Optimization of Wireless Multi-hop Networks with Random Access

1. Optimization of Wireless Multi-hop Networks with Random Access MortezaMardani, Seung-Jun Kim, and Georgios B. Giannakis ECE Department, University of Minnesota Acknowledgments: NSF grants no. CCF-0830480, 1016605 EECS-0824007, 1002180 June 29, 2011

2. Motivation • Random access is a simple MAC with no central coordination • Probabilistic model to design utility-optimal MAC [LCC’07] • Better efficiency and fairness than the contention graph model • Joint design of multi-pathrouting and random access for wireless multi-hop networks • Pathselectionandtrafficsplitting • Routing can avoid interference prone areas of the network • Related work • Joint random access and flow control [YG’08], [WK’06] • Joint random access, routing and flow control, [SS’09], [CLCD’06] • Our goal: Joint optimization of routing, random access and flow control in a distributed way 2

3. System model • Wireless multi-hop network with directed graph • : set of incoming links at node , : set of outgoing links at node • Node generates the single commodity traffic at rate and forward it through its outgoing link with rate • Interference model: simultaneous receptions at the receiver • : set of nodes causing interfering • to link , : set of links • interfered by transmission of node • Example: 3

4. Random access control Slotted Aloha with a single shared channel Node randomly decides to transmit w.p. Active node chooses one of its outgoing links w.p. s.t. • An outgoinglink is active w.p. s.t. • The average achievable MAC layer rate over link Link capacity Prob. that the competing nodes are silent • MAC layer rate constraint 4

5. Network and transport layers • Flow conservation constraint for queue stability set of incoming links except those emanating from destinations • Flow control to adjust the source rate at node based on the collision statistics • Node is awarded a utility to deliver the traffic at rate • Utility function: an increasing and concave function, e.g., • : fairness controlling parameter • =1: proportional fairness • =2: harmonic-mean fairness 5

6. Problem statement • Seek for the optimal random access parameters , the routing variables , and the source rates which • maximize the total network utility • satisfy the MAC and net. layer constraints • Formulation • (P1) is inherently nonconvex due to MAC layer rate constraint Flow rates are bounded in practice 6

7. Successive convex approximation • Theorem [MW’78]: consider the nonconvex problem (P0) convex nonconvex • Approximate the nonconvex functions to s.t. 1) 2) 3) : feasible set of kth convex problem : optimal solution of kth convex problem • By successively updating the convex problem, the solution converges to a KKT point of (P0) 7

8. Single condensation method • After rearranging routing constraint • Based on arithmetic-geometric mean inequality • Tight surrogates for routing constraints Optimal solution in previous iteration Approximation elements 8

9. Convex problem(P2) • Logarithmic change of variables • Solve (P2) at kth iteration (*) (**) Proposition 1: (P2) is convex provided that β ≥ 1 9

10. Distributed solution • Solve (P2) at the network nodes using only limited message exchange with the local nodes • Difficulty: coupling among the routing constraints at different nodes • Solution: keep a local copy of the rate of the outgoing links at each node • Introduce the auxiliary variables • Add to (P2) the constraints The outgoing links not connected to the destinations • Regularization term to ensure feasibility of the converged solution of dual method 10

11. Partial Lagrangian • Relax the MAC layer rate constraints and the constraints on the local copies • Separable over MAC and higher layers at different nodes • Lagrangian associated with MAC layer The price paid for the rate constraint • Lagrangian associated with higher layers The dual variable for constraints on local copies The outgoing links connected to the destinations 11

12. Dual problem • Dual function (**) in which is replaced with • Dual optimization problem 12

13. MAC-layer subproblem • Optimization problem at node n (P4) • Similar problem in [LCC’07] for single-hop networks • Persistence probabilities for node n and its outgoing links • Remarks • Higher the price paid, higher is the channel access • Higher prices to less interfering links Message exchange: If (P4) solved at TX(l), only need 13

14. Higher-layers subproblem • The optimization problem at node n (P5) l.h.s of (***) 14

15. Solution Proposition 2: Denoting , and , the optimum of (P5) for β=1is a) If b) If • Closed-form solutions • Suitable for wireless sensor networks • Share the total outgoing flow • in proportion to and 15

16. Cont’d • If β>1 and a) If satisfies • Remarks • Flow conservation is enforced by finding numerically • A simple root finding method e.g., the bisection method • Remarks • Lower rates for the links with higher MAC competition • Message passing only with neighbors. Node n only needs to receive 16

17. Dual update • Subgradient projection method • Dual iterations • Simple projection computable in closed-form Proposition 4: Dual method converges to the optimum of (P2) if and . • Remarks • Local update of the approximation elements • A global timer to stop (P3) distributed algorithm 17

18. Numerical tests • Network example: 15 nodes, 52 links • dc = di =0.35, ε=1e-3, rmin=1e-5, rmax=10 cl=10 • Monotonic increase • of the utility • Coincidence with the • global optimum 80% • of trials • Existing [YG’07]: prespecified routes • Routing avoids interference around the destination • Net. Utility 0.32vs-0.74 18

19. Concluding summary Cross-layer design of random access, routing and flow control for wireless ad hoc networks Thank You! • Successive convex approximation approach to find a KKT point • Distributed algorithms derived based on the dual method • Closed-form solutions reducing implementation complexity • After few outer iterations the algorithm converges to a point, which often coincides with the global optimum • Optimized collision-aware routing enhances the network utility by avoiding the interference prone areas of the network • Future work: extension to the multi-commodity flows and the asynchronous implementation 19

20. Key references [LCC’07] J.-W. Lee, M. Chiang, and A. R. Calderbank, “Utility optimal random-access control,” IEEE Trans. Wireless Commun., vol. 6, no. 7, pp. 2741–2751, Jul. 2007. [YG’08] Y. Yu and G. B. Giannakis, “Cross-layer congestion and contention control for wireless ad hoc networks,” IEEE Trans. Wireless Commun., vol. 7, no. 1, pp. 37–42, Jan. 2008. [WK’06] X. Wang and K. Kar, “Cross-layer rate optimization for proportional fairness in multi-hop wireless networks with random access,” IEEE J. Sel. Areas. Commun., pp. 1548–1559, Aug. 2006. [SS’09] S. Supittayapornpong and P. Saengudomlert, “Joint flow control, routing and medium access control in random access multi-hop wireless networks,” in Proc. of Intl. Conf. on Comm., Dresden, Germany, pp. 1–6, Jun. 2009. [CLCD’06] L. Chen, S. H. Low, M. Chiang, and J. C. Doyle, “Cross-layer congestion control, routing and scheduling design in ad hoc wireless networks,” in Proc. of the INFOCOM Conf., pp. 1–13, Apr. 2006. [MW’78] B. R. Marks, and Gordon P. Wright, “A General Inner Approximation Algorithm for Nonconvex Mathematical Programs”, Operations research, vol. 26, no. 4, Jul.—Aug. 1978.