1. WIRELESS MESH NETWORKS Ian F. AKYILDIZ* and Xudong WANG** * Georgia Institute of Technology BWN (Broadband Wireless Networking) Lab & ** TeraNovi Technologies

2. 2 2 8. NETWORK CAPACITY

3. 3 Capacity of Multihop Wireless Networks A flow consumes bandwidth at each hop A flow consumes bandwidth at each hop Also, transmission at each hop interferes with the other hops of same flow. Also, transmission at each hop interferes with the other hops of same flow. Different flows also interfere with each other Different flows also interfere with each other Model assumptions: Model assumptions: –randomly placed n nodes, –transmit range sufficient to make network connected, –each node has a flow to a random destination. 3

4. 4 Capacity Enhancement Protocol enhancement would provide marginal improvements Protocol enhancement would provide marginal improvements (should not ignore them though) Capacity limitation – Fundamental: Capacity limitation – Fundamental: –Spatial interference and Spectrum availability Spatial interference: could be handled through effective use of space Spatial interference: could be handled through effective use of space –Directional antenna; MIMO; Transmission Power Control Spectrum availability: enhance channel utilizations Spectrum availability: enhance channel utilizations –Multiple channels; Multiple radios 4

5. 5 From the analytical results of asymptotic lower and upper bounds of network, it follows that the throughput capacity per node reduces significantly when the node density increases. From the analytical results of asymptotic lower and upper bounds of network, it follows that the throughput capacity per node reduces significantly when the node density increases. An important implication is derived as guidelines to improve the capacity of ad hoc networks: An important implication is derived as guidelines to improve the capacity of ad hoc networks: A node should only communicate with nearby nodes. A node should only communicate with nearby nodes. 5Guidelines Gupta P and Kumar PR, The capacity of wireless networks. IEEE Trans. Information Theory, 2000

6. 6 Capacity Increased by Mobility Network capacity of ad hoc networks can be increased by utilizing the node mobility. Network capacity of ad hoc networks can be increased by utilizing the node mobility. When a node needs to send packets to another node, it will not send until the destination node is close to the source node. When a node needs to send packets to another node, it will not send until the destination node is close to the source node. Thus, via the node mobility, a node only communicates with its nearby nodes. Thus, via the node mobility, a node only communicates with its nearby nodes. The scheme has a limitation: the delay may become large and the required buffer for a node may be infinite. The scheme has a limitation: the delay may become large and the required buffer for a node may be infinite. 6 Tse DNC and Grossglauser M, Mobility increases the capacity of ad hoc wireless networks. IEEE/ACM Trans. Networking, 2002.

7. 7 Hybrid Network Architecture Hybrid Network Architecture In the hybrid architecture, nodes only communicate with nearby nodes. In the hybrid architecture, nodes only communicate with nearby nodes. If they need to communicate with nodes with many hops away, base stations or access points are used to relay packets via wired networks. If they need to communicate with nodes with many hops away, base stations or access points are used to relay packets via wired networks. 7 Liu B, Liu Z and Towsley D, On the capacity of hybrid wireless networks. In Proc. IEEE INFOCOM 2003 Zhou P, Wang X and Rao R, Capacity of infrastructure wireless mesh networks. IEEE Transactions on Mobile Computing, 2008.

8. 8 Hybrid Network Architecture Hybrid Network Architecture The capacity of ad hoc networks is improved, however, wired connections between BSs do not exist in many ad hoc networks and neither in WMNs. The capacity of ad hoc networks is improved, however, wired connections between BSs do not exist in many ad hoc networks and neither in WMNs. Thus, capacity analysis is carried out for an infrastructure WMN in which the wireless links between mesh routers has limited bandwidth. Thus, capacity analysis is carried out for an infrastructure WMN in which the wireless links between mesh routers has limited bandwidth. 8

9. 9 Tradeoff between Capacity and Delay In a multi-hop wireless network, capacity is not the single concern of performance. In a multi-hop wireless network, capacity is not the single concern of performance. Delay is another performance metric. Delay is another performance metric. Networks under a certain scheduling scheme can provide high capacity, but the delay may also be significantly large. Networks under a certain scheduling scheme can provide high capacity, but the delay may also be significantly large. 9 Neely MJ, Modiano Wand Rohrs CE, Dynamic power allocation and routing for time varying wireless networks. In Proc. of IEEE INFOCOM 2005 Gamal AE, Mammen J, Prabhakar B and Shah D, Optimal throughput-delay scaling in wireless networks–part I: the fluid model. IEEE Transactions on Information Theory, 2006

10. 10 Open Challenges Most of the existing analytical approaches are based on asymptotic analysis. Most of the existing analytical approaches are based on asymptotic analysis. However, several research problems still open. However, several research problems still open. –To improve network capacity by considering new technologies such as network coding. –More performance metrics need to be taken into account for tradeoff with capacity, i.e., only delay-throughput tradeoff is not sufficient. 10

11. 11 Limitations of the Asymptotic Analysis The upper or lower capacity bounds do not reveal the exact capacity of a multi-hop wireless network with a given number of nodes. The upper or lower capacity bounds do not reveal the exact capacity of a multi-hop wireless network with a given number of nodes. The networking protocols have not been fully captured by the analysis. The networking protocols have not been fully captured by the analysis. –power control –routing protocols 11

12. 12 Analysis Taking Advantage of the Low Mobility Feature The analysis can be simplified by taking advantage of the low mobility feature of WMNs. The analysis can be simplified by taking advantage of the low mobility feature of WMNs. However, the assumptions used by the analysis are not necessarily valid for a WMN. However, the assumptions used by the analysis are not necessarily valid for a WMN. 12 Jun J and Sichitiu ML, The nominal capacity of wireless mesh networks. IEEE Wireless Communications, 2003.

