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### Capacity of ad-hoc wireless Networks

Vicky Sharma

Introduction

- Ad hoc Networking has been an area of active research during the past decade.
- There has been a drastic increase in application scenarios for ad hoc networking (e.g. defense applications)
- A number of routing protocols have been proposed.
- Such protocols should have the following features:-

* Reliable data delivery

* Robust to dynamic channel conditions

* Allow for Scaling. Network services (e.g. throughput, delay) should not degrade rapidly as network grows.

Motivation

- For a routing protocol to scale, the protocol designer requires the following information:-

* An upper bound on the total network capacity

that can be achieved.

* An upper bound on the per – node throughput that is possible.

* How do these limits vary as the network grows.

- Hence, the question becomes :-

What is the maximal achievable throughput for ad-hoc wireless networks and how does it scale ?

Basic Definitions

- f(n) = O(g(n))
- f(n) · cg(n) 8 n > n0 > 0 and a constant c

Basic Definitions (ctd)

- f(n) = o(g(n))
- f(n) ¸ kg(n) 8 n > n0 > 0 and a constant k

Basic Definitions (ctd)

- f(n) = (g(n))
- f(n) = O(g(n)) and f(n) = o(g(n))
- kg(n) · f(n) · cg(n) 8 n > n0 > 0 and constants k,c

Gupta-Kumar Bound

- When n identically randomly located nodes, each capable of transmitting at W bps & using a fixed range form a wireless network and if the maximum throughput achievable at each node is denoted by (n). Then:-
- If nodes are optimally placed in a disk of unit area & traffic patterns and ranges optimally assigned, then we have:-

The total network bit-distance product under the optimal conditions is :-

Implications of Gupta-Kumar Bound

- Bad news for protocol designers.
- Network capacity does not scale as fast as network grows. Total capacity scales as (√n)
- Per-node throughput will approach zero as network grows. Throughput does not improve if channel is divided in m sub-channels
- One can keep throughput constant by transmitting over short distances (to the nearest neighbors)
- Clustering and division of labor may be profitable

A Few Definitions

- Feasible Throughput:

A throughput (n) is feasible for a network if 9 T < 1 s.t. every node can send (n)T bits in a time interval

[(i - 1)T,iT] 8 i 2 Z

- Bit-meter:

A network transports 1 bit-meter if 1 bit is moved to 1 meter towards its destination.

- Throughput capacity

The throughput capacity of a class of networks is of order (f(n)) bps if 9 c > 0, c’ < 1 s.t.

limn !1 P((n) = cf(n) is feasible) = 1

limn !1 P((n) = c’f(n) is feasible) < 1

A Few Definitions (ctd)

- Arbitrary Networks

A network where n nodes are arbitrarily placed. Each node has a destination that is chosen arbitrarily. The transmission range of each node can be different and is arbitrarily chosen.

- Random Networks

A network where n nodes are randomly located on a 2D surface (either surface of a sphere S2 or a planar disk R2). Each node has a randomly chosen destination where it sends data at (n) bps. The destinations are independently chosen. The transmission ranges for each node are the same, however.

Interference Models

- Depending on the perspective, 2 models are defined to describe successful reception:-
- Protocol Model

If a node i at position Xi transmits to node j at Xj at some time in a sub-channel m. If another node k at Xk is transmitting in the same sub-channel at the same time, then the condition for node j to receive from i is as follows:-

|Xk – Xj| ¸ (1 + ) |Xi – Xj|

where > 0 is the guard zone

We will denote nodes by their positions in the following slides.

Graphical representation of Protocol Model

- r = |xi – xj|
- x =
- No other node can transmit within a certain range of the sender’s range.

Interference Models (ctd)

- Physical Model

If transmission power of node xi is denoted by Pi and it decays by exponential factor , then a node xj recieves from xi if :-

Where = minimum SIR needed for reception

N = channel noise and > 2

= set of nodes transmitting at the same time in the same sub-channel

Upper Bound on Network Capacity of Arbitrary Networks

- Assumptions

* There are M sub-channels with a sub-channel m capable of Wm bps and m = 1,2 .. M Wm = W

* Network is Multi-hop. Bits may be stored at any relay node before being transmitted to the next hop.

* Transmissions synchronized with slots of length

* Network transports (n) nT bits over T seconds

Using the protocol model

- If a bit b travels from source to destination through h(b) hops where a hop length is rbh, then

Where Lav = average distance between source and destination. Also

Where Im(b,h) is the indicator function for transmission

of bit b on sub-channel m at hop h

Employing the protocol model

k

l

- If a node xr is receiving from xi and xl is receiving from xk in the same time slot and same sub-channel, then we have:-

|xi – xl| ¸ (1 + )|xk – xl| (1)

|xk – xr| ¸ (1 + )|xi – xr| (2)

Also

|xr – xl| ¸ |xr – xk| - |xl – xk| (3)

|xl – xr| ¸ |xl – xi| - |xr – xi| (4)

Hence, we have

|xl – xr| ¸ (/2)(|xk – xl| + |xi – xr|)

i

r

Hence, each successful reception requires no transmission/reception in a disk of radius (/2)range. Each reception uses some fraction of area.

