Stationary distributions for the random waypoint mobility model

In simulations of mobile ad hoc networks, the probability distribution governing the movement of the nodes typically varies over time and converges to a "steady-state" distribution, known in the probability literature as the stationary distribution. Some published simulation results ignore this initialization discrepancy. For those results that attempt to account for this discrepancy, the practice is to discard an initial sequence of observations from a simulation in the hope that the remaining values will closely represent the stationary distribution. This approach is inefficient and not always reliable. However, if the initial locations and speeds of the nodes are chosen from the stationary distribution, convergence is immediate and no data need be discarded. We derive the stationary distributions for location, speed, and pause time for the random waypoint mobility model. We then show how to implement the random waypoint mobility model in order to construct more efficient and reliable simulations for mobile ad hoc networks. Simulation results, which verify the correctness of our method, are included. In addition, implementation of our method for the NS-2 simulator is available.     

The random waypoint mobility model with uniform node spatial distribution

In this paper, we tackle the problem of designing a random mobility model generating a target node spatial distribution. More specifically, we solve a long standing open problem by presenting two versions of the well-known random waypoint (RWP) mobility model in bounded regions generating a uniform steady-state node spatial distribution. In the first version, named temporal-RWP, we exploit the temporal dimension of node mobility and achieve uniformity by continuously changing the speed of a mobile node as a function of its location and of the density function of trajectories in the movement region R. In the second version, named spatial-RWP, we instead exploit the spatial dimension and achieve uniformity by selecting waypoints according to a suitably defined mix of probability density functions. Both proposed models can be easily incorporated in wireless network simulators, and are thus of practical use. The RWP models presented in this paper allow for the first time completely removing the well-known border effect causing possible inaccuracies in mobile network simulation, thus completing the picture of a “perfect” simulation methodology drawn in existing literature.     

Spatial node distribution of the random waypoint mobility model with applications

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.     

Integration of mobility and intrusion detection for wireless ad hoc networks

One of the main challenges in building intrusion detection systems (IDSs) for mobile ad hoc networks (MANETs) is to integrate mobility impacts and to adjust the behaviour of IDSs correspondingly. In this paper, we first introduce two different approaches, a Markov chain‐based approach and a Hotelling's T² test based approach, to construct local IDSs for MANETs. We then demonstrate that nodes' moving speed, a commonly used parameter in tuning IDS performances, is not an effective metric to tune IDS performances under different mobility models. To solve this problem, we further propose an adaptive scheme, in which suitable normal profiles and corresponding proper thresholds can be selected adaptively by each local IDS through periodically measuring its local link change rate, a proposed unified performance metric. We study the proposed adaptive mechanism at different mobility levels, using different mobility models such as random waypoint model, random drunken model, and obstacle mobility model. Simulation results show that our proposed adaptive scheme is less dependent on the underlying mobility models and can further reduce false positive ratio. Copyright © 2006 John Wiley & Sons, Ltd.     

Epoch, length of the random waypoint model in mobile ad hoc networks

In this paper, we derive the probability distribution of the epoch length for the random waypoint model in mobile ad hoc networks. An epoch here is referred to as the movement between two target locations in the mobility model. Such a study is important as the epoch length distribution may be required for the derivation of the link-duration distribution or node spatial distribution for mobile ad hoc networks. The analytical result is then verified via simulation.     

Large intelligent surfaces: Random waypoint mobility and two‐way relaying

In this paper, we first investigate the effect of mobility via the random waypoint (RWP) mobility model on the performance of nonaccess point (non‐AP) or AP large intelligent surfaces (LISs). The theoretical average bit error probability (ABEP) for each of these LISs under mobility is formulated. The presented formulation is complicated to solve; hence, the trapezoidal approximation is employed. Simulation results serve to validate the ABEP. Second, we investigate a two‐way relaying (TWR) network assisted by non‐AP or AP LISs. A network with two source/destination nodes with a single relay node employing decode‐and‐forward placed between these nodes is considered. The transmission interval is broken into two transmission phases. In the first phase, the two source nodes transmit information blocks to the relay node assisted by LISs. On receiving these information blocks, the relay node decodes the two information blocks and encodes these into a single information block via the use of network coding. In the second phase, the relay node forwards the network‐coded information block to the destination nodes assisted by LISs, where the intended information block is decoded via network coding. The theoretical ABEP is formulated for the proposed non‐AP and AP LIS‐assisted TWR schemes and is validated by simulation results. RWP mobility is also demonstrated for the proposed TWR schemes.     