13. 13 Analysis Taking Advantage of the Low Mobility Feature As a result, the applicability of the theoretical results on practical network architectures still remains unclear. As a result, the applicability of the theoretical results on practical network architectures still remains unclear. –For example, a close match between the theoretical results in (Gupta and Kumar 2000) and IEEE 802.11 based ad hoc networks is reported in (Li et al. 2001). –The study assumes that the traffic pattern in a large ad hoc network tends to be local and nodes usually communicate with nearby nodes. –This assumption is not always valid in a network unless it is intentionally designed so. 13

14. 14 Capacity Analysis Next we will focus on the asymptotic analysis of network capacity of multi-hop wireless networks. Next we will focus on the asymptotic analysis of network capacity of multi-hop wireless networks. Major results in this research area are presented based on concise explanations about how they are derived. Major results in this research area are presented based on concise explanations about how they are derived. 14

15. 15 Notations and Terms O(·): A Landau symbol. Given f = O(φ(n)), where φ(·) is a positive function, there exists a constant c such that |f| < cφ(n) for all values of n. In other words, the order of f is not higher than that of φ. 15

16. 16 Notations and Terms o(·): A Landau symbol. Given f = o(φ(n)), where φ(·) is a positive function, f/φ → 0 as n → ∞. In other words, the order of f is strictly lower than that of φ. 16

17. 17 Notations and Terms Ω Ω Ω (·): This is the inverse of Landau symbol O(·) and they have the following relationship: f(n) ∈ O(g(n)) ⇔ g(n) ∈ Ω (f(n)). ω(·): This is the inverse of Landau symbol o(·). f(n) ∈ o(g(n)) ⇔ g(n) ∈ ω(f(n)). ΘΘ Θ Θ (·): Given g(n) = Θ (f(n)), it is not much better than f(n) but also not much worse than f(n),i.e., Θ (f(n)) = O(f(n)) ∩ (f(n)). 17

18. 18 Feasible Throughput A throughput of λ(n) is feasible A throughput of λ(n) is feasible if there exists a spatial and temporal scheme for scheduling transmissions in a multi-hop wireless network, if there exists a spatial and temporal scheme for scheduling transmissions in a multi-hop wireless network, such that every node can send λ(n) bits per second on average to its destination node. such that every node can send λ(n) bits per second on average to its destination node. 18

19. 19 Throughput Capacity A wireless network has a throughput capacity of order of θ(f(n)) if there exist deterministic constants c > 0 and c′ 0 and c′ < ∞ such that:and 19

20. 20 Capacity of Ad Hoc Networks without Mobility The space of the network is scaled into a The space of the network is scaled into a region of area 1m 2, in which n nodes are region of area 1m 2, in which n nodes are located. located. In the network, each node can transmit W In the network, each node can transmit W bits/s over a common wireless channel. bits/s over a common wireless channel. 20

21. 21 Capacity of Ad Hoc Networks without Mobility Same results will apply to a network with Same results will apply to a network with subchannels of capacity of W 1,W 2,...,W M, subchannels of capacity of W 1,W 2,...,W M, as long as = W. as long as = W. Communications between source and destination are Communications between source and destination are carried out in a multi-hop fashion, and thus carried out in a multi-hop fashion, and thus packets may wait for transmission at some packets may wait for transmission at some intermediate nodes. intermediate nodes. 21

22. 22 Capacity of Ad Hoc Networks without Mobility Two typical setups of multihop ad hoc networks are defined: arbitrary networks and random networks Two typical setups of multihop ad hoc networks are defined: arbitrary networks and random networks In an arbitrary network, nodes are located arbitrarily in a disk of unit area in the plane. In an arbitrary network, nodes are located arbitrarily in a disk of unit area in the plane. 22

23. 23 Capacity of Ad Hoc Networks without Mobility In a random network, nodes are randomly located according to an independent and uniform distribution, either in a unit disk in the plane or on the surface S 2 of a three-dimensional sphere. In a random network, nodes are randomly located according to an independent and uniform distribution, either in a unit disk in the plane or on the surface S 2 of a three-dimensional sphere. The nodes in a random network is homogeneous, i.e., all transmissions use the same transmission range. The nodes in a random network is homogeneous, i.e., all transmissions use the same transmission range. 23

24. 24 Capacity of Arbitrary Networks The capacity of a wireless network is constrained by how a packet can be sent successfully in the network. The capacity of a wireless network is constrained by how a packet can be sent successfully in the network. There are many reasons for packet loss, e.g., erroneous packets due to interference among different transmissions from multiple nodes or dropped packets due to out of order transmission or timeout. There are many reasons for packet loss, e.g., erroneous packets due to interference among different transmissions from multiple nodes or dropped packets due to out of order transmission or timeout. 24

25. 25 Capacity of Arbitrary Networks In this analysis, only interference related errors can be considered. In this analysis, only interference related errors can be considered. Thus, a packet transmitted from one node to another can be assumed to be successfully received if and only if the received signal level is higher enough than the interference. Thus, a packet transmitted from one node to another can be assumed to be successfully received if and only if the received signal level is higher enough than the interference. 25

26. 26 Capacity of Arbitrary Networks Two models can be used to capture the condition under which a transmission can be successful: Two models can be used to capture the condition under which a transmission can be successful: –the protocol model and the physical model. In the protocol model, interference is avoided by allowing only one transmission in the same space region, and thus a transmission from a node with location X i can be received by a node with location X j if In the protocol model, interference is avoided by allowing only one transmission in the same space region, and thus a transmission from a node with location X i can be received by a node with location X j if |X k − X j | > (1 + ∆)|X i − X j | |X k − X j | > (1 + ∆)|X i − X j | 26

