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Capacity of ad-hoc wireless Networks

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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 transmission/reception in a disk of radius (
- Summing over all slots and channels, we get
- Hence,

- As a result transmission/reception in a disk of radius (
- And we get
- Hence, capacity limit in bit-meters/sec is

Upper limit on throughput using physical Model transmission/reception in a disk of radius (

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

- Summing over all slots, bits, sub-channels and hops we get transmission/reception in a disk of radius (
- Following the same approach as in earlier derivation, we get

- If minimum transmission power (P transmission/reception in a disk of radius (min) 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.

A lower bound on capacity of arbitrary Networks transmission/reception in a disk of radius (

- 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 that are a distance r apart where r = 1/(1 + 2

- 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 that are a distance r apart where r = 1/(1 + 2

- 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] that are a distance r apart where r = 1/(1 + 2

- 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 that are a distance r apart where r = 1/(1 + 2

- 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 that are a distance r apart where r = 1/(1 + 2

- 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) that are a distance r apart where r = 1/(1 + 2

- 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 that are a distance r apart where r = 1/(1 + 2

- 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 that are a distance r apart where r = 1/(1 + 2

- [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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