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## Continuum Equilibria for Routing in Dense Static Ad-hoc Networks

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Citations: | 3 - 0 self |

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3242 | The capacity of wireless networks
- Gupta, Kumar
- 1998
(Show Context)
Citation Context ...taken to be where | · | represents the norm of a vector. Examples for f: c(x, T(x)) = f −1 (|T(x)|) (1) • Using a network theoretic approach based on multi-hop communication, Gupta and Kumar prove in =-=[12]-=- that the throughput of the system that can be transported by the network when the nodes are optimally located is Ω( √ λ), and when the nodes are randomly located this throughput becomes Ω( √ λ √ log ... |

1220 | Mobility Increases the Capacity of Adhoc Wireless Networks,”
- Grossglauser, Tse
- 2002
(Show Context)
Citation Context ...ccess scheme, MSR (Multihop Spatial Reuse Aloha), reaching the Gupta and Kumar bound O( √ λ) which does not require prior knowledge of the node density. • A protocol introduced by Tse and Glosglauser =-=[10]-=- has a capacity that scales as O(λ). However, it does not fall directly within the class of massively dense ad-hoc networks and indeed, it relies on mobility and on relaying for handling disconnectivi... |

772 |
Some theoretical aspects of road traffic research
- Wardrop
- 1952
(Show Context)
Citation Context ...he existing theory of routing in massively dense networks which had been developed within the community of road traffic engineers. Indeed, this approach had already been introduced in 1952 by Wardrop =-=[30]-=- and by Beckmann [4] and is still an active research area among that community, see [6, 7, 14, 16, 32] and references therein. We combine in this paper various approaches from this area as well as fro... |

230 |
Capuzzo Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations.
- Bardi, I
- 1997
(Show Context)
Citation Context ..., under suitable conditions, it is the unique solution of (13). In the case that V j is not everywhere differentiable then, under suitable conditions, it is the unique viscosity solution of (13) (see =-=[3, 8]-=-). There are many numerical approaches for solving the HJB equation. One can discretize the HJB equation and obtain a discrete dynamic programming for which efficient solution methods exist. If one re... |

225 | An Aloha protocol for multihop mobile wireless networks. - Baccelli, Blaszczyszyn, et al. - 2006 |

71 | Closing the gap in the capacity of random wireless networks
- Franceschetti, Dousse, et al.
- 2004
(Show Context)
Citation Context ... transported by the network when the nodes are optimally located is Ω( √ λ), and when the nodes are randomly located this throughput becomes Ω( √ λ √ log λ ). Using percolation theory, the authors of =-=[9]-=- have shown that in the randomly located set the same Ω( √ λ) can be achieved. • Baccelli, Blaszczyszyn and Mühlethaler introduce in [2] an access scheme, MSR (Multihop Spatial Reuse Aloha), reaching ... |

63 | A system and traffic dependent adaptive routing algorithm for ad hoc networks
- Gupta, Kumar
- 1997
(Show Context)
Citation Context ...Motivation.- One popular objective in some routing protocols in ad-hoc networks is to assign routes for packets in a way that each packet follows a minimal cost path (given the others’ paths choices) =-=[11]-=-. This has the advantage of equalizing source-destination delays of packets that belong to the same class, which allows one to minimize the amount of packets that come out of sequence. (This is desira... |

62 | Packetostatics: Deployment of massively dense sensor networks as an electrostatics. problem,” - Toumpis, Tassiulas - 2005 |

60 | On the scalability of ad hoc routing protocols.
- Santivanez, McDonald, et al.
- 2002
(Show Context)
Citation Context ...examples are the following: • In Routing: As the network size increases, routes consists of an increasing number of nodes, and so they are increasingly susceptible to node mobility and channel fading =-=[22]-=-. • In Transmission Scheduling: The determination of the maximum number of nonconflicting transmissions in a graph is a NP-complete problem [29]. • In Capacity of Wireless Networks: As the number of n... |

56 |
Geometry of information propagation in massively dense ad hoc networks
- Jacquet
- 2004
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Citation Context ...of large ad-hoc networks go way beyond those related to statistical-mechanics in which macroscopic properties are derived from microscopic structure. Starting from the pioneering work by Jacquet (see =-=[17]-=-) in that area, a number of research groups have worked on massively dense ad-hoc networks using tools from geometrical optics [17] 1 as well as electrostatics (see e.g. [26, 25, 13], and the survey [... |

52 | Optimal Deployment of Large Wireless Sensor Networks
- Toumpis, Tassiulas
- 2006
(Show Context)
Citation Context ...ring work by Jacquet (see [17]) in that area, a number of research groups have worked on massively dense ad-hoc networks using tools from geometrical optics [17] 1 as well as electrostatics (see e.g. =-=[26, 25, 13]-=-, and the survey [27] and references therein). We shall describe these in the next sections. The physical paradigms allow the authors to minimize various metrics related to the routing. In contrast, H... |

