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

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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

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Outline

• Motivation• System model• Optimal routing and resource allocation• Example• Efficient solution methods• Distributed algorithms• Conclusions and extensions

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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!

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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!

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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

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Network topology

Directed graph with nodes , links

set of outgoing links at node , incoming links at

Incidence matrix

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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

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Multicommodity network flow

Some traditional formulations:

• fixed, minimize total delay:

• fixed, maximize total utility:

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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!

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Example: Gaussian broadcast with FDMA

Communication variables:

Shannon capacity:

Total power, bandwidth constraint on outgoing links

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Concavity of the capacity formula

Claim: capacity formula is jointly concave in powers and bandwidths

Proof: its Hessian

is negative semi-definite

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Simultaneous optimization ofrouting and resource allocation

Solution to optimization problem

We assume that are convex

SRRA is a convex optimization problem, hence readily solved

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Examples

SRRA formulation is very general, includes

Maximum utility routing (QoS)

Minimum power routing

as well as minimum bandwidth, minimax link utilization, etc.

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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

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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)

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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

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Outline

• System model• Optimal routing and resource allocation• Example• Efficient solution methods• Distributed algorithms• Conclusions and extensions

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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”?

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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

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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

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Example: Water-filling

Dual function

Dual problem

Solved by adjusting until power constraint becomes tight .

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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)

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Solving the master dual

Master dual problem

solved using sub-gradient method

step-length parameter, sub-gradient

Multipliers decreased when capacity exceeds traffic

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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

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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

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Outline

• System model• Optimal routing and resource allocation• Example• Efficient solution methods• Distributed algorithms• Conclusions and extensions

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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

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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

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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].

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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