27. 27 Capacity of Arbitrary Networks For every other node X k with a simultaneous transmission in the same channel. For every other node X k with a simultaneous transmission in the same channel. ∆ is a guard zone to ensure no interference between neighboring nodes. ∆ is a guard zone to ensure no interference between neighboring nodes. In the physical model, no guard zone but signal to interference ratio (SIR) is ensured to prevent interference from neighboring nodes. In the physical model, no guard zone but signal to interference ratio (SIR) is ensured to prevent interference from neighboring nodes. 27

28. 28 Capacity of Arbitrary Networks Assume a set of nodes, represented by their locations {X k ; k ∈ T }, simultaneously transmit in the same channel using power level Pk for node X k. Assume a set of nodes, represented by their locations {X k ; k ∈ T }, simultaneously transmit in the same channel using power level Pk for node X k. In the physical model, a transmission from a node Xi to a receiving node X j can be received successfully if In the physical model, a transmission from a node Xi to a receiving node X j can be received successfully if 28 where β is the minimum SIR for a successful reception,αis a power decaying factor Thus, the protocol model contains a more simplified space-reuse model than the physical model.

29. 29 Upper Bound Assume there are n nodes in the network, the network transport bits over T seconds, and the average distance between a source and a destination is. Assume there are n nodes in the network, the network transport bits over T seconds, and the average distance between a source and a destination is. Thus, the transport capacity is bit-meters per second. Thus, the transport capacity is bit-meters per second. 29

30. 30 Upper Bound Considering a bit b, 1 ≤ b ≤ λnT, it traverses h(b) hops from its source and its destination, and the h th hop has a distance of. Considering a bit b, 1 ≤ b ≤ λnT, it traverses h(b) hops from its source and its destination, and the h th hop has a distance of. Thus, for each bit, the average distance from source to destination is Thus, for each bit, the average distance from source to destination is 30

31. 31 Upper Bound Since such a average distance is not less than the average distance Since such a average distance is not less than the average distance Assume all nodes in the network transmit according to TDMA in M subchannels. Assume all nodes in the network transmit according to TDMA in M subchannels. Each subchannel has a capacity of Wm bits per second and the overall capacity of all subchannels is W, i.e., Each subchannel has a capacity of Wm bits per second and the overall capacity of all subchannels is W, i.e., 31

32. 32 Upper Bound Assuming that the time slot is seconds, then the number of bits sent in one time slot in subchannel m is, since at most n/2 nodes can transmit, i.e., Assuming that the time slot is seconds, then the number of bits sent in one time slot in subchannel m is, since at most n/2 nodes can transmit, i.e., 32 where δ h (m, s) = 1 when the transmission in h th hop is using subchannel m and time slot s.

33. 33 Upper Bound Thus, summing over all subchannels and time slots, yields Suppose node X j is receiving from node X i at the same time that node X l is receiving from node X k. Suppose node X j is receiving from node X i at the same time that node X l is receiving from node X k. Considering a protocol model and triangle inequality Considering a protocol model and triangle inequality |X j − X l | ≥ (1 + ∆ ) |X i − X j | − |X l − X k |. |X j − X l | ≥ (1 + ∆ ) |X i − X j | − |X l − X k |. 33

34. 34 Upper Bound Similarly, |X l − X j | ≥ (1 + ∆) |X k − X l | − |X j − X i |. Adding the above two inequalities yields |X l − X j | ≥ 0.5 ∆ (|X k − X l | + |X i − X j |). Adding the above two inequalities yields |X l − X j | ≥ 0.5 ∆ (|X k − X l | + |X i − X j |). 34

35. 35 Upper Bound The above result illustrates that for a disk of radius 2 times the length of the hop is disjoint with another such a disk. The above result illustrates that for a disk of radius 2 times the length of the hop is disjoint with another such a disk. Due to the edge effect, such a disk can be on the periphery of the disk of unit area (called domain). Due to the edge effect, such a disk can be on the periphery of the disk of unit area (called domain). 35

36. 36 Upper Bound In addition, the transmission range larger than the diameter of the domain is unnecessary. In addition, the transmission range larger than the diameter of the domain is unnecessary. Thus, at least one quarter of such a disk overlaps with the domain. Thus, at least one quarter of such a disk overlaps with the domain. 36

37. 37 Upper Bound The total number of bits received by a receiver from a transmitter in time slot s and subchannel m is The total number of bits received by a receiver from a transmitter in time slot s and subchannel m is which is which is Considering all subchannels and time slots, we have Considering all subchannels and time slots, we have 37

38. 38 Upper Bound Define, we have Define, we have Thus, Thus, Then, we obtain Then, we obtain 38

39. 39 Capacity of Arbitrary Networks n Transport capacity bound (bit-meters/s): n For the physical model, similar results can be derived P. Gupta and P.R. Kumar, "The capacity of wireless networks", IEEE Trans. on Information Theory, March 2000. … From [1] n This gives the transport capacity bound:

40. 40 Capacity of Arbitrary Networks n From the physical model we have, Consider a special case: –P max /P min <β –Simultaneous Tx by the pairs (X i -> X j ) and (X i -> X j ) To get the physical model in the same format as the protocol model, substitute Δ as,

41. 41 A Constructive Lower Bound The lower bound on transport capacity for the protocol model can be derived as, where –Disk of unit area is centered at origin –Tx and Rx are placed at locations |j+k| = even and |j+k|=odd, respectively –No interference between two Tx-Rx pairs –Total number of Tx-Rx pairs are ≥ n/2 and each pair can send W bits/s

42. 42 A Constructive Lower Bound Similarly, for the physical model, the lower bound can be derived as,

