Presentation Transcript

1. Wireless Continuum Networks

   Stavros Toumpis Department of Informatics Athens University of Economics and Business SNOW 2012
2. Scope Nodes communicating exclusively over a shared wireless medium Nodes are so many as to form a continuum: There is a node at practically every location Distance between neighbors is much smaller than network dimensions Other terms: massive, massively dense, dense, etc.
3. Emergence of Two Spatial Scales Microscopic scale: Distances on the order of nearest neighbor separation Protocols of Interest: PHY, MAC, power control, next hop selection. Macroscopic scale: Distances on the order of typical network dimensions Protocols of interest: Routing, Load Balancing Constitutive Relations
4. Physics Example: Electromagnetism Microscopic scale: Distances on the order of atomic scale Singular charges interacting with each other according to material properties (conducting, semiconducting, insulating, etc.) Macroscopic scale: Distances on the order of 10-5 meters and larger Electric field E, Magnetic field intensity H, etc. Constitutive Relations (Ohm’s law, D=εE, etc.)
5. The Standard Roadmap Define a microscopic setting Define a macroscopic setting in terms of macroscopic quantities Find the constitutive relations between the macroscopic quantities describing the macroscopic setting using the microscopic setting Solve an optimization problem on the macroscopic setting Excellent modeling tradeoff between keeping complexity down and results useful/interesting!
6. Applications of the Roadmap ‘Packetostatics’: Node placement optimization and analogies with Electrostatics ‘Packetoptics’: Route optimization and analogies with Optics Minimax Traffic Load Balancing Cooperative Broadcast

7. 1. ‘Packetostatics’

   [KS04, TT05, TG05, TT06, SA10, SA10a]
8. Example Setting Wireless Sensor Network: Sense the data at the source Transport the data from the sources to the sinks Deliver the data to the sinks Problem: Minimize number of nodes needed What is the best placement for the wireless nodes? What is the traffic flow it induces?
9. Macroscopic Quantities Node Density Functiond(x,y), measured in nodes/m2 In area of size dA centered at (x,y) there are d(x,y)dA nodes Information Density Functionρ(x,y), measured in bps/m2 If ρ(x,y)>0 (<0), information is created (absorbed) with rate ρdA over an area of size dA, centered at (x,y) Traffic flow function T(x,y), measured in bps/m Traffic through incremental line segment is |T(x,y)|dl
10. Gauss’s Law The net amount of information leaving a surface A0 through its boundary B(A0), must be equal to the net amount of information created in that surface: Taking |A0|→0, we get the requirement:
11. The Constitutive Relation Nodes only need to transfer data from sources to sinks They do not need to sense them at the sources They do not need to deliver them to the sinks once their location is reached The traffic flow function and the node density function are related by:
12. Traffic Must Be Irrotational Remember that we must minimize the number of nodes If the constitutive relation is satisfied, then the traffic must be irrotational:
13. Motivation of Constitutive Relation n=ε2d(x,y) nodes are placed randomly in square of side ε Power decays according to power law Transmissions (with rate W), successful only if SINR exceeds threshold A ‘highway system’ on the order of Θ(n½)=Θ[ε(d(x,y))½] highways going from left to right can be created[GK01, FD04]
14. ‘Packetostatics’ The traffic flow T and information density ρmust satisfy: In free space, the electric field Eand the charge density ρ are uniquely determined by: Therefore, the optimal traffic distribution is the same with the electric field when we substitute the sources and sinks with positive and negative charges!
15. Analogy is Uncanny!
16. General Setting Let be the density of nodes (or, more generally, the cost) needed to support the sensing/transport/delivery Optimization Problem: Minimization over all possible traffic flows T(x,y) that satisfy the constraint Standard tool for such problems: Calculus of Variations
17. Result The traffic flow is given by: where the potential function φis given by the scalar non-linear partial differential equation: together with appropriate boundary conditions, and G’, H, properly defined functions
18. Example: Gupta/Kumar
19. Example: Super Gupta/Kumar
20. Example: Sub Gupta/Kumar
21. Example: Mixed case
22. Alternative Microscopic Layers UWB Physical Layer [NR04] When nodes are mobile, optimization must take place across space and time [SA10] When nodes use directional antennas, network is anisotropic, and things become complicated [SA10b] Analogies with macroscopic road traffic engineering [B52] And the spoilsports: The hierarchical cooperation scheme of [OL07] is incompatible to our formulation

