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148
Stochastic Geometry and Random Graphs for the Analysis and Design of Wireless Networks
"... Wireless networks are fundamentally limited by the intensity of the received signals and by their interference. Since both of these quantities depend on the spatial location of the nodes, mathematical techniques have been developed in the last decade to provide communicationtheoretic results accoun ..."
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Cited by 240 (42 self)
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Wireless networks are fundamentally limited by the intensity of the received signals and by their interference. Since both of these quantities depend on the spatial location of the nodes, mathematical techniques have been developed in the last decade to provide communicationtheoretic results accounting for the network’s geometrical configuration. Often, the location of the nodes in the network can be modeled as random, following for example a Poisson point process. In this case, different techniques based on stochastic geometry and the theory of random geometric graphs – including point process theory, percolation theory, and probabilistic combinatorics – have led to results on the connectivity, the capacity, the outage probability, and other fundamental limits of wireless networks. This tutorial article surveys some of these techniques, discusses their application to model wireless networks, and presents some of the main results that have appeared in the literature. It also serves as an introduction to the field for the other papers in this special issue.
Closing the gap in the capacity of wireless networks via percolation theory
 IEEE TRANS. INFORMATION THEORY
, 2007
"... An achievable bit rate per source–destination pair in a wireless network of � randomly located nodes is determined adopting the scaling limit approach of statistical physics. It is shown that randomly scattered nodes can achieve, with high probability, the same Ia � � transmission rate of arbitrari ..."
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Cited by 238 (8 self)
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An achievable bit rate per source–destination pair in a wireless network of � randomly located nodes is determined adopting the scaling limit approach of statistical physics. It is shown that randomly scattered nodes can achieve, with high probability, the same Ia � � transmission rate of arbitrarily located nodes. This contrasts with previous results suggesting that a Ia � � �� � � reduced rate is the price to pay for the randomness due to the location of the nodes. The network operation strategy to achieve the result corresponds to the transition region between order and disorder of an underlying percolation model. If nodes are allowed to transmit over large distances, then paths of connected nodes that cross the entire network area can be easily found, but these generate excessive interference. If nodes transmit over short distances, then such crossing paths do not exist. Percolation theory ensures that crossing paths form in the transition region between these two extreme scenarios. Nodes along these paths are used as a backbone, relaying data for other nodes, and can transport the total amount of information generated by all the sources. A lower bound on the achievable bit rate is then obtained by performing pairwise coding and decoding at each hop along the paths, and using a time division multiple access scheme.
On the Broadcast capacity in multihop wireless networks: Interplay of power, . . .
, 2007
"... In this paper we study the broadcast capacity of multihop wireless networks which we define as the maximum rate at which broadcast packets can be generated in the network such that all nodes receive the packets successfully within a given time. To asses the impact of topology and interference on t ..."
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Cited by 103 (5 self)
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In this paper we study the broadcast capacity of multihop wireless networks which we define as the maximum rate at which broadcast packets can be generated in the network such that all nodes receive the packets successfully within a given time. To asses the impact of topology and interference on the broadcast capacity we employ the Physical Model and Generalized Physical Model for the channel. Prior work was limited either by density constraints or by using the less realistic but manageable Protocol model [1], [2]. Under the Physical Model, we find that the broadcast capacity is within a constant factor of the channel capacity for a wide class of network topologies. Under the Generalized Physical Model, on the other hand, the network configuration is divided into three regimes depending on how the power is tuned in relation to network density and size and in which the broadcast capacity is asymptotically either zero, constant or unbounded. As we show, the broadcast capacity is limited by distant nodes in the first regime and by interference in the second regime. In the second regime, which covers a wide class of networks, the broadcast capacity is within a constant factor of the bandwidth.
The transport capacity of wireless networks over fading channels
, 2005
"... We consider networks consisting of nodes with radios, and without any wired infrastructure, thus necessitating all communication to take place only over the shared wireless medium. The main focus of this paper is on the effect of fading in such wireless networks. We examine the attenuation regime ..."
