1. Cross layer design is wireless multi-hop network
歐永俊

2. Outline Introduction Problem formulation System Model Simulation
Reference

3. Introduction
The overall communication network is modeled by a generalized network utility maximization problem, each layer corresponds to a decomposed subproblem, and the interfaces among layers are quantified as functions of the optimization variables coordinating the subproblems.
Vertical decomposition into functional modules such as congestion control, routing, scheduling and power control.
Horizontal decomposition into distributed computation.
the master problem sets the price for the resources to each subproblem, which has to decide the amount of resources to be
used depending on the price.

4. Introduction Primal problem: Dual problem: dual decomposition
the problem has a coupling constraint such that, when relaxed, the optimization problem decouples into several subproblems

5. Introduction
Dual problem:
Primal problem:
dual decomposition

6. Problem formulation
In traditional framework, each link provides a fixed coding and modulation scheme in the physical layer, and that each user’s utility is only a function of local source rate.
In many practical systems, utility for each user depends on both transmission rate and signal quality
A higher throughput can be obtained on a link at the expense of lower decoding reliability, which in turn lowers the end-to-end signal quality for sources traversing the link and reduces users’ utilities, thus leading to an intrinsic tradeoff between rate and reliability
the Pareto optimal tradeoff curves between rate and reliability.

7. System Model End-to-end reliability constraint
is the end-to-end error probability
is the error probability of source s at link l
is the code rate of source s at link l
is an increasing convex function reflecting the rate-reliability trade-off
is the set of links used by source s

8. System Model Capacity constraint
is the information data rate of source s
is the transmission data rate of source s at link l
is the set of sources using link l
is the capacity of link l
We assumes

9. System Model maximize subject to source problems link problems
: “congestion price”, the price per unit rate to use link l
: “reliability price”, the price per unit reliability that the source s must pay to the network
: with an interpretation of “end-to-end congestion price” on source s
: with an interpretation of “aggregate reliability price” paid by sources using link l

10. System Model at each source s The Lagrange dual function is maximize
subject to
separable
parallel
on each link l
maximize
subject to
The dual problem is
minimize
subject to

11. System Model
Local information exchange

12. System Model
Note that when each link can provide different reliabilities(code rates) for the incoming traffic of different sources, such as there exist two class of users (primary users and secondary users),the problem becomes a GP. By change of variables ,
decomposition method still applies.
maximize
subject to
posynomials

13. Simulation a linear topology consisting of four links and eight users
accounts for scalarization ,changes form 0 to 1 in step size 0.1

14. Simulation
a=0
a=1
Figure 1. rate reliability trade-off among users, each source has same
code rate for a link

15. Simulation
a=0
a=1
Figure 2. rate reliability trade-off among users, each source can have different code rate for a link

16. Reference
[1] Lee, J.-W.; Mung Chiang; Calderbank, A.R., "Price-based distributed algorithms for rate-reliability tradeoff in network utility maximization," Selected Areas in Communications, IEEE Journal on , vol.24, no.5, pp , May 2006
[2] M. Chiang “Balancing transport and physical layer in wireless multihop networks: Jointly optimal congestion control and power control,” IEEE J. Sel. Areas Commun., vol. 23, pp. 104, Jan
[3] Mung Chiang; Low, S.H.; Calderbank, A.R.; Doyle, J.C., "Layering as Optimization Decomposition: A Mathematical Theory of Network Architectures," Proceedings of the IEEE , vol.95, no.1, pp , Jan. 2007
[4] Palomar, D.P.; Mung Chiang, "A tutorial on decomposition methods for network utility maximization," Selected Areas in Communications, IEEE Journal on , vol.24, no.8, pp , Aug. 2006
[5] Lee Jang-Won; Tang Ao; Huang Jianwei; Mung Chiang; Robert, A., "Reverse-Engineering MAC: A Non-Cooperative Game Model," Selected Areas in Communications, IEEE Journal on , vol.25, no.6, pp , August 2007
[6] Jang-Won Lee; Mung Chiang; Calderbank, A.R., "Utility-Optimal Random-Access Control," Wireless Communications, IEEE Transactions on , vol.6, no.7, pp , July 2007
[7] Jang-Won Lee; Chiang, M.; Calderbank, R.A., "Jointly optimal congestion and contention control based on network utility maximization," Communications Letters, IEEE , vol.10, no.3, pp , Mar 2006