43. 43 Summary of Throughput Capacity for Arbitrary Networks In an arbitrary network, transport capacity can be converted into throughput capacity if – Each node is treated equally – Each source could have a destination 1 m away Protocol Model Physical Model Lower Bound Θ(W√n) Upper Bound Θ(W√n) Θ(Wn (α-1)/ α) )

44. 44 Capacity of Random Networks n In the physical model Tx is successful if, For the protocol model, Tx of X i can be received by X j if, –|X i -> X j | ≤ r –|X k -> X j | > (1+Δ)r for every other node X k –Tx power P is same for all nodes

45. 45 A Constructive Lower Bound Voronoi Tessellation – Given a set of p points V p = a 1, a 2,..., a p in surface S 2, the Voronoi cell V(a i ) is the set of all points that are closer to ai than to any points in set Vp – V(a i ) := x ∈ S 2 : |x − a i | = min 1≤j≤p |x − a j |. All points in shaded cell closer to Voronoi point a 1 than any other voronoi point a1a1 voronoi point voronoi cell

46. 46 A Constructive Lower Bound P. Gupta and P.R. Kumar, "The capacity of wireless networks", IEEE Trans. on Information Theory, March 2000. For every Є>0, there is a disk of radius Є contained in the cell, which in turn is contained in a disk of radius 2Є –If Є = ρ(n) is the radius of disk of area 100logn/n then each cell is contained in a disk of radius 2ρn –If Tx range r= 8n, then nodes within a cell or adjacent cells are reachable –Two points in different cells separated by (2+Δ)r(n) are interfering neighbors –It can be proved that each cell has no more than c 1 interfering neighbors, where c 1 depends on Δ and grows linearly in (1+ Δ) 2

47. 47 Maximum Tx Rate in Each Cell Protocol model: –Each cell (out of the complete set v n ) in the voronoi tessellation can get a slot from (1+c 1 ) slots to transmit without interference Physical model: –Same schedule as the protocol model –SIR β must satisfy the following constraint (assuming Δ is large): Max. Rate for transmission in either case is W/(1+c 1 )

48. 48 Needed Tx Rate in Each Cell Depends on routes between source-destination pairs in the voronoi cells Based on uniform convergence in the weak law of large numbers, each cell contains at least one node with probability exceeding For cells intersecting a straight line, packets can be relayed from one cell to the other with the last hop being closest to the destination

49. 49 Needed Tx Rate (ζ) in Each Cell Mean number of routes served in each cell is c 5 (nlogn) 1/2 –c 5 is a constant There is a δ’(n)->0 such that, - Pr(sup(No of st. line segs. intersecting cell V)≤ δ’(n)), for each cell V in the set of voronoi cells v n.. Hence, - Pr(sup(No of st. line segs. intersecting cell V)≤ c 5 (nlogn) 1/2 )≥1- δ’(n)), for each cell V in the set of voronoi cells v n.. Hence, - Pr(sup(ζ)≤δ’(n)), - Pr(sup(ζ)≤ c 5 (nlogn) 1/2 )≥1- δ’(n)), for each cell V in the set of voronoi cells v n, where ζ = Traffic needing to be carried and λ(n) is the traffic rate in bits/s - i.e. ζ < c 5 λ(n) (nlogn)1/2

50. 50 Lower Bound on Throughput Capacity λ(n) From the previous steps, we know c 5 λ(n) (nlogn) 1/2 ≤ W/(1+c 1 ) –c 5 is a constant As c 1 grows no faster than (1+Δ) 2 we can find c>0 such that –c is independent of n, Δ, W –This is applicable for both protocol/physical models but must satisfy the SIR constraint for the latter

51. 51 Upper Bound on Throughput Capacity For random networks, there exists a deterministic constant c’<+infinity: –c’ is independent of n, Δ, W and N = total nodes in the network For the physical model, there exists a deterministic sequence Є(n)->0, not dependent on N, W, α, β such that,

52. 52 Summary of Throughput Capacity Capacity Bounds Arbitrary Networks Random Networks

53. 53 Capacity of Mobile Ad Hoc Networks M. Grossglauser and D. Tse, "Mobility Increases the Capacity of Adhoc Wireless Networks", IEEE/ACM Transactions on Networking, August, 2002 For a fixed wireless network, there is no effective solution to reduce the number of hops between source and destination For mobile wireless ad hoc network, there are 2 important questions: – Given the mobility model, what will be the min. number of hops that can achieve highest throughput? – How should the scheduling scheme be designed so that the required number of hops can be achieved?

54. 54 Models The network, traffic, and physical models used for the study of throughput improvement are described first Network consists of n nodes that are located in a disk of unit area – Location of i th node is X i (t) at time t, and {X i (t)} is assumed to be a stationary and ergodic random process with a uniform distribution – Different nodes move independently as I.I.D. processes Each source i always has data to send to its destination d(i)

55. 55 Models For the physical model, node j can successfully receive packets from node i if –where the channel gain from node i to node j is Υ ij – Transmit power at node i at time t is P i (t) –β is the SIR requirement – No is the background noise power – L is the processing gain of the system

56. 56 Number of Hops vs Throughput Improvement Case I: Mobile networks without relaying nodes –Single hop between source and destination –Source buffers packets till destination is within communication range –Large buffering time may cause a quick fall in throughput Derivation of network capacity for the ad hoc network: –Total number of nodes =n –Transmission rate of a node =R –For c larger than

57. 57 Number of Hops versus Throughput Improvement Hence, the throughput for each source-destination pair drops to zero as fast as Thus, single-hop Tx results in very low throughput and at least one relaying node should be there between source and destination We get

