On the capacity of mobile ad hoc wireless networks

1. On the capacity of mobile ad hoc wireless networks Michele Garetto – Università di Torino, Italy Paolo Giaccone - Politecnico di Torino, Italy Emilio Leonardi – Politecnico di Torino, Italy

2. Outline • Introduction and motivation • Capacity properties of finite networks • Application to experimental traces • Asymptotic network capacity • Application of asymptotic analysis

3. Introduction • The sad Gupta-Kumar result: In static ad hoc wireless networks with n nodes, the per-node throughput behaves as P. Gupta, P.R. Kumar, The capacity of wireless networks, IEEE Trans. on Information Theory, March 2000   

4. Introduction • The happy Grossglauser-Tse result: In mobile ad hoc wireless networks with n nodes, the per-node throughput remains constant M. Grossglauser and D. Tse, Mobility Increases the Capacity of Ad Hoc Wireless Networks, IEEE/ACM Trans. on Networking, August 2002

5. Introduction • Node mobility can be exploited to carry data across the network • Store-carry-forward communication scheme S R D • Drawback: large delays(minutes/hours) • Delay-tolerant networking

6. Mobile Ad Hoc (Delay Tolerant) Networks • Have recently attracted a lot of attention • Examples • pocket switched networks (e.g., iMotes) • vehicular networks (e.g., buses, taxi) • sensor networks (e.g., disaster-relief networks, wildlife tracking) • Internet access to remote villages (e.g., IP over usb over motorbike) • Key issue: how does network capacity depend on the node mobility pattern ?

7. The impact of mobility • In their original paper, Grossglauser and Tse assume that the mobility pattern of each node produces a uniform distribution of node presence over the network area • results have later been extended to a restricted mobility model in which each node moves over a random great circle on the sphere Per-node throughput is still! S. Diggavi, M. Grossglauser, and D. Tse, Even One-Dimensional Mobility Increases Ad Hoc Wireless Capacity, IEEE Trans. on Information Theory, November 2005

8. The general (unanswered) problem • What properties in the mobility pattern of nodes allow to avoid the throughput decay of static networks ? • What are the sufficient conditions to obtain per-node throughput ? • Are there intermediate cases in between extremes of static nodes (Gupta-Kumar ’00) and fully mobile nodes (Grossglauser-Tse ’01) ? Under which conditions ?

9. Our work • We take one step forward in addressing the general problem • We obtain basic results for arbitrary (finite) networks • We provide a general framework to compute the asymptotic transport capacity of mobile networks with anisotropic mobility patterns • We apply the proposed framework to a class of mobile networks comprising heterogeneous nodes and spatial inhomogeneities

10. Outline • Introduction and motivation • Capacity properties of finite networks • Application to experimental traces • Asymptotic network capacity • Application of asymptotic analysis

11. Arbitrary (finite) networks • Main result: Under very general assumptions, the capacity (in networking sense) of a mobile wireless network depends on the mobility process only through the joint stationary distribution of nodes over the area The details on how nodes move (change of speed, direction, group movements) have no impact on network capacity

12. Another possible transmission configuration One possible transmission configuration Arbitrary (finite) networks • Assumptions: • n nodes moving according to a stationary and ergodic mobility process (possibly correlated among the nodes) • A source node s generates traffic for destination d according to a stationary and ergodic process with rate sd • Transmissions between pairs of nodes occur at fixed rate r • The set of transmissions that can be scheduled successfully at timetdepends only on the instantaneous node positions

13. Joint scheduling-routing formulation • Scheduling policyS: selects π(t) among all feasible transmission configurations at time t • According to S, between any pair of nodes ( i, j ) there exist a virtual link of capacity • Routing policyR: described by , denoting the average fraction of the traffic flow from node s to node d, which is routed through link ( i, j ) • Quantities satisfy the classic flow-conservation constraints:

