1. 1 Transport Bandwidth Allocation, Intro to Network Layer 4/3/2012

2. Admin. r Programming assignment 2 status 2

3. 3 Recap: TCP CC Signaling TCP/RenoTCP/Vegas Congestion signal Dynamics Equilibrium loss rate p queueying delay T queueing

4. Recap: Global Resource Allocation Examples ObjectiveAllocation (x1, x2, x3) TCP/Reno0.260.74 TCP/Vegas1/32/3 Max Throughput011 Max-min½½½ Max sum log(x)1/32/3 Max sum of -1/(RTT 2 x)0.260.74 4 x1x1 x2x2 x3x3 C= 1

5. 5 Recap: Resource Allocation Frameworks r Forward (design) engineering: m how to determine objective functions m given objective, how to design effective alg r Reverse (understand) engineering: m understand current protocols (what are the objectives of TCP/Reno, TCP/Vegas?)

6. 6 Recap: Nash Bargain Solution Axioms m Pareto optimality m symmetry m invariance of linear transformation m independence of irrelevant alternatives => R T s x1x1 x2x2

7. 7 Roadmap: Resource Allocation Frameworks r Forward (design) engineering: how to determine objective functions m given objective, how to design effective alg r Reverse (understand) engineering: m understand current protocols (what are the objectives of TCP/Reno, TCP/Vegas?)

8. 8 A Two-Slide Summary of Constrained Convex Optimization Theory g(x) f(x) f(x) concave g(x) linear S is a convex set q S -Map each x in S, to [g(x), f(x)] -For each slope q (>=0), computes f(x) – q g(x) of all mapped [f(x), g(x)] q

9. 9 A Two-Slide Summary of Constrained Convex Optimization Theory g(x) f(x) f(x) concave g(x) linear S is a convex set q1q1 q2q2 S -D(q) is called the dual; q (>= 0) are called prices in economics - According to optimization theory: when D(q) achieves minimum over all q (>= 0), then the optimization objective is achieved. -D(q) provides an upper bound on obj. q3q3

10. 10 Dual of the Primal x1x1 x2x2 x3x3 1 1

11. 11 Decomposition r Assume each link l has non-negative congestion signal q l, consider the dual D(q) x1x1 x2x2 x3x3 1 1

12. 12 Distributed Optimization: User Problem r Given price signal per unit rate p f (=sum of q l along the path) flow f chooses rate x f to maximize: r Using the price signals, the optimization problem of each user is independent of each other!

13. 13 Distributed Optimization: User Problem At equilibrium (i.e., at optimal), x f satisfies: How should flow f adjust x f locally?

14. 14 Interpreting Congestion Measure x f (t) q l (t) q 2 (t)

15. 15 Distributed Optimization: Network Problem The network (i.e., link l) adjusts the link signals q l (assume after all flows have picked their optimal rates given congestion signal)

16. 16 Distributed Optimization: Network Problem how should link l adjust q l locally?

17. 17 Decomposition r SYSTEM(U): r USER f : r NETWORK:

18. Outline r Bandwidth allocation framework m framework m Nash Bargaining Solution (NBS) m distributed computation m TCP/Reno, TCP/Vegas revisited 18

19. 19 TCP/Reno Dynamics

20. 20 TCP/Vegas Dynamics

21. 21 Summary: TCP/Vegas and TCP/Reno r Pricing signal is queueing delay T queueing r Pricing signal is loss rate p

22. 22 Summary: Resource Allocation Frameworks r Forward (design) engineering: m how to determine objective functions m given objective, how to design effective alg r Reverse (understand) engineering: m understand current protocols (what are the objectives of TCP/Reno, TCP/Vegas?) r Additional pointers: m http://www.statslab.cam.ac.uk/~frank/pf/

23. 23 Network Layer r Transport packet from source to dest. r Network layer in every host, router Most basic functions: r Control plane: path determination and call setup m determine routes taken by packets from sources to destinations r Data plane: forwarding m move packets from router’s input port to router output port

24. 24 Internet Network Layer: Protocols forwarding Network layer functions: Routing protocols path selection e.g., RIP, OSPF, BGP Network layer protocol (e.g., IP) addressing conventions packet format packet handling conventions Control protocols error reporting e.g. ICMP Transport layer Link layer physical layer Network layer Control protocols - router “signaling” e.g. RSVP