58. 58 Number of Hops versus Throughput Improvement Case II: Mobile networks with relaying nodes –Throughput in two-hop source destination network is evaluated –It is shown that the achievable throughput of the network is θ(n) –i.e. for each source-destination pair, the throughput is θ(1) Two phase scheduling scheme assumptions –Sender density θ Є (0,1) –Number of senders in each slot n s = nθ –Each node transmits to nearest neighbor with unit Tx power

59. 59 Number of Hops versus Throughput Improvement N t our of n s Tx-Rx pairs have successful transmission N t is a random process and it can be proved its expected value is θ(n), i.e., The probability that nodes i and j are scheduled to as a Tx- Rx pair is θ(1/n)

60. 60 Number of Hops versus Throughput Improvement :::: n-1 routes Relay Nodes Destination Node Direct Transmission Phase 1Phase 2

61. 61 Number of Hops versus Throughput Improvement Phase I: – n s nodes are selected as sources – For each source, if a receiver is the destination, the packets are directly sent from the source the destination with one-hop transmission – Otherwise, the receiver acts as a relaying node

62. 62 Number of Hops versus Throughput Improvement Phase II: – n R nodes are selected as senders (can be both relaying nodes or sources) – For each sender, only packets whose final destination is this receiver can be Tx – For other packets, another receiver should be considered

63. 63 Number of Hops versus Throughput Improvement Packet can have direct route or n-2 two hop routes via relaying nodes –Locations are I.I.D stationary and ergodic –This gives long-term throughput between any two nodes as θ(1/n), thus departure rate at the source is θ(1/n) –Hence, the direct route throughput is θ(1/n) –For the n-2 two hop routes, each server is a relaying node with rate θ(1), thus departure rate at each relay node is θ(1/n) –At destination, overall throughput of n-1 flows is θ(1) From the above, throughput per source-dest. pair is θ(1)

64. 64 Discussions Throughput of a mobile ad hoc network can be improved by – Reduction of the number of hops by a two-phase Tx scheme – Using multiple relay nodes for the same destination (multiuser diversity) Multiuser diversity improves throughput capacity but may increase delay –Other scheduling schemes may be devised

65. 65 Capacity of Ad Hoc Networks with Infrastructure Support Capacity of multi-hop networks can be improved by having relay nodes or clustering nodes into groups Limited infrastructure support may also increase capacity of ad hoc networks [1,2] with different bounds and scaling laws [ 2] A. Zemlianov, G. de Veciana, “Capacity of ad hoc wireless networks with infrastructure support, “ IEEE JSAC, March 2005 [ 1] U. C. Kozat, L. Tassiulas, “Capacity of ad hoc wireless networks with infrastructure support,” ACM Mobicom, 2003

66. 66 Regularly Placed Infrastructure Nodes and Randomly Located Ad Hoc Nodes n nodes are placed in random (independent and uniform) locations in a disk of unit area –The disk is divided into a hexagon tessellation –One infrastructure node is placed at the center of each hexagon cell –The infrastructure nodes are connected by wires and it has no bandwidth or power constraints Infrastructure nodes only relay traffic, and are not sources or destinations

67. 67 Regularly Placed Infrastructure Nodes and Randomly Located Ad Hoc Nodes Network Assumptions –The Tx rate of the ad hoc node is W bits/s and its intra-cell, uplink, and downlink Tx rates are W 1, W 2, and W 3, respectively where W=W 1 +W 2 +W 3 Routing Strategy –K-nearest cell routing: If source-destination pair are located within k-nearest cells, then only ad hoc networking is needed –Probabilistic routing: For each node, its Tx in infrastructure mode has probability p and (1-p) in the ad hoc mode

68. 68 Regularly Placed Infrastructure Nodes and Randomly Located Ad Hoc Nodes Results: Probabilistic Routing Per-node throughput capacity is: W2/W->0 implies that almost no communication with infrastructure nodes occurs, as in the case of pure ad hoc networks

69. 69 Regularly Placed Infrastructure Nodes and Randomly Located Ad Hoc Nodes Results: Probabilistic Routing Per node throughput capacity is:

70. 70 Implications and Discussions When probabilistic routing is selected and the number of infrastructure nodes scales slower than (n/logn) 1/2 then the hybrid network has same capacity as pure ad hoc network If this number scales faster than (n/logn) 1/2 then throughput capacity is higher than the case of pure ad hoc networks When the number of infrastructure nodes is the same order of magnitude as √n, per node throughput capacity in k-nearest cell routing is the same order of magnitude as the case when all nodes transmit with rate W

71. 71 Implications and Discussions Two critical factors improve capacity of the hybrid networks – The number of infrastructure nodes must be large enough – A proper scheduling scheme is needed, as neigher k-nearest cell or probabilistic routing is an optimal routing scheme

72. 72 Randomly Placed Infrastructure Nodes and Ad Hoc Nodes Capacity of such a hybrid network is different under two different connectivity conditions Strong Connectivity: Here ad hoc nodes form a connected topology graph with high probability –Ad hoc nodes can form a stand-alone network Weak Connectivity: Here ad hoc nodes can have partitions but they are still connected via infrastructure nodes – The ad hoc nodes stay connected in the overall topology graph, even though they are partitioned

73. 73 Randomly Placed Infrastructure Nodes and Ad Hoc Nodes Both the N ad hoc and M infrastructure nodes are uniformly distributed on a disk of area A R =ΠR 2, with the following bound: For each ad hoc node, traffic to another node is bits/s with total bandwidth being W bits/s Protocol interference model is used and the guard distance is Δ i.e., interference-free Tx must be at distances > (1+Δ)r T

