Stochastic properties of the random waypoint mobility model
TL;DR: The random waypoint model is a commonly used mobility model for simulations of wireless communication networks and some of its fundamental stochastic properties are investigated, including the transition length and time of a mobile node between two waypoints, the spatial distribution of nodes, and the direction angle at the beginning of a movement transition.
Abstract: The random waypoint model is a commonly used mobility model for simulations of wireless communication networks. By giving a formal description of this model in terms of a discrete-time stochastic process, we investigate some of its fundamental stochastic properties with respect to: (a) the transition length and time of a mobile node between two waypoints, (b) the spatial distribution of nodes, (c) the direction angle at the beginning of a movement transition, and (d) the cell change rate if the model is used in a cellular-structured system area.
The results of this paper are of practical value for performance analysis of mobile networks and give a deeper understanding of the behavior of this mobility model. Such understanding is necessary to avoid misinterpretation of simulation results. The movement duration and the cell change rate enable us to make a statement about the "degree of mobility" of a certain simulation scenario. Knowledge of the spatial node distribution is essential for all investigations in which the relative location of the mobile nodes is important. Finally, the direction distribution explains in an analytical manner the effect that nodes tend to move back to the middle of the system area.
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TL;DR: In this paper, the authors presented a detailed analytical study of the spatial node distribution generated by random waypoint mobility and derived an exact equation of the asymptotically stationary distribution for movement on a line segment and an accurate approximation for a square area.
Abstract: The random waypoint model is a commonly used mobility model in the simulation of ad hoc networks It is known that the spatial distribution of network nodes moving according to this model is, in general, nonuniform However, a closed-form expression of this distribution and an in-depth investigation is still missing This fact impairs the accuracy of the current simulation methodology of ad hoc networks and makes it impossible to relate simulation-based performance results to corresponding analytical results To overcome these problems, we present a detailed analytical study of the spatial node distribution generated by random waypoint mobility More specifically, we consider a generalization of the model in which the pause time of the mobile nodes is chosen arbitrarily in each waypoint and a fraction of nodes may remain static for the entire simulation time We show that the structure of the resulting distribution is the weighted sum of three independent components: the static, pause, and mobility component This division enables us to understand how the model's parameters influence the distribution We derive an exact equation of the asymptotically stationary distribution for movement on a line segment and an accurate approximation for a square area The good quality of this approximation is validated through simulations using various settings of the mobility parameters In summary, this article gives a fundamental understanding of the behavior of the random waypoint model
1,122 citations
01 Jan 2003
TL;DR: This article considers a generalization of the random waypoint model in which the pause time of the mobile nodes is chosen arbitrarily in each waypoint and a fraction of nodes may remain static for the entire simulation time and derives an exact equation of the asymptotically stationary distribution for movement on a line segment and an accurate approximation for a square area.
1,104 citations
Cites background from "Stochastic properties of the random..."
...vmaxˇvmin [ 5 ]. Thus, it follows immediately that...
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...Only recently have some papers appeared that study its stochastic properties and warn researchers of pitfalls that might occur when using this model (see [2], [ 5 ], [7], [8], [10], [28], [31], [34])....
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13 Mar 2005
TL;DR: A generic mobility model for independent mobiles that contains as special cases the random waypoint on convex or non convex domains, random walk with reflection or wrapping, city section, space graph and other models is defined.
Abstract: We define "random trip", a generic mobility model for independent mobiles that contains as special cases: the random waypoint on convex or non convex domains, random walk with reflection or wrapping, city section, space graph and other models. We use Palm calculus to study the model and give a necessary and sufficient condition for a stationary regime to exist. When this condition is satisfied, we compute the stationary regime and give an algorithm to start a simulation in steady state (perfect simulation). The algorithm does not require the knowledge of geometric constants. For the special case of random waypoint, we provide for the first time a proof and a sufficient and necessary condition of the existence of a stationary regime. Further, we extend its applicability to a broad class of non convex and multi-site examples, and provide a ready-to-use algorithm for perfect simulation. For the special case of random walks with reflection or wrapping, we show that, in the stationary regime, the mobile location is uniformly distributed and is independent of the speed vector, and that there is no speed decay. Our framework provides a rich set of well understood models that can be used to simulate mobile networks with independent node movements. Our perfect sampling is implemented to use with ns-2, and it is freely available to download from http://ica1www.epfl.ch/RandomTrip.
