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# The Capacity of Wireless Networks

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1. The Capacity of Wireless Networks Piyush Gupta and P. R. Kumar Presented by Zhoujia Mao

2. Outline Arbitrary networks • Two models: protocol and physical • An upper bound on transport capacity • Constructive lower bound on transport capacity Random networks • Two models: protocol and physical • Constructive lower bound on throughput capacity Conclusions

3. Arbitrary Networks • n nodes are arbitrary located in a unit area disc • Each node can transmit at W bits/sec over the channel • Destination is arbitrary • Rate is arbitrary • Transmission range is arbitrary • Omni directional antenna • When does a transmission received successfully ? Allowing for two possible models for successful reception over one hop: The protocol model and the Physical model

4. Protocol Model • Let Xidenote the location of a node • A transmission is successfully received by Xjif: • For every other node Xksimultaneously transmitting • D is the guarding zone specified by the protocol

5. Physical Model • Let Be a subset of nodes simultaneously transmitting • Let Pkbe the power level chosen at node Xk • Transmission from node Xiis successfully received at node Xj if:

6. Transport Capacity of Arbitrary Networks • Network transport one bit-meter when one bit transported one meter toward its destination • Main result : Under the Protocol Model the transport capacity is ( as n ) if nodes are optimally placed, the traffic pattern is optimally chosen and the range of each transmission is optimally chosen

7. Arbitrary Network – upper bound on transport capacity Assumptions: • There are n nodes arbitrarily located in a disk of unit area on the plane • The network transport lnT bits over T seconds • The average distance between source and destination of a bit is L

8. Theorem 2.1 • In the protocol model, the transport capacity lnL is bounded as follows: • In the physical model,

9. Remarks • The upper bound in Protocol Model only depends on dispersion in the neighborhood of the receiver • The upper bound in Physical Model improves when α is large, i.e., when the signal power decays more rapidly with distance • When the domain is of A squares meters rather than 1 m^2, then all the upper bounds above are scaled by

10. Arbitrary Network – constructive lower bound • Theorem 3.1: There is a placement of nodes and an assignment of traffic patterns such that the network can achieve under protocol model • Proof –define r := Place transmitters at locations: Place receivers at locations:

11. A constructive lower bound on capacity of arbitrary network r Dr >(1+D)r (( )) (( )) (( )) r 2Dr (( ))

12. Random Networks • n nodes are randomly located on S2 (the surface of a sphere of area 1sq m) or in a disk of area 1sq m in the plane • Each node has randomly chosen destination to send l(n) bits/sec • All transmissions employ the same nominal range or power • Two models: Protocol and Physical

13. Protocol Model • Let Xidenote the location of a node and r the common range • A transmission is successfully received by Xjif: For every other Xk simultaneously transmitting

14. Physical Model • Let Be a subset of nodes simultaneously transmitting • Let Pbe the common power level • Transmission from node Xiis successfully received at node Xj if:

15. Throughput Capacity of Random Networks • Feasible throughput: λ(n) bits per second is feasible if there is a spatial and temporal scheme for scheduling transmissions such that every node can send λ(n) bits per second on average to its chosen destination • Throughput capacity: throughput capacity of the class of random network is of order θ(f(n)) bits per second if there are constants c > 0, c’ < ∞ such that

16. Spatial tessellation • Let {a1,a2,….ap} be a set of p points on S2 • The Voronoi cell V(ai) is the set of all points which are closer to ai than any of the other aj’s i.e.: Point ai is called the generator of the Voronoi cell V(ai)

17. A Voronoi tessellation of S2

18. Tessellation properties • For each e>0, There is a Voronoi tessellation such that Each cell contains a disk of radius e and is contained in a disk of radius 2e We will use a Voronoi tessellation for which : • Every Voronoi cell contains a disk of area 100logn/n . Let r(n) be its radius • Every Voronoi cell is contained in a disk of radius 2r(n)

19. Adjacency and interference • Adjacent cells are two cells that share a common point. • We will choose the range of transmission r(n) so that: With this range, every node in a cell is within a distance r(n) from every node in its own cell or adjacent cell 8r(n) 2r(n)

20. Theorem 4.1 • For Random Networks on in the Protocol Model, there is a deterministic constant c > 0 such that bits per second is feasible whp • For Physical Model, there are c’, c” such that is feasible whp

21. Proof: • Lemma: in the Protocol Model there is a schedule for transmitting packets such that in every (1+ ) slots, each cell in the tessellation gets one slot in which to transmit, and such that all transmissions are successfully received within a distance r(n) from their transmitters • From the above lemma, the rate at which each cell gets to transmit is W/(1+ ) per second

22. Lemma: There is a δ’(n)→0 such that Prob ( (Traffic needing to be carried by cell V) ≤ c5λ(n) ) ≥ 1- δ’(n) • From the above lemma, the rate at which each cell needs to transmit is less than c5λ(n) whp. With high probability, this rate can be accommodated by all cells if it is less than the available rate, i.e., if

23. Within a cell, the traffic to be handled by the entire cell can be handled by any one node in the cell, since each node can transmit at rate W bits per second whenever necessary • Lemma: Every cell in has no more than interfering neighbors. depends only on ∆ and grows no faster than linearly in (1+∆)^2 • Thus, for Protocol Model, is feasible whp

24. Lemma: if ∆ is chosen to satisfy then the above result of Protocol Model also holds for Physical Model • Plug the expression of ∆ into , we get

25. Theorem 5.1 • For Random Networks on under the Protocol Model, there is a deterministic c’ < ∞ such that

26. Proof: • Lemma: the number of simultaneous transmission on any particular channel is no more than in the Protocol Model • Let L denote the mean length of the path of packets, then the mean number of hops taken by a packet is at least

27. Since each source generates λ(n) bits per second, there are n sources and each bit needs to be relayed on average by at least nodes, so the total number of bits per second needs to be at least . Also, each transmission over a single channel is of W bits per second, so from the above lemma the number of bits can not be more than bits per second, so

28. Lemma: the asymptotic probability that graph G(n, r(n)) has an isolated node and is disconnected is strictly positive if and . • By the definition of feasible throughput, the absence of isolated node is a necessary condition for feasibility of any throughput. Thus, is necessary to guarantee connectivity whp. We obtain the upper bound for Protocol Model

29. Conclusion • Implication for design • Number of nodes • Signal decay rate • … • Not considered • Delay • Mobility • …

30. Thanks ~ 

31. A graph of degree no more than c1 can have its vertices colored by using no more than (1+c1) colors • So color the graph such that no two interfering neighbors have the same color, so in each slot all the nodes with the same color transmit

32. If V’ is an interfering neighbor of V, then V’ and similarly every other interfering neighbor, must be contained within a common large disk D of radius 6r(n)+ (2+D)r(n)