74. 74 Randomly placed Infrastructure Nodes and Ad Hoc Nodes Results: Throughput connectivity under strong connectivity condition is given by [1]: [1] U. C. Kozat, L. Tassiulas, “Capacity of ad hoc wireless networks with infrastructure support,” ACM Mobicom, 2003 The throughput capacity of a random hybrid network is Thus, compared to pure random ad hoc network this capacity is signficantly improved

75. 75 Throughput Capacity Under Weak Connectivity Condition The strong connectivity does not fully utilize the infrastructure network However, a weak connectivity can be formed via infrastructure nodes. With weak connectivity, the radius r T is different from the case with strong connectivity. – –As N → ∞, the radius r T can be smaller when weak connectivity is concerned. – –c 4 : any positive finite number

76. 76 Throughput Capacity Under Weak Connectivity Condition Traffic rate from one ad hoc node to another Thus, we cannot find a constant c > 0 such that   The per-node throughput capacity of θ(W) cannot be achieved with probability of one

77. 77 Throughput Capacity Under Weak Connectivity Condition We assume Then. the achievable upper bound of the throughput capacity is

78. 78 Capacity and Delay Tradeoff In a wireless multi-hop network, the network capacity, usually represented by throughput, is not a single concern for users  In fact, QoS is equivalently important. Usually QoS metrics include delay, delay jitter, and packet loss ratio. A. E. Gamal, J. Mammen, B. Prabhakar, and D. Shah, “ Optimal throughput-delay scaling in wireless networks–part I: the fluid model, “ IEEE Transactions on Information Theory, 2006

79. 79 Capacity and Delay Tradeoff Packet Loss – –In all theoretical analysis, it is assumed that, as long as the constraint defined in the interference model is satisfied, a packet can be sent and received successfully without any packet error. – –Thus, packet loss has not been taken into account as a key factor when capacity analysis is carried out for wireless networks.

80. 80 Capacity and Delay Tradeoff Delay Jitter – –Delay jitter has not been considered either in any existing research work. – –However, delay has already been considered as key performance metric when the capacity of a wireless multi-hop network is analyzed.

81. 81 Capacity and Delay Tradeoff Existing Approaches – –Throughput of a random network with both static and fixed nodes is analyzed in [1], where the routing algorithm is proposed to optimize throughput. – –Based on such a framework, delay is studied. – –In [2], throughput and delay tradeoff is analyzed using i.i.d mobility model. – –In [3], the throughput-delay tradeoff has been analyzed for both static networks and mobile networks. – –For mobile networks, the random walk (RW) model is assumed in the analysis. [1] A. Acharya A, A. Misra A and S. Bansal, “High-performance architectures for IP-based multihop 802.11 Networks,” IEEE Wireless Communications, 2003 [2] M. J. Neely, W. Modiano, and C. E. Rohrs, “Dynamic power allocation and routing for time varying wireless Networks,” In Proc. of IEEE INFOCOM 2005. [3] A. E. Gamal, J. Mammen, B. Prabhakar, and D. Shah, “ Optimal throughput-delay scaling in wireless networks–part I: the fluid model, “ IEEE Transactions on Information Theory, 2006

82. 82 Capacity and Delay Tradeoff In order to increase the network capacity, two rules can be followed: 1) reducing the number of hops that a packet shall travel 2) the interference range of a transmission.  However, scheduling schemes following these rules usually improves throughput but increases delay. The Need for Capacity-Delay Tradeoff – –Thus, given a scheduling scheme, it is necessary to investigate whether or not it causes unacceptable delay. – –It is also interesting to perform throughput-delay tradeoff when a scheduling scheme is designed for wireless multi-hop networks. A. E. Gamal, J. Mammen, B. Prabhakar, and D. Shah, “ Optimal throughput-delay scaling in wireless networks–part I: the fluid model, “ IEEE Transactions on Information Theory, 2006

83. 83 Analytical Models and Definitions Network Model – –n nodes are assumed to be uniformly distributed into a unit torus, and are randomly split into n/2 source-destination pairs. – –Time is slotted into unit length for packetized transmission. – –The unit torus is divided into n square cells, i.e., the unit torus becomes a √n × √n discrete torus.

84. 84 Analytical Models and Definitions Mobility Model – Random Walk – –Each node walks randomly on the two-dimensional √n × √n discrete torus. – –That means, if a node is cell (i, j), its next location will be in the four neighbor cells (i − 1, j), (i, j − 1), (i + 1, j), (i, j + 1) with equal probability. – –Thus, the given random walk model assumes each node moves 1/√n in unit time, – –Moreover, all nodes move independently according to a uniform stationary and ergodic distribution. – –The random walk is preferred due to its Markovian nature, since the present position of node determines the distribution of its future position.

85. 85 Protocol Model – –Applied to determine if a transmission is successful. – –When a node can send packets successfully, it uses a transmission rate of W bits per second. – –In a time-slotted system, usually packets being transmitted in a time slot need to be scheduled  These issues complicate the analysis of capacity and delay in a multi-hop wireless network. Analytical Models and Definitions

86. 86 [Fluid Model] In order to simplify the problem, a fluid mode is adopted for packet transmission in a time slot. – –In this model, as long as time remains in a time slot, a fraction of a packet can be sent in the time slot. – –Thus, a packet may be sent in two different time slots. – –Similarly, a time slot may be used to send multiple packets. When the packet size is constant, then the fluid model is not applicable. Analytical Models and Definitions

87. 87 Definitions of Throughput-Delay Optimality Given a scheduling scheme Π n in a network with n nodes, its throughput T Π (n) can be defined as follows. 1. Given a specific variable C(n), a random variable A Π (n) is defined as the event that the average number of bit rate for the source-destination pair i is not less than C(n), i.e., – –i : a source-destination pair i, 1 ≤ i ≤ n/2 – –B Πn (i, t): the number of bits transferred in t time slots If Pr(A(n)) → 1 when n → ∞, then the scheme Π = {Π n } is said to achieve a throughput of C(n), i.e., T(n) = C(n).