503 citations
TL;DR: In this paper, the stationary spatial distribution of a node moving according to the random waypoint model in a given convex area is analyzed, which is in the form of a one-dimensional integral giving the density up to a normalization constant.
Abstract: The random waypoint model (RWP) is one of the most widely used mobility models in performance analysis of ad hoc networks. We analyze the stationary spatial distribution of a node moving according to the RWP model in a given convex area. For this, we give an explicit expression, which is in the form of a one-dimensional integral giving the density up to a normalization constant. This result is also generalized to the case where the waypoints have a nonuniform distribution. As a special case, we study a modified RWP model, where the waypoints are on the perimeter. The analytical results are illustrated through numerical examples. Moreover, the analytical results are applied to study certain performance aspects of ad hoc networks, namely, connectivity and traffic load distribution.
375 citations
13 Mar 2005
TL;DR: It is established that any piecewise linear movement applied to a user preserves the uniform distribution of position and direction provided that users were initially uniformly throughout the space with equal likelihood of being pointed in any direction.
Abstract: A number of mobility models have been proposed for the purpose of either analyzing or simulating the movement of users in a mobile wireless network. Two of the more popular are the random waypoint and the random direction models. The random waypoint model is physically appealing but difficult to understand. Although the random direction model is less appealing physically, it is much easier to understand. User speeds are easily calculated, unlike for the waypoint model, and, as we observe, user positions and directions are uniformly distributed. The contribution of this paper is to establish this last property for a rich class of random direction models that allow future movements to depend on past movements. To this end, we consider finite oneand two-dimensional spaces. We consider two variations, the random direction model with wrap around and with reflection. We establish a simple relationship between these two models and, for both, show that positions and directions are uniformly distributed for a class of Markov movement models regardless of initial position. In addition, we establish a sample path property for both models, namely that any piecewise linear movement applied to a user preserves the uniform distribution of position and direction provided that users were initially uniformly throughout the space with equal likelihood of being pointed in any direction.
334 citations
References
More filters
Book•
01 Jan 1965
TL;DR: This chapter discusses the concept of a Random Variable, the meaning of Probability, and the axioms of probability in terms of Markov Chains and Queueing Theory.
Abstract: Part 1 Probability and Random Variables 1 The Meaning of Probability 2 The Axioms of Probability 3 Repeated Trials 4 The Concept of a Random Variable 5 Functions of One Random Variable 6 Two Random Variables 7 Sequences of Random Variables 8 Statistics Part 2 Stochastic Processes 9 General Concepts 10 Random Walk and Other Applications 11 Spectral Representation 12 Spectral Estimation 13 Mean Square Estimation 14 Entropy 15 Markov Chains 16 Markov Processes and Queueing Theory
13,886 citations
"Stochastic properties of the random..." refers background in this paper
...In general, the expected value of a variable g (L, V ) can be expressed in terms of the joint pdf fLV (l, v) as [23] E {g (L, V )} = ∫ ∞ −∞ ∫ ∞ −∞ g (l, v) fLV (l, v) dl dv ....
[...]
...Example plots of the beta distribution can be found, for instance, in [23,28]....
[...]
Book•
01 Jan 2002
TL;DR: In this paper, the meaning of probability and random variables are discussed, as well as the axioms of probability, and the concept of a random variable and repeated trials are discussed.
Abstract: Part 1 Probability and Random Variables 1 The Meaning of Probability 2 The Axioms of Probability 3 Repeated Trials 4 The Concept of a Random Variable 5 Functions of One Random Variable 6 Two Random Variables 7 Sequences of Random Variables 8 Statistics Part 2 Stochastic Processes 9 General Concepts 10 Random Walk and Other Applications 11 Spectral Representation 12 Spectral Estimation 13 Mean Square Estimation 14 Entropy 15 Markov Chains 16 Markov Processes and Queueing Theory
12,407 citations
25 Oct 1998
TL;DR: The results of a derailed packet-levelsimulationcomparing fourmulti-hopwirelessad hoc networkroutingprotocols, which cover a range of designchoices: DSDV,TORA, DSR and AODV are presented.