23. 2. Packetoptics

    [JA04, CT07, CT09, ST12]
24. Motivation Problem: find route between (0,0) and (0,200) with minimum cost Nodes distributed according to spatial Poisson process Cost per hop increases quadratically with hop distance
25. Question: what happens in the limit?
26. Single Macroscopic Quantity Cost Function: Cost of route C that starts at A and ends at B: Macroscopic Problem: Find route from A to B that minimizes cost
27. Relation to Optics Fermat’s Principle: To travel from A to B, light will take the route that locally minimizes the integral: Therefore we have the following analogy: Index of refraction n(r) becomes the cost function c(r) Rays of light become minimum-cost routes
28. Microscopic Model [CT09] Node placement: spatial Poisson process with density λ(r) Cost per hop: Proportional to distance covered: CH(d)=d Conserving bandwidth: CH(d)=d2 Conserving energy: CH(d)=adb+f Routing rule: Greedy routing Forward packet to node for which cost to progress ratio is minimized
29. Computing the Cost Function Tools: spatial Poisson processes, law of large numbers, some approximations Results: (Minimizes distance covered) (Conserves bandwidth) (Conserves energy)
30. Choice of Cost Function Important! R1: c(r)=[λ(r)]½ [JA04] R2: c(r)=const R3: c(r)= 1/[λ(r)]½ R4: c(r)=f(λ(r))
31. Optics Analogy →The mathematics of Optimal Routing Unicast routing: Optimal routing paths r satisfy the equation Broadcast routing: Wave fronts of optimally broadcast packets satisfy the eikonal equation:
32. Broadcast Routing
33. Any Practical Gain by Knowing Limit? With finite but many nodes, the optimum route is hard to find So let us find the optimum route in the macroscopic limit, and use it to create a near optimum route
34. The Elephant in the Room We did not prove that optimal routes converge to the optical limit We assumed that they converge to a limit, and showed it is the optical one (400 nodes) (2000 nodes) (10000 nodes)
35. Related Work Using trajectories other than a straight line has already been proposed (TBF) [NN03]. Now we know optimum! Cost created by static external interference investigated in [BB06] The eikonal equation can be used to predict how a packet propagates throughout the network Of particular interest in Delay Tolerant Networks [JM10], [ST12]

36. 3. Minimax Macroscopic Traffic Load Balancing

    [PP03, PR07, HV07, DK08, HV09]
37. Setting Until now, we supposed only one type of traffic, or at most a few In general case, if there are n nodes, there will be n(n-1) distinct traffics (and that ignoring multicasting!) We assume that everyone is interested in sending traffic to everyone else, and we would like to minimize the maximum of the traffic flows experienced at all locations
38. Macroscopic Formulation (1/2) Location r1 creates traffic for location r2 with traffic generating rate λ(r1,r2), measured in bps/m4 Set of all paths is P Distance between r1andr2usingP is s(P, r1, r2) Total packet generation rate is Λ and mean packet length is l, where
39. Macroscopic Formulation (2/2) Traffic through location rwith direction θhas angular fluxΦ(P,r,θ), measured in bps/m/rad Total volume that passes through location r is given by scalar flux Φ(P,r): Problem: Find optimal distribution of paths, so that maximum traffic load is minimized:
40. Results (1/2) Lower bounds: where A1, A2 are subdivisions of A created by some curve of length L Simplified formulas for the scalar flux in special topologies and routing classes
41. Results (2/2) Structure of the optimal traffic flow Optimal routes are uniquely defined in a bottleneck area ‘Field-line’ routing suffices to achieve optimality Optimization is possible in terms of single scalar function Load Balancing in the Unit Disk:
42. Related Results One-turn Rectilinear Routing Optimization in [DK08] Multipath routing on a disk explored in [PP03] Optics analogy and ‘Curveball Routing’ shown in [PR07] The ‘Grand’ Open Problem: given an arbitrary 2D connected shape and a traffic generating rate λ(r1,r2), find Φopt andPopt

43. 4. Cooperative Broadcast in the continuum limit

    [SS05, SS06, SS07, KS09, SC10]
44. Toy Setting [SS05] Topology: source placed on left side of strip, destination placed on right side of strip, relays are placed in strip, Poisson distributed Reception model: nodes susceptible to thermal noise, power decays with distance as pr(d)=kd-2, reception successful if SINR>γ Protocol: We slot time. In first slot, source transmits. In i-th slot, everyone transmits if he received for first time in previous slot. Transmission powers add up at potential receivers
45. What the Simulations Say For sufficiently low threshold, a wave is formed that propagates along the strip After a while, wave achieves fixed width and goes on for ever For high threshold, wave eventually dies out, irrespective of how many nodes initially had the packet Position of initial relays critical
46. The Continuum Assumption Analysis very hard because of random placement of nodes Assumption: We have so many nodes, that there is a node practically everywhere Not interested in which node receives the packet in i-th slot Interested in which region of space receives the packet in i-th slot
47. Result: evolution of the strip widths can be predicted in straightforward manner Extensions: Various channel models (fading, etc.) Multiple sources of data traffic
48. Conclusions New framework for studying problems, based on macroscopic approach Many optimization problems with a pronounced spatial aspect can be handled Some detail is sacrificed, but solutions are insightful Math often borrowed from Physics An important open problem: we do not have convergence rates!
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