The Capacity of Wireless Networks

1. The Capacity of Wireless Networks Danss Course, Sunday, 23/11/03

2. Wireless ad hoc network • No wired backbone • No centralized control • Nodes may cooperate in routing each other’s data packets • At the Network Layer – problems are in routing, mobility of nodes and power constraints • At the MAC layer – problems with protocols such as TDMA, FDMA,CDMA • At the Physical layer – problems in power control

3. Lecture Minutes 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

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

5. 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

6. 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:

7. Transport Capacity of Arbitrary Networks • Network transport one bit-meter when one bit transported one meter toward its destination • Main result 1: Under the Protocol Model the transport capacity is • Main result 2: Under the Physical Model, is feasible While is not

8. 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, i.e. each node generate bits at rate l • The average distance between source and destination of a bit is L • Transmissions are slotted into synchronized slots of length t sec

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

10. Arbitrary Network – constructive lower bound • 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 • Main result 1: Under the Protocol Model the order of the throughput capacity • Main result 2: Under the Physical Model, is feasible While is not

16. Random Networks: A constructive lower bound on capacity We will show a scheme such that each source-destination pair can be guaranteed a channel of capacity With probability approaching 1 as Steps • Define the Voronoi tessellation • Bound the number of interfering neighbors of a Voronoi cell • Bound the length of an all-cell transmission schedule • Define the routes of a packet on the Voronoi tessellation • Prove that each cell contains at least one node • Calculate the expected routes that pass through a cell and infer the expected traffic of each node

17. 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 of the other aj’s i.e.: • Point ai is called the generator of the Voronoi cell V(ai)

18. A Voronoi tessellation of S2

19. 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)

20. 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)

21. A bound on the number of Interfering cells • Two cells are interfering neighbors if there is a point in one cell which is within a distance of (2+D)r(n) of some point in the other cell • Lemma – Every cell in Vn has no more than c1 interfering cells. c1 grows no faster than linearly in (1+D)2 Proof –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)

22. A bound on the length of an all-cell inclusive transmission schedule • Lemma - In the protocol model, there is a schedule for transmitting packets such that in every (1+c1) slots, each cell in Vn gets one slot in which to transmit Proof – 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 • There is a schedule also for the physical model….

23. The routes of packets • Source – destination pairs –let Yibe a randomly chosen location such that Xi and Yi are independent. The destination Xdest(i) is chosen as the node Xj which is closest to Yi • Corollary: The random sequence {Li} = {straight line connecting Xi and Yi } is i.i.d. • Routes of packets will be choose to approximate these straight line segments • Final destination will be one hop away from Yi , with high probability

24. Each cell contains at least one node Definition 1: Let Fbe a set of subset. A finite set of points A is said to be shattered by F if for every subset B of A there is a set F in Fsuch that Definition 2: The VC-dimension of F , denoted by VC- dim(F) , is defined as the supremum of the sizes of all finite sets that can be shattered by F

25. Vapnic-Chervonenkis Theorem If F is a set of finite VC dimension d and {Xi}is a sequence of i.i.d. random variables with common probability distribution P, then for every e,d > 0,

26. VCdim of the set of disks in R2 x2 x1 x3 x4

27. A cell contains at least one node Let F denote the class of disks of area 100logn/n. So VCdim(F) is 3. Let V be a cell contained in a disk D. Hence:

28. Mean number of routes served by each cell • First calculate the probability that a line Li or great circle intersect a cell V Lemma: for every line Li and cell V • So the expected number of lines Li that intersect a cell is bounded as: • The same as for great circles !

29. Actual traffic served by each cell • We bounded the mean number of routes passing through each cell. However, we need to bound the actual random number of routes served by each cell !! • Remember the sequence {Xi ,Yi} is i.i.d. • Therefore , we can appeal to uniform convergence • We will show that each great circle that intersect a disc D, can be mapped to a point on the band F(D) that is equidistant from the center of D • Then we can bound the VCdim of the band and so of the great circles

30. Transforming great circles intersecting disks into points lying in equatorial bands Z C F(D)

31. Lower bound on throughput capacity • Because of uniform convergence, we obtain:

32. Lower bound on throughput capacity • We have shown that there is a schedule for transmitting packets such that in every (1+c1) slots, each cell can transmit. • Thus the rate at which each cell transmit is W/(1+c1) bits/sec • On the other hand, the rate a cell needs to transmit is less than: • So with high probability, and because c1 is grow linearly with (1+D)2 we have:

33. Conclusions • Designers may want to consider designing networks with small number of nodes • Communication with nearby nodes at constant bit rates can be provided in a dense clusters of nodes, since the source – destination distance shrink as

34. Questions ?