Network connectivity of one-dimensional MANETs with random waypoint movement

An analysis of network connectivity of one-dimensional mobile ad hoc networks with a particular mobility scheme is presented, focusing on the random waypoint mobility scheme. The numerical results are verified using simulation to show their accuracy under practical network conditions. Observations on RWP properties further lead to approximations and an eventual simple network connectivity formula.     

Random waypoint mobility model in cellular networks

In this paper we study the so-called random waypoint (RWP) mobility model in the context of cellular networks. In the RWP model the nodes, i.e., mobile users, move along a zigzag path consisting of straight legs from one waypoint to the next. Each waypoint is assumed to be drawn from the uniform distribution over the given convex domain. In this paper we characterise the key performance measures, mean handover rate and mean sojourn time from the point of view of an arbitrary cell, as well as the mean handover rate in the network. To this end, we present an exact analytical formula for the mean arrival rate across an arbitrary curve. This result together with the pdf of the node location, allows us to compute all other interesting measures. The results are illustrated by several numerical examples. For instance, as a straightforward application of these results one can easily adjust the model parameters in a simulation so that the scenario matches well with, e.g., the measured sojourn times in a cell.     

The node distribution of the random waypoint mobility model for wireless ad hoc networks

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.     

A Statistical Analysis of the Long-Run Node Spatial Distribution in Mobile Ad Hoc Networks

In this paper, we analyze the node spatial distribution of mobile wireless ad hoc networks. Characterizing this distribution is of fundamental importance in the analysis of many relevant properties of mobile ad hoc networks, such as connectivity, average route length, and network capacity. In particular, we have investigated under what conditions the node spatial distribution resulting after a large number of mobility steps resembles the uniform distribution. This is motivated by the fact that the existing theoretical results concerning mobile ad hoc networks are based on this assumption. In order to test this hypothesis, we performed extensive simulations using two well-known mobility models: the random waypoint model, which resembles intentional movement, and a Brownian-like model, which resembles non-intentional movement. Our analysis has shown that in Brownian-like motion the uniformity assumption does hold, and that the intensity of the concentration of nodes in the center of the deployment region that occurs in the random waypoint model heavily depends on the choice of some mobility parameters. For extreme values of these parameters, the uniformity assumption is impaired.     

Ad Hoc网络移动模型节点空间分布研究

节点空间分布是Ad Hoc网络移动模型研究的重点,也是对移动模型评价的重要指标.对RW,RD,RWP,GM,SMS移动模型分别在不同条件下处于矩形及圆仿真区域中节点空间分布进行了仿真,并对其进行比较研究,得出了不同速度、速度服从不同分布,暂停时间服从不同分布和不同仿真区域对移动模型节点空间分布的影响,为移动模型的选择、仿真区域的选择及模型中各类参数的设置提供了重要的依据.     

Stochastic Properties of Mobility Models in Mobile Ad Hoc Networks

The stochastic model assumed to govern the mobility of nodes in a mobile ad hoc network has been shown to significantly affect the network's coverage, maximum throughput, and achievable throughput-delay trade-offs. In this paper, we compare several mobility models, including the random walk, random waypoint, and Manhattan models on the basis of the number of states visited in a fixed time, the time to visit every state in a region, and the effect of the number of wandering nodes on the time to first enter a set of states. These metrics for a mobility model are useful for assessing the achievable event detection rates in surveillance applications where wireless-sensor-equipped vehicles are used to detect events of interest in a city. We also consider mobility models based on Correlated Random Walks, which can account for time dependency, geographical restrictions, and nonzero drift. We demonstrate that these models are analytically tractable by using a matrix-analytic approach to derive new, closed-form results in both the time and transform-domains for the probability that a node is at any location at any time for both semi-infinite and finite 1D lattices. We also derive first entrance time distributions for these walks. We find that a correlated random walk 1) covers more ground in a given amount of time and takes a smaller amount of time to cover an area completely than a random walk with the same average transition rate, 2) has a smaller first entrance time to small sets of states than the random waypoint and random walk models, and 3) leads to a uniform distribution of nodes (except at the boundaries) in steady state.     

