AbstractâIt is generally foreseen that the number of wirelessly connected networking devices will increase in the next decades, leading to a rise in the number ...
NetDetect: Neighborhood Discovery in Wireless Networks Using Adaptive Beacons Venkatraman Iyer, Andrei Pruteanu, Stefan Dulman Delft University of Technology, The Netherlands Emails: {v.g.iyer, a.s.pruteanu, s.o.dulman}@tudelft.nl
Abstract—It is generally foreseen that the number of wirelessly connected networking devices will increase in the next decades, leading to a rise in the number of applications involving largescale networks. A major building block for enabling self-* system properties in ad-hoc scenarios is the run-time discovery of neighboring devices and somewhat equivalently, the estimation of the local node density. This problem has been studied extensively before, mainly in the context of fully-connected, synchronized networks. In this paper, we propose a novel adaptive and decentralized solution, the NetDetect algorithm, to the problem of discovering neighbors in a dynamic wireless network. The main difference with existing state of the art is that we target dynamic scenarios, i.e., multihop mesh networks involving mobile devices. The algorithm exploits the beaconing communication mechanism, dynamically adapting the beacon rate of the devices in the network based on local estimates of neighbor densities. We evaluate NetDetect on a variety of networks with increasing levels of dynamics: fully-connected networks, static and mobile multi-hop mesh networks. Results show that NetDetect performs well in all considered scenarios, maintaining a high rate of neighbor discoveries and good estimate of the neighborhood density even in very dynamic situations. More importantly, the proposed solution is adaptive, tracking changes in the local environment of the nodes without any additional algorithmic reconfiguration. Comparison with existing approaches shows that the proposed scheme is efficient from both convergence time and energy perspectives.
I. I NTRODUCTION Technological advances in Embedded Systems have led to an increase in the number of small-sized sensing and communicating devices equipped with wireless interfaces. Networks of such devices find applicability in monitoring and surveillance activities for Wireless Sensor Networks (WSNs) [16], [21], vehicular traffic policing [11], [17], human social networks [15], etc. It is generally foreseen that the number of devices will increase during the next decades, leading to a rise in the number of application scenarios involving large networks. A fundamental requirement for enabling self-* system properties of wirelessly connected systems is the need of maintaining neighborhood information locally on each node (meaning an estimate of the local node density and the identifiers of its neighbors). The simplest application example is that of routing in wireless sensor networks [5], where every node maintains a list of neighbors to forward its packets to. Maintaining neighbor state has been used in mobile networks as well, such as for instance, in docking applications where mobile
nodes offload relevant tracking information to a static gateway device. In recent years, the emergence of gossip-based network aggregation protocols (such as Push-Sum [8]) has fostered an active interest to their applicability to both static and mobile wireless networks. Particularly, the class of protocols discussed in [8] demands every node to maintain knowledge of its neighboring nodes, in order to aggregate their values. With the increase in the number of deployed wireless devices, sharing the communication medium becomes a major concern. Especially for the case of dense networks, message collisions can reduce throughput to an extent such as to inhibit any meaningful network coordination. Furthermore, with the growing popularity of mobile applications where links are constructed and taken down on-the-fly, scheduling communication between participating nodes becomes increasingly difficult. From the perspective of achieving neighborhood discovery, we pose and answer the following research questions: 1) What is the best discovery performance that can be obtained in a distributed (mobile) network? 2) Is there a general strategy for achieving neighborhood discovery in both static multi-hop and mobile networks? 3) Can the discovery strategy self-adapt to changing circumstances in per-node neighborhood sizes, a common behavior in mobile networks? The answer to Question 1 gives a bound over what is achievable in terms of communication throughput and the number of neighbors that are discovered. The answer to Question 2 is constructive, in the sense that we provide a distributed algorithm that approaches the boundary pertaining to Question 1. The answer to Question 3 takes the form of an adaptive mechanism that modifies transmission rates on the nodes, in response to changes in neighborhood size. In this paper, we focus on the problem of efficiently discovering neighbors in a wireless network of nodes. Considering the class of protocols that rely on probabilistic transmissions, such as [14] and [20], we focus on maximizing the number of discoveries per time unit, and look for strategies that allow nodes to discover their neighbors as fast and as efficient as possible. Existing approaches target mainly static, fullyconnected networks [20] and suffer from a bootstrap problem as all nodes need to start at the same moment in time. The main result of the paper is the design and implementation of an adaptive and decentralized strategy, named NetDetect, that solves the local neighborhood discovery problem