88. 88 Definitions of Throughput-Delay Optimality 2. For the same source-destination pair, the average delay per packet is : the delay of packet j, for the source destination pair i under the scheduling scheme Π n Considering all source-destination pairs, the average delay is Consequently, the delay for the scheme, denoted by D(n) is defined as the expectation of the average delay over all source-destination pairs

89. 89 Definitions of Throughput-Delay Optimality 3. With the above defined throughput and delay, the throughput-delay optimality can be specified. – –Given a throughput-delay pair (T(n),D(n)), it is throughput-delay optimal if there exists a scheduling scheme such that – –but for any other scheme ′, It should be noted that, given a network, there could exist multiple pairs of (T (n),D(n) that achieve throughput-delay tradeoff.

90. 90 Throughput-Delay Tradeoff in Static Networks Assuming the unit torus is divided into square cells of area a(n), then the number of such cells is 1/a(n). Consider these cells in the unit torus, the following properties are held – –Property 1: Consider n nodes in the network, if a(n) ≥ 2 log n/n, then each cell has at least one node with high probability. – –Property 2: Under the protocol interference model, given any cell, the number of cells that causes interference to is bounded above by a constant c 1 that is independent of n. – –Thus, each cell can start transmission every (1 + c 1 ) time slots, which is actually an interference- free schedule. – –Property 3: If a(n) = Ω(log n/n), the number of source-destination lines passing through a cell is O(n√a(n) ) with high probability.

91. 91 Throughput-Delay Tradeoff in Static Networks The scheduling scheme is developed based on these properties, as described below: – –For 1/a(n) square cells on the unit torus, two conditions must be checked 1) Cell is empty 2) The number of source-destination lines passing through each cell is at most – –If neither condition is satisfied, then each of the n/2 sources just send packets to its destination directly in a round-robin fashion.

92. 92 Throughput-Delay Tradeoff in Static Networks – –Otherwise, the following policy is applied by utilizing property 2. 1) 1)For each source-destination pair, a straight line is used to connect the source and the destination. The packets go through cells that are intersected by the straight line. 2) Each cell becomes active every (1 + c 1 ) time slots. Cells that are sufficiently far apart transmit simultaneously in the same time slot. 3) When a cell becomes active in a time slot, one packet from each source-destination pair whose straight line intersects the cell is sent according to a TDM scheme.

93. 93 Throughput-Delay Tradeoff in Static Networks Given the above scheduling scheme, the throughput and delay can be derived as follows. The probability that neither condition is satisfied is vanishingly low as n → ∞, because of properties 1 and 3 are held by the static random network. – –Thus, the throughput and delay results are nearly contributed by the scheduling scheme when both conditions are satisfied, i.e., no cell is empty and the number of source-destination lines passing through each cell is at most c 2 n√a(n).

94. 94 Throughput-Delay Tradeoff in Static Networks From property 2 – –Each cell can transmit at least a packet every (1 + c 1 ) time slots, where c 1 is independent of n. – –Thus, the cell throughput is θ(1). – –According to the scheduling scheme, such a throughput is contributed by all source-destination lines in the cell. From property 3 – –The number of such lines is O(n√a(n) )  Thus, a throughput of each source-destination line is

95. 95 Throughput-Delay Tradeoff in Static Networks Given a source-destination pair i, the number of hops for each packet is θ(n√a(n) ) – –d i is the distance of the straight line the given source-destination pair. Thus, the average number of hops for a packet averaged over all source-destination pairs is When n is large, the average distance of source-destination lines is

96. 96 Throughput-Delay Tradeoff in Static Networks Thus, the average number of hops that a packet travels is θ(1/√a(n) ) For each hop, the delay of a packet is at most c 1 time slots, because this packet should be sent out within the current time slot or the one after the next c 1 time slots is over. Thus, the delay of a packet after all hops is at most c 1 times the number of hops. In other words, the delay is

97. 97 Throughput-Delay Tradeoff in Static Networks For the given scheduling scheme, if a(n) ≥ 2 logn/n, the throughput and delay of the static random network are and, respectively. Thus, for it can be proved that such a throughput-delay pair is optimal, i.e., for any other scheme that can achieve the same throughput, T(n), as that in the given scheduling scheme, then delay D(n) will be Ω(nT (n)).

98. 98 Throughput-Delay Tradeoff in Mobile Networks It has been proved in [1] that a throughput of θ(1) can be achieved if a two-hop scheduling scheme is adopted to a mobile ad hoc network. However, such a scheme can result in a large delay.  Needs schemes that can achieve a better tradeoff between throughput and delay by analyzing the delay of scheduling schemes [1] D. N. C. Tse and M. Grossglauser, “Mobility increases the capacity of ad hoc wireless networks,” IEEE/ACM Trans. Networking, 2002

99. 99 Delay of Mobile Networks with Throughput of θ(1) The key concept is that the scheduling scheme must consider two constraints: 1) 1)at most two hops are allowed for a packet from its source to its destination; 2) packets of the same source-destination pair are relayed by different nodes in the network.

100. 100 Delay of Mobile Networks with Throughput of θ(1) With these constraints in mind, the scheduling scheme is described as follows. – –The unit torus is divided into n square cells each with an area of 1/n. – –Each cell can become active once in every (1 + c 1 ) time slots. – –When a cell becomes active, only nodes in the same cell can have transmissions to each other.

101. 101 Delay of Mobile Networks with Throughput of θ(1) – –Given an active cell, its time slot is split into two subslots: subslot A and subslot B 1. 1. Subslot A: A source node is selected at random. With a probability of p 1, this source node sends a packet to another randomly chosen node, which can be a relay node or a destination. Thus, with probability 1 − p 1, the source node does nothing.