The Random Trip Model: Stability, Stationary Regime, and Perfect Simulation

We define "random trip", a generic mobility model for random, independent node motions, which contains as special cases: the random waypoint on convex or nonconvex domains, random walk on torus, billiards, city section, space graph, intercity and other models. We show that, for this model, a necessary and sufficient condition for a time-stationary regime to exist is that the mean trip duration (sampled at trip endpoints) is finite. When this holds, we show that the distribution of node mobility state converges to the time-stationary distribution, starting from the origin of an arbitrary trip. 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, thus closing a long standing issue. We show that random walk on torus and billiards belong to the random trip class of models, and establish that the time-limit distribution of node location for these two models is uniform, for any initial distribution, even in cases where the speed vector does not have circular symmetry. Using Palm calculus, we establish properties of the time-stationary regime, when the condition for its existence holds. We provide an algorithm to sample the simulation state from a time-stationary distribution at time 0 ("perfect simulation"), without computing geometric constants. For random waypoint on the sphere, random walk on torus and billiards, we show that, in the time-stationary regime, the node location is uniform. Our perfect sampling algorithm is implemented to use with ns-2, and is available to download from http://ica1www.epfl.ch/RandomTrip     

Evaluating inter-arrival time in general random waypoint mobility model

In the study of wireless ad hoc networks, the Random Waypoint (RWP) mobility model is extensively used to describe the movement pattern of the hosts. In this paper, we extend our discussion to the general RWP mobility model, where the waypoints may not be uniformly distributed, and hosts may use different distributions to generate their waypoints. In particular, we study a useful property, namely the Inter-Arrival Time (IAT) of hosts to a given area in a network, when the mobility of such hosts is modeled using the random waypoint mobility model. We derive the value of IAT analytically. Three schemes are used to estimate its value. The correctness of our analysis and estimations are verified through simulations. Case studies are also carried out in this paper to show how IAT could be used in the study and applications of mobile wireless networks.     

The critical transmitting range for connectivity in mobile ad hoc networks

In this paper, we have investigated the critical transmitting range for connectivity in mobile ad hoc networks. We have proven that, in the presence of bounded and obstacle free mobility, the CTR in the mobile case is at least as large as the CTR in the case of uniformly distributed points (asymptotically). For the case of RWP mobility, we have proven a more accurate characterization of the CTR and shown that, if the pause time is 0, there is an asymptotic gap between the mobile and uniform scenario. We have verified the quality of our results through simulation. We have also presented a formula that, given the value of the CTR in the uniform case, provides a good approximation of the CTR in the most extreme case of RWP mobility, i.e., when the pause time is set to 0. We want to remark that the approach presented in this paper can be easily extended to other mobility models: If the expression of the pdf f/sub m/ that resembles the long-term node distribution is known and satisfies certain properties, it is sufficient to compute the minimum value of f/sub m/ on R to determine the value of the critical range for connectivity. We believe that the results presented in this paper provide a better understanding of the behavior of a fundamental network parameter in the presence of mobility and, in particular, of RWP mobility. From a practical point of view, our results can be used to improve the accuracy of RWP mobile ad hoc networks simulation, which is commonly used to evaluate the performance of ad hoc networking protocols.     

Throughput analysis for interference limited TDMA and TDMA/CDMA wireless ad hoc networks

The node throughput, which is defined as the total rate received at each node, is evaluated for the interference limited TDMA and TDMA/CDMA wireless ad hoc networks. In the TDMA wireless ad hoc network, there is only one transmission link connected to each node in the same time slot, whereas in the TDMA/CDMA wireless ad hoc network there are multiple transmission links connected to each node in the same time slot. We first derive the node throughput for these two wireless ad hoc networks and then make a comparison of the node throughput between them. The theoretical results and simulation results both reveal that the TDMA wireless ad hoc network outperforms the TDMA/CDMA wireless ad hoc network in the node throughput. Copyright © 2008 John Wiley & Sons, Ltd.     