Abstract: An ad hoc networkis a collwtion of wirelessmobilenodes dynamically forminga temporarynetworkwithouttheuseof anyexistingnetworkirrfrastructureor centralizedadministration.Dueto the limitedtransmissionrange of ~vlrelessnenvorkinterfaces,multiplenetwork“hops”maybe neededfor onenodeto exchangedata ivithanotheracrox thenetwork.Inrecentyears, a ttiery of nelvroutingprotocols~geted specificallyat this environment havebeen developed.but little pcrfomrartwinformationon mch protocol and no ralistic performancecomparisonbehvwrrthem ISavailable. ~Is paper presentsthe results of a derailedpacket-levelsimulationcomparing fourmulti-hopwirelessad hoc networkroutingprotocolsthatcovera range of designchoices: DSDV,TORA, DSR and AODV. \Vehave extended the /~r-2networksimulatorto accuratelymodelthe MACandphysical-layer behaviorof the IEEE 802.1I wirelessLANstandard,includinga realistic wtrelesstransmissionchannelmodel, and present the resultsof simulations of net(vorksof 50 mobilenodes.
5,147 citations
"Stochastic properties of the random..." refers methods in this paper
...It is implemented in the network simulation tools ns-2 [27] and GloMoSim [11] and used in several performance evaluations of ad hoc networking protocols [ 7 ,9,16]....
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TL;DR: This article considers the empirical data and then reviews the main approaches to modeling pedestrian and vehicle traffic, including microscopic (particle-based), mesoscopic (gas-kinetic), and macroscopic (fluid-dynamic) models.
Abstract: Since the subject of traffic dynamics has captured the interest of physicists, many surprising effects have been revealed and explained. Some of the questions now understood are the following: Why are vehicles sometimes stopped by ``phantom traffic jams'' even though drivers all like to drive fast? What are the mechanisms behind stop-and-go traffic? Why are there several different kinds of congestion, and how are they related? Why do most traffic jams occur considerably before the road capacity is reached? Can a temporary reduction in the volume of traffic cause a lasting traffic jam? Under which conditions can speed limits speed up traffic? Why do pedestrians moving in opposite directions normally organize into lanes, while similar systems ``freeze by heating''? All of these questions have been answered by applying and extending methods from statistical physics and nonlinear dynamics to self-driven many-particle systems. This article considers the empirical data and then reviews the main approaches to modeling pedestrian and vehicle traffic. These include microscopic (particle-based), mesoscopic (gas-kinetic), and macroscopic (fluid-dynamic) models. Attention is also paid to the formulation of a micro-macro link, to aspects of universality, and to other unifying concepts, such as a general modeling framework for self-driven many-particle systems, including spin systems. While the primary focus is upon vehicle and pedestrian traffic, applications to biological or socio-economic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are touched upon as well.
3,117 citations
"Stochastic properties of the random..." refers background in this paper
...Surveys and classifications on this topic can be found in [4, 16,19,20,29]....
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...Moreover, well–known motion models from physics and chemistry—such as random walk or Brownian motion—and models from transportation theory [8, 15,16] are used in simulations of mobile networks....
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01 Aug 2000
TL;DR: GLS combined with geographic forwarding allows the construction of ad hoc mobile networks that scale to a larger number of nodes than possible with previous work, and compares favorably with Dynamic Source Routing.
Abstract: GLS is a new distributed location service which tracks mobile node locations. GLS combined with geographic forwarding allows the construction of ad hoc mobile networks that scale to a larger number of nodes than possible with previous work. GLS is decentralized and runs on the mobile nodes themselves, requiring no fixed infrastructure. Each mobile node periodically updates a small set of other nodes (its location servers) with its current location. A node sends its position updates to its location servers without knowing their actual identities, assisted by a predefined ordering of node identifiers and a predefined geographic hierarchy. Queries for a mobile node's location also use the predefined identifier ordering and spatial hierarchy to find a location server for that node.Experiments using the ns simulator for up to 600 mobile nodes show that the storage and bandwidth requirements of GLS grow slowly with the size of the network. Furthermore, GLS tolerates node failures well: each failure has only a limited effect and query performance degrades gracefully as nodes fail and restart. The query performance of GLS is also relatively insensitive to node speeds. Simple geographic forwarding combined with GLS compares favorably with Dynamic Source Routing (DSR): in larger networks (over 200 nodes) our approach delivers more packets, but consumes fewer network resources.
1,769 citations
"Stochastic properties of the random..." refers background in this paper
...There are two reasons for studying the cell change rate: first, our major motivation is that some network services in a mobile ad hoc network, like the Grid Location Service [21], assume that the system area shows a grid structure and nodes have to send some signaling messages whenever they move from one cell to another one....
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