Layering as Optimization Decompositionlich1539/fn/Layering.pdfLayering as optimization decomposition...
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TLEN7000/ECEN7002: Analytical Foundations of Networks
Layering as Optimization Decomposition
Lijun Chen 11/29/2012
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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
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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
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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
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Internet
IP
TCP/AQM
Physical
Application Diverse
Diverse
Highly conserved
core mechanisms
Layered architecture
MAC
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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)
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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
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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
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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
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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
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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
−
+
⎧ =⎪⎨
+ = + −⎪⎩
∑
∑
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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
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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 ≤
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Problem formulation
Network resource allocation:
Π∈
≤
∑
ffAxHts
xUs
ssfx
)()( ..
)( max,
constraint for routing
constraint from wireless interference
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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
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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
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vertical decomposition ⎣ ⎦+−−=+ )))}]((()))((({)([)1( tpxHtpfAtptp tγ
q Congestion update:
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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
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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,
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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
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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
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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
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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
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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
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❒ 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 ……