The Capacity of Wireless Networks
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1 The Capacity of Wireless Networks Piyush Gupta & P.R. Kumar Rahul Tandra --- EE228 Presentation
2 Introduction We consider wireless networks without any centralized control. Try to analyze the capacity of such networks. We consider two types of networks: a) Arbitrary networks b) Random networks
3 Arbitrary Networks n nodes are arbitrarily located in a disc of unit area. Traffic pattern is completely arbitrary. Range or power level is also arbitrary. Two models for successful reception of a transmission over one hop: a) Protocol Model b) Physical Model
4 The protocol Model Suppose X i transmits over the m-th sub-channel to a node X j. This transmission is successfully received by node X j if X k X (1 + ) j for every node X k simultaneously transmitting over the same sub-channel. The quantity > 0 models situations where a guard zone is specified by the protocol. X i X j
5 The Physical Model { } Let X k ;k Τ be the subset of nodes simultaneously transmitting at some time instant over a certain sub-channel. Let P k be the power level chosen by node X k. Then the transmission from a node X i is successfully received by a node X j if N + X i k Τ k i P X i X k j P α k X j α β
6 Transport Capacity of Arbitrary Networks We say that the network transports one bit-meter when one bit has been transported a distance of one meter towards it s destination. This sum of products of bits and the distances over which they are carried is called the transport capacity of the network.
7 Main Results for Arbitrary Networks Main Result 1: The transport Capacity of an Arbitrary Network under the protocol Model is Θ( W n) bit-meters/sec if the nodes are optimally placed, the traffic pattern and the range are optimally chosen. If the transport capacity is equitably divided between all W the n nodes, then each node would obtain Θ ( ) bitmeters/sec. n Further if, each node has its destination about the same distance of 1 meter away, then each node would W obtain a throughput capacity of Θ( ) bits/sec. n
8 Results cont. Main Result 2: For the Physical Model, cw n α 1 bit-meters/sec is feasible, while ' α c Wn bit-meters/sec is not, for appropriate constants c and c.
9 Arbitrary Networks: An upper bound on transport capacity We consider the setting on a planar disk of unit area with the following assumptions: (A1) There are n nodes arbitrarily located in disk of unit area on the plane. (A2) The network transports λnt bits over T seconds. (A3) The average distance between the source and the destination is L.
10 Assumptions cont. (A4) Each node can transmit over any subset of M sub-channels with capacities W m bits/sec,, where W. 1 m M W (A5) Transmissions are slotted into synchronized τ slots of length. M m=1 m =
11 Upper bound for the Transport Capacity Theorem 1: In the Protocol model the transport capacity is bounded as follows: λnl 8 π 1 W n bit-meters/sec
12 Lower bound on the Transport Capacity Theorem 2: There is a placement of nodes and an assignment of traffic patterns such that the network can W n achieve bit-meters/sec under the n + Protocol Model. 8π
13 Random Networks n nodes are randomly, i.e., independently and uniformly located on the surface of a sphere of surface area 1 sq. meter. Each node randomly chooses a destination to transmit. The destination node is independently chosen to be the node nearest to this randomly located point. Also, we assume that all nodes are homogeneous ( same power or range).
14 The Protocol Model All nodes employ a common range r for transmission. When node X i transmits to node X j, the transmission is successfully received by X j if: (i) X i X j r (ii) For every other node X k transmitting over the same sub-channel X k X j ( 1+ ) r
15 The Physical Model All nodes choose a common power level P for transmission. Let { X k ;k Τ} be the subset of nodes simultaneously transmitting at some time instant over a certain sub-channel. Then the transmission from a node X i is successfully received by a node X j if N + X i k k Τ i P X X k j α P X j α β
16 The Throughput Capacity of Random Networks Main Result 3: The order of the throughput capacity for the Protocol model is λ( n) = Θ bits/sec nlog n Main Result 4: For the Physical Model a cw λ( n) = Throughput Capacity of nlog n bits/sec is c' W feasible, while λ( n) = bits/sec is not for n appropriate constants c and c W
17 Outline of the Proof We use Voronoi tessellation on the surface S 2 of the sphere. Let {a 1,a 2,,a p } be a set of p points on S 2. The Voronoi cell V(a i ) is the set of point which are closer to a i than to any of the other a j s. Note that the distances are measured on the surface of S 2 by segments of great circles connection two points.
18 ε Lemma 1: For ever > 0, there is a Voronoi tessellation of S 2 with the property that every Voronoi cell contains a disk of radius ε and is contained in a disk of radius 2 ε. In the rest of the proof we will use a Voronoi tessellation V n for which: (a) Every Voronoi cell contains a disk of area (100logn)/n. Let ρ(n) be the radius of this disk. (b) Every Voronoi cell is contained in a disk of radius 2 ρ(n)
19 Adjacency and interference We say that two cells are adjacent, if they share a common point. Let us choose the range r(n) of each transmission so that r( n) = 8ρ( n) Lemma 2: Every node in a cell is within a distance of r(n) from every node in its own cell or adjacent cell. We say that two cells are interfering neighbors if there is a point in one cell which is within a distance 2 + ) r( of some point in the other cell. ( n)
20 Bound on the number of interfering neighbors of a cell Lemma 3: Every cell in V n has no more than c 1 interfering neighbors. c 1 depends only on 2 and grows no faster than linearly in ( 1+ ) Lemma 4: In the Protocol model there is a schedule for transmitting packets such that in every (1+c 1 ) slots, each cell in the tessellation V n gets one slot in which to transmit, and such that all transmissions are successfully received within a distance of r(n) from their transmitters.
21 The source-destination Pairs Each node wishes to communicate with the node nearest to a randomly chosen location. Let Y i be a randomly chosen location such that X i and Y i are independently and uniformly distributed on S 2, and the sequence {(X i,y i )} is i.i.d.
22 Some intermediate results They prove that each cell contains atleast one node with high probability (whp). Routing is done through cells. The traffic carried per each route is λ(n) The number of routes intersecting each cell is of the order if c nlog n whp. Hence we have cλ( n) nlog n W 1+ c 1
23 Some possible implications The throughput of a wireless network goes to zero as the number of nodes increases. Perhaps designers should target their efforts on networks with small number of users. A feasible scenario is where nodes communicate only with nearby neighbors.
24 Implications cont. In Random networks it is observed that one can group the nodes into small clusters or cells, where in each cell one can designate a specific node to relay the multi-hop traffic, if so desired. This would reduce the total transmission power consumed by the nodes. Also note that dividing the channel into subchannels does not change any of the results.
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