CAPACITY OF POWER CONSTRAINED AD-HOC NETWORKS Prof. Rohit Negi Arjunan Rajeswaran Presented by Jin Heo * The slides are from the authors and modified by Jin Heo

Background – what has been done, how, results ? Outline Introduction – what, why is it a hard problem ? Background – what has been done, how, results ? Recent Research – why, novelty, results ?

Introduction ad hoc wireless networks Infrastructure Last hop is wireless High Cost, maintenance Data rates known Reliable Ad hoc Multi-hop wireless Low cost, maintenance Supported data rate ? Reliability ? Cellular : Key point: Wireless is simply a last-hop link in a primarily wired (telephone/internet) network. Basically for un-tethering. High cost in deployment, signaling billing etc. Ad-hoc : key point : All wireless links – possibly many links till reach infrastructure or not at all. : NO infrastructure. Many variations : mobility , capabilities, traffic patterns, node placement.

Introduction ad hoc wireless networks Consider model with low spectral efficiency Arbitrary large bandwidth Power constrained  Interference becomes relatively negligible! Two applications UWB Sensor network The result is that throughput increases with node enter the network

Capacity of UWB ad-hoc wireless networks Summary Capacity of UWB ad-hoc wireless networks Previous Result : bits/s Our result : bits/s under alternate Communication model (UWB) n is the node density distance loss exponent

Design problem overview source 1 destination 1 This shared wireless channel is scheduled into links by the MAC (link layer) Routes : set of links from each source to destination Created by the routing layer destination 2 source 2 Aim : maximize the uniform throughput per node r (bits/s) ( fairness, uniform capabilities ) Uniform throughput – the throughput achieved by all nodes in the network Two Control Parameters to maximize throughput – Routing and MAC Routing (choosing sets of links) – MAC (scheduling access of shared channel, generates the links)  Strong interaction Definition of uniform throughput – the throughput achieved by all nodes in the network : fairness Now the aim is to maximize this uniform throughput We have the MAC and Routing to control. Variables MAC : scheduling access of the shared channel to the competing nodes. MAC creates the links over which routing may optimize. Routing – scheme to relay data packets form the source to destination in many hops – all wireless ? Strong interaction because the routes decide the link capacities so let us look at a simpler problem – only MAC

Design problem MAC Original ad-hoc network 1 2 3 Flow contention graph Link Contention MAC (in simplest case) ~ MAC scheduling problem to a Flow contention graph: Graph Coloring NP complete Complicated due to Routing - MAC interaction Formally : optimization problem – intractable Exact solution : very hard Need some kind of order bound as a function of number of nodes! MAC problem – the scheduling problem may be made explicit by a conversion to a Flow contention graph where each link is represented by a node and the contention by an edge between them. So now we need a coloring of the flow contention graph which would be the schedule solving the MAC problem, where the number of colors for a node proportional to the required capacity of the link. So, how do we analyze this problem ?? The idea is rather than find an exact value we need some kind of order bound ?? So we can at least say it is increasing or decreasing as a function of some variable ( number of nodes n )??

Background1 – what has been done, how, results ? Outline Introduction – what, why is it a hard problem ? Background1 – what has been done, how, results ? Recent Research – why, novelty, results ? 1. P.Gupta, P.R. Kumar, “Capacity of Wireless Networks,” IEEE Trans. Information Theory, Vol.46, March 2000.

Background overview Assumptions Nodes(n) location Xi : identical, independent uniform Unit area : sphere’s surface Homogenous nodes : rate r(n) , power P Destination : random independent Control : MAC, routing Here n nodes are assumed to be iid on the surface of a sphere. Thus it is a random network !! Surface of a sphere is assume to avoid the edge effects you would see on a plane – later it is shown that the same results hold for a plane too Identical nodes are assumed so it is typical of sensor networks …. So if the network is random what is under our control in the design – MAC and routing Network is a particular sample form the underlying distribution of nodes on the unit area. The network is random so is the capacity and hence capacity is a random variable – note the definition With this definition we shall now on call throughput capacity as capacity ( fro simplicity) note IT IS NOT SHANNON CAPACITY. Aim : provide order bounds on the capacity as a function of n . c>0 c’<infinity It is the time average bits/s per source destination pair.

Background overview Assumptions Metric Nodes(n) location Xi : identical, independent uniform Homogenous nodes : rate r(n), power P Unit area : sphere’s surface Destination : random independent Control : MAC, routing Metric Finding order bounds on the UNIFORM THROUGHPUT CAPACITY r(n) It is the time average bits/s per source destination pair. Here n nodes are assumed to be iid on the surface of a sphere. Thus it is a random network !! Surface of a sphere is assume to avoid the edge effects you would see on a plane – later it is shown that the same results hold for a plane too Identical nodes are assumed so it is typical of sensor networks …. So if the network is random what is under our control in the design – MAC and routing Network is a particular sample form the underlying distribution of node son the unit area. The network is random so is the capacity and hence capacity is a random variable – note the definition With this definition we shall now on call throughput capacity as capacity ( fro simplicity) note IT IS NOT SHANNON CAPACITY. Aim : provide order bounds on the capacity as a function of n . c>0 c’<infinity It is the time average bits/s per source destination pair.

