Lecture 10: Wireless Network Capacity

Anish Arora

CIS788.11J

Introduction to Wireless Sensor Networks

2

Goals

• Transmission Rates

• Information Capacity versus Network Capacity

• Impact of Wireless Link Model: use short links• Impact of Traffic Pattern: use beamforming

use local traffics• Impact of Mobility: spread across nodes• Impact of Duty Cycling: spread across

time

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Slides use some material from

Rahul Mangaram

Nitin Vaidya

Roger Watenhofer

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Bit Error Rate

• BER = Errors / Total number of bits Error means reception of “1” when “0” transmitted, or vice versa

• Noise is the main factor of BER performance – signal path loss, circuit noise, …

• Packet Error/Reception Rate incorrectly received data packets / total # of received packets for packet of length n bits, this probability is

assuming bit errors are independent of each other

• For small bit error probabilities, approximately

Bit errors and SINR

Bit errors depend essentially on strength of received signal compared to the corruption sources Captured by signal to noise and interference ratio (SINR)

SINR allows to compute bit error rate (BER) for a given modulation Also depends on data rate (# bits/symbol) of modulation E.g., for simple DPSK, data rate corresponding to bandwidth:

For QPSK and AWGN noise,

where  Eb/N0 is energy per bit to noise power spectral density ratio, erfc(z)=

66

Thermal Noise

• Thermal Noise white noise since it contains the same level of power at all

frequencies kTB, where

k is the Boltzmann’s constant = 1.381e-21 W / K / Hz, T is the absolute temperature in Kelvin, and B is the bandwidth

• At room temperature, T = 290K, thermal noise power spectral density kT = 4.005e-21 W/Hz or

–174 dBm/Hz

77

Receiver Sensitivity

• The minimum input signal power needed at receiver input to provide adequate SNR at receiver output to do data demodulation

• SNR depends on Received signal power

Background thermal noise at antenna (Na)

Noise added by the receiver (Nr)

• Pmin = SNRmin ×(Na +Nr)

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Noise Figure

Noise Figure (F) quantifies the increase in noise caused by the noise source in the receiver relative to input noise

F = SNRinput/SNRoutput = (Na + Nr)/Na

Pmin = SNRmin×(Na + Nr) = SNRmin×F ×Na

Example: if SNRmin = 10 dB, F = 4 dB, BW = 1 MHz

Pmin= 10 + 4 -174 + 10×log(106) = -100 dBm

99

802.15.4 - Modulation Scheme

• 2.4 GHz PHY 250 kb/s (4 bits/symbol, 62.5 kBaud) Data modulation is 16-ary orthogonal O-QPSK 16 symbols are ~orthogonal set of 32-chip PN codes

• 868 MHz/915 MHz PHY Symbol rate

868 MHz band: 20 kbps (1bit/symbol, 20 Kbaud) 915 MHz band: 40 kbps (1bit/symbol, 40 Kbaud)

Spreading code is 15-chip Data modulation is BPSK

868 MHz: 300 Kchips/s 915 MHz: 600 Kchips/s

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802.15.4 - PHY Communication Parameters

• Transmit power Capable of at least 0.5 mW

• Transmit center frequency tolerance ±40 ppm

• Receiver sensitivity (packet error rate < 1%) −85 dBm @ 2.4 GHz band −92 dBm @ 868/915 MHz band

• Receiver Selectivity 2.4 GHz: 5 MHz channel spacing, 0 dB adjacent channel requirement

• Channel Selectivity and Blocking 915 MHz and 2.4 GHz band: 0 dB rejection of interference from

adjacent channel 30 dB rejection of interference from alternate channel

• Rx Signal Strength Indication Measurements Packet strength indication Clear channel assessment Dynamic channel selection

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802.15.4: Receiver Noise Figure Calculation

• Channel Noise bandwidth is 1.5 MHz• Transmit Power is 1mW or 0 dBm• Thermal noise floor is –174 dBm/Hz X 1.5 MHz = –112 dBm• Total SNR budget is 0 dBm –(–112 dBm) = 112 dBm • To cover ~100 ft. at 2.4 GHz results in a path loss of 40 dB

i.e. Receiver sensitivity is –85 dBm

• Required SNR for QPSK is 12.5 dB 802.15.4 packet length is 1Kb Worst packet loss < 1%, (1 –BER)1024= 1 –1%, BER = 10–5

• Receiver noise figure requirement

NF = Transmit Power – Path Loss – Required SNR – Noise floor = 0 + 112 –40 –12.5 = 59.5 dB

• The design spec is very relaxed• Low transmit power enables CMOS single chip solution at low cost

and power!