by exploiting the beaconing mechanism. The approach is to have nodes estimate the local neighborhood size from the number of errors detected on the communication channel with a maximum likelihood estimator. The node interaction leads to a self adaptive mechanism, where the beaconing probability swiftly converges to the desired optimum. We evaluate the algorithm on a variety of networks with increasing levels of dynamics: a fully-connected network, static and mobile multihop mesh network. Results show that NetDetect performs well in all considered scenarios, maintaining a high rate of neighbor discoveries and good estimates of the neighborhood densities even in very dynamic situations. Comparison with existing approaches shows that the NetDetect scheme is efficient from both the convergence time and energy perspective. In this paper we make the following contributions: 1) we identify a strategy of maximizing discovery efficiency in a distributed (mobile) network, 2) we propose, design and implement NetDetect, a distributed algorithm wherein nodes adapt their probabilities of transmission based on locally measured throughput, and 3) we evaluate NetDetect in comparison to existing neighborhood discovery protocols and showcase its efficiency and robustness. We chose to use the errors on a communication channel as the main source of information for estimating the local neighborhoods, as this information is always present at every node in a wireless communication environment. This leads to an elegant solution only exploiting already existing information. Measuring channel contention in order to change the transmission channel frequency [10], or adapt contention window sizes [6] has already been researched. Nevertheless, to the best of our knowledge, this is the first adaptive algorithm exploiting channel errors information for deriving information regarding the available neighbors, working in a distributed environment and being able to track changes without additional user actions. While we designed NetDetect specifically for low power wireless networks, the algorithm is directly applicable to other systems that can be modeled using Poisson processes, and that are in need of a self-adaptive behavior. The lightweight communication overhead of the algorithm combined with its applicability on a broad range of scenarios characterized by different network topologies and its robustness, make NetDetect an attractive solution. The paper is organized as follows: in Section II, we cover the related work on neighborhood discovery in wireless networks. Section III introduces the NetDetect algorithm, and describes the components that enable nodes to adapt their transmission probabilities in response to observed channel contention. Section IV details the sensitivity of the adaptive algorithm to control parameters that are explained in Section III. It then proceeds to compare the performance of the NetDetect algorithm with a state of the art protocol for neighborhood discovery, and also showcases the adaptivity part with specific test cases. Finally, we conclude the paper in Section V with
remarks on future research and extensions to our work. II. R ELATED W ORK Neighborhood discovery in wireless networks has been an actively researched topic over the years. Broadly, we can classify the related work as being either algorithmic [13], [14], [20], or more practical [4], [7], [19]. In the following paragraphs, we cover the literature on each of these works. Algorithms for neighborhood discovery are either deterministic [9], rely on probabilistic approaches [2], [14], [20] or use group testing strategies [12], [13]. Probabilistic approaches for neighborhood discovery apply to the class of random-access protocols, and assume slotted behavior for communication. Participating nodes send and receive messages at every slot with probabilities that are either fixed or adaptive, depending on chosen policies. For example, McGlynn et al. [14] address the problem of energy-efficient neighborhood discovery at deployment time for a static WSN. As a performance metric, they consider the fraction of links that are discovered per time unit in a clique of N nodes. Their algorithm, the socalled probabilistic round robin (PRR) performs best when the probability of transmission for every node is set to N1 . Vasudevan et al. [20] model neighborhood discovery as a coupon collector’s problem, and formulate two algorithms, one that is Aloha-like and the other that relies on collision detection mechanisms. Both the algorithms are adaptive in nature, and typically involve nodes decreasing their transmission probabilities progressively over expanding intervals of time. Discovery convergence, i. e., the time it takes a node to find all its neighbors is bounded by Θ(n ln(n)) for the Alohalike protocol and Θ(n) for the algorithm that uses collision detection. However, the algorithms use specific terminating conditions that work only for static network topologies. The work by [2] derives the optimal settings for transmission probability and message length, per node, that maximize the average number of packet receptions in unit time. However, the algorithm requires the nodes to have an initial estimate of their neighborhood size. In contrast to probabilistic transmission, discovery algorithms that use group testing are designed for identifying a small subset of nodes as neighbors from a much larger set. The algorithms in [12], [13] work by a simple principle of elimination, wherein a central node initiates a series of socalled signatures from its neighbors. A signature from a neighbor is typically a collection of binary responses, which are logically ORed to infer proximity. Nodes that are unresponsive are considered as not being neighbors. These algorithms score over the random access discovery schemes mainly on latency. However, the disadvantages with these protocols are a need for strict time synchronization, and knowledge of the total number of nodes in the network. There have been approaches in the field of ad-hoc networks and WSNs that achieve energy-efficient neighborhood discovery. Minimizing power consumption is a typical concern for duty cycling sensor motes. Particularly, the research objective in these works is directed towards discovering neighbors in
