1. Capacity of Multi-Channel Wireless Networks: Impact of Number of Channels and Interfaces
Pradeep Kyasanur
Nitin H. Vaidya
Presented by Chen, Chun-cheng
University of Illinois at Urbana-Champaign

2. Multi-hop Wireless Networks
Wireless is popular in local area networks
Possible use in multi-hop networks as well
Mesh networks
Ad hoc networks

3. Need for more capacity
Per-node capacity decreases with increased node density
Effect of shared wireless medium
New applications require higher throughput
Streaming video, games
Use more spectrum

4. Multiple channels
Typically, available frequency spectrum is split into multiple channels
Large number of channels may be available
26 MHz
100 MHz
200 MHz
150 MHz
2.45 GHz
915 MHz
5.25 GHz
5.8 GHz
3 channels
8 channels
4 channels
250 MHz
500 MHz
1000 MHz
61.25 GHz
GHz
122.5 GHz

5. Multiple Interfaces Radio interfaces typically support one channel
We assume a half-duplex transceiver
Nodes may be equipped with multiple interfaces
Common case may be small number of interfaces
Number of interfaces per node expected to be smaller than number of channels

6. Example configuration
IEEE has multiple channels
12 in IEEE a
Devices can be equipped with multiple interfaces
Fewer interfaces than channels
Soekris net4521 device

7. Interface and Channel Model1 m
m interfaces per node
c identical channels are available
Total bandwidth W, per channel bandwidth W/c 1 c W

8. Related Work
[Gupta&Kumar] have studied the capacity of single channel networks
Applicable to multi-channel networks when c = m 1 OR m
c = m

9. How does the network capacity scale with large number of channels?
Capacity Problem
Denser network  Lower capacity 1 c m
How does the network capacity scale with large number of channels?

10. Contributions Established tight bounds
In arbitrary and random networks
Shown capacity depends on ratio of c to m
Derived insights from constructions
Capacity-optimal routing and scheduling strategies
Other work has developed practical protocols

11. Arbitrary network Nodes can be located anywhere on the torus
Traffic patterns can be arbitrarily chosen
Measure of capacity – bit-meters/sec
maximize total bit-meters/sec

12. Interference constraint
Interference among simultaneous transmissions
Must have minimum distance between transmitters
Limits number of simultaneous transmissions
[Gupta & Kumar]:
r(1+ ) r A B C D
 is a guard parameter

13. Interface constraint Throughput is limited by number of interfaces
Fewer interface & more channels  lower node throughput can be supported
Maximum node throughput = m * W/c 1 m
n nodes in the neighborhood
 total throughput <= n* m * W/c

14. Arbitrary network – Region 1
Capacity constrained by interference

15. Arbitrary network – Region 2
Capacity constrained by interfaces

16. Random network [Gupta&Kumar]
n nodes in the network
Nodes placed uniformly at random on the torus
Each node sets up a flow to a randomly chosen destination
Measure of capacity – minimum of flow throughputs

17. Random Network Capacity
Capacity limited by a set of constraints
The constraints result in different capacity region s

18. Connectivity constraint
Flows can be scheduled only when path between source-destination is connected
This establishes the minimum transmission range
[Gupta & Kumar]:
Not connected
Connected A D

19. Interference constraint
Interference among simultaneous transmissions
Must have minimum distance between transmitters
Limits number of simultaneous transmissions
[Gupta & Kumar]:
r(1+ ) r A B C D
 is a guard parameter

20. Destination bottleneck constraint
Destinations randomly chosen
 single node may be destination of multiple flows
Per-flow throughput limited by total number of incoming flows & node throughput
Lemma 3. The maximum number of fows for which a node in the network is a destination, D(n), is w.h.p.
If node throughput, T <= m*W/c
Per-flow throughput <= T/f
f incoming
flows D
We have n nodes  capacity in

21. Random network – Region 1
Capacity constrained by connectivity + interference
No dependence on m and c

22. Random network – Region 2
Capacity constrained by interfaces + interference

23. Random network – Region 3
Capacity constrained by destination bottleneck

24. Achieving capacity lower bound (LB)
Single construction applicable for all regions
Start with:
Node locations, source-destination pairs A B C D
Consider two source-destination pairs
AB, CD

25. LB Step 1: Partition into cells
Divide torus into square cells of area a(n)
Lemma 4. If a(n) > , then each cell has nodes per cell, with high probability. A B C D

26. LB Step 2: Routing
Route through cells on the straight line joining source and destination A B C D E F

27. LB Step 2: Routing Balance route assignment within each cell
Not required in single channel networks A B C D E F
Loses throughput
Optimal strategy

28. LB Step 3: Flow Scheduling
Goal: Ensure node is not scheduled to receive or send simultaneously
Half-duplex transceiver constraint
Build a routing graph D D F F E G B E G A A B
1) Vertices are nodes in the network
2) One edge for every hop C C

29. LB Step 3: Flow SchedulingA E G C D F B
Edge color graph
Divide every second into multiple hop slots
Number of slots = Number of edge colors of routing graph
Schedule each hop in some slot
One second interval
EG, FD
AE, GB, CF

30. LB Step 4: Node Scheduling
Goal: Ensure transmissions from a node do not interfere with other nodes
Build an interference graph D D F F E G E G A B A B C C
1) Vertices are nodes in the network
2) One edge between each pair of interfering nodes

31. LB Step 4: Node scheduling
Each hop slot uses a node schedule
Four node colors are needed D D
Vertex color graph F F E G E G A B A B C C
One second interval
EG, FD
AE, GB, CF G E
A, B, D
C, F
Channel 1
Channel 2

32. Derivation of lower bound
Lemma 5. The maximum number of source-destination lines that intersect any cell is
From lemma 4, each cell has nodes  each node has flows  can be colored with edge colors  divide 1sec into each of sec
From lemma 4, each cell has nodes  each node has interfering nodes can be colored with  divide each edge color slot into mini-slots each of sec
In each mini-slot bits can be transmitted  remove ceiling func., we have bits can be transmitted  

33. Derivation of lower bound
Recall
and capacity is
Use the above three condition to derive the lower bound to be the same as the upper bound.

34. Conclusion
Studied the capacity of multi-channel networks with varying number of interfaces
Single interface per node often suffices
Up to log(n) channels, 1 interface is sufficient

35. Future Work Impact of switching delay has to be better studied
Is switching required at all?
Capacity under other switching constraints – switch among only a subset of channels
Analyze capacity of deterministic networks
Other papers in this session look at this question

36. Arbitrary Network: Upper bound
Interference constraints [Gupta&Kumar]: Each pair of simultaneous receivers must have minimum separation
Separation depends on transmission radius
Bounds the number of simultaneous transmissions
Interface constraint: Only m interfaces available
n nodes, each node transmits at rate W/c bits/sec