Background communication model Wireless link : T bits/s Interference Model : Protocol : interference radius, guard zone of Transmissions is successful if every other receiver is out of interference range of the receiver Physical : propagation loss , If SNR > threshold, the receiver can receive packets Transmitter Simultaneous Transmitter The wireless link has w bits/s and so is a communication theoretic perspective Interference models tell us when a node Xk interferes with a transmission form node Xi to Xj The guard zone of delta works for the high SNR threshold case since then the interference region is greater than the transmission region and so 1+delta is valid. Also works when you have some threshold of SNR for operation and assuming a single interferer. Receiver

Background intuition r(n) is MAC vs Routing tr(n) - transmission range/power L – mean source-destination distance MAC vs Routing Reduces to Generates bits/s Available capacity Generated Traffic Hence Mean number of hops to destination Capacity loss due to interferences Here provide the intuition about how MAC and the Routing need to be traded off Only then can we justify the voronoi tessellation and the chosen sizes for uniform convergence. Let L be the mean source destination distance can be figured out right – a constant So now number of hops = l/ Tr trafiic generated is l* r/tr Each node has T ? No relay has T/tr^2 and so the tradeoff Upper bound is a limit on tr – above a threshold – else implies loss of connectivity

traffic to be carried available capacity Background capacity Lower bound (achievable) : use the a cellular like scheme traffic to be carried available capacity Upper bound : using transmission radius transmission radius lower bound for connectivity Order Bound : r(n) is Describe the bound briefly –

Background – what has been done, how, results ? Outline Introduction – what, why is it a hard problem ? Background – what has been done, how, results ? Recent Research – why, novelty, results ?

Recent Research overview Communication model : UWB (ultra wideband systems) limited power large bandwidth MAC : CDMA is optimal At least as good as any other algorithm in this communication model Routing : power constrained routing R. Negi and Arjunan – The Capacity of power constrained ad-hoc networks Note the difference an increasing function of n

Recent Research communication model power spectral density Power : constrained to P0 Medium loss Bandwidth: arbitrarily large W large w.r.t data rate – low spectral effeciency Examples : UWB (802.15.3a), sensors Link operates at Shannon Capacity Power and data rate adaptation according to the link condition Bandwidth scaling is one and the large is another As a function of n you will require bandwidth to make interference negligible This tells you growth rate as a function of n But to make it hold it needs to be large – how large – approximation of shannon capacity should hold This gives absolute values.

Recent Research PHY - interference SINR = Signal / (Noise + Interference) Noise = noise density * bandwidth In bandwidth-constrained scenario, SINR is dominated by interference In low spectral efficiency, SINR is mainly affected by ambient noise And also that Xi is transmitter and Xj is receiver and the rest are interferers

Recent Research PHY - interference Noise interference Bandwidth scaling : Makes W arbitrarily large  interference becomes negligible w.r.t ambient noise Interference negligible  no scheduling ! CDMA MAC is optimal, compared to FDMA/TDMA And also that Xi is transmitter and Xj is receiver and the rest are interferers

Recent Research PHY –link capacity What is the link capacity – The Shannon capacity for a link with Gaussian noise and interference sources: W log(1+SINR) Capacity is LINEAR in allotted power Rate/Capacity : adapts to link The link capacity is bounded due to the power constraint And also that Xi is transmitter and Xj is receiver and the rest are interferers

Recent Research routing Relaxed power constraint (upper bound) : routing can be decoupled from power constraint of each node Power on a route : Optimal Route : r(n) can be very large : each hop arbitrarily high capacity ! Number of hops is limited : hop length is bounded Note mention that the relaxed power constraint is assumed to derive an upper bound Also that this implies a power adaptation / control and since power and rate are connected a rate adaptation : results in route decoupling So basically each source destination attempts to use minimum power : resulting in minimum overall power for the network since no constraint on any node. Thus this solution maybe scaled for the appropriate maximum uniform throughput capacity.

Recent Research capacity Upper bound : relaxed power constraint Power is bounded by the average Number of hops (K) : bounded by tessellation Distance term : bounded by average source destination distance Lower bound : extend the previous scheme: traffic to be carried < available capacity Order bound : (soft order neglecting log terms) r(n) is Here in the bounding of the distance the problem is that Pessimistically – the number of nodes could be just one and so the distance is order 1^alpha - very bad for capacity. Optimistically – it could be all nodes and then the distance term is (1/n)^alpha * n too good for capacity So what do MOST or rather almost all routes actually see – calculate the number of nodes and bound it by the average distance between source and destination. IN these cases mostly the routes are all close to the average …………… We are designing for the worst case …to ensure that ever node will satisfy its minimum criterion that is why it is hard.

Capacity of UWB ad-hoc wireless networks Conclusion Capacity of UWB ad-hoc wireless networks Previous Result : bits/s Our result : bits/s under alternate Communication model (UWB) n is the node density distance loss exponent GAIN The gain comes two fold : 1. The infinite bandwidth washes away interference ( in the previous case basically washes away noise) in our case the uwb washes away interference. The second gain is that the large bandwidth gives us an explicit relationship between the capacity and power ie we exploit the fact that a node close by could be transmitting at high rate and so we need not be designing it for the farthest node like in the previous communication model Gives us the gain of \sqrt(n) ^alpha.

Information order bounds Intuitively f grows no faster than g Make a note as a function of naturals etc Basically growth rates Intuitively f,g grow at the same rate