the shortest possible time, for a fixed radio duty cycle. For example, Zheng et al. [22] propose a (k 2 + k + 1, k + 1, 1) design for discovering neighbors in a network, where each node has a duty cycle of (k + 1)/(k 2 + k + 1). The design is suited for symmetric neighbor discoveries, i. e., every node operates at the same duty cycle. Dutta et al. [4] propose Disco, a protocol that bounds the pairwise discovery between nodes, using pairs of prime numbers (p1 , p2 ), to duty cycle their radio. Therefore, a node wakes up every p1 and p2 slots to communicate with its neighbors, such that pairwise discovery is bounded by Θ(p1 · p2 ). Disco involves a complicated choice of the right pairs of primes for a required duty-cycle, that minimizes the discovery latency. Quorumbased protocols [19] divide discrete time slots into blocks of size n × n slots, where n is a global parameter. Every node selects a random row and column from the square matrix, for communication. Since every pair of rows and columns has at least two intersection points, discovery is guaranteed in Θ(n2 ) time. Kandhalu et al. [7] implemented UConnect, a discovery protocol for sensor nodes that uses only a single prime number p, as the duty cycling parameter. Unlike Disco [4], the algorithm is composed of a Low-power Listening mode, wherein nodes wakeup every p slots to receive messages, and a Low-power Transmit mode, wherein nodes wakeup every p2 slots, and transmit a message with a preamble that lasts p+1 slots. Pairwise discovery between nodes is guaranteed 2 with a latency in the order of Θ(p2 ) slots. Uconnect scores over Disco and Quorum-based protocols on the power-latency product, a metric that captures the trade off between average power consumption and worst-case discovery latency. An observation that runs common to most of the works detailed above is that nodes in the network use a set of parameters, such as radio duty cycle or transmission probability, that are mostly fixed at compile-time. An exception is perhaps, the work by Vasudevan et al. [20], in which nodes progressively adapt their transmission probabilities until the procedure terminates. However, for dense neighborhoods (> 20 nodes), or for mobile networks in which the neighborhood fluctuates over a period of time, setting these parameters to a single value will not always yield the best result in terms of either energy consumption or discovery latency. In comparison, our work abstracts away from the requirements of energy conservation, and focuses on adaptively changing the probabilities of message transmission, in response to changing neighborhoods. Wireless links in mobile networks are mostly transient, i. e., links between neighboring nodes exist only for a short duration. Related works [1], [3], [18] have focused on this so-called link persistence, and have attempted to characterize it for various mobility models. While such efforts have been largely targeted towards estimating residual link duration for routing in MANETs, one of the possible extensions to our approach is that of exploiting the knowledge of link persistence, to dynamically maintain a neighbor table at each node in a mobile environment.
Algorithm 1 NetDetect(pri , di ) 1: ⊲ ph i represents the set of last received probabilities 2: ⊲ dh i represents the set of inter-arrival times between messages 3: ⊲ α is a constant ∈ (0, 1) 4: ⊲ compute the current neighborhood size estimator h 5: n ˆ i ← MLE(pri , di , ph i , di ) 6: ⊲ compute current probability 7: pˆi ← 1/ˆ ni 8: pi ← (1 − α)pˆi + αpri 9: ⊲ return value 10: pi
III. NetDetect A LGORITHM In this section we will introduce the main contribution of the paper, the NetDetect algorithm. After defining the target we want to achieve, we introduce the neighborhood size estimator and the adaptive algorithm approaching it. A. General Concepts In the following subsection, for simplicity of exposition, we consider a discrete time model in which communication takes place in discrete time slots. We focus only on the case in which the devices in the network can find themselves in one of the two states: transmitting or receiving information. We consider that if two or more devices transmit in the same time slot, a collision will occur and the neighboring nodes will not be able to distinguish any of the information being sent. We do not distinguish between an empty slot and a slot with collisions. Identifying one of the two cases is feasible with today’s radio technology and will bring a minor improvement to the estimator presented next. Nevertheless, in this paper we target the overall analysis of the novel technique, and will not explore incremental optimizations of the algorithm. If only one node transmits in a slot, we assume its neighbors will be able to receive the message - these slots will be referred to as successful slots. As previously described, we approach the problem from the perspective of a periodic beaconing mechanism. Each node i has a local probability pi [k] which can be modified with time (k indexes the time slots). At each moment k, the node i will transmit a packet with probability pi . The content of a successful slot consists of the node identifier (used for identifying the neighborhood) and the probability with which the node transmitted the packet. It is well known that, for a uniform distribution of independent beaconing probabilities, the expected number of transmissions in a given time slot follows a Poisson process. For our problem, we are interested in the specific case of having only one successful transmission in a given time slot. As shown in previous work [14], for a clique of n nodes, the maximum number of successful slots is attained for all nodes transmitting with the same probability pi = n1 (see Figure 1). In this optimal case, the ratio of successful slots converges to e. This means, that, on average in the best case scenario, we will have one successful slot occurring every 2.718 slots.
