1. Distributed Network Utility Maximization in Multi-hop Wireless Networks: Noisy Feedback, Lossy Channel and Stability
Junshan Zhang
Department of Electrical Engineering
Arizona State University
Jan. 30, 2007

2. Outline
Overview of our research on wireless ad hoc networks and sensor networks
Distributed Network utility maximization in multi-hop wireless networks.

3. Wireless Ad-Hoc/Sensor Networks
Potential applications:
Battlefield wireless networks,
Monitoring chemical/biological warfare agents,
Homeland security.
Basic network models:
(1) Many-to-one networks,
(2) Multi-hop wireless networks,
(3) Sensory relay networks.

4. Network Models
(2)
(1)
(3)

5. Unique Features Stochastic nature of ad-hoc networks:
time-varying fading, co-channel interference, hostile jamming, …
mobility, dynamic network topology, …
Key features of sensor networks:
node cooperation
data correlation
energy constraints
QoS metrics: e.g., distortion, energy, event detection

6. Challenges and Opportunities
Challenge: How to transmit data/information over ad-hoc/sensor networks efficiently and reliably?
Network environments are not “friendly”;
Heterogeneous wireless applications:
wireless Internet access, battlefield multi-hop networks, sensor networks
Diverse requirements: high-bandwidth video and data, low-bandwidth voice and data
Every challenge, if taken properly, is an opportunity!

7. Why and How to do Cross-Layer Design
Goals: reliable communication-on-the-move in dynamic environments, QoS provisioning, energy efficiency
Why:
There exists direct coupling between physical layer and upper protocol layers in wireless networks;
Cross-layer design is perhaps single most promising approach to meet the fast-growing demands in many cases, e.g., CDMA/HDR, n, Flarion OFDM
Misconceptions:
Cross-layer design means to get rid of protocol layers.
Cross-layer design requires to integrate all protocol layers.

8. Research Thrust I: Cross-Layer Optimization and Design
Holistic approach to devise adaptive protocol modules to exploit interplay between layers
For network models (1)-(3), we explore cross-layer design for joint optimization of transport layer, routing, MAC and physical layer.

9. Wireless Network Architecture
Data Link/MAC
Network
Transport
Session
Presentation
Application
Physical
A holistic
approach
to wireless
networking
Cross-layer Design: adaptive modules
Application Driven
our goal: To understand overriding principles for cross layer design

10. Projects on Cross-layer Design
Self-similarity of multi-access interference in CDMA networks with bursty traffic
Traffic-aided opportunistic scheduling
MIMO ad-hoc Networks
Joint clustering and routing in sensor networks
Joint congestion control and MAC design in multi-hop wireless networks
Distributed opportunistic scheduling for ad-hoc communications

11. Research Thrust II: Cooperative Relaying and Networking
Capacity bounds of MIMO relay channel
Power allocation in wireless relay networks
Scaling laws of wideband sensory relay networks;
Technical details can be found in our recent papers:
B. Wang, JZ & Host-Madsen (IT 05);
Host-Madsen & JZ (IT 05);
B. Wang, JZ & L. Zheng (IT 06)

12. Distributed Network Utility Maximization in Multi-hop Wireless Networks
Random access
Bursty traffic
Channel fading
Distributed NUM in multi-hop wireless networks:
Noisy feedback,
Lossy channel,
Stochastic stability

13. Outline of Distributed NUM
I: Distributed NUM under noisy feedback
Deterministic P-D algorithm via Lagrangian dual;
Stochastic P-D algorithm under noisy feedback:
Unbiased case: stochastic stability, rate of convergence;
Biased case: contraction region, stochastic stability.
II: Distributed NUM under lossy channel:
Effective NUM over wireless networks.
Impact of UDP.
Acknowledgement:
Part I: based on joint work with Dong Zheng & Prof. Mung Chiang;
Part II: based on joint work with Qinghai Gao.

14. Basic NUM ² C o n s i d e r a w l t k m c g p h G = ( N ; E ) { L , by - v f S u x . U M : P j 8 1

15. Decentralized solution to NUM
Distributed solutions are key in making the NUM framework attractive in practical networks.
General form for flow control [Low-Srikant 04]:
Key observation:
Some level of message passing is always needed!
Reverse engineering perspective: Above network control method can be viewed as distributed solution to optimization problems; implemented via (sub)-gradient algorithm. x s ( t + 1 ) = F ; q 8 2 S l G y L w h e r a n d o m - g i v f u c , p k .