25. Data Plane: Forwarding routing and call setup

26. 26 Control Plane: Routing Graph abstraction for the routing problem: r graph nodes are routers r graph edges are physical links m links have properties: delay, capacity, $ cost, policy Goal: determine “good” paths (sequences of routers) thru network from source to dest. Routing A E D CB F 2 2 1 3 1 1 2 5 3 5

27. 27 Routing Design Space r Routing has a large design space m who decides routing? source routing: end hosts make decision network routing: networks make decision m how many paths from source s to destination d? multi-path routing single path routing m does the route(s) provide QoS? QoS best effort m will routing adapt to network traffic demand? adaptive routing static routing

28. 28 Routing Design Space: User-based, Multipath, Adaptive r Routing has a large design space m who decides routing?  source routing: end hosts make decision network routing: networks make decision m how many paths from source s to destination d?  multi-path routing single path routing m does the route(s) provide QoS? QoS best effort m will routing adapt to network traffic demand?  adaptive routing static routing m …

29. 29 User Optimal, Multipath, Adaptive r User optimal: users pick the shortest routes (selfish routing) Braess’s paradox

30. 30 Bound on Price of Anarchy For a network with linear latency functions  total latency of user (selfish) routing for given traffic demand ≤ 4/3 total latency of network optimal routing for the traffic demand

31. 31 Bound on Price of Anarchy r For any network with continuous, non- decreasing latency functions  total latency of user (selfish) routing for given traffic demand ≤ total latency of network optimal routing for twice traffic demand

32. 32 Routing Design Space: Adaptive Routing r Routing has a large design space m who decides routing? source routing: end hosts make decision network routing: networks make decision m how many paths from source s to destination d? multi-path routing single path routing m does the route(s) provide QoS? QoS best effort m will routing adapt to network traffic demand? adaptive routing static routing

33. 33 Routing: Single-Path, Adaptive Routing r Assume link costs reflect current traffic D A B C 0 2+e 1+e 1 0 0 t=2 Solution: Link costs are a combination of current traffic intensity (dynamic) and topology (static). To improve stability, the static topology part should be large. Thus less sensitive to traffic; thus non- adaptive. D A B C 2+e 0 0 0 1+e 1 t=1 0 Link costs reflect current traffic intensity D A B C 1 1+e e 0 e 1 1 0 0 t=0 0 0 D A B C 2+e 0 e 0 1+e 1 t=3 0 D A B C e 1 1

34. 34 r Routing has a large design space m who decides routing? source routing: end hosts make decision  network routing: networks make decision - (applications such as overlay and p2p are trying to bypass it) m how many paths from source s to destination d? multi-path routing  single path routing (with small amount of multipath) m does the route(s) provide QoS? QoS best effort m will routing adapt to network traffic demand? adaptive routing  static routing (mostly static; adjust in larger timescale) m … Routing Design Space: Internet

35. 35 QoS Support of Network Routing Arch Network Architecture Internet ATM Service Model best effort CBR VBR ABR UBR Bandwidth none constant rate guaranteed rate guaranteed minimum none Loss no yes no Order no yes Timing no yes no Congestion feedback no (inferred via loss/delay) no congestion no congestion yes no Guarantees ? r Internet model being extended: Intserv, Diffserv m multimedia networking ATM: Asynchronous Transfer Mode; CBR: Constant Bit Rate; V: Variable; A: available; U: User

36. 36 Outline  Recap  Network overview  Routing overview  Distance vector protocols

37. Distance Vector Routing r Positive costs assigned to network links r The path from a source to a destination chosen by the routing protocol is a shortest path among all possible paths r Basis of RIP, IGRP, EIGRP routing protocols r Conceptually, runs for each destination separately

38. Example: Cisco Proprietary Recommendation on Link Cost r Link metric: By default, k1=k3=1 and k2=k4=k5=0. The default composite metric for EIGRP, adjusted for scaling factors, is as follows: EIGRP : Enhanced Interior Gateway Routing Protocol