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This sets the goal for designing our algorithm. The problem can be formulated now as: assume a network with unknown topology and an initial set of random beaconing probabilities for the nodes. The objective is to find an algorithm that will ensure all the nodes converge to the locally optimum beaconing probability. Particularly for a clique, this equals the reciprocal of their neighborhood size; so the algorithm will automatically estimate the local neighborhood size as well. Apart from the slotted approach and the beaconing mechanism described above, nothing else is assumed (no synchronization information, no additional messages being exchanged between the nodes, no central authority or special nodes in the network, no information on average densities, topologies or mobility patterns). The solution we propose, the NetDetect algorithm, is presented in Algorithm 1. Conceptually, it contains two mechanisms that work in parallel (in practice the mechanisms overlap, as detailed in the next sections). The first mechanism is a form of distributed consensus: nodes spatially close to each other will converge towards an average of each-other’s probabilities via a mechanism similar to gossiping [8]. The second mechanism will push these local values towards the reciprocal of the local density of nodes, using a maximum likelihood estimator based on the distribution of successful slots, achieving the desired goal. B. Consensus The first mechanism we address is the distributed consensus. To illustrate it, we consider a clique in which each of the n nodes holds a (random) probability pi . In Algorithm 1, at line 8, assume that the constant α = 0.4 and that only the second term in the right hand side is considered. Thus, the current probability of the node i is then updated only based on the probabilities heard from the successful slots pri [k], pri [k − 1], ... as a weighted average. It is easy to
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Convergence of the Consensus Algorithm in a clique of 100 nodes. Horizontal axis represents time, while the vertical axis represents the variance in the exchanged probability values. The lines indicate that the nodes agree on a common value exponentially fast for low values of α. Fig. 2.
see that this mechanism maps onto a distributed gossip-based consensus (see Figure 2). The difference with the averaging algorithms presented in [8], is that the value to which the aggregate converges cannot be predicted beforehand - it is a value situated between the minimum and maximum pi , but the order in which the nodes advertise their values in the successful slots will influence it. Being from the same class as the gossiping algorithms, the consensus algorithm is assured to rapidly converge in fully-connected and mobile mesh networks. The convergence in static meshes will be slow at global level but fast for local neighborhoods - while in some algorithms this is a disadvantage, in our case it actually helps, being able to support networks in which the local densities are not constant. The constant α can be used to control the trade off between the rate of convergence for the consensus mechanism and the MLE estimator. It does not play a crucial role, and influences only moderately the speed of convergence. We found that any value in the range of 0.2-0.4 works fine for all topologies and mobility models considered in this paper. C. Maximum Likelihood Estimator In this section we introduce the derivation of the estimator for the local probabilities pi (i being the node index i ∈ {1, 2, ..., n}). The estimator works by taking into account the spacing in time between the successfully transmitted packets. To illustrate its underlying mechanism, we consider having the network in the steady state in which the probabilities on all nodes have stabilized to the same value. If each of the n nodes transmits in the current slot with the probability pi = p, it follows that the probability of having only one node transmitting in the current slot is npi (1 − pi )n−1 . Let d be a random variable describing the inter-arrival time of the successful packets. Based on the process from which the packets are generated, d is an exponential random variable, conditioned on the local probabilities pi = p via the
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Fig. 3. Mean estimated Neighborhood Size (left), and Standard deviation (right) for the NetDetect algorithm. The horizontal axes represent varying neighborhood sizes and history queue sizes, while the vertical axis represents the mean estimated values and their corresponding variance. The variance in the estimated values drop rapidly as the history window size is increased.
