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Optimization of Wireless Multi-hop Networks with Random Access

Morteza Mardani, 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

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Motivation

Our goal: Joint optimization of routing, random access and flow control in a distributed way

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-path routing and random access for wireless multi-hop networks

Path selection and traffic splitting 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]

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System model

Example:

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

A

B

C

D

E

1

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5

46

7

8

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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 outgoing link is active w.p. s.t. The average achievable MAC layer rate over link

MAC layer rate constraint

Prob. that the competing nodes are silentLink capacity

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Network and transport layers Flow conservation constraint for queue stability

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

set of incoming links except those emanating from destinations

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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

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Successive convex approximation Theorem [MW’78]: consider the nonconvex problem (P0)

convex nonconvex

Approximate the nonconvex functions to s.t.

By successively updating the convex problem, the solution converges to a KKT point of (P0)

1)

2)

3)

: feasible set of kth convex problem : optimal solution of kth convex problem

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Single condensation method After rearranging routing constraint

Tight surrogates for routing constraints

Based on arithmetic-geometric mean inequality

Approximationelements

Optimal solution in previous iteration

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Convex problem(P2)

Proposition 1: (P2) is convex provided that β ≥ 1

Logarithmic change of variables Solve (P2) at kth iteration

(*)

(**)

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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

Regularization term to ensure feasibility of the converged solution of dual method

The outgoing links not connected to the destinations

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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

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Dual problem Dual function

Dual optimization problem

(**) in whichis replaced with

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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

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Higher-layers subproblem The optimization problem at node n (P5)

l.h.s of (***)

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SolutionProposition 2: Denoting , and , the optimum of (P5) for β=1 is

a) If b) If

Share the total outgoing flow in proportion to and

Closed-form solutions Suitable for wireless sensor networks

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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

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Dual update Subgradient projection method

Simple projection computable in closed-form

Dual iterations

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

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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 0.32 vs -0.74

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Concluding summary Cross-layer design of random access, routing and flow control

for wireless ad hoc networks 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

Thank You!

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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.