14. Traffic sustainability • Z(t):network backlog, i.e., the amount of traffic (in bits) already generated by sources that has not yet been delivered to destinations at time t • traffic matrix is sustainable if there exists a scheduling policy S and a routing strategy R, such that: • Capacity region: the set of all sustainable traffic matrices

15. Main theorem • Maximum throughput is achieved by the “simple” class of scheduling policies based only on position information (stateless and memoryless policy) • No gain in terms of throughput can be achieved by complex scheduling policies considering: • Speed and direction of nodes • History of encounters • Queue lengths • Age of stored information • Proof’s idea: if a given traffic matrix can be sustained by some arbitrarily complex scheduling, there exist a simple scheduling policy that sustains it • Corollary: the capacity region of a mobile wireless network is convex

16. Outline • Introduction and motivation • Capacity properties of finite networks • Application to experimental traces • Asymptotic network capacity • Application of asymptotic analysis

17. DieselNet-Umass experiment • 30 buses running the campus transport service in Spring 2005 (60 days) • traces provide the amount of data transferred through TCP connections using WLAN 802.11 MAC access • mij measured in bytes

18. Infocom2005-iMotes experiment • iMotes carried by 41 volunteers attending the conference (5 days) • traces provide the radio contact duration between any two iMotes using Bluetooth MAC 802.15 access • mij measured in seconds

19. Experimental traces • Virtual capacities are directly measured for each pair of nodes (i,j) • contact graph: a capacitated graph in which • each vertex corresponds to a mobile node • edge (i,j) is weighted by • The maximum throughput is computed by solving a standard multi-commodity flow problem (for a given flow assignment)

20. “minDC” flow assignment “MaxDC” flow assignment Experimental contact graphs • significant asymmetries and inhomogeneous capacities • few edges contribute most of the capacity: class 3-4 contribute for 80% (Umass) and 60% (iMotes) of the overall transport capacity class 1 class 2 class 3 class 4 capacity x 10x 100x

21. Theoretical performance of experimental networks Probability Number of hops

22. Outline • Introduction and motivation • Capacity properties of finite networks • Application to experimental traces • Asymptotic network capacity • Application of asymptotic analysis

23. l l l l l l Assumptions • n nodes moving over unit-area closed connected region • independent, stationary and ergodic mobility processes • uniform permutation traffic matrix: each node is origin and destination of a single traffic flow with rate l (n) bits/sec • all transmissions employ the same nominal range or power • all transmissions occur at common rate r • single-radio, omni-directional antennas • interference described by protocol model (next slide)

24. Protocol Model • Let dijdenote the distance between node i and nodej, andRT the common transmission range • A transmission from i to j at rater is successful if: for every other node k simultaneously transmitting RT (1+Δ)RT j i k

25. Asymptotic analysis • We say that the per-node capacity isΘ( l(n) ) if there exist two constants c and c’such that • Equivalently, we say that the network capacity in this case is Θ( n l(n) )

26. Uniformly dense networks • We define the local asymptotic node density ρ(XO) at point XO as: Where is the disk centered in XO , of radius • A network is uniformly dense if:

27. Properties of uniformly dense networks • Theorem: the maximum network capacity is achieved by scheduling policies forcing the transmission range to be • Corollary: simple scheduling policies leading to link capacities are asymptotically optimal, i.e., allow to achieve the maximum network capacity (in order sense)

28. The scheduling policy S* • Transmission from i to j is enabled when • Theorem: S* is optimal, i.e., link capacities resulting from S* satisfy (1+Δ)RT (1+Δ)RT note: it complies with the protocol interference model i j

29. Outline • Introduction and motivation • Capacity properties of finite networks • Application to experimental traces • Asymptotic network capacity • Application of asymptotic analysis

