Simultaneous Routing and Resource Allocation in Wireless Networks
Mikael Johansson Signals, Sensors and Systems, KTH
Joint work with Lin Xiao and Stephen Boyd, Stanford University
2
About this talk
Pedagogical motivation• To convey ideas and techniques from distributed convex optimization
Technological motivation• Wireless ad-hoc networks promising emerging technology
Intellectual motivation• Will ad-hoc networks deliver the required performance (capacity)? • Compute the optimal parameters for a given network configuration• Devise simple, distributed protocols that ensure efficient network operation
Control-theoretic motivation• Distributed resource allocation problems roots of distributed control theory• New technological challenges/problems may inspire theoretical advances
3
Outline
• Motivation• System model• Optimal routing and resource allocation• Example• Efficient solution methods• Distributed algorithms• Conclusions and extensions
4
Motivation: cross-layer optimization
Standard (OSI) network model• Physical/radio link layer, network layer (routing), transport...
Wireless data network• Optimal routing of data depends on link capacities• Link capacities are determined by resource allocation
Efficient operation requires coordination of layers!
5
Disclaimer
This talk only considers orthogonal channel models,• simple and elegant theory
Interference-limited systems require other techniques• High signal-to-noise ratio: convex approximation [JXB:03]• Low signal-to-noise ratio: scheduling, integer programming [JX:03]
In practice, time-varying channels and delays fundamental limitations
Very active area of research, many open problems!
6
System model
We assume• fixed topology• fixed coding, modulationand optimize
• rates, routing & resource allocation
We model• multiple data flows• influence of resource allocation
on link capacities • local & global resource limits
7
Network topology
Directed graph with nodes , links
set of outgoing links at node , incoming links at
Incidence matrix
8
Network flow model
Model average data rates, multiple source/destination pairs
Identify flows by destination
– source flows flow from node to node – link flows flow on link to node
Flow conservation laws
9
Multicommodity network flow
Some traditional formulations:
• fixed, minimize total delay:
• fixed, maximize total utility:
10
Communications model
Capacities determined by resource (power, bandwidth) allocation
Communications model
Where – is a vector of resources allocated to link , e.g., – is concave and increasing– resource limits local (power at node) or global (total bandwidth)
Many (most?) channel models satisfy these assumptions!
11
Example: Gaussian broadcast with FDMA
Communication variables:
Shannon capacity:
Total power, bandwidth constraint on outgoing links
12
Concavity of the capacity formula
Claim: capacity formula is jointly concave in powers and bandwidths
Proof: its Hessian
is negative semi-definite
13
Simultaneous optimization ofrouting and resource allocation
Solution to optimization problem
We assume that are convex
SRRA is a convex optimization problem, hence readily solved
14
Examples
SRRA formulation is very general, includes
Maximum utility routing (QoS)
Minimum power routing
as well as minimum bandwidth, minimax link utilization, etc.
15
Numerical example
– 50 nodes, 340 links (transmitters)– 5 nodes exchange data (i.e., 20 source-destination pairs)– transmitters use FDMA, power limited in each node– goal: maximize network utility
16
Numerical example: details
Topology:• node locations drawn from uniform distribution on a square• two nodes can communicate if distance smaller than threshold• source and destination nodes chosen randomly
Radio layer properties• bandwidth allocation fixed, power constraint at each node • quadratic path loss model• noise power drawn from uniform distribution
Optimization problem has 2060 variables (1720 flows, 340 powers)
17
Numerical example: results
Routing to node 1 Aggregate data flows Power allocation
Result: Result: 35% improvement over routing w. uniform power allocation
Note: log-utility gives diminishing returns, throughput improvements much larger
18
Outline
• System model• Optimal routing and resource allocation• Example• Efficient solution methods• Distributed algorithms• Conclusions and extensions
19
Solution methods
Small problems readily solved using ”off-the-shelf” software
Real-world problems: hundreds of nodes, thousands of links
Size proportional to , often large!
Can we do better? Can we do it distributedly, in ”real-time”?
20
Duality
”Primal” problem ”Dual” problem
Convex duality: • optimal values of both problems equal¹• can solve original problem via its dual²
Lagrange decomposition: multipliers for “critical” constraints only• Decompose dual into subproblems that are easy to solve• Can give very efficient overall optimization
21
Example: Water-filling
Consider the following convex optimization problem
(equivalent to maximizing weighted total utility)
Total power constraint destroys separable structure!
Solution approach– introduce Lagrange multiplier for this constraint only– solve dual problem– recover optimal solution
22
Example: Water-filling
Dual function
Dual problem
Solved by adjusting until power constraint becomes tight .
23
Dual decomposition of SRRA
Introduce multipliers for capacity constraints only
Problem decomposes into• Uncapacitated network flow problems (one per commodity)• Resource allocation problem (often solved by water-filling)
24
Solving the master dual
Master dual problem
solved using sub-gradient method
step-length parameter, sub-gradient
Multipliers decreased when capacity exceeds traffic
25
Convergence of dual method
Convergence of dual method vs. number of iterations
An alternative approach, the analytic-center cutting plane method,• has better convergence• requires considerably more computations per iteration• appears hard to implement distributedly
26
Economics interpretation
Interpret dual variables as ”price per unit traffic on each link”
Network layer: Network layer: minimizes network loss + cost of capacities used
Radio control layer:Radio control layer: allocates resources to maximize revenue
Price updates:Price updates: follow laws of supply and demand
27
Outline
• System model• Optimal routing and resource allocation• Example• Efficient solution methods• Distributed algorithms• Conclusions and extensions
28
Distributed algorithms
Simplified model for fixed routing
The matrix indicates what flows traverse what links
Note: relation to TCP Vegas (e.g., [LPW:02]) over wireless links
29
Distributed algorithms
Consider a dual approach
The first subproblem admits closed-form solution-solved locally by sources, if they know their “total path cost”
The second subproblem solved using water-filling in each node
30
Distributed algorithms
Dual problem can be solved using subgradient method
Note that multipliers can be computed locally for each link
A distributed algorithm:
Transport layer:Transport layer: sources maxmimize utility minus resource cost
Radio layerRadio layer: nodes allocate resources to maximize revenue
Link prices: Link prices: follow supply and demand
Convergence follows along the lines of [LL:99].
31
Conclusions
Conclusions:• Optimal cross-layer coordination in wireless data networks• Simultaneous optimization of routing & resource allocation• Convex optimization problem, hence readily solved• Very efficient solution methods by exploiting structure• Distributed methods for adaptive resource allocation
More info at
http://www.s3.kth.se/~mikaelj