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
1
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-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]
 Our goal: Joint optimization of routing, random access and flow
control in a distributed way
2
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
1
C
7
A
2
3
 Example:
E
6
4
B
5
8
D
3
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.
 An outgoing link is active w.p.
s.t.
s.t.
 The average achievable MAC layer rate over link
Link capacity
Prob. that the competing nodes are silent
 MAC layer rate constraint
4
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
collision statistics
at node
based on the
 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
Problem statement
 Seek for the optimal random access parameters
routing variables , and the source rates which
, the
 maximize the total network utility
 satisfy the MAC and net. layer constraints
 Formulation
Flow rates are bounded
in practice
 (P1) is inherently nonconvex due to MAC layer rate constraint
6
Successive convex approximation
 Theorem [MW’78]: consider the nonconvex problem (P0)
convex
nonconvex

Approximate the nonconvex functions to
1)
2)
3)
s.t.
: 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
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
Convex problem(P2)
 Logarithmic change of variables
 Solve (P2) at kth iteration
(*)
(**)
Proposition 1: (P2) is convex provided that β ≥ 1
9
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
The outgoing links not connected
each node
to the destinations
 Introduce the auxiliary variables
 Add to (P2) the constraints
 Regularization term to ensure feasibility of the converged
solution of dual method
10
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
Dual problem
 Dual function
(**) in which
is replaced with
 Dual optimization problem
12
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
Higher-layers subproblem
 The optimization problem at node n (P5)
l.h.s of (***)
14
Solution
Proposition 2: Denoting
, and
, the optimum of (P5) for β=1 is
a) If
 Share the total outgoing flow
in proportion to
and
b) If
 Closed-form solutions
 Suitable for wireless sensor networks
15
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
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
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
18
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!
19
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 multihop 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.
20