30. A realistic mobility model for DTNs • Realistic mobility processes are characterized by : Restricted mobility of individual nodes: Non-uniform density due to concentration points From: J.H.Kang, W.Welbourne, B. Stewart, G.Borriello, Extracting Places from Traces of Locations, ACM Mobile Computing and Communications Review, July 2005. From: Sarafijanovic-Djukic, M. Piorkowski, and M. Grossglauser, Island Hopping: Efficient Mobility-Assisted Forwarding in Partitioned Networks,, IEEE SECON 2006

31. Heterogeneous nodes with restricted mobility “home-points” of the nodes

32. Restricted mobility • The shape of the spatial distribution of each node is according to a generic, decreasing function s(d) of the distance from the home-point s(d) d

33. Anisotropic node density (clustering) • Achieved through the distribution of home-points Clustered model: nodes randomly assigned to m = nν clusters uniformly placed over the area. Home-points within disk of radius r from the cluster middle point Uniform model: home-points randomly placed over the area according to uniform distribution n = 10000

34. We assume that: Scaling the network size 10 nodes……100 nodes…..1000 nodes ? Moreover: node mobility process does not depend on network size

35. Asymptotic capacity results logn [(n)] Uniform Model 0 per-node capacity -1/2 Independently of the shape of s(d) ! -1 0 1/2 Recall:

36. Asymptotic capacity results ? ? logn [(n)] Clustered Model 0 per-node capacity -1/2 Lower bound : in case s(d) has finite support “Super-critical regime”: mobility helps Lower bound : in case s(d) has finite support -1 “Sub-critical regime”: mobility does not help 0 1/2 Recall: #clusters

37. Super-critical regime • Let (m = n in the Uniform Model) • When we are in the super-critical regime • Theorem: in super-critical regime a random network realization is uniformely dense w.h.p. • Transmission range is optimal • Scheduling policy S* is optimal

38. Mapping over Generalized Random Geometric Graph (GRGG) • Link capacities can be evaluated in terms of contact probabilities: which depend only on the distance dijbetween the homepoints of i and j • Wecan construct a random geometric graph in which • Vertices stand for homepoints of the nodes • edges are weighted by • Network capacity is obtained by solving the maximum concurrent flow problem over the constructed graph

39. Upper bound : network cut 1 0 1/2 1

40. Average/random network flows • The “average” flow through the cut is easily computed as • fundamental question: Answer: YES ! 1 Proof’s idea: Consider regular tessellation where squarelets have area γ(n) Take upperandlowerbounds for number of homepoints falling in each squarelets, combined, respectively, with lower and upper bounds of distances between homepoints belonging to different squarelets 0 1/2 1

41. An optimal routing scheme • The above routing strategy sustains per-node traffic Routing strategy: Consider a regular tessellation where squarelets have area s Create a logical route along sequence of horizontal/vertical squarelets, choosing any node whose home-point lie inside traversed squarelet as relay d

42. Sub-critical regime • Restricted mobility does not allow nodes whose home-points belong to different clusters to communicate using • Nodes have to use • System behaves as network of m static node

43. Model variations • Mobility of nodes dependent on network size • Spatial distributions not symmetric around home-point. E.g.: mobility restricted on lines with random orientation • Different classes of nodes (e.g.: fully mobile nodes + nodes with restricted mobility) Can be studied using the same techniques (e.g. computation of network flow through physical cut)

44. Best solution: Worst solution: n1/2-β n-β/2 n-1/2 n-β/2 network capacity is Θ( n3/2-β) network capacity is Θ( n1-β/2) An interesting case • Nodes contrained to move uniformly over rectangles of area n-β(1/2 < β < 1), with minimum edge n-1/2and random orientation • In general, network capacity can depend on the geometry of the space visited by the nodes

45. Conclusions • One step forward in the capacity analysis of mobile wireless networks with general node mobility processes • Some results of general validity for finite and infinite number of nodes • Mapping over maximum concurrent flow problem over geometric random graphs • Application to a general class of mobile networks with heterogenous nodes and clustering behavior

46. ...the end... Questions ? Comments ?