16. Distributed NUM via Lagrange Dual
Classification: Primal; dual; primal-dual
Distributed NUM via Lagrangian dual decomposition L ( x ; ) = X s U + l c f a g r n e d u t i o : Q m M h p b D - 1 [ ] 8

17. Example 1: Joint flow Control and MAC DesignC o n s i d e r a w l t k m c g p h G = ( N ; E ) M u U x y f b . j 8 2 L I [ v , '
Goal: fair rate control through joint congestion control and MAC design in multi-hop random access networks.
Approach: network utility maximization
Treat rate control as a utility maximization problem
Different layers function “cooperatively” to achieve the optimum point (equilibrium point).

18. Example 1: NUM for Cross-layer Rate Control: m a x P s 2 S U ( ) u b j e c t o i ; p Q k N I 1 8 = M w h r f d , v g n l y [ W ] I n g e r a l , P o b m i s - c v x d p .

19. Example 1: Change of Variablest ~ x s = l o g ( ) , w h a v P : m f 2 S U u b j c i ; p k N 1 8 M r . O b s e r v a t i o n : P l m c x d p f 1 .

20. Example 1: Lagrange Dual ApproachT h e L a g r n i f u c t o s ( ~ x ; p ) = 8
< : X U j l @ 2 S 1 A 9 + Y k N I P , d Q m M b v y D

21. Example 1: Distributed Primal-dual Alg.T h e s o u r c a t p d b y ~ x ( n + 1 ) = 2 6 4 @ _ U X i ; j L P S A | { z } , 3 7 5 M : w l g k N I m f

22. Distributed NUM under Noisy Feedback
Implementation of distributed algorithms hinges on message passing among network elements
Unfortunately, in practical systems feedback is often obtained using error-prone measurement mechanisms and also suffers from other random errors in its transmissions.
Feedback signals are stochastically noisy in nature.
Examples:
ECN for Internet congestion control: congestion price is obtained via probabilistic packet marking technique.
Wireless networks with random access: flow rate is based on sample-path measurements and feedback signal is transmitted over lossy wireless channel.
Congestion price is affected by stochastic UDP

23. An Example on Exponential MarkingA s u m e x p o n t i a l r k g d f b c , w h v - y q = P ( ; j ) 2 L S T N K . ^ B ~ E [ ]

24. Stochastic Primal Dual Alg.I n t h e p r s c o f i y d b a k , g L x ( ) l . + 1 = ^ ; D 8
Fundamental open questions on impact of noisy feedback
Would distributed NUM alg., in the presence of stochastic noisy feedback, converge to the optimal point of deterministic algorithms?
If yes, under what conditions and how quickly does it converge?
In case not, would the distributed NUM alg. be stable at all?

25. Stochastic Stability
Feedback information is needed to estimate subgradients;
Unfortunately, the feedback is noisy in practical systems!
Stochastic stability is pertinent to following issues:
number of users/queuing length remain finite;
algorithms converge in some stochastic sense.
Goal: obtain a clear understanding of impact of stochastic noisy feedback on stability.

26. Related Work Related work on stability
Connection-level stability: [Bonald-Massoulie 01], and [Lin-Shroff-Srikant 06] [Eryilmaz-Srikant05] [Neely-Modiano-Li 05] [Stolyar 05] and many more ;
Delay stability: [Low-Srikant 04] [Ranjan-La-Abed 06] …
Joint flow control/routing/MAC/PHY design in wireless networks: [Lin-Shroff 05], [Wang-Kar 05], [Chiang 04] [Chen-Low-Chiang-Doyle 06]
Deterministic feedback error: [Mehyar-Spanos-Low 04]
Rate of convergence around the equilibrium points: [Kelly-Malloo-Tan 98], [Kelly-03], … ;

