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)

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  • 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.


  • 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
Basic Definitions

  • f(n) = O(g(n))

  • f(n) · cg(n) 8 n > n0 > 0 and a constant c

Basic definitions ctd
Basic Definitions (ctd)

  • f(n) = o(g(n))

  • f(n) ¸ kg(n) 8 n > n0 > 0 and a constant k

Basic definitions ctd1
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
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
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
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
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
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
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
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
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
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
Employing the protocol model



  • 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)


    |xr – xl| ¸ |xr – xk| - |xl – xk| (3)

    |xl – xr| ¸ |xl – xi| - |xr – xi| (4)

    Hence, we have

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



  • 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
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

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

Strategies to design a scalable network
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
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
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
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
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
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