39. Example: EIGRP Link Cost r BW min is in kbps and the sum of delays are in 10s of microseconds. r Example r The bandwidth and delay for an Ethernet interface are 10 Mbps and 1ms, respectively. r The calculated EIGRP BW metric is as follows: m 256 × 10 7 /BW = 256 × 10 7 /10,000 m = 256 × 10000 m = 256000

40. Backup Slides 40

41. 41 Distance Vector Routing: Basic Idea r At node i, the basic update rule where - d i denotes the distance estimation from i to the destination, - N(i) is set of neighbors of node i, and - d ij is the distance of the direct link from i to j; assume positive i j destination

42. 42 Outline  Recap  Control plane: routing overview  Distance vector protocols  synchronous Bellman-Ford (SBF)

43. 43 Synchronous Bellman-Ford (SBF) r Nodes update in rounds: m there is a global clock; m at the beginning of each round, each node sends its estimate to all of its neighbors; m at the end of the round, updates its estimation A E D CB 7 8 10 2 1 2 A B E C D

44. 44 Outline  Network overview  Control plane: routing overview  Distance vector protocols  synchronous Bellman-Ford (SBF) - SBF/ 

45. 45 SBF/  r Initialization (time 0): A E D CB 7 8 1010 2 1 2

46. 46 Example A E D CB 7 8 1010 2 1 2 A B C E D d(0)     0 Consider D as destination; d(t) is a vector consisting of estimation at each node at round t d(1)   2 2 0 d(2) 12 3 2 2 0 d(3) 10 3 2 2 0 d(4) 10 3 2 2 0 Observation: d(0)  d(1)  d(2)  d(3)  d(4) =d*

47. 47 A Nice Property of SBF: Monotonicity r Consider two configurations d(t) and d’(t) r If d(t)  d’(t) m i.e., each node has a higher estimate in one scenario (d) than in another scenario (d’), r then d(t+1)  d’(t+1) m i.e., each node has a higher estimate in d than in d’ after one round of synchronous update.

48. 48 Correctness of SBF/   Claim: d i (h) is the length L i (h) of a shortest path from i to the destination using ≤ h hops m base case: h = 0 is trivially true  assume true for ≤ h, i.e., L i (h)= d i (h), L i (h-1)= d i (h-1), …

49. 49 Correctness of SBF/   consider ≤ h+1 hops: since d i (h) ≤ d i (h-1)

50. 50 Bellman Equation r We refer to these equations as Bellman equations (BE): where d D = 0. - SBF/  yields a shortest path solution to BE. - Does BE have a unique solution (i.e., the shortest path one)?

51. 51 Uniqueness of Solution to BE r Assume another solution d, we will show that d = d* case 1: we show d  d* Since d is a solution to BE, we can construct paths as follows: for each i, pick a j which satisfies the equation; since d* is shortest, d  d* 1 1 1 1 Dest. A E D CB 7 8 1010 2 1 2 2 2 3 10

52. 52 Uniqueness of Solution to BE Case 2: we show d ≤ d* assume we run SBF with two initial configurations: m one is d m another is SBF/  (d  ), -> monotonicity and convergence of SBF/  imply that d ≤ d*

53. 53 Example: Latency Optimal Routing r Assume each link has a latency function l e (x): latency of link e when x amount of traffic goes through e, m e.g., prop. + 1/(  –x), where prop. is link propagation delay and  the link capacity r Objective: min  e x e l e (x e ) st l(x) = 1 l(x) = x total demand = 1

54. 54 Network problem r As if the network maximizes a logarithmic utility function, but with constants (w f, f  F) chosen by the users

55. 55 Three Optimization Problems r SYSTEM(U f ): r USER f (U f ; p f ) r NETWORK(w f )

56. 56 Decomposition Theorem r There exist vectors p, w and x such that 1. w f = p f x f for f  F 2. w f solves USER f (U f ; p f ) 3. x solves NETWORK(w) r The vector x then also solves SYSTEM(U).

57. 57 Another Decomposition r SYSTEM(U): r USER f (U f ; p f ) r NETWORK(w f )

58. 58 Decomposition Theorem r There exist vectors p, w and x such that 1. w f = p f x f for f  F 2. w f solves USER f (U f ; p f ) 3. x solves NETWORK(w) r The vector x then also solves SYSTEM(U).