1. 10/24-10/27.2004MWCN 20041 Theoretical Capacity of Multi-hop Wireless Ad Hoc Networks Yue Fang A.Bruce McDonald R-WIN Lab ECE Department Northeastern University

2. 10/24-10/27.2004MWCN 20042 Outline Introduction Network Saturation Capacity Maximum Instantaneous Capacity Discussion

3. 10/24-10/27.2004MWCN 20043 Outline Introduction Network Saturation Capacity Maximum Instantaneous Capacity Discussion

4. 10/24-10/27.2004MWCN 20044 Wireless Ad Hoc Networks Easy to setup, no wiring required Provides support of mobile (and ubiquitious) computing Limited resources Lower capacity Dynamic characteristics

5. 10/24-10/27.2004MWCN 20045 Capacity Analysis of Wireless Ad Hoc Networks The capacity of wireless network (Gupta & Kumar) Theoretical maximum throughput of 802.11 Channel capacity of multi-hop wireless ad hoc network

6. 10/24-10/27.2004MWCN 20046 Network Capacity The ability of data exchange the whole network can bear at any time. No universal semantic is available. Two interpretations of network capacity Maximum instantaneous capacity (MIC) Ideal routing and scheduling Network saturation capacity (NSC) Uniformly distributed nodes and traffic independent of routing and scheduling.

7. 10/24-10/27.2004MWCN 20047 Topology Generation Network topology is generated by repeating specific patterns to avoid unnecessary randomness. n avg =3 n avg =6

8. 10/24-10/27.2004MWCN 20048 Outline Introduction Network Saturation Capacity Maximum Instantaneous Capacity Discussion

9. 10/24-10/27.2004MWCN 20049 Previous Work Novel concepts --- deferral set and equivalent competitor are proposed to facilitate multi-hop capacity analysis. Deferral set: the set of all nodes and links that will affect the ongoing communication Equivalent Competitor: The amount of competition faced by the ongoing communication in terms of single node.

10. 10/24-10/27.2004MWCN 200410 Previous Work (2) Node being two hop neighbor depends on whether it has a neighbor which is direct neighbor of ongoing communication. Only the communication from two hop neighbor to one hop neighbor will affect the ongoing communication. Channel capacity (S chan ) is derived based on node behavior model. F I G H A C E D B Only communication between C and F will affect the communication between A and B, thus the equivalent competitor is 1/3. X X

11. 10/24-10/27.2004MWCN 200411 Network Saturation Capacity (NSC) It is necessary to study the relation between the capacity and node location. n avg NNSC(Mb/s)n avg NNSC(Mb/s) N l x S chan Sim- ulation N l x S chan Sim- ulation 3491.751.7794641.561.636 6811.061.528810.9571.17 11750.4170.8612810.4050.82

12. 10/24-10/27.2004MWCN 200412 Boundary Condition Nodes that close to the boundary of the network have fewer neighbors. Hence less channel contention, consequently, greater available capacity. Boundary zone is defined as the doughnut shaped region occupied by all nodes that are more closer to the boundary (X i < 2r, where X i is the distance from node i to the network boundary).

13. 10/24-10/27.2004MWCN 200413 Phantom Node Nodes in the boundary zone (A, B) tend to have higher capacity than the nodes in the center of the network. Nodes in the shaded area are called phantom nodes In order to have an accurate estimation of network saturation capacity, the percentage of phantom node to the number of nodes in the network should below a threshold. A B 2r-d d α

14. 10/24-10/27.2004MWCN 200414 NSC: How big is big? Boundary condition effect can be regarded as negligible when then network radius is at least 10 times of transmission range. Additional parameters may affect network capacity: such as spatial and temporal variation of distribution of nodes, traffic, channel quality, mobility, etc. Number of phantom nodes vs. R/r Percentage of phantom nodes to Num Number of Phantom Nodes # of phantom nodes/ total nodes

15. 10/24-10/27.2004MWCN 200415 Outline Introduction Network Saturation Capacity Maximum Instantaneous Capacity Discussion

16. 10/24-10/27.2004MWCN 200416 Maximum Instantaneous Capacity (MIC) MIC reflects the bottleneck throughput between any set of sources and destinations. MIC can only be achieved under ideal scheduling - -- every link either transmitting/receiving, or in deferral state.

