1. 5/21/20151 Mobile Ad hoc Networks COE 549 Capacity Regions Tarek Sheltami KFUPM CCSE COE http://faculty.kfupm.edu.sa/coe/tarek/coe549.htm

2. Outline 5/21/20152  Capacity of Wireless Ad Hoc Networks  Why do we want to know the capacity of the network?  Basic Rate Matrices  Definition of Convex Hull  Convex Combinations of Basic Rate Matrices  Successive Interference Cancellation  Comparison Between Different Schemes

3. 5/21/20153 Capacity of Wireless Ad Hoc Networks  Assume perfect coordination among nodes:  Perfect medium access control  Perfect routing  Perfect queuing  Every node knows exactly what to do  How much traffic can the network support?

4. 5/21/20154 Why do we want to know the capacity of the network? If we know the theoretical limits, we can compare them with the performance of protocols we design and know how much we could improve If we know how the theoretical limits are achieved, we can improve our protocols Unfortunately, we still do not have any good answers

5. 5/21/20155 A Special Case Assumptions: All nodes can transmit with rate W bps Any two nodes can communicate directly Any two transmissions will not collide What is the network capabilities?

6. Transmission Schemes and Rate Matrices

7. 5/21/20157 Rows represent original data source. Columns represent receiver or transmitter of information. Negative entries represent the rate of bits sent. Positive entries represent the rate of bits received. Transmission Schemes  Basic Rate Matrices

8. 5/21/20158 K 1 = 0.5 R 1 + 0.5 R 2 K 1 = 0.75 R 1 + 0.25 R 2 Transmission Schemes  Basic Rate Matrices..

9. 5/21/20159 Transmission Schemes..

10. 5/21/201510 How many transmission schemes are there? There are finitely many transmission schemes: Each node is either transmitting or staying quiet Each transmitter is transmitting data to one out of n nodes The precise number of transmission schemes depends on capabilities of nodes: Can nodes forward other nodes’ packets? Is spatial reuse allowed? Is power control allowed?

11. Properties of Rate Matrices 5/21/201511  Sum of elements along same row is 0.  We have n 2 − n = n(n − 1) degrees of freedom.  Each of the n(n − 1) non-diagonal element corresponds to one of the n(n − 1) source-destination pairs.  Rate matrices describe the information exchange in a convenient format.

12. 5/21/201512 Convex Combinations of Basic Rate Matrices (time division routing)

13. 5/21/201513 Definition of Convex Hull  With words: Convex hull of a collection of vectors (or matrices!) is all their convex combinations (weighted sums where all the coefficients are positive and sum up to 1).  With math:

14. 5/21/201514 Does it make sense?  A node must receive a packet before it transmits it (unless the node is also the source)  Therefore rate matrices with negative off-diagonal components must be excluded  All other rate matrices make sense

15. 5/21/201515 Capacity Region Definition  With words: All the convex combinations of the basic rate matrices {R i }, provided all off-diagonal components are non- negative.  With math: P n is the subset of all n × n matrices with all their off-diagonal components non-negative

16. 5/21/201516 Feasibility Problem Feasibility Problem: Given a set of end-to-end communication rates R ij, we want to know if these rates are supportable by the network.

17. 5/21/201517 Specifying the Basic Rate Matrices  So far: capacity region is convex hull of basic rate matrices.  Still need to specify them!  Each network has a repertoire of basic rate matrices depending on:  Multiple hops allowed?  Multiple transmissions allowed?  Power Control?  Successive Interference Cancellation?  More capabilities  more basic rate matrices  larger capacity region.

18. 5/21/201518 Five Sets of Rules

19. 5/21/201519 Successive Interference Cancellation (SIC)  Without SIC, each node treats signals intended for other users as noise.  With SIC, nodes may decode interference signals and subtract them out.  Advantage: Subtracting interference increases the data rate that the receiver can handle.  Disadvantage: SIC imposes a constraint on the rate of the interfering link  Disclaimer: The decoded interfering signal is not forwarded

20. 5/21/201520 Network Model  Single channel (no frequency division), half-duplex nodes.  All nodes transmit with maximum power P (no power control).  No multicasting/broadcasting (each created packet has a single destination).  Channel described by gain matrix G = [G ij ].  Receivers are hampered by thermal noise with power  Given the SINR, the received rate will have to be less than:

21. 5/21/201521 Network Model..

22. 5/21/201522 Example

23. 5/21/201523 Example..

24. 5/21/201524 Capacity Region

25. References 5/21/201525 [1] Stavros Toumpis and Andrea Goldsmith, “Capacity Regions for Wireless Ad Hoc Networks”