1. Networking Technologies for Cloud Computing USTC-INY5316 Instructor: Chi Zhang Fall 2015 Welcome to

2. Today’s agenda Methods –25-words-or-less model –back-of-the-envelope calculation Scaling laws for wireless networks –P. Gupta & P. R. Kumar, “The Capacity of Wireless Networks,” IEEE Trans. on Information Theory, 2000. –M. Grossglauser & D. Tse, “Mobility Increases the Capacity of Ad-hoc Wireless Networks,” IEEE INFOCOM, 2001. –A. El Gamal, J. Mammen, B. Prabhakar, D. Shah, “Throughput-delay Trade-off in Wireless Networks,” IEEE INFOCOM, 2004. 2

3. 3 The Capacity of Wireless Networks P. Gupta & P. R. Kumar IEEE Trans. on Information Theory, 2000.

4. My plan ***Top down approach*** Background knowledge Basic idea –The story told in this paper –25-words-or-less model –back-of-the-envelope calculation Mathematical details Some discussion 4

5. My plan ***Top down approach*** Background knowledge –Critical transmission range Basic idea –The story told in this paper –25-words-or-less model –back-of-the-envelope calculation Mathematical details Some discussion 5

6. Questions first! Who read the following papers? –P. Gupta & P. R. Kumar, “Critical power for asymptotic connectivity in wireless networks,” in Stochastic Analysis, Control, Optimization and Applications, 1998. –S. Kulkarni & P. Viswanath, “A deterministic approach to throughput scaling in wireless networks,” IEEE Trans. Information Theory, 2004. –M. Franceschetti, O. Dousse, D. Tse, and P. Thiran, “Closing the gap in the capacity of wireless networks via percolation theory,” IEEE Trans. Information Theory, 2007. 6

7. Asymptotic notation 7

8. Critical transmission range (CTR) CTR Problem –n nodes randomly distributed in a bounded region A –transmission range is r –which is the minimum value r c of the transmitting range that ensures connectivity w.h.p. (with prob  1 as n  )? Related problems –Gilbert disc model (1961), magic # 6 for packet radio networks –Penrose (1996) on continuum percolation, random geometric graph 8

9. Two models Dense network model –the side l of the deployment region A is fixed (e.g. A is the unit square) –the value of the CTR is characterized as the node density d=n/l 2 grows to infinity Extended network model –the side l of the deployment region A increases as –the node density d=n/l 2 keeps as a constant c Scaling technique –from dense to extended model 9

10. Critical number of neighbors (CNN) Why considering critical number of neighbors? –For two models above, this number keeps the same –CNN = Graph evolution –Consider a process in which all the network nodes have initially transmission range r = 0, and then increase their transmission range simultaneously –As the ranges are increased, new edges are added to the communication graph 10

11. Three cases in graph evolution 11 Two critical values –critical value for percolation kp –critical value for connectivity kc

12. Two critical values Critical value for percolation kp –Rigorous bound: 4.508 < kp < 4.515 (Bollobas 2005) –Non-rigorous bound: 4.51218 < kp < 4.51228 –Magic # 6 Critical value for connectivity kc –kc =  (logn) (Penrose 1996) Critical phenomena – 物理学中的临界现象 12

13. Three cases in graph evolution Subcritical case –E[# of neighbors] < kp –The largest component is of size O(log n) Supercritical case –kc > E[# of neighbors ] > kp –The largest component is of size O(n). The cluster with size O(n) is unique, and is called the giant cluster. Connected case –E[# of neighbors] > kc –There is only one cluster 13

14. Three cases in graph evolution 14 对比规则分布时的情形，用随机涨落来理解结论 –critical value for percolation kp = 4.5 –critical value for connectivity kc = logn

15. Discrete and continuum percolation See examples Relationship 15

16. Poisson point process Problem of “n nodes randomly distributed…” –weak dependency Definition of PPP 16

17. De-Poissonization technique Let be independent and uniformly distributed random points on a bounded region A Uniform n-point process Let be a Poisson random variable with mean, independent of It can be shown that the point process is a PPP with mean Then as 17

18. Takeaways For random and dense network model To guarantee the connectivity, the minimum r c should be n can be understood as “the # of deployed nodes” or “the mean of the PPP” holds for dense or extended network model 18

19. My plan ***Top down approach*** Background knowledge Basic idea –The story told in this paper –25-words-or-less model –back-of-the-envelope calculation Mathematical details Some discussions 19

