1. 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. 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. 3 Outline Motivation System model Optimal routing and resource allocation Example Efficient solution methods Distributed algorithms Conclusions and extensions

4. 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. 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. 6 System model We assume fixed topology fixed coding, modulation and optimize rates, routing & resource allocation We model multiple data flows influence of resource allocation on link capacities local & global resource limits

7. 7 Network topology Directed graph with nodes, links set of outgoing links at node, incoming links at Incidence matrix

8. 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. 9 Multicommodity network flow Some traditional formulations: fixed, minimize total delay: fixed, maximize total utility:

10. 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. 11 Example: Gaussian broadcast with FDMA Communication variables: Shannon capacity: Total power, bandwidth constraint on outgoing links

12. 12 Concavity of the capacity formula Claim: capacity formula is jointly concave in powers and bandwidths Proof: its Hessian is negative semi-definite

13. 13 Simultaneous optimization of routing and resource allocation Solution to optimization problem We assume that are convex SRRA is a convex optimization problem, hence readily solved

14. 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. 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. 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. 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. 18 Outline System model Optimal routing and resource allocation Example Efficient solution methods Distributed algorithms Conclusions and extensions

19. 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. 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. 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. 22 Example: Water-filling Dual function Dual problem Solved by adjusting until power constraint becomes tight.

23. 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. 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. 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. 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. 27 Outline System model Optimal routing and resource allocation Example Efficient solution methods Distributed algorithms Conclusions and extensions

28. 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. 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. 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 layer Radio layer: nodes allocate resources to maximize revenue Link prices: Link prices: follow supply and demand Convergence follows along the lines of [LL:99].

31. 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