TLEN7000/ECEN7002: Analytical Foundations of Networks

Layering as Optimization Decomposition

Lijun Chen 11/29/2012

2

The Internet

Complexity is ever increasing

❒  Large in size and scope ❒  Enormous heterogeneity ❒  Incomplete information ❒  Uncertain environments ❒  Emerging technologies ❒  New applications ❒  New design dimensions ❒  ……

Design (& understanding) is increasingly dominated by

❒  Efficiency (optimality) ❒  Manageability ❒  Reliability & Security ❒  Economic viability ❒  Scalability ❒  Evolvability ❒  …… emerging, global properties

3

Components

Systems requirements:

functional, efficient, robust, secure, evolvable, …

architecture

Constraints that deconstrain

❒  Organizational principles, including abstractions and interfaces

❒  Highly conserved core resource allocation and management mechanisms

4

Components

Systems requirements:

functional, efficient, robust, secure, evolvable, …

architecture

❒  Understand architecture and main mechanisms of existing networks

❒  Design architecture and

main mechanisms for emerging networks

5

Internet

IP

TCP/AQM

Physical

Application Diverse

Diverse

Highly conserved

core mechanisms

Layered architecture

MAC

6

Internet architecture

IP

TCP/AQM

Physical

Application

MAC

Little quantitative understanding ❒  Why? Optimal? In what sense?

Lots of problems ❒  Efficiency, security, mobility,

accountability, …

fixes (middle boxes & overlays)

7

Components

Systems requirements:

functional, efficient, robust, secure, evolvable, …

architecture

❒  Rigorous foundations and new methodologies for understanding & designing architecture and various mechanisms

❒  Employ and develop techniques in ❒  constrained optimization ❒  game theory ❒  distributed control

8

Cross-layer design in ad hoc wireless networks

IP

TCP/AQM

Physical

Application

MAC

❒  Network performance can be improved if network layers are jointly designed

❒  Most works ❒  design based on intuition,

evaluated by simulations ❒  unintended consequences

9

Cross-layer design in ad hoc wireless networks

IP

TCP/AQM

Physical

Application

MAC

A “theory-based” approach

❒  Capture global structure of the problem

❒  Derive the design from the

distributed decomposition of certain optimization problem

❒  design objective ❒  constraints

10

Protocol decomposition: TCP/AQM

Duality model: TCP/AQM as distributed primal-dual algorithm over network to maximize aggregate utility (Kelly ’98 , Low ’99, ’03)

cRx

xUs

ssx

≤

∑≥

s.t.

)( max0

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠

⎞⎜⎝

⎛− ∑∑ ∑

≥≥l

ll

s lllssss

xpcppRxxU

s

)( max min00

Primal: Dual

121 cxx ≤+ 231 cxx ≤+

⎟⎟⎠

⎞⎜⎜⎝

⎛≤

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛=

2

1

3

2

1

100111

cc

xxx

Rx

1x

3x2x

2c1c

11

Protocol decomposition: TCP/AQM

Duality model: TCP/AQM as distributed primal-dual algorithm over the network to maximize aggregate utility (Kelly ’98 , Low ’99, ’03)

cRx

xUs

ssx

≤

∑≥

s.t.

)( max0

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠

⎞⎜⎝

⎛− ∑∑ ∑

≥≥l

ll

s lllssss

xpcppRxxU

s

)( max min00

Primal: Dual

horizontal decomposition

' 1( ) ( )

( 1) [ ( ) ( ( ) )]

s s ls ll

l l ls s ls

x t U R p

p t p t R x t cγ

−

+

⎧ =⎪⎨

+ = + −⎪⎩

∑

∑

12

Cross-layer design/optimization

cRxxUs

ssx

≤∑≥

s.t. )( max0

Network

Transport

Physical

Application

Link/MAC

❒  Extend to include decision variables and constraints of other layers

❒  Derive cross-layer design from decomposition of extended utility maximization

13

Cross-layer design/optimization

Network model

Π∈

Π

∈

∈

ffc

Nl

Nl

;,

d

s

each source s: sending rate and utility )( ss xUsx

routing of service requirement allocation of service capacity

)(xH)( fA

)()( fAxH ≤

14

Problem formulation

Network resource allocation:

Π∈

≤

∑

ffAxHts

xUs

ssfx

)()( ..

