Capacity of ad-hoc wireless Networks

1. Capacity of ad-hoc wireless Networks Vicky Sharma

2. 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.

3. 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 ?

4. Basic Definitions • f(n) = O(g(n)) • f(n) · cg(n) 8 n > n0 > 0 and a constant c

5. Basic Definitions (ctd) • f(n) = o(g(n)) • f(n) ¸ kg(n) 8 n > n0 > 0 and a constant k

6. 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

7. 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 :-

8. 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

9. 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

10. 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.

11. 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.

12. Graphical representation of Protocol Model • r = |xi – xj| • x =  • No other node can transmit within a certain range of the sender’s range.

13. 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

14. 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

15. 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

16. Summing over all m and time slots, we get

17. 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

18. 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.

19. Hence, we get • Summing over all slots and channels, we get • Hence,

20. As a result • And we get • Hence, capacity limit in bit-meters/sec is

21. Upper limit on throughput using physical Model • Using the physical model definition and previous notations we get:- • we get

22. Summing over all slots, bits, sub-channels and hops we get • Following the same approach as in earlier derivation, we get

23. If minimum transmission power (Pmin) 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.

24. A lower bound on capacity of arbitrary Networks

25. 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

26. Strategies to design a scalable Network • 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

27. Use of mobility • 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.

28. Use of directional Transmission/Reception[3] • 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))

29. Use of bit-error rate • 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

30. Employing Relay nodes – Hybrid Networks • 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.

31. Employing Relay nodes – Hybrid Networks (ctd) • 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]

32. Conclusion • 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.

33. References • [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.