parameter λ. Its punctual distribution function is thus: fd (d|λ) = λe−λd H(d)
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When the network has “stabilized” (i. e., the probability on each node has converged to the same value), each node has for a series of m consecutive successful slots, access to a list of tuples {pj , dj }. Here, pj denotes the probability heard from the nodes that generated the successful slots, and dj the distance, in slots, between the successful message reception events. The notation j denotes an iterator over the number of successful slots, j ∈ {1, 2, ..., m}. Knowing the punctual distribution function for the dj random variable, the question we ask is finding an estimator n ˆ j for the number of nodes in the network that generated the outcomes dj . Once n ˆ j has been determined, each node can choose the probability of transmitting in the next round as pj+1 = nˆ1j (line 7 of Algorithm 1). As a solution, we determine n ˆ j using the maximum likelihood estimation technique. First we introduce the likelihood function: m Y (3) λj e−λj dj L(dj |λj ) = j=1
with λj defined by λj = npj (1 − pj )n−1 . The log-likelihood function is then given by: ˆl(dj |n) = 1 log L(dj |λj ) = m m 1 X (log λj − λj dj ) = m j=1
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The value n ˆ j we are searching is the value that maximizes the ˆl(dj |λj ) expression - λj is a function of n thus ˆl(dj |λj ) =
ˆl(dj |n). n ˆ j is to be found through the extremum values of the ˆl(dj |λj ) function. In other words, n ˆ j is to be found in the solution set of: ∂ ˆl(dj |n) =0 (5) ∂n This equation expands to: m X 1 ∂λj ( − dj ) =0 λ ∂n j j=1
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The values n for which this equation holds cannot be computed with a closed-form expression. Thus, the procedure is to obtain a numerical estimation of the solution. n ˆ j will be chosen as the solution that maximizes Equation 4. The MLE estimator provides a very good result. As it is proven that MLE estimators, when they exist, achieve the Cramer-Rao bound (they are the unbiased estimators achieving the lowest possible standard deviation), this is the best approach one could choose from the class of unbiased estimators. The presented MLE is a one-shot estimator - in one step it offers a value with an accuracy that depends on the number of samples considered (the tuples {pj , dj }). In the description of Algorithm 1, the notations phi and dhi denote the buffers for these samples. D. Details on Implementation An implementation of the NetDetect algorithm on the nodes requires a setting of an appropriate window size to record a history of inter-arrival times between received messages. We denote the size of the history window by Wsize , and set its value depending upon an exploration of the controller’s performance, whose details are covered in Section IV. The Maximum Likelihood Estimation procedure performs an estimation of n over a range of values. For now, we set the range up to an arbitrary large number. While such a provision accounts for the maximally expected network size, it tends to
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Graph showing the Latency in initial estimation (number of time slots), for varying history window sizes. The increase in latency is linear over increasing window sizes. Moreover, the latency is practically constant across various neighborhood sizes, as shown by the different lines. Fig. 5.
Variance in Beaconing Probability for the NetDetect algorithm. The horizontal axes represent varying neighborhood sizes and history queue sizes, while the vertical axis represents the variance in the estimated probability. Fig. 4.
delay the average latency for the estimator. Therefore, we have to implement an optimization over the estimation process that is essentially a search over the space of valid n-s. We believe this optimization makes the algorithm viable for use on real mote platforms, as it reduces the average computation time. IV. E XPERIMENTAL R ESULTS AND D ISCUSSION We split the evaluation of NetDetect into three sections. The first section analyzes the accuracy of our adaptive controller in predicting the neighborhood density in a clique, and its sensitivity to specific parameters, namely the history size. Next, we evaluate NetDetect on a static multihop network, and compare the discovery evolution against that achieved by the algorithm in [20], which we shall refer to as the Coupon Collector algorithm. Finally, we run NetDetect on a mobile network with uniform density, and quantify the performance in terms of average network throughput and fraction of encounters that are discovered. Recall that the authors of the Coupon Collector algorithm [20] propose two algorithms for neighbor discovery; one that is Aloha-like and the other that relies on a mechanism to detect collisions. We make no presumptions regarding the capability of the nodes to detect collisions, and therefore compare NetDetect with the Aloha variant of the Coupon Collector algorithm. As a brief context, the Aloha-like algorithm proceeds in phases, and nodes decrease their beaconing probabilities exponentially as they advance from the current to the next phase. For simplicity of exposition, nodes start the discovery process at the same time, such that phase synchronization is assumed. Although the authors claim that the assumption of phase synchronization can be relaxed, we claim and show via simulations that different bootstrapping times of nodes can have an impact of an increased average discovery latency: the time taken by a node to discover all of its neighbors. More importantly, the Coupon Collector algorithm was designed
