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.     

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.     

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.     

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.     

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.     

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.     

Superdiffusive Behavior of Mobile Nodes and Its Impact on Routing Protocol Performance

Mobility is the most important component in mobile ad hoc networks (MANETs) and delay-tolerant networks (DTNs). In this paper, we first investigate numerous GPS mobility traces of human mobile nodes and observe superdiffusive behavior in all GPS traces, which is characterized by a “faster-than-linear” growth rate of the mean square displacement (MSD) of a mobile node. We then investigate a large amount of access point (AP) based traces, and develop a theoretical framework built upon continuous time random walk (CTRW) formalism, in which one can identify the degree of diffusive behavior of mobile nodes even under possibly heavy-tailed pause time distribution, as in the case of reality. We study existing synthetic models and trace-based models in terms of the capability of producing various degrees of diffusive behavior, and use a set of Lévy walk models due to its simplicity and flexibility. In addition, we show that diffusive properties make a huge impact on contact-based metrics and the performance of routing protocols in various scenarios, and that existing models such as random waypoint, random direction model, or Brownian motion lead to overly optimistic or pessimistic results when diffusive properties are not properly captured. Our work in this paper, thus, suggests that the diffusive behavior of mobile nodes should be correctly captured and taken into account for the design and comparison study of network protocols.     

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.     

Queuing network models for delay analysis of multihop wireless ad hoc networks

In this paper we analyze the average end-to-end delay and maximum achievable per-node throughput in random access multihop wireless ad hoc networks with stationary nodes. We present an analytical model that takes into account the number of nodes, the random packet arrival process, the extent of locality of traffic, and the back off and collision avoidance mechanisms of random access MAC. We model random access multihop wireless networks as open G/G/1 queuing networks and use the diffusion approximation in order to evaluate closed form expressions for the average end-to-end delay. The mean service time of nodes is evaluated and used to obtain the maximum achievable per-node throughput. The analytical results obtained here from the queuing network analysis are discussed with regard to similarities and differences from the well established information-theoretic results on throughput and delay scaling laws in ad hoc networks. We also investigate the extent of deviation of delay and throughput in a real world network from the analytical results presented in this paper. We conduct extensive simulations in order to verify the analytical results and also compare them against NS-2 simulations.     

Gamma random waypoint mobility model for wireless ad hoc networks

Random waypoint (RWP) mobility model is widely used in ad hoc network simulation. The model suffers from speed decay as the simulation progresses and may not reach the steady state in terms of instantaneous average node speed. Furthermore, the convergence of the average speed to its steady state value is delayed. This usually leads to inaccurate results in protocol validation of mobile ad hoc networks modeling. Moreover, the probability distributions of speed vary over the simulation time, such that the node speed distribution at the initial state is different from the corresponding distribution at the end of the simulation. In order to overcome these problems, this paper proposes a modified RWP mobility model with a more precise distribution of the nodes' speed. In the modified model, the speeds of nodes are sampled from gamma distribution. The results obtained from both analysis and simulation experiments of the average speed and the density of nodes' speed indicate that the proposed gamma random waypoint mobility model outperforms the existing RWP mobility models. It is shown that a significant performance improvement in achieving higher steady state speed values that closely model the pre‐assumed average speeds are possible with the proposed model. Additionally, the model allows faster convergence to the steady state, and probability distribution of speed is steady over the simulation time. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

Bandwidth estimation in high mobility scenarios of IEEE 802.11 infrastructure‐less mobile ad hoc networks

Bandwidth estimation in mobile ad hoc networks (MANET), where each node can move randomly and is capable of frequently changing its link with other nodes, is a challenging task. Motivation of this work is in contrast with TCP new‐reno which decreases the congestion window both in the event of link failure and congestion, which in the case of packet loss due to link failure should be close to available channel bandwidth. The proposed novel approach capture the node's mobility behavior in broadcast and unicast scenarios of IEEE 802.11 standard to efficiently estimate the sender's window size. This proposal introduces a data structure and source‐to‐destination path stability metric to imitate the mobility behavior of network and presents the analytic characterization of steady‐state throughput as a function of packet loss, round trip time, and path stability over IEEE 802.11 infrastructure‐less MANET. The performance is evaluated over random‐walk, random‐waypoint, and Gauss‐Markov mobility models in 2D and 3D environments using QualNet 7.4 network simulator. The proposed analytical model is also evaluated through two‐tailed statistical test. Analytical, statistical, and simulation‐based comparisons demonstrate the effectiveness of proposed method in high‐mobility scenarios.     

