Lecture 10 Wireless Network Capacity Anish Arora CIS

Lecture 10: Wireless Network Capacity Anish Arora CIS 788. 11 J Introduction to Wireless Sensor Networks

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: • Impact of Duty Cycling: spread across nodes spread across time 2

Slides use some material from Rahul Mangaram Nitin Vaidya Roger Watenhofer 3

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 4 4

Bit errors and SINR Bit errors depend essentially on strength of received signal compared to the corruption sources q 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 q E. g. , for simple DPSK, data rate corresponding to bandwidth: q q For QPSK and AWGN noise, where Eb/N 0 is energy per bit to noise power spectral density ratio, erfc(z)=

Thermal Noise • Thermal Noise § white noise since it contains the same level of power at all frequencies § k. TB, where - k is the Boltzmann’s constant = 1. 381 e-21 W / K / Hz, - T is the absolute temperature in Kelvin, and - B is the bandwidth • At room temperature, T = 290 K, thermal noise power spectral density § k. T = 4. 005 e-21 W/Hz or – 174 d. Bm/Hz 6 6

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) 7 7

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 d. B, F = 4 d. B, BW = 1 MHz Pmin= 10 + 4 -174 + 10×log(106) = -100 d. Bm 8 8

802. 15. 4 - Modulation Scheme • 2. 4 GHz PHY § 250 kb/s (4 bits/symbol, 62. 5 k. Baud) § 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 (1 bit/symbol, 20 Kbaud) - 915 MHz band: 40 kbps (1 bit/symbol, 40 Kbaud) § Spreading code is 15 -chip § Data modulation is BPSK - 868 MHz: 300 Kchips/s - 915 MHz: 600 Kchips/s 9 9

802. 15. 4 - PHY Communication Parameters • Transmit power § Capable of at least 0. 5 m. W • Transmit center frequency tolerance § ± 40 ppm • Receiver sensitivity (packet error rate < 1%) § − 85 d. Bm @ 2. 4 GHz band § − 92 d. Bm @ 868/915 MHz band • Receiver Selectivity § 2. 4 GHz: 5 MHz channel spacing, 0 d. B adjacent channel requirement • Channel Selectivity and Blocking § 915 MHz and 2. 4 GHz band: 0 d. B rejection of interference from adjacent channel § 30 d. B rejection of interference from alternate channel • Rx Signal Strength Indication Measurements § Packet strength indication § Clear channel assessment § Dynamic channel selection 10

802. 15. 4: Receiver Noise Figure Calculation • • • Channel Noise bandwidth is 1. 5 MHz Transmit Power is 1 m. W or 0 d. Bm Thermal noise floor is – 174 d. Bm/Hz X 1. 5 MHz = – 112 d. Bm Total SNR budget is 0 d. Bm –(– 112 d. Bm) = 112 d. Bm To cover ~100 ft. at 2. 4 GHz results in a path loss of 40 d. B § i. e. Receiver sensitivity is – 85 d. Bm • Required SNR for QPSK is 12. 5 d. B § 802. 15. 4 packet length is 1 Kb § 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 d. B • • The design spec is very relaxed Low transmit power enables CMOS single chip solution at low cost and power! 11 11

Information (or Channel or Transmission) Capacity maximizes time average bit rate, optimizing over all coding strategies 12

Information Theoretic Concept of Capacity L Capacity Region Λ = Set of all end-to-end rate vectors (or matrices) achievable over a network l • Results known for point-to-point links • Results known for small 1 -hop systems (broadcast/MAC) 13

In terms of SNR 14

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). 15

Shannon’s Theorem: Example • For SNR of 0, 10, 20, 30 d. B, 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 log 2(1 + 1000) = (3400)(9. 97) ≈34 kbps 16 16

Protocol Model (k can send reliably when j sends if) 17

Physical (SINR) Model Power level of sender u Noise Received signal power from all other nodes (=interference) Received signal power from sender Path-loss exponent Minimum signal-to -interference ratio Distance between two nodes 18

Example: Protocol vs. Physical Model C B A 4 m 1 m D 2 m Assume a single frequency Is spatial reuse possible? NO Protocol Model YES With power control Let =3, and N=10 n. W Transmission powers: PB= -15 d. Bm and PA= 1 d. Bm SINR of A at D: SINR of B at C: 19

Terminology 20

From Roger Watenhofer 21

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? 22

Why make the distinction? 23

Transport Capacity 24

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 25

Transport Capacity: Example sender receiver 26

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 sourcedestination pairs more carefully? No, having a sender-receiver pair at distance d inhibits another receiver within distance up to 2 d from the sender. In other words, it kills an area of (d 2) d We want to maximize n transmissions with distances d 1, d 2, …, 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 27

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) 28

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 29

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 12 th Annual International Conference on Mobile Computing and Networking (Mobi. Com '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 30

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 31

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 S(t) – Set of source nodes scheduled for successful transmission 32

Mobile nodes without relaying • Without relaying the achievable throughput per S-D pair goes to 0 at least as fast as Distance attenuation factor 33

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 34

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 35

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) 36

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 37

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) 38

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 39

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 40 Delay Tradeoffs, " IEEE Transactions on Mobile Computing, 99(1), 2011

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 41

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 42

Asymptotic Scalability for Local Traffics • Per node capacity with power-law distributed traffic with exponent greater than 2 scales as O(1) 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) 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 • “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 43

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] 44

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 45

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 (MC 2 R), 2007 46