to operate on static networks only; it imposes a terminating condition specific to the number of discoveries registered per phase. Such a policy, while being ideal for the case of a static network, will not apply for a mobile network where not only the neighborhood size undergoes continuous changes, but also the neighborhood membership changes constantly. Furthermore, a natural extension of the Coupon Collector algorithm to discover mobile neighbors is not straightforward; as several alternatives regarding probabilistic transmissions can be envisaged. Therefore, we restrict the comparison of the NetDetect algorithm with the Coupon Collector to the case of static networks. In contrast, the NetDetect algorithm is applicable to both static and mobile networks, solving all previously mentioned shortcomings, as will be shown in the following subsections. A. Distributed Controller Evaluation In order to analyze the prediction accuracy of the NetDetect algorithm, we conducted a series of simulations on cliques with sizes ranging from 10 to 100. For each clique, we repeated the tests for different history sizes that were varied in the range of 10 to 1000 messages. Results of each configuration were averaged over 100 runs. Figure 3 shows the average estimated neighborhood size and the variance in the estimation, normalized to the mean value. The estimated values closely match the actual neighborhood sizes, as shown by the linear nature of the graph. The variance in the estimates drops rapidly as the history size is increased, and is negligible at values greater than 600. This is to be expected as a smaller window of observation would exhibit a larger variance in the distribution of time gaps between message receptions. Nevertheless, we notice that the variance in the estimates is always within 25% of the mean value, even at window sizes as small as 10 or 20 samples. Figure 4 shows the corresponding variance in beaconing probabilities for the nodes, as a result of the density estimation. Although we do not show the
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values, the mean beaconing probabilities reflect the actual neighborhood density even for small history sizes (< 100). The variance in the beaconing probabilities go as high as only one-fourth the mean value. Since our focus is on efficient neighborhood discovery, we would like to keep the discovery latency as low as possible. This requires us to operate the NetDetect algorithm at a window size that is not too large; for example 50 samples. This setting comes at an expense of an increased variance in beaconing probability. However, as we will show in the following subsections, the trade off does not degrade the discovery performance. An observation that results from Figure 5 is that the latency in deriving an initial estimate does not change across networks of varying neighborhood sizes. To validate this result, we ran NetDetect with a history size of 50 samples on a clique of 500 nodes, and found the algorithm to predict the number of neighbors in as few as 400 time slots. This demonstrates the ability of NetDetect to predict the number of communicating nodes in extremely large neighborhoods (> 100 nodes) within bounded time. The variance of the neighborhood size estimator decreases progressively as the sample size of the distribution is increased. However, incorporating a larger sample size results in an increased delay for the maximum likelihood prediction, as shown in Figure 5. As a first conclusion, we see that the estimation of the NetDetect algorithm is accurate up to 25% of the actual neighborhood size at comparatively smaller window sizes (50 messages), and involves minimal latency. Also, the NetDetect algorithm is able to predict different neighborhood sizes with almost constant latency, for a given size of the history window. Both these features are favorable towards self adaptation of node transmissions in changing neighborhoods, and estimating the neighborhood size, respectively. In the following subsections, we quantify these features via simulations on both static and mobile networks.
B. Discovery Process Evaluation in Static Networks In this subsection, we quantify the performance of the NetDetect algorithm in terms of neighborhood discovery in a static network. In order to study the effect of adaptive beaconing on the speedup in neighbor discovery, we conducted a series of simulations on both a fully-connected network and a multihop mesh network. Figure 6 shows the discovery evolution of NetDetect in comparison to the Coupon Collector algorithm on a complete network, with sizes ranging from 15 to 50 nodes. The horizontal axis represents the simulation time, and the vertical axis shows the fraction of neighbors that are discovered on average. Every simulation run lasted for a duration of 1000 time slots. The values for each configuration of neighborhood size are averaged over 50 runs. The staircase nature of the graphs results from a constant interpolation of the points, as was done in Dutta et al. [4]. We observe that apart from the delayed start, NetDetect achieves discovery in nearly the same order of magnitude in latency as the Coupon Collector algorithm. The delayed start for NetDetect is mainly attributed to a multiplicative decrease policy of NetDetect. Since nodes start the NetDetect algorithm with random values of beaconing probabilities, it could happen that no communication occurs on the channel for a prolonged time period, owing to channel contention and message collisions. To improve the convergence speed of the estimation process, we implemented a multiplicative decrease policy on every node, whereby a node reduces its beaconing probability by a factor of 2, if it does not receive any message over a time interval of Wsize slots. Such a recourse allows nodes to converge quickly 2 to the desired probability of beaconing, without affecting the accuracy of the estimator. Referring back to the latency for NetDetect shown in Figure 6, we believe it could be reduced further by imposing an exponentially increasing time interval over repeated invocations of the multiplicative decrease, as implemented by [20].