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Information (or Channel or Transmission) Capacity

Capacity maximizes time average bit rate, optimizing over all coding strategies

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Information Theoretic Concept of Capacity

• Results known for point-to-point links• Results known for small 1-hop systems (broadcast/MAC)

Capacity Region Λ = Set of all end-to-end rate vectors (or matrices) achievable over a network

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In terms of SNR

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Shannon-Hartley Theorem

• channel capacity , the tightest upper bound on information rate (excluding error correcting codes) of arbitrarily low bit error rate data that can be sent with a given average signal power S through an additive white Gaussian noise channel of power N, is:

• C is the channel capacity in bits per second• B is the bandwidth of the channel in hertz • S is the total received signal power over bandwidth, in watts• N is the total noise or interference power over bandwidth, in watts • S/N is the signal-to-noise ratio (SNR) expressed as a linear power ratio

(not as logarithmic decibels).

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Shannon’s Theorem: Example

• For SNR of 0, 10, 20, 30 dB, one can achieve C/B of 1, 3.46, 6.66, 9.97 bps/Hz, respectively

• Example: Consider the operation of a modem on an ordinary telephone

line. The SNR is usually about 1000. The bandwidth is 3.4 KHz. Therefore:

C = 3400 X log2(1 + 1000)

= (3400)(9.97)

≈34 kbps

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Protocol Model (k can send reliably when j sends if)

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Physical (SINR) Model

Minimum signal-to-interference

ratio

Power level of sender u Path-loss exponent

Noise

Distance betweentwo nodes

Received signal power from sender

Received signal power from all other nodes (=interference)

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Example: Protocol vs. Physical Model

1m

Assume a single frequency

Let =3, =3, and N=10nW

Transmission powers: PB= -15 dBm and PA= 1 dBm

SINR of A at D:

SINR of B at C:

4m 2m

A B C D

Is spatial reuse possible? NO Protocol Model

YES With power control

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Terminology

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From Roger Watenhofer

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Network Capacity Measures

Throughput capacity Number of packets successfully delivered per time Dependent on the traffic pattern E.g.: What is the maximum achievable rate, over all

protocols, for a random node distribution and a random destination for each source?

Transport capacity A network transports one bit-meter when one bit has

been transported a distance of one meter What is the maximum achievable rate, over all node

locations, and all traffic patterns, and all protocols?

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Why make the distinction?

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Transport Capacity

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Transport Capacity

• n nodes are arbitrarily located in a unit disk

• We adopt the protocol model with R=2, that is a transmission is successful if and only if the sender is at least a factor 2 closer than any interfering transmitter. In other words, each node transmits with the same power, and transmissions are in synchronized slots

• Quiz: What configuration and traffic pattern will yield the highest transport capacity?

• Idea: Distribute n/2 senders uniformly in the unit disk. Place the n/2 receivers just close enough to senders so as to satisfy the threshold

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sender

receiver

Transport Capacity: Example

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Transport Capacity: Understanding the example

• Sender-receiver distance is (1/√n).