20. Every paper tells a story what is the “elevator pitch” of this paper’s story? 20

21. Multi-hop wireless networks Communication networks formed by nodes with radios –Spontaneously deployable anywhere –Automatically adaptive to number of nodes, traffic requirements, locations “Multi-hop transport” –Nodes relay packets until they reach their destinations 21

22. Questions asked in this paper How much traffic can wireless networks carry? –Or what is the capacity of wireless networks? And how should information be transferred in wireless networks? 22

23. Wireless capacity: research history Shannon capacity: between 2 nodes 3 nodes? n nodes? Network information theory? Information Theory 中的 统计物理 Newton 力学：两体问题 三体问题？ n 体问题？ Statistical Physics 统计物理 23

24. Two fundamental properties of wireless medium It is subject to fading and attenuation –Signals get distorted –Time varying channel –Unreliable It is a shared medium –Users share the same spectrum –Users are located next to each other –Transmissions can interfere with each other –So users need to cooperate to use the medium 24

25. 主要矛盾 : Shared nature of wireless medium Packets can “collide” destructively –Destructive interference –Nothing can be decoded from two concurrent transmissions in same region Transmissions consume area 25

26. Spatial reuse of spectrum Spatial reuse of frequency in cellular systems 26

27. 25-words-or-less model (random) Node distribution –uniform n-point process – or PPP with mean n Interference model –physical model –or protocol model Traffic pattern –uniformly distributed  (n) unicast flows Static network Perfect scheduling & routing 27

28. Back-of-the-envelope calculation Capacity upper bound Connectivity constraint: S-D distance constraint: Interference constraint: 28

29. Back-of-the-envelope calculation Transmissions consume area –T time slots –Total # of transmissions possible –If is feasible –Total # of transmissions needed 29

30. Key parameter: r n To increase the throughput/capacity –Reduce the multi-hop burden  increase r n –Increase spatial concurrency and frequency reuse  decrease r n –two objectives are in conflict Consider both issues together, we need keep r n as small as possible –The loss from increasing r n is quadratic due to the area of the conflict involved However, there is a limit to how small one can make r n (i.e., connectivity constraint) 30

31. My plan ***Top down approach*** Background knowledge Basic idea –The story told in this paper –25-words-or-less model –back-of-the-envelope calculation Mathematical details Some discussions 31

32. Mathematical details Interference model –Physical model –Protocol model –Generalized physical model Capacity upper bound –Exclusion disk Constructive lower bound –Achievability result: to show capacity bound is tight –Techniques from distributed computing Handling edge effects 32

33. Interference model Physical model –specifies the condition of successful transmission in terms of SINR Protocol model –specifies the condition of successful transmission in terms of geometry Generalized physical model –data rate is generalized to be continuous in SINR, based on Shannon's capacity formula 33

34. Physical model 34

35. Protocol model 35 Protocol model 是 physical model 在一定条件下的近似

36. Generalized physical model Data rate is generalized to be continuous in SINR, based on Shannon’s capacity formula for the additive Gaussian noise channel. 36

37. Throughput capacity Fairness constraint –Average capacity or sum capacity are not good metrics A per-flow throughput is said to be feasible/achievable if every node can send at least at a rate of packets/time slot to its chosen destination. We denote by the maximum feasible throughput as the throughput capacity for the network. 37

38. Interference constraint system 38

39. Exclusion disk 39

40. Exclusion disk 40 exclusion disk around each receiver node Sum of areas of exclusion disks is upper- bounded by |A|

41. Exclusion disk 41 Condition that exclusion disks are disjoint is only the necessary condition for successful concurrent transmissions, not the sufficient condition

42. Achievability Give a scheme, to show is achievable. is a tight bound We use techniques from distributed computing –Simplest proof –Show connections between different disciplines –S. Kulkarni & P. Viswanath, “A deterministic approach to throughput scaling in wireless networks,” IEEE Trans. Information Theory, 2004. 42

43. Discretization 就像研究 continuum percolation 一样，把连续域 的问题转化为离散域的问题 再利用离散域中已知的结论（ distributed computing 中关于 2-D arrays 的结论） 43

44. Known results about 2-D arrays 44

45. Torus partition 45

46. Torus partition 46 Each cell contains O(logn) nodes w.h.p. No empty cell w.h.p. 123

47. L-shaped routing 47 Row routing Column routing

48. Scheduling Cell scheduling Packet scheduling –Each cell contains O(logn) nodes w.h.p. –No empty cell w.h.p. 48

49. Cell scheduling 49

50. Put all together 50

51. Handling edge effects To directly prove that edge/border effects can be safely ignored Sphere: 有限无边 Ring: Torus: 51

52. My plan ***Top down approach*** Background knowledge Basic idea –The story told in this paper –25-words-or-less model –back-of-the-envelope calculation Mathematical details Some discussions 52