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Fig. 7. Discovery Evolution in a Multihop Network for varying neighborhood sizes (left) and phase offsets (right). We tested for the case of a uniformly distributed neighborhood sizes of 20, 25 and 30 nodes, and drew phase offsets δφ uniformly over the intervals (0, 0.1 · Tmax ), (0, 0.5 · Tmax ), and (0, 0.9 · Tmax ), Tmax being the duration of the experiment (1000 slots). The tests for varying phase offsets were conducted on networks with an average neighborhood size of 20 nodes.
In a real world environment, not all nodes in a wireless network bootstrap at the same time; they start communicating from a random offset in time from a fixed origin. This random distribution in start times is likely to affect the convergence of a neighbor discovery, particularly for the Coupon Collector algorithm that operates in time phases. In our attempt to understand how the discovery procedure evolves under the influence of different bootstrap times, we simulated both NetDetect and the Coupon Collector algorithm on a clique of 15 nodes, with phase offsets of the nodes drawn uniformly up to a limit of 0.1 · Tmax , 0.5 · Tmax , and 0.9 · Tmax , where Tmax is the duration of the experiment. Figure 6 shows that the discovery latency is delayed by an offset as the average phase difference is increased. Interestingly, we find that the Coupon Collector algorithm lags behind NetDetect in terms of discovery latency, for large offsets such as 0.9 · Tmax . This is largely attributed to the neighbor count-specific termination policies that are detailed in [20]. In particular, nodes that bootstrap earlier and run the Coupon Collector algorithm are likely to terminate the adaptive beaconing process prematurely. In contrast, NetDetect never terminates, but continuously adapts the probability of transmission in response to measured communication efficiency. As real-world networks typically extend over several communication hops, we also performed a comparison between NetDetect and the Coupon Collector algorithm on a uniformly distributed multihop setup of nodes. Figure 7 shows the discovery evolution in a multihop network for varying values of average node densities and phase offsets. We present the averaged results only for nodes having neighborhood densities of 20, 25 and 30 nodes. As for the case of a complete network, the discovery evolution for NetDetect lags behind the Coupon Collector algorithm only by a fixed offset, and takes almost the same order of magnitude in latency for registering all of a node’s neighbors. Furthermore, NetDetect outperforms the Coupon Collector marginally when nodes start beaconing at
offsets that are drawn uniformly from a large time window, showcasing the applicability of our algorithm to realistic cases of network bootstrapping. C. Discovery Process Evaluation in a Mobile Setting We also consider the case of mobile networks in which the per-node neighborhood is constantly changing. In particular, we look at two metrics, namely the receiver efficiency or the number of messages processed per unit time, and the fraction of total node-to-node encounters that are registered on average by a mobile node. We used the RandomWalk Mobility Model to mimic the case of large groups of people moving at a speed of roughly 1.25 meter per second, and simulate networks with average neighborhood densities of 15, 20, 25 and 30. Figures 8 and 9 show respectively, the receiver efficiency across varying node densities and the corresponding discovery success ratios. Each point in the graphs represents an average computed over 50 runs. We conducted the simulations with history window sizes of 50, 100 and 150, and found the results not to vary significantly. Referring to Figure 8, we note that for node densities such as 15, the receiver throughput is recorded to be within 25% of the optimum throughput of 1e . The average throughput increases slowly as the node density increases, and nearly flattens at 0.32, which is roughly within 13% of the optimum value. This is a strong result; mainly because the NetDetect algorithm tunes the beaconing probabilities on every node continuously in order to sustain an appreciable average throughput. Figure 9 indicates that across the evaluated node densities, every node on average registers at least about 60% of its total neighbors. Note that it would be interesting to study the optimum discovery fraction that can be achieved in a mobile network, and to relate the performance of the NetDetect algorithm to the optimum bound. Another interesting direction would be to inspect the duration of the node-to-node encounters, and to map the registered encounters onto the so-called link persistence, in order to assess the
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Fig. 8. Average Receiver Throughput achieved in a Mobile Network
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Neighbor Discovery in a Mobile Network based upon the RandomWalk Mobility Model (Maximum node velocity = 1.25 meters per second). Horizontal axis represents the average neighborhood sizes, while the vertical axis represents the discovery success ratio, or the average fraction of total encounters that were registered. Fig. 9.
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viability of our discovery scheme for routing protocols in MANETs. We leave these two directions for future work.