27 |
Mother Nature Knows Best: A Survey of Recent Results on Wireless Networks Based on Analogies with
- Toumpis
- 2008
(Show Context)
Citation Context ...]) in that area, a number of research groups have worked on massively dense ad-hoc networks using tools from geometrical optics [17] 1 as well as electrostatics (see e.g. [26, 25, 13], and the survey =-=[27]-=- and references therein). We shall describe these in the next sections. The physical paradigms allow the authors to minimize various metrics related to the routing. In contrast, Hyytia and Virtamo pro... |

25 |
Bits-per-joule capacity of energy-limited wireless networks
- Rodoplu, Meng
- 2007
(Show Context)
Citation Context ...m 4deployment of nodes, where each node has data to send to another randomly selected node, the capacity (in bits per Joule) has the form f(λ) = Ω ( (λ/ log λ) (q−1)/2) where q is the path-loss, see =-=[21]-=-. The cost is then obtained using (1). 3 Preliminary In the work of Toumpis et al. ([26, 25, 13, 28, 27]), the authors address the problem of the optimal deployment of Wireless Sensor Networks by a pa... |

24 |
On load balancing in a dense wireless multihop network
- Hyytia, Virtamo
- 2006
(Show Context)
Citation Context ...erences therein). We shall describe these in the next sections. The physical paradigms allow the authors to minimize various metrics related to the routing. In contrast, Hyytia and Virtamo propose in =-=[15]-=- an approach based on load balancing arguing that if shortest path (or cost minimization) arguments were used, then some parts of the network would carry more traffic than others and may use more ener... |

18 |
Optimal design and operation of massively dense wireless networks: Or how to solve 21st century problems using 19th century mathematics
- Toumpis
- 2006
(Show Context)
Citation Context ...etworks, researchers frequently stumble on the scalability problem that can be summarized in the following sentence: “As the number of nodes in the network increases, problems become harder to solve” =-=[26]-=-. The sentence takes its meaning from several issues. Some examples are the following: • In Routing: As the network size increases, routes consists of an increasing number of nodes, and so they are in... |

12 |
Continuum Modeling of Transportation Networks
- Dafermos
- 1980
(Show Context)
Citation Context ...n the community of road traffic engineers. Indeed, this approach had already been introduced in 1952 by Wardrop [30] and by Beckmann [4] and is still an active research area among that community, see =-=[6, 7, 14, 16, 32]-=- and references therein. We combine in this paper various approaches from this area as well as from optimal control theory in order to formulate models for routing in massively dense networks. We furt... |

11 | The mathematics of routing in massively dense ad-hoc networks
- Altman, Silva, et al.
- 2008
(Show Context)
Citation Context ... a model that uses directional antennas. The approach that we follow is inspired by the work of Dafermos (see [6]) on road traffic. An alternative approach based on road traffic tools can be found in =-=[1, 23]-=-. For energy efficiency, it is assumed that each terminal is equipped with one or with two directional antennas, allowing transmission at each hop to be directed either from North to South or from Wes... |

11 |
Variational Inequalities and discrete and continuum models of network equilibrium protocols,” Mathematical and Computer Modelling 35:689–708
- Daniele, Maugeri
- 2002
(Show Context)
Citation Context ...y in the context of road traffic engineering. The use of a continuum network approach was already introduced on 1952 by Wardrop [30] and by Beckmann [4]. For more recent papers in this area, see e.g. =-=[6, 7, 14, 16, 32]-=- and references therein. We formulate it below and obtain some of its properties. Motivation.- One popular objective in some routing protocols in ad-hoc networks is to assign routes for packets in a w... |

11 |
Optimal Placement of Nodes in Large Sensor Networks under a General Physical Layer Model
- Toumpis, Gupta
- 2005
(Show Context)
Citation Context ...ring work by Jacquet (see [17]) in that area, a number of research groups have worked on massively dense ad-hoc networks using tools from geometrical optics [17] 1 as well as electrostatics (see e.g. =-=[26, 25, 13]-=-, and the survey [27] and references therein). We shall describe these in the next sections. The physical paradigms allow the authors to minimize various metrics related to the routing. In contrast, H... |

11 |
A Review of the Two-Dimensional Continuum Modeling Approach to Transportation Problems
- Ho, Wong
- 2006
(Show Context)
Citation Context ...n the community of road traffic engineers. Indeed, this approach had already been introduced in 1952 by Wardrop [30] and by Beckmann [4] and is still an active research area among that community, see =-=[6, 7, 14, 16, 32]-=- and references therein. We combine in this paper various approaches from this area as well as from optimal control theory in order to formulate models for routing in massively dense networks. We furt... |