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Cited by 89 (4 self)
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We consider networks consisting of nodes with radios, and without any wired infrastructure, thus necessitating all communication to take place only over the shared wireless medium. The main focus of this paper is on the effect of fading in such wireless networks. We examine the attenuation regime where either the medium is absorptive, a situation which generally prevails, or the path loss exponent is greater than 3. We study the transport capacity, defined as the supremum over the set of feasible rate vectors of the distance weighted sum of rates. We consider two assumption sets. Under the first assumption set, which essentially requires only a mild time average type of bound on the fading process, we show that the transport capacity can grow no faster than (), where denotes the number of nodes, even when the channel state information (CSI) is available noncausally at both the transmitters and the receivers. This assumption includes common models of stationary ergodic channels; constant, frequencyselective channels; flat, rapidly varying channels; and flat slowly varying channels. In the second assumption set, which essentially features an independence, time average of expectation, and nonzeroness condition on the fading process, we constructively show how to achieve transport capacity of ( ) even when the CSI is unknown to both the transmitters and the receivers, provided that every node has an appropriately nearby node. This assumption set includes common models of independent and identically distributed (i.i.d.) channels; constant, flat channels; and constant, frequencyselective channels. The transport capacity is achieved by nodes communicating only with neighbors, and using only pointtopoint coding. The thrust of these results is that the multihop strategy, toward which much protocol development activity is currently targeted, is appropriate for fading environments. The low attenuation regime is open.
Optimal ThroughputDelay Scaling in Wireless Networks  Part I: The Fluid Model
"... Gupta and Kumar (2000) introduced a random model to study throughput scaling in a wireless network with static nodes, and showed that the throughput per sourcedestination pair is Θ ( 1 / √ n log n). Grossglauser and Tse (2001) showed that when nodes are mobile it is possible to have a constant thr ..."
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Cited by 80 (2 self)
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Gupta and Kumar (2000) introduced a random model to study throughput scaling in a wireless network with static nodes, and showed that the throughput per sourcedestination pair is Θ ( 1 / √ n log n). Grossglauser and Tse (2001) showed that when nodes are mobile it is possible to have a constant throughput scaling per sourcedestination pair. In most applications delay is also a key metric of network performance. It is expected that high throughput is achieved at the cost of high delay and that one can be improved at the cost of the other. The focus of this paper is on studying this tradeoff for wireless networks in a general framework. Optimal throughputdelay scaling laws for static and mobile wireless networks are established. For static networks, it is shown that the optimal throughputdelay tradeoff is given by D(n) = Θ(nT (n)), where T (n) and D(n) are the throughput and delay scaling, respectively. For mobile networks, a simple proof of the throughput scaling of Θ(1) for the GrossglauserTse scheme is given and the associated delay scaling is shown to be Θ(n log n). The optimal throughputdelay tradeoff for mobile networks is also established. To capture physical movement in the real world, a random walk model for node mobility is assumed. It is shown that for throughput of O ( 1 / √ n log n) , which can also be achieved in static networks, the throughputdelay tradeoff is the same as in static networks, i.e., D(n) = Θ(nT (n)). Surprisingly, for almost any throughput of a higher order, the delay is shown to be Θ(n log n), which is the delay for throughput of Θ(1). Our result, thus, suggests that the use of mobility to increase throughput, even slightly, in realworld networks would necessitate an abrupt and very large increase in delay.
Multicast capacity of wireless ad hoc networks
 IEEE/ACM Trans. Netw
, 2009
"... Abstract—We study the multicast capacity of largescale random extended multihop wireless networks, where a number of wireless nodes are randomly located in a square region with side length a = p n, by use of Poisson distribution with density 1. All nodes transmit at a constant power P, and the powe ..."