Assuming channel bandwidth W [bits], transport capacity is (W√n) [bit-meter], or per node: (W/√n) [bit-meter]

• Can we do better by placing the source-destination pairs more carefully? No,having a sender-receiver pair at distance dinhibits another receiver within distance upto 2d from the sender. In other words, it killsan area of (d2)

• We want to maximize n transmissions with distances d1, d2, …, dn given

that the total area is less than a unit disk. This is maximized if all di =

(1/√n). So the example is asymptotically optimal

d

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More capacity results

The throughput capacity of an n node random network is

I.e., there exist constants c and c’ such that

Transport capacity: Per node transport capacity decreases with Maximized when nodes transmit to neighbors

Throughput capacity: For random networks, decreases with Near-optimal when nodes transmit to neighbors

Result improved by Franceschetti et al to : (W/√n)

0]log

'Pr[lim

1]log

Pr[lim

feasible is

feasible is

nnW

c

nnW

c

n

n

)log

(nn

W

n1

nn log1

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Convergecast Capacity

• Single sink/collector node (potential bottleneck)

• Information theoretic network transmission capacity (node capacity) scales not as Θ(1) but as Θ(log (n))

• Idea: Each node talks to closely located nodes, which is efficient given node

density Relay nodes cooperate to transmit the information to collector using a

beamformer, to get logarithmic increase in received power, and therefore, the capacity

• H. El Gamal, "On the Scaling Laws of Dense Wireless Sensor Networks: The Data Gathering Channel," IEEE Trans. Inform. Theory, vol. 51, no. 3, pp. 1229-1234, Mar. 2005

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Broadcast Capacity

• Network transport capacity scales not as Θ(1), but as Θ(log (n))

• Similar idea as convergecast: two phases

(i) source broadcasts the message;

(ii) close-by neighbors of source retransmit the message

with log (n) scaling factor

• A. Keshavarz-Haddad, V. Ribeiro, and R. Riedi, “Broadcast Capacity in Multihop Wireless Networks”, Proceedings of the 12th Annual International Conference on Mobile Computing and Networking (MobiCom '06). ACM, New York, NY, USA, 239-250, 2006

• B. Sirkeci-Mergen, Michael Gastpar, ``On the Broadcast Capacity of High Density Wireless networks'', 2007 Information Theory and Applications Workshop, San Diego, CA, January 2007

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Capacity in the presence of mobility

• Results are based on an idealized setup

• Assume a central scheduler At time t, scheduler chooses the senders and their power levels

• Goal: under random motion patterns Show that long term throughput remains constant as number

of users increases

Caveat: long term throughput averaged over node mobility time-scale

delays of same order can occur

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Mobile Nodes w/o Relaying

• Can mobile nodes achieve a throughput of O(1) per S-D pair by not relaying at all?

• Answer: number of simultaneous long range communications is limited by interference in physical model

Positions of nodes t,j at time t

)(

2/2|)()(|tSi

ji

LtXtX

S(t) – Set of source nodes scheduled for successful transmission

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Mobile nodes without relaying

• Without relaying the achievable throughput per S-D pair goes to 0 at least as fast as

2/1

1

an

Distance attenuation factor

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Mobile nodes with relaying

• What is the problem with direct transmission to S-D pairs? Transmissions are long range => interference limits the

number of concurrent transmissions

• How can we increase throughput? Constrain transmission to nearest neighbors

Use lower transmission power to avoid interference Cannot wait for nearest neighbor to come close by, time 1/n –

vanishes at time goes by

• Spread out packets along a large number of relay nodes Nodes temporarily buffer packets while they move Ensure that every node will have packets to send to its nearest

neighbor at any time Cannot do this with direct transmission alone

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Main idea

• spread traffic stream between s and d over large number of intermediate relay nodes (all others can be relays)

• each packet goes through a relay node that temporarily buffers the packet until final delivery to d is possible

• as node location processes are independent, stationary, and ergodic, it is sufficient to relay only once

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Scheduling Policy & Theorem

• Assume that time is divided into slots• Fix a sender density parameter

• Select the sender receiver pairs where interference is small enough to make transmission possible

• Theorem

The number of feasible sender-receiver pairs is O(n)

)1,0(

receivers and senders R SS nnnnn

Sn

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2-phase scheduling policy

Apply a 2-phase interleaved scheduling policy:1) Source sends to relay (odd slots)2) Relay sends to destination (even slots)

Direct transmission to destination is also allowed if destination is close enough

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1 hop vs. 2-hop routes

Theorem: Number of feasible sender receiver pairs is O(n)

