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

2. Wireless ad hoc network No wired backbone No wired backbone No centralized control No centralized control Nodes may cooperate in routing each other s data packets 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 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 MAC layer – problems with protocols such as TDMA, FDMA,CDMA At the Physical layer – problems in power control At the Physical layer – problems in power control

3. Lecture Minutes Arbitrary networks 1. Two models: protocol and physical 2. An upper bound on transport capacity 3. Constructive lower bound on transport capacity Random networks 1. Two models: protocol and physical 2. Constructive lower bound on throughput capacity Conclusions

4. Arbitrary Networks n nodes are arbitrary located in a unit area disc n nodes are arbitrary located in a unit area disc Each node is can transmit at W bits/sec over the channel Each node is can transmit at W bits/sec over the channel Destination is arbitrary Destination is arbitrary Rate is arbitrary Rate is arbitrary Transmission range is arbitrary Transmission range is arbitrary Will later add some assumptions on the network Will later add some assumptions on the network When does a transmission received successfully ? 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 X i denote the location of a node Let X i denote the location of a node A transmission is successfully received by X j if: A transmission is successfully received by X j if: For every other node X k simultaneously transmitting is the guarding zone specified by the protocol is the guarding zone specified by the protocol

6. Physical Model Let Let Be a subset of nodes simultaneously transmitting Let P k be the power level chosen at node X k Let P k be the power level chosen at node X k Transmission from node X i is successfully received at node X j if: Transmission from node X i is successfully received at node X j if:

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

8. Arbitrary Network – upper bound on transport capacity Assumptions: There are n nodes arbitrarily located in a disk of unit area on the plane There are n nodes arbitrarily located in a disk of unit area on the plane The network transport nT bits over T seconds, i.e. each node generate bits at rate The network transport nT bits over T seconds, i.e. each node generate bits at rate The average distance between source and destination of a bit is L The average distance between source and destination of a bit is L Transmissions are slotted into synchronized slots of length sec Transmissions are slotted into synchronized slots of length sec

9. Theorem In the protocol model, the transport capacity nL is bounded as follows: In the protocol model, the transport capacity nL is bounded as follows: In the physical model, 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 There is a placement of nodes and an assignment of traffic patterns such that the network can achieve under protocol model Proof – define r := Proof – define r := Place transmitters at locations: Place receivers at locations:

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

12. Random Networks n nodes are randomly located on S 2 (the surface of a sphere of area 1sq m) or in a disk of area 1sq m in the plane n nodes are randomly located on S 2 (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 (n) bits/sec Each node has randomly chosen destination to send (n) bits/sec All transmissions employ the same nominal range or power All transmissions employ the same nominal range or power Two models: Protocol and Physical Two models: Protocol and Physical

13. Protocol Model Let X i denote the location of a node and r the common range Let X i denote the location of a node and r the common range A transmission is successfully received by X j if: A transmission is successfully received by X j if: For every other X k simultaneously transmitting

14. Physical Model Let Let Be a subset of nodes simultaneously transmitting Let P be the common power level Let P be the common power level Transmission from node X i is successfully received at node X j if: Transmission from node X i is successfully received at node X j if:

15. Throughput Capacity of Random Networks Main result 1: Main result 1: Under the Protocol Model the order of the throughput capacity Main result 2: Main result 2: Under the Physical Model, While is not is feasible

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 Define the Voronoi tessellation Bound the number of interfering neighbors of a Voronoi cell Bound the number of interfering neighbors of a Voronoi cell Bound the length of an all-cell transmission schedule Bound the length of an all-cell transmission schedule Define the routes of a packet on the Voronoi tessellation Define the routes of a packet on the Voronoi tessellation Prove that each cell contains at least one node 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 Calculate the expected routes that pass through a cell and infer the expected traffic of each node

17. Spatial tessellation Let {a 1,a 2, ….a p } be a set of p points on S 2 Let {a 1,a 2, ….a p } be a set of p points on S 2 The Voronoi cell V(a i ) is the set of all points which are closer to a i than of the other a j s i.e.: The Voronoi cell V(a i ) is the set of all points which are closer to a i than of the other a j s i.e.: Point a i is called the generator of the Voronoi cell V(a i ) Point a i is called the generator of the Voronoi cell V(a i )

18. A Voronoi tessellation of S 2

19. For each >0, There is a Voronoi tessellation such that Each cell contains a disk of radius and is contained in a disk of radius 2 For each >0, There is a Voronoi tessellation such that Each cell contains a disk of radius and is contained in a disk of radius 2 We will use a Voronoi tessellation for which : 1.Every Voronoi cell contains a disk of area 100logn/n. Let (n) be its radius 2.Every Voronoi cell is contained in a disk of radius 2 (n) Tessellation properties

20. Adjacency and interference Adjacent cells are two cells that share a common point. Adjacent cells are two cells that share a common point. We will choose the range of transmission r(n) so that: 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 8 (n) 2 (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+ )r(n) of some point in the other cell Two cells are interfering neighbors if there is a point in one cell which is within a distance of (2+ )r(n) of some point in the other cell Lemma – Every cell in V n has no more than c 1 interfering cells. c 1 grows no faster than linearly in (1+ ) 2 Lemma – Every cell in V n has no more than c 1 interfering cells. c 1 grows no faster than linearly in (1+ ) 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 6 (n)+ (2+ )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+c 1 ) slots, each cell in V n gets one slot in which to transmit Lemma - In the protocol model, there is a schedule for transmitting packets such that in every (1+c 1 ) slots, each cell in V n gets one slot in which to transmit Proof – A graph of degree no more than c 1 can have its vertices colored by using no more than (1+c 1 ) 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 …. There is a schedule also for the physical model ….

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

24. Each cell contains at least one node Definition 1: Let F be 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 F such 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 {X i }is a sequence of i.i.d. random variables with common probability distribution P, then for every > 0,

26. VCdim of the set of disks in R 2 x1x1 x2x2 x3x3 x4x4

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 L i or great circle intersect a cell V First calculate the probability that a line L i or great circle intersect a cell V Lemma: for every line L i and cell V So the expected number of lines L i that intersect a cell is bounded as: So the expected number of lines L i that intersect a cell is bounded as: The same as for great circles ! 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 !! 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 {X i,Y i } is i.i.d. Remember the sequence {X i,Y i } is i.i.d. Therefore, we can appeal to uniform convergence 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 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 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: 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+c 1 ) slots, each cell can transmit. We have shown that there is a schedule for transmitting packets such that in every (1+c 1 ) slots, each cell can transmit. Thus the rate at which each cell transmit is W/(1+c 1 ) bits/sec Thus the rate at which each cell transmit is W/(1+c 1 ) bits/sec On the other hand, the rate a cell needs to transmit is less than: On the other hand, the rate a cell needs to transmit is less than: So with high probability, and because c 1 is grow linearly with (1+ ) 2 we have: So with high probability, and because c 1 is grow linearly with (1+ ) 2 we have:

33. Conclusions Designers may want to consider designing networks with small number of nodes 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 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 ?