In order to demonstrate the adaptive nature of our algorithm to changes in a node’s neighborhood size, we simulated NetDetect on a clique of 50 nodes, for 2000 time slots. Changes in the neighborhood size were simulated as follows; 70 nodes left the neighborhood at time t = 1000s, and 50 nodes rejoined the neighborhood at time t = 1500s. Figure 10 shows the estimation on every node over time. The values represent the averaged outcome of 50 runs. It can be seen that while there is a time lag between nodes joining the network and the NetDetect recording the change in neighborhood size, the estimation is fairly accurate with a certain variance that we detailed in Section IV-A. While the accuracy of our estimates can be improved by increasing the history window size, it must be noted that the time to adapt to neighborhood changes will increase proportionally, as it would take a while before the measurements from the history queue are replaced. However, as we noted earlier in sections IV-A and IV-B, the relative inaccuracy in the estimation does not adversely impact the discovery process. We have so far, demonstrated the ability of the NetDetect algorithm to discover neighbors in both static and mobile networks in a time-efficient manner, by adapting to changes in neighborhood sizes. Our results show that the average discovery latency in a static network does not differ significantly from that achieved by a state of the art algorithm for neighborhood discovery, namely the Coupon Collector algorithm. Further, we tested NetDetect on a mobile network, and showed that average receiver throughput is within 25% of the maximum throughput that can be achieved. Although we restricted our simulations to the case of a slotted Aloha link layer protocol, we expect NetDetect to perform with comparable results on other classes of Aloha networks. We
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plan to extend our evaluation of NetDetect with experiments on real nodes, and to compare the resulting performance with protocols such as [4], [7] that have been tested on real mote platforms. V. C ONCLUSIONS
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F UTURE W ORK
In this paper, we propose a novel adaptive and decentralized solution, namely the NetDetect algorithm, to the problem of time-efficiently discovering neighbors in a dynamic wireless network. The main difference with existing work is that we target dynamic scenarios that also includes mobility of devices. The algorithm exploits the beaconing communication mechanism, dynamically adapting the beacon rate of the devices in the network based on a local estimate of the network density. We evaluate the algorithm on a variety of networks with increasing levels of dynamics: a fully-connected network, a static and a mobile multi-hop mesh network. Results show
that NetDetect performs well in all considered scenarios, maintaining a high rate of neighbor discoveries and good estimate of the neighborhood density even in very dynamic situations. Comparison with a state of the art algorithm showed that our proposed scheme is efficient from both a convergence time and energy perspective. The lightweight communication overhead of the algorithm combined with its applicability to a broad range of scenarios characterized by different network topologies and its robustness, make NetDetect an attractive solution. As directions for future research, we identified problems that relate strongly to the problem of channel efficiency and neighborhood discovery. Firstly, we note that discovering neighbors in a mobile network is more problematic because of the random duration of wireless links between nodes. It would be interesting to analyze the persistence of wireless links in a mobile neighborhood, and to derive an optimum bound on neighborhood discovery and assess how close is NetDetect to this optimum. Secondly, we would like to investigate possible beaconing strategies for diverse application requirements, such as for example, best-effort delivery of packets from a node to a base-station versus a scheme where fairness between nodes is ensured. VI. ACKNOWLEDGMENTS Venkatraman Iyer is supported by the ALwEN project, a Point-One project funded by SenterNovem. Andrei Pruteanu is supported by the FREE project funded by SenterNovem. R EFERENCES [1] F. Bai, N. Sadagopan, B. Krishnamachari, and A. Helmy. Modeling path duration distributions in MANETs and their impact on reactive routing protocols. Selected Areas in Communications, IEEE Journal on, 22(7):1357–1373, 2004. [2] S. Borbash, A. Ephremides, and M. McGlynn. An asynchronous neighbor discovery algorithm for wireless sensor networks. Ad Hoc Networks, 5(7):998–1016, 2007. [3] S. Cho and J. Hayes. Impact of mobility on connection in ad hoc networks. In Wireless Communications and Networking Conference, 2005 IEEE, volume 3, pages 1650–1656. IEEE, 2005. [4] P. Dutta and D. Culler. Practical asynchronous neighbor discovery and rendezvous for mobile sensing applications. In Proceedings of the 6th ACM conference on Embedded network sensor systems, pages 71–84. ACM, 2008. [5] O. Gnawali, R. Fonseca, K. Jamieson, D. Moss, and P. Levis. Collection tree protocol. In Proceedings of the 7th ACM Conference on Embedded Networked Sensor Systems, pages 1–14. ACM, 2009.
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