53. Discussions Dense and extended network model –Using scaling technique Uniform n-point process and Poisson point process –Using de-Poissonization technique Arbitrary network model Improved capacity bound via percolation theory 53

54. Arbitrary network model Node location is arbitrary: –n nodes are arbitrarily located in a disk of area A in the plane. Traffic pattern is arbitrary: –each node has an arbitrarily chosen destination to which it wishes to send traffic at an arbitrary rate. Transmission power is arbitrary: –each node can choose an arbitrary range or power level for each transmission. 54

55. Transport Capacity of arbitrary networks 55

56. Capacity upper bound 56

57. Feasibility of Θ(  n) bit-meters/sec 57

58. Improved capacity bound Gupta & Kumar: with one transmission range Capacity bound is possible with two transmission ranges M. Franceschetti, O. Dousse, D. Tse, and P. Thiran, “Closing the gap in the capacity of wireless networks via percolation theory,” IEEE Trans. Information Theory, 2007. 58

59. How to do it? 59 Capacity upper bound Connectivity constraint: S-D distance constraint: Interference constraint:

60. Percolation theory Supercritical case –kc>E[# of neighbors]>kp –The largest component is of size O(n). The cluster with size O(n) is unique, and is called the giant cluster. Two phase routing with different transmission ranges –Draining phase with –Highway phase with 60

61. 61 Mobility Increases the Capacity of Ad-hoc Wireless Networks M. Grossglauser & D. Tse IEEE INFOCOM, 2001

62. Gupta & Kumar’s results 62 Gupta & Kumar’s Upper Bound on Throughput –Interference constraints –Transmission consumes area –Throughput upper bound –Delay lower bound Ref: [Gupta & Kumar 00]

63. Mobility increases capacity 63 Gupta & Kumar’s Upper Bound on Throughput –Interference constraints –Transmission consumes area –Throughput upper bound –Delay lower bound Grossglauser & Tse’s Scheme –2-hop relay –Throughput –Mobility increases throughput –Delay –Also increases delay! Ref: [Grossglauser & Tse 01] Throughput-Delay Tradeoff

64. Basic idea 64

65. Basic idea 65

66. Basic idea 66

67. Mobility model Grossglauser & Tse use fast mobility model

68. Transmission model and cell scheduling

69. Analysis 69

70. Analysis Delay under slow mobility 70

71. 71 Throughput-delay Trade-off in Wireless Networks A. El Gamal, J. Mammen, B. Prabhakar, D. Shah IEEE INFOCOM, 2004.

72. Throughput-delay tradeoffs Throughput-Delay Tradeoff Gupta & Kumar’s Upper Bound on Throughput –Interference constraints –Transmission consumes area –Throughput upper bound –Delay lower bound Grossglauser & Tse’s Scheme –2-hop relay –Throughput –Mobility increases throughput –Delay –Also increases delay!

73. Throughput-delay tradeoffs 73 For static random network Feasible area Optimal tradeoff curve

74. Throughput-delay tradeoffs 74 For mobile random network (slow mobility) （ Delay 有 logn 的差异，因为 mobility model 不同）

75. For low delay cases 75 For static random network For mobile random network For low delay applications, mobility is in fact a hindrance

76. Chasing the destination for low delay 76

77. For high delay cases As soon as mobility is used to boost the throughput beyond, the delay jumps up to Θ(n log log n). Even though the throughput increases to Θ(1), the delay only increases to Θ(n log n). If mobility is used to boost the throughput even slightly beyond that in static wireless networks then the delay shoots up to its highest value. There is almost no trade-off between throughput and delay for this range of high throughputs. 77

78. Chasing only when close enough 78 Chasing the destination

79. Optimality of the tradeoffs t(n): mean distance traveled by wireless transmissions d(n): mean distance traveled by relay nodes carrying Achievable throughput is bounded by 79

80. Controllable flooding and redundancy Wireless network resources are shared by all users/ devices and scarce –Redundancy should be controllable Two methods to control packet redundancy –Control number of hops each packet will take from source to destination –Control total number of copies (replicas) of each original packet in the network

81. Throughput-delay tradeoffs with redundancy Proof: [1] [Grossglauser & Tse 01] Proof: [2] & [3] [Neely & Modiano 05] a b

82. Throughput-delay tradeoffs with redundancy Proof: [1] [El Gamal 04] Proof: [2] & [3] [Our paper 09]

83. Takeaways 83/52

84. 84 Last words…