Stavros Toumpis, University of Cyprus Inter-perf 2006, Pisa, Oct. 14 2006

1. Optimal Design and Operation of Massively Dense Wireless Networks(or How to Study 21st Century Problems with 19th Century Math) Stavros Toumpis, University of Cyprus Inter-perf 2006, Pisa, Oct. 14 2006

2. Scope • Massively Dense1 (i.e. very, very large) Wireless Networks • Any other setting that satisfies the following: • Involves an optimization • Has a strong spatial component • Permits continuity arguments • For example: • Transportation optimization • Large IC design 1 P. Jacquet, “Geometry of information propagation in massively dense ad hoc networks, in Proc. ACM MobiHOC, Roppongi Hills, Japan, 2004.

3. Appetizer • 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 length:

6. Question: what happens in the limit?

7. Limiting case predicted by Optics

8. Second Appetizer • Source on left sends packets to sink on right with help of wireless nodes. • Problem: find optimal placement of wireless nodes, so that minimum number is needed. • Solution: place nodes at intersections of lines.

9. Distribution of nodes as data volume increases • Data flow resembles an electrostatic field.

10. Central Idea of this Talk: Macroscopic View • Many problems in wireless networking (and elsewhere) are way too complicated to be solved without proper abstractions. • Standard approach is based on microscopic quantities: individual node placement, individual link properties, etc. • We can take a novel macroscopic approach, using macroscopic quantities: node density, data creation density, etc.

11. Central Idea (cont’) • Macroscopic quantities are connected with each other through ‘constitutive laws’. • Microscopic considerations enter only through these laws. • Approach opens gateway to new (or old, depending on how you look at it) Math: • Calculus of Variations, Partial Differential Equations, Optics, Electrostatics, etc. • Results are not as detailed as with standard approach, but detailed enough to remain useful.

12. Contents • Introduction • “Packetostatics” • “Packetoptics” • Cooperative Transmissions • Energy Efficient Routing • Load Balancing

13. 2. “Packetostatics” G. Gupta, IIT New Delhi L. Tassiulas, University of Thessaly S. Toumpis, University of Cyprus

14. Setting • Wireless Sensor Network: • Sense the data at the source. • Transport the data from the source locations to the sink locations. • 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?

15. 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.

16. What goes in, must come out • 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:

17. Special Case • Nodes only need to transfer data from the locations of the sources to the locations of the 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 the following constitutive law:

18. Motivation of (2): Back to microscopic level • 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 created1. 1 M. Franceschetti, O. Dousse, et. Al., ‘Closing the gap in the capacity of random wireless networks,’ in Proc. ISIT, Chicago, IL, June-July 2004

19. Traffic must be irrotational • We must minimize the number of nodes • If (2) is satisfied, then the traffic must be irrotational: • Easy proof by contradiction.

20. ‘Packetostatics’ • The traffic flow T and information density ρmust satisfy: • In uniform dielectrics (e.g., 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!

21. Example: A point source and a linear sink

22. The Potential Function • In Electrostatics, the potential function U(·) is defined as follows: • What is the interpretation of the potential function in our setting? Let a route from A to B be parallel to T: • Therefore, U(A)-U(B) is the number of hops needed to go from A to B.

23. Nonhomogeneous propagation environments • Motivation: different parts of the network may have different properties (available bandwidth, propagation characteristics, etc.) • Modeling: space is partitioned in ppropagation regionsPi , • Theorem: The traffic flow function satisfies the same equations with the electric displacement vector D, when the propagation regions are replaced with dielectrics with dielectric constant

24. Example: Two propagation regions

25. Thomson’s Theorem • Consider a number of perfect conductors Ci, each infused with some electric charge Qi. Conductors are placed within a region of space that may contain dielectrics. • Basic Question: How do the charges distribute themselves on the conductors? • Thomson’s Theorem: The charges are distributed so as to minimize the electrostatic energy of the setting:

26. Sources and sinks with freedom in placement • Motivation: In some cases, we have some freedom on the placement of the sources/sinks. • Modeling: Consider t traffic regions Ti, i=1,…,t each associated with an information rate Qi. • Question: What is the optimal deployment of Qi in Ti? • Answer: By Thomson’s theorem, the optimal deployment of Qi makes the traffic flow look like the electrostatic field when the Qi are charges and Ti are perfect conductors.

27. Example: Distributed sink over the infinite plane • We have a point source and a distributed sink that we can place in any way we like along the plane. • The best placement causes the sink to be distributed as charge would do in a plane conductor.

28. Limitations of the Special Case • So far, we have assumed that nodes only need to transport the data. • They also need to sense them at the source locations, and deliver them at the sinks once the transport is over. • We assumed that which only makes sense for a particular type of physical layer: bandwidth limited, and capacity achieving. • Equation makes no sense when, for example, we have infinite bandwidth

29. Generalized Problem • Let be the density of nodes needed to support the sensing/transport/delivery • Optimization Problem: • The minimization should be performed over all possible traffic flows T(x,y) that satisfy the constraint. • Standard tool for such problems: Calculus of Variations.

30. 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.

31. Example: Gupta/Kumar physical layer

32. Example: Super Gupta/Kumar

33. Example: Sub Gupta/Kumar

34. Example: Mixed case

35. A final look at the optimization problem • The integrant can have alternative interpretations: delay, energy, etc. • This is a problem in optimal transportation.

36. 3. “Packetoptics” P. Jacquet, INRIA R. Catanuto, G. Morabito, University of Catania S. Toumpis, University of Cyprus

37. Problem • Find optimal route in the limit of a very large number of nodes.

38. Macroscopic formulation • Cost Function: • Cost of route C that starts at A and ends at B: • Problem: Find route from A to B that minimizes cost.

39. 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.

40. The advantages of Optical Routing • We can use the rich body of math that already exists in Optics for our setting. • For example, we know that light satisfies the following equations: • We can use the intuition that already exists. • For example, we know that rays of light bend toward optically denser materials.

41. The cost function of Jacquet • is the spatial node density. • The cost function was used implicitly, with no proof. • This selection of cost function is justified when: • We want to minimize the number of hops. • Communication is only permitted between nearest neighbors.

42. Bandwidth limited cost function • This is justified when bandwidth is limited so nodes need to share. • Intuitively clear: better to go through thickly populated regions, to avoid transmitting over long distances, because such transmissions have large footprint.

43. Energy limited cost function • Let the energy dissipated per transmission be • Then the cost function is

44. Other cost functions • Minimize delay • Minimize congestion • Dynamic cost functions • Avoid high levels of interference (Baccelli, Bambos, Chan, Infocom 2006) • Very versatile!!!

45. Choice of cost function very important! R1: Jacquet, R2: Constant cost, R3: Energy limited, R4: Bandwidth limited

46. Trajectory Based Forwarding (TBF) • With Optics, we get a macroscopic route description • With TBF, we get the microscopic details that we miss • Many options possible

47. What the Optics-Networking Analogy does not tell us • How does the source know the initial angle with which the packet/ray should be launched? • In some nonhomogeneous environments, there are multiple rays connecting two points • All of them local minimums • One of them global minimum

48. Route Discovery • Basic idea: Nodes launch multiple rays • Intersection points notify pairs of node

49. 4. Cooperative Transmissions B. Sirkeci-Mergen, A. Scalione, Cornell University

50. Setting • 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.