs )  xs p s
s
ms  x s  M s
ps 
p
l L ( s )
BW price along path of s
l
s
 Given p sources can max own benefit individually
s
s
 xs ( p ) indeed primal optimal if p is dual optimal
 Solve dual problem!
Jeonghoon Mo
October 2004
34
Distributed Solution (cont…)
min D(p)

Dual problem:

Grad projection alg:

Update rule:
p 0
p(t  1)  [ p(t )  D( p(t ))]
pl (t  1)  [ pl (t )   ( xl (t )  cl )]
xs (t  1)  U s'1 ( p s (t ))

A distributed computation system to solve the dual
problem by gradient projection algorithm
Jeonghoon Mo
October 2004
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Source Algorithm
xs ( p s ) = U s '-1 ( p s )
ps
s
Decentralized: Source s needs only U s ' ( xs ) and p
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October 2004
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Router (Link) Algorithm
pl (t  1)  [ pl (t )   ( x l (t )  cl )]
aggregate
source
rate
Decentralized
 Rule of supply and demand
 Any work-conserving service discipline
 Simple

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October 2004
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Random Exponential Marking (REM)

Source algorithm
- Identical but does not communicate source rate

Link algorithm
- At update time t, sets price to a fraction of buffer occupancy:

Theorem: Synchronous convergence
Under same conditions (with possibly smaller
- Price update maintains descent direction
- Gradient estimate converge to true gradient
- Limit point
is primal-dual optimal

Jeonghoon Mo
October 2004
):
38
RED
- Idea: early warning of congestion
- Algorithm
Link:
Source (Reno):
marking
window
1
queue
B
Jeonghoon Mo
October 2004
time
39
RED
- Idea: marks for estimation of shadow price
- Algorithm
Link
Source
marking
rate
1
queue
Q
1
fraction
of marks
Global behavior of network of REM: stochastic
gradient algorithm to solve dual problem
Jeonghoon Mo
October 2004
40
Window-based Model [Mo,Walrand]
d1
w1
q11
c1 x1
q21 c
2
x2 q
12
x3 q
23
d2
d3
xi  0, i = 1, 2, 3
qi  0, i = 1, 2,
x1 + x2  c1
q1(c1 - x1 - x2) = 0
Q = diag{qi }; X = diag{xi }.
A’x  c
Q(c - A’x) = 0
w = X(d + qA)
w1 = x1 d1 + x1 q1 + x1 q2
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October 2004
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Window-based Algorithm
Theorem: [Mowlr98]
Let
dwi
ds
=-k i i
dt
ti wi
si := wi - xi di - pi
ti := end-to-end delay
Then x(t) -> unique weighted -fair point x*
Proof:
The function i (si /wi ) 2 is a Lyapunov function
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October 2004
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Extensions
-
Aggregated Flow Control
Quality of Service
Wireless Network
Maxnet and Sumnet
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October 2004
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Aggregate Flow Control
Motivations:
- High Capacity of Optical Fiber
 Idea:
- player are core routers and access routers.

access router: regulates the rate of aggregated
flow
 core router: provide feedbacks to access routers

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October 2004
44
Quality of Service
Only bandwidth is modeled.
 QoS is affected by
- loss and delay also
 How to incorporate other parameters?

Jeonghoon Mo
October 2004
45
Non-Convex Utility Function (Lee04)


Considered sigmoidal
utility function
Non-convex optimization
problem =>duality gap
(source: J. Lee et. al. Non-convexity Issues, INFOCOM 04)
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October 2004
46
Non-Convex Utility Functions
Dual Algorithm with
Self-Regulating Property
Without Self-Regulation
With Self-Regulation
(source: J. Lee et. al. Non-convexity Issues, INFOCOM 04)
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October 2004
47
Wireless Ad-Hoc Network [RAD04]

Physical Model:Rate r is an increasing function of SINR.
MAC : Each time slot determines power pn,which
determines rate xn

Routing matrix R and flow to path matrix F are given.

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October 2004
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Random Topology Results
100m x100m grid
12 random node, with 6 pairs of transmissions
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October 2004
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In the wireless Ad-hoc Networks

The max-min fair rate allocation of any
network has all rates equal to the worst node.

The capacity maximization objective leads to
starving users.

Proportional Fair Allocation give reasonable
trade-off between fairness and efficiency.
- The worst node does not starve.
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October 2004
50
MaxNet and SumNet
Source takes max(d1,d2,…, dN) in the maxnet
architecture
 Source takes sum(d1,d2,…,dN) in the sumnet
architecture.

Max
Si
Source
Jeonghoon Mo
October 2004
d1
Link 1
d2
Link 2
dN
Link N
Destination
51