Title: ICOM 6505: Wireless Networks Wireless Ad Hoc Networks
1ICOM 6505 Wireless Networks- Wireless Ad Hoc
Networks -
- By Dr. Kejie Lu
- Department of Electronic and Computer Engineering
- Spring 2008
2Outline
3References
- 1 P. Gupta and P. R. Kumar, The capacity of
wireless networks, IEEE Transaction on
Information Theory, vol. 46, No. 2, pp. 388-404,
2000 - 2 M. Franceschetti, O. Dousse, D. Tse, and P.
Thiran, Closing the gap in the capacity of
wireless networks via percolation theory, IEEE
Transaction on Information Theory, Vol. 53, No.
3, pp. 1009-1018, 2007 - 3 J. Liu, D. Goeckel, and D. Towsley, Bounds
on the gain of network coding and broadcasting in
wireless networks, in Proc. IEEE Infocom, 2007 - 4 Kejie Lu, Shengli Fu, and Yi Qian, "Capacity
of Random Wireless Networks Impact of
Physical-layer Network Coding," to appear, IEEE
ICC 2008, Beijing, China, May 2008
4Motivation
- The performance of wireless ad hoc networks is
very difficult to analyze - Layers
- Physical
- Antenna
- Interference
- MAC
- Scheduling
- Network
- Topology control
- Routing
- Transport
- Others
- Traffic pattern
- Infrastructure
- Mobility
- Etc.
5Performance Metrics
- Throughput capacity
- The total data rate that can be supported
- Distance may also be considered
- Delay
6Notations
- Big O
- Used mainly in two areas
- In mathematics for asymptotic behavior
- We will use this in the capacity analysis
- In computer science for computational complexity
- Definition
- f(x) O(g(x)) when x tends to infinity if and
only if - There exist x0gt0 and Mgt0 such that
- f(x) lt Mg(x) for xgt x0
- Example
- 3x3-6x29x O(x3)
7Notations
- f(x) O(g(x))
- g(x) is the asymptotic upper bound of f(x)
- f(x) ?(g(x))
- g(x) is the asymptotic lower bound of f(x)
- f(x) ?(g(x))
- g(x) is the asymptotic tight bound of f(x)
8Network Model
- Scaling scenario
- Model 1 (Dense network) n nodes are located in a
unit area and n goes to infinity - Model 2 (Extended network) The density of nodes
is fixed but the area of network goes to infinity - Randomness scenario
- Arbitrary network the placement of nodes and the
traffic pattern can be determined - Random network nodes are randomly located and
the traffic pattern is random
9Network Model
- Dimension
- Two-dimension
- The most common one
- One-dimension
- In 3 and 4
- Three-dimension
- In 3
10Transmission Model
- Protocol model
- Suppose Xi is the location of node i
- Each node transmit with fixed rate W
- At a certain time epoch, node i is transmitting
to node j - For arbitrary network, the location of all
transmitting nodes shall be - For random network, the location of transmitting
nodes shall be
Impact of interference
Transmission range
11Transmission Model
- Physical model
- Suppose Xi is the location of node i, Pi be the
transmission power of node i - Let TXk be the set of nodes that are
transmitting at a certain time epoch - Node i is transmitting to node j
- The location of all transmitters shall be
12Main Results Arbitrary Network
- 2-D Arbitrary network
- In 1, the authors proved that the capacity of
arbitrary network scales with - In 2, the authors shown that the same bound is
feasible for the physical model, with certain
constraints
13Main Results Random Network
- 2-D random networks
- In 1, the authors proved that the per node
capacity for random wireless network is - However, the authors in 2 proved that the above
bound is not correct and they shown that the
bound is still
14Main Results Network Coding
- In 3, the authors considered the same scenarios
used in 1-2, but they consider network coding - They proved that network coding does not change
the scaling law - For 1-D network, they proved that the per node
throughput is
15Main Results Physical-Layer Network Coding
- In our recent study 4, we prove that
physical-layer network coding can improve the
capacity of wireless network, but cannot change
the scaling law - Specifically, we proved that, if ?lt2, then the
per node throughput can be - The interesting point is that is the impact of
interference is removed
16Network Coding
Node
A
R
B
Wireless links
(b) Transmission Schedule
(a) Network Scenario
17Proofs
- 2-D arbitrary networks
- 2-D random networks
- 1-D random networks
18Capacity of 2-D Arbitrary Networks
- Upper bound
- Protocol model
- Physical model
- Lower bound
19Upper Bound for the Protocol Model
- Suppose each node can send data to one
destination at rate ? - The transmission data rate is Wgt ?
- Then for a certain long duration T, we have
- Total number of bits delivered is ?nT
- Let b be a bit and its hop distance is h(b)
- Because at each time instance there are at most
n/2 transmissions, we have
20Upper Bound for the Protocol Model
- If L is the average distance between source and
destination, we also have - Here r is the transmission distance
21Upper Bound for the Protocol Model
- Now consider the transmission distance
- Below is a simple scenario, in which node i is
transmitting to node j, while node k is
transmitting to node j - According to the protocol model, we have
22Upper Bound for the Protocol Model
- Similarly, we have
- Combining the above two inequalities, we have
- But what does that mean?
23Upper Bound for the Protocol Model
- Answer for each transmission, there is a region,
in which there are no other receivers
kl
kl?/2
24Upper Bound for the Protocol Model
- Remember we are considering a unit area, which
means the summation of the area of each
transmission at the same time shall be less than
1 - Taking into account the boundary effect, we have
- Note that 1/W is the duration for transmitting
one bit - Let 1/W be the duration of a slot, then we have
WT slots
25Upper Bound for the Protocol Model
- The previous inequality can be rewritten as
- Since function f(x)x2 is convex, we have
- And then
For example, (a/2b/2)2 lt (a2b2)/2
26Upper Bound for the Protocol Model
27Upper Bound for the Physical Model
28Upper Bound for the Physical Model
- Summing all transmitter-receiver pairs, we have
- Summing for all bits, we have
29Upper Bound for the Physical Model
- If xgt0, x? is convex
- So we can use the same technique to prove that
30Upper Bound for the Physical Model
- A special case, when is bounded
- Consider again the following scenario
- We have
31Upper Bound for the Physical Model
- Then we have
- So the upper bound for the physical model is the
same as the upper bound for the protocol model
32Lower Bound for the Protocol Model
- Constructive lower bound
- Lattice
Transmitter
Receiver
r
2?r
33Capacity of 2-D Random Networks
- Communication range
- Connectivity
- Minimum cut
34Communication Range
- Protocol model
- Range must be at least in the order of
sqrt(logn/n) for the dense network - Physical model
- Range can be in the order of sqrt(1/n) for the
dense network
35Minimum Cut
- The upper bound of the cut capacity is
proportional to W/r
36Capacity of 1-D Networks
- Main results
- Traditional transmission scheme
- Network coding scheme
- Physical network coding scheme
37Schedule for Network Coding
?2-
38Schedule for Physical Layer Network Coding
?2-
39Discussion
- How can we improve the performance?
- Existing approaches
- Directional antenna
- Hybrid network
- Add base stations
- Mobility
- Not suitable for delay-required traffic