11 |
A First Course
- Weinberger
- 1995
(Show Context)
Citation Context ...2 ∂ + k2 2φ ∂x2 1 = 0. Let ki = K 2 i . Divide the above equation by k1k2. One obtains 1 K 2 1 ∂2φ ∂x2 + 1 1 K2 ∂ 2 2φ ∂x2 2 18 = 0.Following the classical way of analyzing the Laplace equation, (see=-=[31]-=-) we attempt a separation of variables according to We then get that φ(x1, x2) = F1(K1x1)F2(K2x2) . F ′′ 1 (K1x1) F1(K1x1) = −F ′′ 2 (K2x2) F2(K2x2) = s2 . In that formula, since the first term in ind... |

10 | Optical routing in massively dense networks: Practical issues and dynamic programming interpretation
- Toumpis, Catanuto, et al.
- 2006
(Show Context)
Citation Context ...ode, the capacity (in bits per Joule) has the form f(λ) = Ω ( (λ/ log λ) (q−1)/2) where q is the path-loss, see [21]. The cost is then obtained using (1). 3 Preliminary In the work of Toumpis et al. (=-=[26, 25, 13, 28, 27]-=-), the authors address the problem of the optimal deployment of Wireless Sensor Networks by a parallel with Electrostatic. Consider in the two dimensional plane X1 ×X2, the continuous information dens... |

9 | Numerical solutions of continuum equilibria for routing in dense ad-hoc networks
- Silva, Bernhard, et al.
- 2008
(Show Context)
Citation Context ... a model that uses directional antennas. The approach that we follow is inspired by the work of Dafermos (see [6]) on road traffic. An alternative approach based on road traffic tools can be found in =-=[1, 23]-=-. For energy efficiency, it is assumed that each terminal is equipped with one or with two directional antennas, allowing transmission at each hop to be directed either from North to South or from Wes... |

7 |
Variational inequalities and applications to a continuum model of transportation network with capacity constraints
- Idone
(Show Context)
Citation Context ...n the community of road traffic engineers. Indeed, this approach had already been introduced in 1952 by Wardrop [30] and by Beckmann [4] and is still an active research area among that community, see =-=[6, 7, 14, 16, 32]-=- and references therein. We combine in this paper various approaches from this area as well as from optimal control theory in order to formulate models for routing in massively dense networks. We furt... |

4 |
A continuum model of transportation,” Econometrica 20:643–660
- Beckmann
- 1952
(Show Context)
Citation Context ... routing in massively dense networks which had been developed within the community of road traffic engineers. Indeed, this approach had already been introduced in 1952 by Wardrop [30] and by Beckmann =-=[4]-=- and is still an active research area among that community, see [6, 7, 14, 16, 32] and references therein. We combine in this paper various approaches from this area as well as from optimal control th... |

4 |
J.J.Sun and B.P.Y.Loo, “Sensitivity analysis for a continuum traffic equilibrium problem,” Ann Reg Sci 40:493– 514
- Wong
- 2006
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Citation Context |

1 |
Vector Calculus, Third Edition, W.H. Freeman and Company
- Marsden, Tromba
- 1988
(Show Context)
Citation Context ... introducing the standard attribute (plus or minus) to a path according to the direction of the movement along it. The definition is different than in [6] (which we used in Section 4). Definition 6.1 =-=[18]-=-. (i) Let C be some simple closed curve surrounding some region R. Then C + corresponds to a counterclockwise movement; more precisely, it corresponds to moving so that the region R is to our left. Th... |

1 |
Eitan Altman, “Jointly Optimal Power Control and Hop Length for a Single Cell
- Ramaiyan, Kumar
- 2007
(Show Context)
Citation Context ...in a single cell of 802.11 IEEE wireless LAN one can improve the energy consumption by using multiple hops, as it has been shown not to be efficient in terms of energy consumption to use a single hop =-=[20]-=-. Alternatively, the cost can take into account the scaling of the nodes (as we had in Subsection 2.1) that is obtained when there are energy constraints. As an example, assuming random 4deployment o... |

1 |
Truong A. Ephremides, “Scheduling broadcasts in multihop radio networks
- V
- 1990
(Show Context)
Citation Context ...asingly susceptible to node mobility and channel fading [22]. • In Transmission Scheduling: The determination of the maximum number of nonconflicting transmissions in a graph is a NP-complete problem =-=[29]-=-. • In Capacity of Wireless Networks: As the number of nodes increases, the determination of the precise capacity becomes an intractable problem. ∗ INRIA, B.P.93, 2004 Route des Lucioles, 06902 Sophia... |

1 | Merouane Debbah, “Tools from Physics and Road-traffic Engineering for Dense Ad-hoc Networks,” available at http://www-sop.inria.fr/mistral/personnel/ Eitan.Altman/mobile.html - Altman, Silva |

1 | A continuum equilibrium model for estimating market are competitive facilities with elastic demand and market externality,” Transp - Yang, Wong - 2000 |