17. 10/24-10/27.2004MWCN 200417 Maximum Instantaneous Capacity (MIC) The objective is to find a sequence of simultaneously active links --- aggregate link set that cover the connected work. MIC is the bottleneck of the aggregate link set. MIC can be approximated in two steps: Find the maximum aggregate link set --- NP problem Find the bottleneck c1 simultaneous links c2 simultaneous links c3 simultaneous links c1>= c2>=c3 Bottleneck is the MIC: c3

18. 10/24-10/27.2004MWCN 200418 412 35 NP completeness By appropriate means, the problem of finding maximum aggregate link set in the network can be transformed to classic maximum independent set problem [6]. (1,2) (4.5) (3,4)(2,5) (2,3)

19. 10/24-10/27.2004MWCN 200419 MIC: The greedy Algorithm 1. List all the feasible deferral sets in ascending order by the number of links in each set. 2. Pick the first deferral set in the list, transmission along corresponding link can be granted. 3. If more than one deferral set have same size, the tie is broken by activating the link with minimum LOS (line-of- sight) distance. 4. Update the candidate deferral list. 5. Update the size of remaining feasible deferral sets. 6. Repeat 1-5 until the candidate deferral set list is empty.

20. 10/24-10/27.2004MWCN 200420 Random Link Selection Randomly select the link to be activated. Faster than greedy heuristic Results are of the same order. NConcurrent Links GreedyRandom 432311899 530411987 7376139108 9528163115 navg

21. 10/24-10/27.2004MWCN 200421 Bottleneck Aggregate Link Set Every link has to be activate at least once. The MIC is the bottleneck the maximum aggregate link set. The optimal solution is NP-complete. Greedy algorithm has polynomial bounded number of iterations. Random selection.

22. 10/24-10/27.2004MWCN 200422 Experimental Result NGreedy AlgorithmRandom Selection RoundBottleneckRoundBottleneck 43231161114594 53042411109582 7376591128178101 95281899152348107navg Number of requited iterations and corresponding lower bound

23. 10/24-10/27.2004MWCN 200423 Outline Introduction Network Saturation Capacity Maximum Instantaneous Capacity Discussion

24. 10/24-10/27.2004MWCN 200424 Discussion The semantic of network capacity itself is analyzed to provide a clearer understanding and basis for comparison. The analysis is central of a broad cross-layer framework Extensible in terms of access protocols, generalization and application to real control problems. The results, while sub-optimal and on worst-case analysis improve on the most often cited results from [2]

25. 10/24-10/27.2004MWCN 200425 Discussion (2) From [2], using protocol model, the capacity of a random network is Number of concurrent active link in a saturated network can be approximated by the number of non-overlapping level-1 interference sets. Which can be obtained by: Network capacity then is:

26. 10/24-10/27.2004MWCN 200426 Conclusion The extension of the channel capacity analysis [?] The semantic of network capacity is discussed with two interpretation --- NSC and MIC. Agreement of the results reported in [2] mutually validated the two models Provides insight regarding how to more effectively leverage available network capacity.

27. 10/24-10/27.2004MWCN 200427 Reference [1] Theoretical channel capacity in multi-hop ad hoc networks by Yue Fang and A. Bruce McDonald [2] The capacity of wireless networks by P. Gupta and P.R. Kumar [3] Finding a maximum independent set by R. E. Tarjan and A. E. Trojanowski. [4] Theoretical Maximum Throughput of IEEE 802.11 and its Applications by Jangeun Jun, Pushkin Peddabachagari and Miail Sichitiu