Long-term throughput between any two nodes = probability that 2 nodes are a feasible node pair O(1/n) per theorem

Throughput over direct route O(1/n) Single hops routes alone O(1/n)

In 2-hop routes there are n-2 routes Total average throughput per S-D pair is O(1)

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Main result

Theorem: The two-phased algorithm achieves a throughput per S-D pair of O(1) i.e. there exists a constant c>0 such that

1feasible} is )(Pr{lim

cRnn

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Capacity and Delay Tradeoffs

• There is a minimum critical delay to achieve capacity results

• Capacity achieving strategy yields O(N) delay

• Redundant transmission protocol can achieve O( N )

delay at expense of reducing throughput to O(1 / N)

• M. J. Neely and E. Modiano, “Capacity and Delay Tradeoffs for Ad-Hoc Mobile Networks”, Proceedings of the First International Conference on Broadband Networks (BROADNETS), 2004

• X. Wang, L. Fu, X. Tian, Y. Bei, Q. Peng, X. Gan, H. Yu, J. Liu, "Converge-Cast: On the Capacity and Delay Tradeoffs," IEEE Transactions on Mobile Computing, 99(1), 2011

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Network Capacity in Directional Link Network

• Higher fidelity of the physical layer yields better by allowing antenna sharing for coherent relaying and interference subtraction or for MIMO beamforming

• With a sender gain of A and receiver gain of B, an AB gain is possible

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Network Capacity with MIMO Links

• Create nulls for up to N-2 other nodes to increase capacity

• R. Mudumbai, D.R. Brown, U. Madhow, and H.V. Poor, “Distributed Transmit Beamforming: Challenges and Recent Progress”, Communications Magazine, 47, 2, 102-110. February 2009

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Asymptotic Scalability for Local Traffics

• Per node capacity with power-law distributed traffic with exponent greater than 2 scales as O(1)

• “Scalability of Mobile Ad Hoc Networks: Theory vs. Practice”, by R. Ramanathan, R. Allan, P. Basu, J. Feinberg, G. Jakllari, V. Kawadia, S. Loos, J. Redi, C. Santivanez and J. Freebersyse, in The 2010 Military Communications Conference

if exponent is

•< 1, then GK result for uniform traffic

•= 1, then it is O(ln(n)/√n)

•< 2, then it is O( )

• = 2, then O(1/ln(n))

•> 2 then scales as O(1)

2

2

n

J. Li, C. Blake, D. S. J. De Couto, C. Hu, H. I. Lee, and R. Morris. Capacity of ad hoc wireless networks. In In ACM Mobicom, pages 61–69, 2001

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Duty Cycled Transport Capacity

• Ignore short links, assume all links are global i.e., network is 1-hop

• each node is up with duty cycle

• per node throughput capacity bps provided

• i.e., each node gets a private copy of the channel until the network capacity is reached

[Jing Li, Wenjie Zeng, A]

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Relationship between Capacity and Network Overhead

• “Hierarchical Cooperation achieves Optimal Capacity Scaling in Ad hoc Networks”, by A. Özgür, O. Lévêque, D. N. C. Tse, , IEEE Trans. Inf. Theory, 2007

Fixed Size

Capacity NLO NLO/Capacity

Traditional P2P O n

2O n

3 2O n

Long Link — Arbitrary Traffic 1O 2nO 2nO

Long Link — Broadcast O n 2nO O n

Virtual Hierarchy O n 3 2log( )O n n log( )O n n

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References

• P. Gupta and P. R. Kumar, “The Capacity of Wireless Networks,” IEEE Transactions on Information Theory, vol. 46, no. 2, pp. 388-404, Mar. 2000

• “Scaling Laws for Ad Hoc Wireless Networks: An Information Theoretic Approach” by F. Xue and P. R. Kumar, in Foundations and Trends in Networking, vol. 1, no. 2, 2006, pp. 145-270

• "Mobile Ad hoc Networking and the IETF — IETF 69", by I. D. Chakeres and J. P. Macker, in ACM SIGMOBILE Mobile Computing and Communications Review (MC2R), 2007