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Cited by 67 (22 self)
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Abstract—We study the multicast capacity of largescale random extended multihop wireless networks, where a number of wireless nodes are randomly located in a square region with side length a = p n, by use of Poisson distribution with density 1. All nodes transmit at a constant power P, and the power decays with attenuation exponent> 2. The data rate of a transmission is determined by the SINR as B log(1 + SINR), where B is the bandwidth. There are ns randomly and independently chosen multicast sessions. Each multicast session has k randomly chosen terminals. n We show that when k 1 and ns (log n) 2n 1=2+, the capacity that each multicast p session can achieve, with high proban bility, is at least c8 p, where 1, 2, and c8 are some special conn k stants and> 0 is any positive real number. We also show that for k = O( n), the perflow multicast capacity under Gaussian log n p n channel is at most O ( p) when we have at least ns = (log n) n k random multicast flows. Our result generalizes the unicast capacity for random networks using percolation theory.
The capacity of wireless networks: Informationtheoretic and physical limits
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 2009
"... It is shown that the capacity scaling of wireless networks is subject to a fundamental limitation which is independent of power attenuation and fading models. It is a degrees of freedom limitation which is due to the laws of physics. By distributing uniformly an order of users wishing to establish ..."
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Cited by 61 (2 self)
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It is shown that the capacity scaling of wireless networks is subject to a fundamental limitation which is independent of power attenuation and fading models. It is a degrees of freedom limitation which is due to the laws of physics. By distributing uniformly an order of users wishing to establish pairwise independent communications at fixed wavelength inside a twodimensional domain of size of the order of , there are an order of communication requests originating from the central half of the domain to its outer half. Physics dictates that the number of independent information channels across these two regions is only of the order of , so the peruser information capacity must follow an inverse squareroot of law. This result shows that informationtheoretic limits of wireless communication problems can be rigorously obtained without relying on stochastic fading channel models, but studying their physical geometric structure.
The multicast capacity of large multihop wireless networks
 In Proc. of ACM MobiHoc ’07
, 2007
"... We consider wireless ad hoc networks with a large number of users. Subsets of users might be interested in identical information, and so we have a regime in which several multicast sessions may coexist. We first calculate an upperbound on the achievable transmission rate per multicast flow as a fun ..."
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Cited by 49 (2 self)
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We consider wireless ad hoc networks with a large number of users. Subsets of users might be interested in identical information, and so we have a regime in which several multicast sessions may coexist. We first calculate an upperbound on the achievable transmission rate per multicast flow as a function of the number of multicast sources in such a network. We then propose a simple combbased architecture for multicast routing which achieves the upper bound in an order sense under certain constraints. Compared to the approach of constructing a Steiner tree to decide multicast paths, our construction achieves the same orderoptimal results while requiring little location information and no computational overhead.
Capacity of large scale wireless networks under gaussian channel model
 in Mobicom08
, 2008
"... In this paper, we study the multicast capacity of a large scale random wireless network. We simply consider the extended multihop network, where a number of wireless nodes vi(1 ≤ i ≤ n) are randomly located in a square region with sidelength a = √ n, by use of Poisson distribution with density 1. ..."
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Cited by 49 (21 self)
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In this paper, we study the multicast capacity of a large scale random wireless network. We simply consider the extended multihop network, where a number of wireless nodes vi(1 ≤ i ≤ n) are randomly located in a square region with sidelength a = √ n, by use of Poisson distribution with density 1. All nodes transmit at constant power P, and the power decays along path, with attenuation exponent α> 2. The data rate of a transmission is determined by the SINR as B log(1 + SINR). There are ns randomly and independently chosen multicast sessions. Each multicast has k rann domly chosen terminals. We show that, when k ≤ θ1 (log n) 2α+6, and ns ≥ θ2n 1/2+β, the capacity that each multicast session can n achieve, with high probability, is at least c8 √ , where θ1, θ2, ns k and c8 are some special constants and β> 0 is any positive real number. Our result generalizes the unicast capacity [3] for random networks using percolation theory.