Nonlinearities compensator based on SORBFNN in ultra-long-haul high-capacity CO-OFDM systems

The ultraviolet (UV) scattering communication can be applied in military networked on-the-move and unattended ground sensor networks. This paper focuses on the connectivity properties of unmanned aerial vehicles (UAVs) network based on UV communication that ensures the secret communications between UAVs. UAVs network is consisting of a group of UAVs, each of which moving according to a particular mobility model. We discuss random waypoint (RWP) mobility model and circle movement based model (CMBM), which can describe the actual movement of UAVs, respectively. In this paper, the approximations of the probability that the network is k-connected are provided with consideration of transmission using on–off keying and pulse position modulation. More precisely, we evaluate the effects of node density, transmission power, and data rate on k-connectivity. The numerical examples show that the mobility degrades the connectivity probability. When the numbers of nodes \(n=500\) and data rate \(R_b =10\,\hbox {kbps}\), the required transmission power for nodes moving according to RWP is lower than CMBM in order to achieve 2-connectivity of the UAVs network, but it is about twice that for uniformly distributed nodes.     

A unified mobility model for analysis and simulation of mobile wireless networks

We propose a novel mobility model, named Semi-Markov Smooth (SMS) model, to characterize the smooth movement of mobile users in accordance with the physical law of motion in order to eliminate sharp turns, abrupt speed change and sudden stops exhibited by existing models. We formulate the smooth mobility model by a semi-Markov process to analyze the steady state properties of this model because the transition time between consecutive phases (states) has a discrete uniform distribution, instead of an exponential distribution. Through stochastic analysis, we prove that this model unifies many good features for analysis and simulations of mobile networks. First, it is smooth and steady because there is no speed decay problem for arbitrary starting speed, while maintaining uniform spatial node distribution regardless of node placement. Second, it can be easily and flexibly applied for simulating node mobility in wireless networks. It can also adapt to different network environments such as group mobility and geographic constraints. To demonstrate the impact of this model, we evaluate the effect of this model on distribution of relative speed, link lifetime between neighboring nodes, and average node degree by ns-2 simulations.

Adaptive backoff scheme for ad hoc networks based on IEEE 802.11

The performance of backoff scheme plays an important role in designing efficient Medium Access Protocols for ad hoc networks. In this paper, we propose an adaptive backoff scheme and evaluate the performance of the proposed scheme for ad hoc networks. The backoff mechanism devised by us grants a node access to the channel based on its probability of collision for a transmitted frame in comparison to the nodes in the two‐hop contention area. We use both an analytical model and simulation experiments to evaluate the performance of our adaptive backoff mechanism in an ad hoc network. The results show that our protocol exhibits a significant improvement in power saving, end‐to‐end goodput, packet delivery ratio, and hop‐put, compared with the existing IEEE 802.11 DCF. Copyright © 2010 John Wiley & Sons, Ltd.     

Two-Dimensional Modeling and Analysis of Generalized Random Mobility Models for Wireless Ad Hoc Networks

Most important characteristics of wireless ad hoc networks, such as link distance distribution, connectivity, and network capacity are dependent on the long-run properties of the mobility profiles of communicating terminals. Therefore, the analysis of the mobility models proposed for these networks becomes crucial. The contribution of this paper is to provide an analytical framework that is generalized enough to perform the analysis of realistic random movement models over two-dimensional regions. The synthetic scenarios that can be captured include hotspots where mobiles accumulate with higher probability and spend more time, and take into consideration location and displacement dependent speed distributions. By the utilization of the framework to the random waypoint mobility model, we derive an approximation to the spatial distribution of terminals over rectangular regions. We validate the accuracy of this approximation via simulation, and by comparing the marginals with proven results for one-dimensional regions, we find out that the quality of the approximation is insensitive to the proportion between dimensions of the terrain.