Wireless Network Capacity. Nitin Vaidya

Transcription

1 Wireless Network Capacity Nitin Vaidya

2 Wireless Networks Why use multi-hop routes to delivery data? Is this optimal? What s the best performance achievable? Capacity analysis can help answer such questions 2

3 Capacity of Wireless Networks [Gupta-Kumar] Exact capacity is difficult to analyze Easier to analyze trends in capacity We will consider a few such results Two network models: Arbitrary network Random network 3

4 Network Model 4

5 Arbitrary Networks: Interference Constraint : Protocol Model d(i,j) = distance between nodes i & j Parameter Δ (constant) i j (node i is transmitting to node j) If node k transmits simultaneously (interference) then i j transmission is reliable iff 5

6 Protocol Model p i y x j k d(i,j) (1+Δ)d(i+j) 6

7 Arbitrary Networks: Interference Constraint Interference model is an approximation of reality Approximation adequate to derive results on trends in capacity as a function of number of nodes How does network capacity change when number of nodes is increased? How to define capacity? 7

8 Arbitrary Networks n = number of nodes in a unit area W = rate of successful transmission (interpreted as a measure of bandwidth ) Arbitrary Place n nodes wherever desired Choose any flows as desired (each flow has a source & a destination) Schedule transmissions as desired (under interference constraint) 8

9 Arbitrary Network Placing transmitter-receiver close to each other, n/2 nodes can transmit reliably simultaneously Aggregate throughput can increase linearly with n 9

10 Arbitrary Network When transmitter-receiver placed close, information doesn t travel very far Throughput alone is not interesting metric Throughput * Distance a more interesting metric Analogy: Work = Force * Distance 10

11 Arbitrary Networks Transport capacity = max Throughput * Distance Bit-meter/second Transmitter-receiver too close small distance reduced transport capacity Transmitter-receiver too far lower spatial reuse due to interference reduced transport capacity 11

12 Arbitrary Networks How does transport capacity scale with n? Results 12

13 Order Notation f(n) is O(g(n)) if there exists a constant d such that, for large enough n f(n) < d g(n) Examples: 5 log n is O(n) 2n is NOT O(log n) 13

14 Order Notation f(n) is Ω(g(n)) if g(n) is O(f(n)) Examples: n is Ω(5 log n ) log n is NOT Ω (n) 14

15 Order Notation f(n) is Θ(g(n)) if f(n) is O(g(n)) & f(n) is Ω(g(n)) Examples: log n is NOT Θ(n) (5000 n + 12) is Θ(n) 15

16 Arbitrary Networks How does transport capacity scale with n? Results Upper bound Lower bound construction Transport capacity is 16

17 Upper Bound: Arbitrary Networks Assume slots of duration One flow at each of the n nodes λ = Average rate (average over n flows) = average distance betwee transmitter-receiver of a bit (averaged over all the bits) = Transport capacity 17

18 Upper Bound: Arbitrary Networks Consider bit b, 1 b λn h(b) = number of hops on route used for bit b = distance between endpoints of h-hop of bit b (1) 18

19 Upper Bound: Arbitrary Networks H = total number of bits transmitted in 1 second by all n nodes combined Each transmission uses two hosts (2) 19

20 A B D C 20

21 Protocol Model For transmissions below to be both reliable: A B we need D C Each transmission consumes area of radius (Δ/2)distance around the receiver 21

22 Since any part of the unit area can be consumed by at most W bits in 1 second: (3) Rewriting this inequality: It can be shown that: (4) 22

23 Since any part of the unit area can be consumed by at most W bits in 1 second: (3) Rewriting this inequality: It can be shown that: (4) Using (1), (2), and rearranging terms: (5) From (1), (2), (5) 23

24 Transport Capacity of Arbitrary Networks: Upper Bound Although we assumed that each transmission uses all the bandwidth, this result (and a similar proof) applies even if the spectrum is divided into multiple channels, provided that each node can use all available channels simultaneously Transport capacity is 24

25 Transport Capacity of Arbitrary Networks: Constructive Lower Bound Divide the unit area into n/8 square cells Side of each cell of length Place 8 nodes in each cell (as shown on next slide) 25

26 26

27 >r(1+δ) 27

28 Transport Capacity of Arbitrary Networks: Constructive Lower Bound n/2 transmissions at rate W Distance between transmitter-receiver = r Throughput-distance = 28

29 Transport Capacity of Arbitrary Networks Throughput-distance = Transport capacity : lower bound Lower bound & upper bound 29

30 Random Network Nodes placed randomly over unit area: Expected number of nodes in area A = na Each node picks as destination the node closest to a randomly chosen location (uniform distribution) Transmission range = r i j reliable if interferers not too close to j i j k 30

31 Random Network: Throughput Capacity Suppose all flows achieve identical rate λ Throughput Capacity = Maximum n λ bits/second Distance not included in throughput capacity, since average distance pre-determined by random topology formation Throughput capacity of random networks 31

32 Multiple Channels 32

33 Multi-Channel Environments Available spectrum Spectrum divided into channels c 33

34 Multiple Channels 8 channels 4 channels 26 MHz 100 MHz 200 MHz 150 MHz 915 MHz 2.45 GHz 5.25 GHz 5.8 GHz IEEE in ISM Band 34

35 Shared Access : Time & Spectrum B D A C One Channel Two Channels Spectrum A B A C A C B C Time Time 35

36 Why Divide Spectrum into Channels? Manageability: Different networks on different channels avoids interference between networks Contention mitigation: Fewer nodes on a channel reduces channel contention 36

37 Why Divide Spectrum into Channels? Lower transmission rate per channel Slower hardware (simpler, cheaper) Reducing impact of bandwidth-independent overhead fixed time data size/rate 37

38 Simple Analysis Consider A transmitting Data to B, followed by Ack from B 38

39 c channels Efficiency = T/R 39

40 Data = 4000 bits, Ack = 100 bits, = 1 s 40

41 Capacity of Multi-Channel Networks Earlier results apply when each node can tune to all available channels simultaneously Now consider the case when this is not feasible 41

42 Interfaces & Channels An interface can only use one channel at a time Channel 1 W cw Channel c CAVEAT: In the course notes, the total bandwidth is specified as W, with per-channel bandwidth W/c. 42

43 Multiple Interfaces Decreasing hardware cost allows for multiple interfaces m interfaces per node 1 m 43

44 Practical Scenario m < c A host can only be on subset of channels 1 1 m m m+1 c c m unused channels at each node 44

45 Interface Constraint Throughput is limited by number of interfaces in a neighborhood M nodes in the neighborhood total throughput M * m * W Interfaces as a resource in addition to spectrum, time and space 45

46 Impact on Routing m = 1, c = 1 or 2 (compare scenario below) 46

47 Mutlti-Channel Throughput Capacity of Random Networks Capacity Channels (c/m) 47

48 CAVEAT: In the course notes, the total bandwidth is specified as W, with per-channel bandwidth W/c. Therefore, the results in the graph on previous slides are obtained by replacing W in the formula in the course notes by (Wc). 48

49 Multi-Channel Networks When number of channels not large, possible to achieve capacity improvement with more channels (c), even if hardware (m) is limited Similar result for arbitrary networks (see course notes) 49

50 Improving Capacity Wireless network capacity can be improved by adding infrastructure Base stations Wired network connecting the base stations (wired links do not compete with wireless) Mobility may also be exploited to improve capacity Reduce wireless channel usage, by exploiting infrastructure and mobility 50