- Due to edge effects, at least a quarter area of the disk is used by a transmission.

Hence, we get

- Summing over all slots and channels, we get
- Hence,

As a result

- And we get
- Hence, capacity limit in bit-meters/sec is

Upper limit on throughput using physical Model

- Using the physical model definition and previous notations we get:-
- we get

Summing over all slots, bits, sub-channels and hops we get

- Following the same approach as in earlier derivation, we get

If minimum transmission power (Pmin) and maximum power transmission (Pmax) are related as Pmax· Pmin, then the physical model reduces to the protocol model with = ( Pmin/Pmax)1/ - 1.

- Hence, the results of the protocol model hold for the physical model as well in such a case.

The topology shown above has a receiver-transmitter pair that are a distance r apart where r = 1/(1 + 2)1/(p(n/4) + p(2))

- There are n/2 possible simultaneous transmissions, each with a range r and throughput W.
- Hence, the network capacity becomes

Strategies to design a scalable Network

- Some assumptions of the multi-hop model used for derivation of the bound:-

* Average hops is of order O(pn)

* reception and transmission is omni-directional

* nodes are stationary

- Hence, packets should be routed over the closest distance possible (i.e. to the next nearest neighbor)
- A small network is desirable. Clustering could be used to get modest improvements (i.e. use of relay nodes)
- Directional reception and transmission may yield some improvement.
- Mobility may be employed to scale throughput

Use of mobility

- If number of hops is reduced to O(1) and the transmission takes place over a small range, then the throughput should not depend on n.
- Mobility of nodes can be used[2] to reduce the number of hops and transmission range
- Basic idea: The source can transmit the packet to the nearest neighbor (relay node). The relay node will store the packet until it is close enough to the destination
- However, delay will become large and would be dependent on the rate at which node change their positions.
- Not practical for delay-intolerant applications.

Use of directional Transmission/Reception[3]

- Number of simultaneous transmissions is restricted as a successful transmission requires that no other transmissions/receptions occur in a disk centered at the receiver.
- If directional reception is used, the “interference-area” can be reduced by (/2) where = reception width
- If directional transmission is used, number of interfering transmitters is reduced. Let = transmission width.
- The improvements obtained are p(2/) and p(2/) respectively
- However, we cannot improve beyond a certain limit. (An extremely narrow transmission ray won’t provide a significant improvement. The limit is O(W))

Use of bit-error rate

- Gupta-kumar bound assumes zero probability of error.
- We can instead allow a probability of error Pe = > 0.
- In such a case, the per-node throughput (n) for random networks can be expressed as[4] :-

where c is constant

Employing Relay nodes – Hybrid Networks

- A sparse base station network can be provided that is connected by a wired medium.
- The base station network only forwards data.
- Localizes the wireless traffic avoiding long hops.

Employing Relay nodes – Hybrid Networks (ctd)

- A significant improvement is achieved when number of base stations m grows faster than p(n)[5]
- A trade-off between pure ad-hoc networks and cellular structures.
- Cost of base station network is significant. Always need base station networks more than required
- Hybrid networks enable nodes to transmit over a short hop to the nearest base station.
- As a result, number of base station is significant.
- Number of hops that a packet can be carried over through the wireless medium can be bounded by L. This reduces the number of base stations employed with a small decrease in throughput.[6]

Conclusion

- Several information theoretic approaches conclude that the throughput decreases with network size and eventually approaches zero.
- Hybrid-Networks can improve the capacity but a significant cost is involved.
- The bottleneck is due to interference at the receiver.
- Small networks and short hops should be concentrated upon for better throughput.
- Improvement – cost tradeoff for Directional transmission/reception is yet to be studied and may be application dependent.

References

- [1]P. Gupta and P. R. Kumar. The capacity of wireless networks. IEEE Transactions on Information Theory,IT-46(2):388–404,March 2000.
- [2] M. Grossglauser and D. Tse. Mobility increases the capacity of ad hoc wireless networks. In IEEEINFOCOM’01,April 2001.
- [3] Su Yi, Yong Pei and Shivkumar Kalyanaraman. On the Capacity Improvement of Ad Hoc Wireless Networks Using Directional Antennas, MobiHoc’03, June 1–3, 2003,
- [4] Shuchin Aeron and Saligrama Venkatesh. Capacity Scaling in Wireless ad-hoc networks with Pe, ISIT 2004, Chicago, USA, June 27 – July 2, 2004
- [5] Benyuan Liu , Zhen Liu and Don Towsley. On the Capacity of Hybrid Wireless Networks, 2003 IEEE
- [6] Yong Pei & James W. Modestino and Xiaochun Wang. ON THE THROUGHPUT CAPACITY OF HYBRID WIRELESS NETWORKS USING AN L-MAXIMUM-HOP ROUTING STRATEGY, 2003 IEEE.

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