27. Structure of Stochastic Gradients^ L x ( ; ) : O b v = + w m , y E l

28. Structure of Stochastic Gradients (Cont’d)^ L ( ; j ) : O b v x = + w m f , y E l

29. Technical Assumptions1 . W e a s u m t h i o r f g d n b c l y 2 C p z :
> , ! P
< 3 j ( ) ; 8 4 E [ ]

30. Main Result 1: Stability for Unbiased Case
Theorem 1:
Under Conditions the iterates
generated by the stochastic primal-dual algorithm, converge with probability one to the optimal solutions of Problem A 1 4 , f ( x n ) ; = 1 2 : g , . 1 . G o d n e w s : T h t c a i l g r m v p u A { 4 2 C y k W b z , E x f q - N \
"

31. Remarks Biased case: Key ideas for Proof:
Construct Lyapunov function based on saddle point properties for P-D alg.;
Super-martingale method; stochastic analysis.
Biased case:
When the gradient estimator is biased, we cannot hope for almost sure convergence;
Instead, we expect that the iterates return to a neighborhood of the optimal points under certain conditions.
Indeed, we can show that if the biases are uniformly bounded, there exists a “contraction region” such that the iterates return to this region infinitely often w.p.1.

32. Main Result 2: Stability for Biased Case7 . C o n d i t h e b a s r : T x - g v c f ( ; j ) u l m p 8 D e n t h \ c o r a i g
" A s f l w : , ( x ; ) j L ~ p m
< 1
Theorem 2: U n d e r C o i t s A 1 2 , 4 a 7 h f ( x ) ; = : g b y c p m l - u w .

33. Remarks 1 . C o n d i t A 7 e s a l y r q u h b m p c , w k 3 g 2 T \
" v f z - N

34. Rate of Convergence ² T h e r a t o f c n v g i s d w y m p b l z . A, u P - D ( x ; ) D e n U x ( ) , = p a d . C o s t r u c b h i w l f ; g + 2 [ 1 P U n + 2 U n ( t ) U n U n + 1 n n + 1 n + 2 t

35. Assumptions A 5 . L e t µ ( n ) , ~ x ; ¸ a d Á ³ » S u p o s f r y gv m l
> h c = E [ T ] I j ! 1 D H : 6 + 2 w z N b - B k 9

36. Main Result 3: Rate of Convergence (unbiased case)1 3 6 , ( c v g w k l y h u - S p b m x = H + I 2 z ; b ) I f ( x ; i s a n t e r o p h c , l m g U y G u d v w z . c ) I f ( x ; i s o n t h e b u d a r y , l m g p U .

37. Engineering Implications1 ) I n g e r a l , t h i m p o c s y d u G ; z f [ K 9 8 ] . 2 ) T h e l i m t p r o c s w u d b G a n y f . 3 ) I n t u i v e l y s p a k g , h r c o m w d f b z x . 4 ) T h e c o v a r i n m t x f l p s g u d q b , y \
" .

38. Numerical Studies C o n s i d e r l g a t h m u y f c w = 1 .

39. Numerical Examples – Deterministic Case

40. Numerical Examples – Stochastic (unbiased) Case

41. Numerical Examples: Biased Case

42. Towards a Framework for Understanding Impact of Noisy Feedback on Distributed NUM
Impact of noisy feedback on distributed Algorithm is not well understood; applicable to both wireline networks and wireless networks.
We are developing a general theory on stochastic NUM.
Single time scale algorithm (P-D Alg.)
Two time scale algorithm (e.g., based on primal decomposition).
Tradeoff between complexity and robustness.
Asynchronous NUM algorithms.
Correlated measurements: tradeoff between stability and subgradient estimation

43. Two Time-Scale Algorithms
Among alternative decomposition methods, primal decomposition is a useful machinery for problems with coupled variables:
Important issues:
stability of multiple time-scale algorithms under noisy feedback.
complexity and robustness of two time-scale algorithm, compared with single time-scale algorithm?
Optimizing over selected variables first, then the remaining variables; yields multiple time-scale algorithms

44. NUM Problem C o n s i d e r t h f l w g N U M p b m : ¥ a x z P ( ) u2 a x z P ( ) u j c L ; 8 = H F L i n k c a p t e s f l g r u o M A C m , d w b H h .
For ease of exposition, we assume convexity of the above NUM problem.

45. Deterministic AlgorithmF a s t e r ( m l ) i c : I T h o u p d b y x n = g M @ U X 2 L 1 A ; 8 w + 4 S 3 5 C
" k #
Cross-layer message passing is significantly reduced.
The two time-scale algorithm has a lower complexity.