The diameter of mobile ad hoc networks

With the prevalence of mobile devices, it is of much interest to study the properties of mobile ad hoc networks. In this paper, we extend the concept of diameter from static ad hoc network to mobile ad hoc network, which is the expected number of rounds for one node to transmit a message to all other nodes in the network, reflecting the worst end‐to‐end delay between any two node. Specifically, we investigate the diameter of identically and independently mobility model in cell‐partitioned network and random walk mobility model in two‐dimensional torus network, achieving the boundary , when (k=Ω(n)), and O(k log2k), respectively, where n is the number of nodes and k is the number of cells of network and especially under random walk mobility model . A comparison is made among the diameter of mobile ad hoc networks under identically and independently mobility model, random walk mobility model and static ad hoc network, showing that mobility dramatically decreases the diameter of the network and speed is an essential and decisive factor of diameter. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Ensuring strong data guarantees in highly mobile ad hoc networks via quorum systems

Ensuring the consistency and the availability of replicated data in highly mobile ad hoc networks is a challenging task because of the lack of a backbone infrastructure. Previous work provides strong data guarantees by limiting the motion and the speed of the mobile nodes during the entire system lifetime, and by relying on assumptions that are not realistic for most mobile applications. We provide a small set of mobility constraints that are sufficient to ensure strong data guarantees and that can be applied when nodes move along unknown paths and speed, and are sparsely distributed.

In the second part of the paper, we analyze the problem of conserving energy while ensuring strong data guarantees, using quorum system techniques. We devise a condition necessary for a quorum system to guarantee data consistency and data availability under our mobility model. This condition shows the unsuitability of previous quorum systems and is the basis for a novel class of quorum systems suitable for highly mobile networks, called mobile dissemination quorum (MDQ) systems. We also show a MDQ system that is provably optimal in terms of communication cost by proposing an efficient implementation of a read/write atomic shared memory.

The suitability of our mobility model and MDQ systems is validated through simulations using the random waypoint model and the restricted random waypoint on a city section. Finally, we apply our results to assist routing and coordinate the low duty cycle of mobile nodes while maintaining network connectivity.     

Mobility–capacity–delay trade-off in wireless ad hoc networks

We show that there is a trade-off among mobility, capacity, and delay in ad hoc networks. More specifically, we consider two schemes for node mobility in ad hoc networks. We divide the entire network by cells whose sizes can vary with the total number of nodes n, or whose size is independent of the number of nodes. We restrict the movement of nodes within these cells, calculate throughput and delay for randomly chosen pairs of source–destination nodes, and show that mobility is an entity that can be exchanged with capacity and delay. We also investigate the effect of directional antennas in a static network in which packet relaying is done through the closest neighbor and verify that this approach attains better throughput than static networks employing omnidirectional antennas.     

Futuristic speed prediction using auto‐regression and neural networks for mobile ad hoc networks

In this paper, we propose a speed prediction model using auto‐regressive integrated moving average (ARIMA) and neural networks for estimating the futuristic speed of the nodes in mobile ad hoc networks (MANETs). The speed prediction promotes the route discovery process for the selection of moderate mobility nodes to provide reliable routing. The ARIMA is a time‐series forecasting approach, which uses autocorrelations to predict the future speed of nodes. In the paper, the ARIMA model and recurrent neural network (RNN) trains the random waypoint mobility (RWM) dataset to forecast the mobility of the nodes. The proposed ARIMA model designs the prediction models through varying the delay terms and changing the numbers of hidden neuron in RNN. The Akaike information criterion (AIC), Bayesian information criterion (BIC), auto‐correlation function (ACF), and partial auto‐correlation function (PACF) parameters evaluate the predicted mobility dataset to estimate the model quality and reliability. The different scenarios of changing node speed evaluate the performance of prediction models. Performance results indicate that the ARIMA forecasted speed values almost match with the RWM observed speed values than RNN values. The graphs exhibit that the ARIMA predicted mobility values have lower error metrics such as mean square error (MSE), root MSE (RMSE), and mean absolute error (MAE) than RNN predictions. It yields higher futuristic speed prediction precision rate of 17% to 24% throughout the time series as compared with RNN. Further, the proposed model extensively compares with the existing works.     

On the stationary distribution of speed and location of random waypoint

In "Stationary distributions for the random waypoint mobility model" (TMC, Vol. 3, No, 1), Navidi and Camp find the stationary distribution of the random waypoint model, with or without pause on a rectangular area. In this short note, we show that, under the stationary regime, speed and location are independent.     

An Analytical Model for the Node Degree in Wireless Ad Hoc Networks

Node degree is regarded as an important and convenient metric to measure the connectivity of wireless ad hoc networks. Existing studies are mainly based on the assumption that nodes are static and do not provide closed-form expressions for node degree. In this paper, we investigate three fundamental characteristics of a wireless ad hoc network: Its node degree distribution, its average node degree and its maximum node degree experienced by the nodes during their movement. We introduce a novel mathematical model to derive analytical expressions in the presence of radio channel fading. Furthermore, our results reveal that the node degree distribution follows a binomial distribution regardless of the initial distribution of nodes?? location. The results of this paper are useful to study node connectivity and to improve the algorithmic complexity of incentive protocols.