102. 102 Delay of Mobile Networks with Throughput of θ(1) 2. Subslot B: a destination node is selected at random. Then another node, which can be a relay node or a source node, is selected randomly to send a packet to this destination node. However, if the selected node for packet transmission has no packets, it does nothing.  Such a scheduling can achieve a throughput of θ(1).

103. 103 Delay of Mobile Networks with Throughput of θ(1) Given a packet for a source-destination pair, its delay consists of two components: hop delay and mobile delay. – –Hop delay is the time for sending a packet from a source node to a relay node or destination node. Thus, it is independent of n. – –Mobile delay is the time for a relay node to find the destination so that a packet is delivered.   As a result, the delay of a source-destination pair is dominated by the mobile delay.

104. 104 Upper Bound Given the above scheduling scheme, the transmission of a packet can be modeled as a relay queue, in which each source-destination pair has (n − 2) queues to relay packets for this pair. Thus, a packet arrives at the relay queue when – –1) a relay node R is the in the same cell as the source node S – –2) the cell is active; – –3) S and R are selected as a sender-receiver pair; – –4) S actually sends a packet.

105. 105 Upper Bound A packet can depart from the relay queue only when 1) R is in the same cell as the destination node D; 2) the cell is active; 3) R and D are selected as a sender-receiver pair.

106. 106 Upper Bound However, an actual packet departure depends on whether R has packets for D. Thus, without considering available packets in R, the departure is only called potential departure.

107. 107 Upper Bound Considering the above relay queue, the inter-arrival times and inter-departure times are not i.i.d for the following reason. – –Both processes depend on whether a pair of nodes can be selected as a sender-receiver pair. – –However, due to the Markovian nature of the mobility model, two nodes being selected as a sender-receiver pair in the current time slot impacts the probability of any other two nodes being selected as a sender-receiver pair.

108. 108 Upper Bound Without i.i.d. for either arrival and departure process, it is difficult to analyze the relay queue  needs to use different types of queues to approximate the relay queue and then derive the delay.

109. 109 Upper Bound Consider the first queue Q 1, in which an arrival is assumed to occur with probability p 1 whenever S and R meet, irrespective of whether S and R are selected as a sender-receiver pair. – –Without requiring S and R to be a sender-receiver pair, the Markovian nature is eliminated from inter-arrival times  the inter-arrival times of Q 1 is i.i.d.

110. 110 Upper Bound Since the inter-arrival times of Q 1 are stochastically dominated by those of relay queue, the delay of Q 1 provides an upper bound for the delay of the relay queue. – –The potential departure process in Q 1 depends on two other processes: 1) R and D are in the same cell and 2) R and D are selected as a sender-receiver pair  Due to the latter process, the inter-potential-departure process is not i.i.d. anymore.

111. 111 Upper Bound – –However, the probability that a potential departure occurs can be not less than a constant c 2 for large enough n, where c 2 is independent of n. – –Thus, Q 2 is such a queue that its arrival process is the same as Q 1, but for the departure process, R and D are selected as a potential departure instant with probability c 2  In Q 2, the inter-potential-departure process is i.i.d.

112. 112 Upper Bound Furthermore, the inter-potential-departure times of Q 2 is stochastically dominated by those of Q 1. Considering that Q 1 and Q 2 have the same arrival process, the delay of Q 1 is upper-bounded by that of Q 2

113. 113 Upper Bound Q 2 can be further approximated by two queues in tandem,Q 3 andQ 4. – –The arrival process of Q 3 is the same as that of Q 2, but the departure process is an i.i.d. Bernoulli process with parameter 2/3n. – –When Q 3 is not empty, and the arrival to Q 4 is the packet sent from Q 3 ; otherwise, a dummy packet is fed to Q 4. – –Thus, the arrival process of Q 4 is the same as the potential-service process of Q 3. – –It can proved that the delay of the two queues in tandem provides an upper bound on the delay of Q 2 and it is O(n log n)   By considering the construction of all the above queues, we know that the delay in the given scheduling scheme is O(n log n).

114. 114 Lower Bound Considering a relay node in cell (i, j) and its corresponding destination node in (k, l) – –The delay of a packet is at least equal to the time for the two nodes walking into the cell. – –By a difference random walk model, the time is equivalent to the case where the relay node walks from cell (i, j) until arrives at cell (k, l). – –It can be proved that the expectation of this time is θ(n log n). – –Thus, the delay of the given scheduling scheme is lower bounded by θ(n log n).  Combining lower bound and the upper bound, we know that the scheduling scheme that achieve a throughput of θ(1) has a delay of θ(n log n), i.e., D(n) = θ(n log n).

115. 115 A Scheme for Throughput-Delay Tradeoff In order to carry out throughput-delay tradeoff, a different scheduling scheme is needed for the same network but with a different throughput range. Here, two regimes are considered differently: i) sub-scheme A for ii) sub-scheme B for  The scheduling scheme for throughput-delay tradeoff consists of two parts for these two regimes:

116. 116 Open Research Issues: Throughput-delay tradeoff Hybrid networks with infrastructure support can help improve the capacity of multi-hop wireless networks, which is a typical case in WMN. However, the delay performance in such networks is still unknown and not research work has been carried out to study this problem.

117. 117 Open Research Issues: Throughput-Delay tradeoff Besides delay, there are other performance metrics that are critical for users. – –Delay jitter, maximum delay, packet loss ratio all important parameters for determining if a network provides enough quality of service to users. – –Thus, it is meaningful to study the tradeoff between throughput and these parameters.  Up to date, no research result has been reported on this subject.