46. Stochastic Two Time-Scale AlgorithmF a s t e r i m c l : I S A g o h f u p d n x ( ) = M @ U 2 4 X L + 3 5 1 w - , b y . ; C k N

47. Geometric Interpretation
Two possible ways to implement the above two time-scale Algorithm:
One popular method is to let faster time-scale iteration converge first, and then execute slower time-scale loop.
Alternatively, one can set and run algorithms at both time scales;
Method 2 is more robust to noisy feedback. b n = o ( a )

48. Notation F D e ¯ n t , P d m ( ) s u c h · < . l o g y w f r p : M1 i = d m ( ) s u c h
< + . b l o g y w f r p : M ; x X N

49. Distributed NUM Under Lossy Channel
Motivation:
Data transmission over wireless links suffers from lossy channels.
Under-utilization with TCP over wireless channels.
Existing solutions for flow control over wireless channels:
distinguishing packet loss due to lossy channels from that due to congestion.
Concerns: retransmissions incur time out; scalability

50. Problem Formulation
Our approach: effective network utility maximization C h a n e l c p i t y s r d o m ; u f R . T v P ( ) E w k : X U x Y 2 L 1 g b

51. Effective NUM Link Constraints: ENUM Problem: m a x P U ³ Q 1 ¡ p ´ :
< s 2 S ( ) x g
ENUM Problem: m a x f s 2 I g P U Q l L ( ) 1 p S : t C
< ; 8

52. Lagrangian Dual Method
ENUM with logarithm utility: m a x f s 2 I g P ! l o ( ) L t S 1 : ~ c ; 8 w h e r i d n b y k u .
Use Lagrangian Dual method: L ( x ; ) = X s @ ! l o g 2 P t S 1 A + ~ c D m a f I i n

53. ENUM Algorithm link congestion price depends on total flow ratex s ( n + 1 ) =
" ! X l 2 Ã P t S # I 4 @ ~ c A 3 5
link congestion price depends on total flow rate
Link error price depends on total flow rate and link SNR, along the path.

54. Numerical Example 2 links, 3 flows U(x) = log(x) SNR1=10dB, SNR2=20dB
Each link has link constraint: log(1+SNR)
Rayleigh fading
x2 > x1 > x3

55. Injection Rate & Effective Rate

56. ENUM under UDP m a x f s 2 I g P U e p ( l L ) t S + u 1 : ~ c ; 8
Decrease the data rate limit for elastic flows at each link.
Increase the outage probability at each link.

57. ENUM under UDP

58. Conclusions and Ongoing work
Distributed NUM over multi-hop wireless networks.
Distributed NUM under noisy feedback
Deterministic P-D alg. via Lagrangian dual;
Stability of Stochastic P-D algorithm:
Unbiased case: convergence, rate of convergence;
Biased case: contraction region.
Distributed NUM under lossy channel:
Effective NUM over wireless networks.
Impact of UDP.
Combined impact of noisy feedback and lossy channel:
Q) End-to-end approach vs. hop-by-hop approach?

59. Proof for the Main Result 1: U s i n g h o c a L y u v b l T r m , w f d . L e t ( x ; ) b a s d l p o i n . D h y u v f c V w : V ( x ; ) , j 2 + m i n D e n f o r a g i v
> , h b d s t A x ; ) : V (

60. Stochastic Lyapunov FunctionW e s h o w E n [ V ( x + 1 ) ; ] 2 G O j i t u d r a g = T L m . I t s u c e o h w a G ( x n ) ;
< 2 A .

61. A Supermartingale Lemmaf X n g b a R r - v l u d s o c h i p , V ( ) . S Y q m y P j
< 1 w F ; x A E [ + ] = 2 T

62. Step II – Local Analysis: W s a b l i h , v \ o c n y
" r u - d m g f w ( ~ x ) ; = 1 2 A 3 . L k

63. Proof for Main Result 2 N o w l e t ¹ ! , h a v A . 1 ) D e ¯ n A , [. I t c a b s h o w i m p A , f ( x ; ) : s j L o r m e i
< 1 g 2 ) S i m l a r t o h e p f M n R s u 1 , c b w E [ V ( x + ; ] G O : 3 ) S i n c e
< 1 , u s g t h f a G ( x ; o r m p v w = 2 A l - y b Y 8 . N o w l e t ! , h a v A .

64. Thank You!