)( max,

constraint for routing

constraint from wireless interference

15

Protocol decomposition

)}(max))()(( max{min

),()( .. )( max

0

,

:Dual

:Primal

fApxHpxU

ffAxHtsxU

T

f

T

sssxp

sssfx

Π∈≥+−

Π∈≤

∑

∑

Rate control Scheduling Routing

rate constraint schedulability constraint

Cross-layer implementation

q  Rate control:

q  Routing: solved with rate control or scheduling q  Scheduling:

)()()(maxarg))(()( xHtpxUtpxtx T

sssx−== ∑

)()(maxarg))(()( fAtptpftf T

f Π∈==

Network

Transport

Physical

Application

Link/MAC

0min{max ( ) ( )) max ( )} (Dual: T T

s sp x fsU x p H x p A f

≥ ∈Π− +∑

Rate control Scheduling Routing

16

vertical decomposition ⎣ ⎦+−−=+ )))}]((()))((({)([)1( tpxHtpfAtptp tγ

q  Congestion update:

17

Extension to time-varying channel

q  Rate control:

q  Routing: solved with rate control or scheduling

q  Scheduling:

q  Congestion update

)()()(maxarg))(()( xHtpxUtpxtx T

sssx−== ∑

)()(maxarg))(()())((

fAtptpftf T

thf Π∈==

Network

Transport

Physical

Application

Link/MAC

channel state : i.i.d. finite state process with distribution

random

h)(hq

⎣ ⎦+−−=+ )))}]((()))((({)([)1( tpxHtpfAtptp tγ

18

Extension to time-varying channel: Stability and optimality

Theorem (Chen el al ’06; ’10): The Markov chain is stable. Moreover, the cross-layer algorithm solves the following optimization problem

Applicable to any queueing network with interdependent, time- varying, parallel servers

q  optimality holds even with time-varying topologies q  throughput-optimal when flow-level dynamics is considered

)}()(,)()(:{ hhrhrhqrrh

Π∈==Π ∑Π∈

≤

∑

ffAxHts

xUs

ssfx

)()( ..

)( max,

19

Cross-layer design

❒  A Wi-Fi implementation by Rhee et al shows significantly better performance than existing system

❒  This series of work (starting with Kelly-Low model) has rekindled an interest in theory-based network design q  e.g., DARPA CB-MANET program

❒  Layering as optimization decomposition q  see survey article Chiang et al, IEEE Proceedings ’07

20

Generalized utility maximization

❒  Objective function: user application needs and network cost ❒  Constraints: restrictions on resource allocation (could be

physical or economic) ❒  Variables: Under the control of this design ❒  Constants: Beyond the control of this design

)( tosubj

)( max,,,

pXccRx

RxλxU T

iii

pcRx

∈

≤

−∑

Application utility

IP: routing Link: scheduling

Phy: power

Network cost

21

Layering as optimization decomposition

❒  Network generalized NUM ❒  Layers sub-problems ❒  Interface functions of primal/dual variables ❒  Layering decomposition methods

•  Vertical decomposition: into functional modules of different layers

•  Horizontal decomposition: into distributed computation and control over geographically disparate network elements

IP

TCP/AQM

Physical

Application

Link/MAC

22

Layering as optimization decomposition

❒  Network generalized NUM ❒  Layers sub-problems ❒  Interface functions of primal/dual variables ❒  Layering decomposition methods

Exposes the interaction among protocol

layers as different ways to modularize and distributed a centralized computation

Formalizes the common practice of

breaking down the design for a complex system into simpler modules

IP

TCP/AQM

Physical

Application

Link/MAC

23

Layering as optimization decomposition

❒  Network generalized NUM ❒  Layers sub-problems ❒  Interface functions of primal/dual variables ❒  Layering decomposition methods

Provides a top-down approach to design protocol stack

q  explicitly tradeoff design objective q  explicitly model constraints and effects of, e.g.,

new technologies q  provide guidance on how to structure and

modularize different functions q  make transparent the interactions among

different components and their global behaviors

IP

TCP/AQM

Physical

Application

Link/MAC

24

❒  Utility design, i.e., how to model the user or application needs q  Inelastic traffic q  Delay sensitive traffic, time-based utility

❒  Nonconvex problems q  Nonconcave objective functions q  Nonconvex constraints

❒  Stability under delay ❒  Issues related to control plane

q  Implementation and management complexity q  Evolvability q  ……