Link probability, node degree and coverage in three-dimensional networks
Introduction
In this work we investigate fundamental properties of 3D Networks, such as Underwater Sensor Networks (UWSNs), Mobile Ad Hoc Networks (MANETs), and Aeronautical Ad Hoc Networks (AANETs). UWSN has the potential for many applications, such as coast surveillance, ocean monitoring, 4D monitoring (space and time) for ocean and biology studies [1]. AANETs can enhance aircrafts situation awareness. All these applications can benefit from network fundamental properties. First, sensing applications require the deployment of sensor nodes. This can be planned such that they would benefit from network coverage and connectivity in the specified deployment region. Second, while in operation, it would be interesting to guarantee sensing coverage, even when some sensor nodes are operating in a duty cycle that has sleeping and awakening periods. Finally, the application performance can be influenced by the Medium Access Control (MAC) contention, which in turn, depends on the network node degree. Those are examples that demonstrate why it is important to investigate fundamental properties such as link probability, network coverage and node degree.
The model presented in this work can be extended to any 3D MANET. UWSN and AANET are practical examples of the importance of studying 3D MANETs. Previous proposals on MANETs focus on 2D deployments as they were concerned with terrestrial applications. We present this novel study focusing 3D MANETs.
This work was motivated by fundamental problems currently faced in the deployment of UWSNs, more specifically, intrusion detection applications. We are interested in answer the following question: given a region where nodes are deployed and that this region is to be protected, how many sensors should be deployed so that every point in this region is cover by one sensor? This question can be answered by solving the coverage problem. Coverage is important for sensing and surveillance applications. Every point in a surveillance region should be coverage by at least one sensor.
These are important issues in network management and protocol design for scheduling duty cycles and controling transmission power. A proper adjustment of the duty cycle can extend the lifetime of the network. By knowing the node degree, some nodes can sleep, or reduce their transmission power and still guarantee message delivery.
We summarize our main contributions as follows:
- •
We provide closed form analysis on the link probability between two distinct nodes i and j. This is the basic building block that allows us to compute node degree and realize further analysis;
- •
We derive the formulae for node degree for both cases: a single node and for the entire network. By having an estimate of the expected node degree, it is possible to tune applications or protocols for a better performance. One such example is the case of the MAC protocol;
- •
We describe how to compute the network coverage for two possible cases: considering border effects or not. This important result allows us to determine the required number of sensor nodes that guarantee sensing coverage (when considering the sensing radius) and network coverage (when considering the communication radius);
- •
We solve the coverage problem.
- •
We describe how to use these results into a network protocol to illustrate the results’ usefulness.
The results and theories present here can be used to answer the formulated question in surveillance applications. In addition, it may be applied to a variety of scenarios, platforms and applications, other than the one mention here. To demonstrate the usefulness of the results, we mention one in the design of MAC protocols.
The rest of the paper is organized as follows: in Section 2 we describe the MANET physical layer. In Section 3, we compute the analytical probability that two distinct nodes i and j have a common link. In Section 4, we compute the expected node degree. In Section 5, we present the analytical result on one-coverage. In Section 6, we describe some practical applications that can benefit from our results. In Section 7, we discuss the related work. Finally, we present our conclusion in Section 8.
Section snippets
Physical layer
In terrestrial 3D MANETs, the signal propagation is described by the power received (PL) in dB, at a distance d, which is given by: where PL(d) is the path loss at distance d, is the known average path loss over distance d0 and α is the path loss exponent [2].
In UWSNs, unlike terrestrial networks, radio signal is not used due to the high attenuation. Instead, current research proposes acoustic channels [4], [5], which have low bandwidth [3] and large propagation
Link probability
In this section, we compute the analytical probability that two distinct nodes i and j have a common link. We assume the simple ball model. This is a simplified model as only path loss is taken into account. In the following, we derive a general expression for the link probability. After that, we instanciate this result to obtain expressions for the link probability in two specific cases, namely, the uniform and left triangular distributions.
Two nodes have a common link if they are within each
Node degree
Let random variable Li, j be the number of links connecting nodes i and j. Li, j is either 1 or 0. Let be the node degree of node i.
Theorem 2 Consider a network ⟨n, r, ℓ1, ℓ2, ℓ3⟩. Then, we have The expected node degree isand; The expected number of links is
where p is the link probability.
Proof
From Theorem 1, the expected link occurrence E(Li, j) is equal p. By definition, . By linearity of expected values, no matter if Li, j’s are independent or
Expected coverage
Given n nodes deployed in a region R with volume we are interested in computing the expected network k-coverage, that is, the volume of R covered by at least k nodes. This kind of characterizations are useful in several and diverse sensing applications. In the following we preceed to derive an expression for the simplest and more common case, namely, the expected one-coverage. Based on the aforementioned result, we derive an expression for the expected k-coverage. In this section we
Applications
In this section we describe how the previous analysis can be useful in the design of protocols and applications. We describe an uncoordinate node scheduling for MAC protocols.
MAC protocols have to cope with channel contention in order to work properly. For example, in underwater networks, due to the fact that high-frequency waves are rapidly absorbed by water, one must use acoustic channel. This channel is characterized by low and range-dependent bandwidth. Also, slow acoustic signal
Related work
Yen and Yu [7] described the fundamental properties of a 2D wireless ad hoc networks. In their work they consider a wireless Ad Hoc network as ⟨n, r, ℓ1, ℓ2⟩-network with n nodes, with transmission radius r, placed in a ℓ1 × ℓ2 rectangle area. The occurrence probability of a link (i, j) between two distinct nodes i and j is given by Eq. 12. We improve that work for 3D MANETs and considere the effects of the 3D environment. We investigate further into how these results apply to certain
Conclusion
In this work, we considered the fundamental properties of 3D Networks. We showed how to compute the link probability, node degree, and expected coverage. Those are necessary and useful in real applications. Furthermore, we were able to derive analytical results for those properties.
The results presented are useful for many applications. In particular, we showed one such an application: reducing MAC contention. The approach of increasing duty cycle reduces energy consumption and increases
Luiz F. M. Vieira is an Assistant Professor at the Computer Science Department, Universidade Federal de Minas Gerais (UFMG). He received his undergraduate and M.S. degree at the Universidade Federal de Minas Gerais in Belo Horizonte, and Ph.D. degree in Computer Science from the University of California Los Angeles (UCLA). His research interest is in Computer Networking.
References (18)
- et al.
Network coverage using low duty-cycled sensors: random and coordinated sleep algorithms
Proceedings of the Third International Symposium on Information Processing in Sensor Networks (IPSN’04), Berkeley, CA
(2004) - et al.
Building underwater ad-hoc networks and sensor networks for large scale real-time aquatic applications
Proceedings of the IEEE Military Communications Conference (MILCOM’05), Atlantic City, NJ
(2005) Wireless communications: principles and practice
(2002)On the relationship between capacity and distance in an underwater acoustic communication channel
ACM SIGMOBILE Mob. Comput. Commun. Rev.
(2007)- et al.
Practical spreading laws: the snakes and ladders of shallow water acoustics
Proceedings of the 2nd International Conference and Exhibition on Underwater Acoustics, Rhodes, Greece
(2014) - et al.
Underwater acoustic communication channels: propagation models and statistical characterization
IEEE Commun. Mag.
(2009) - et al.
Expected k-coverage in wireless sensor networks
Ad Hoc Net.
(2006) - et al.
Link probability, network coverage, and related properties of wireless ad hoc networks
Proceedings of the IEEE International Conference on Mobile Ad-hoc and Sensor Systems (MASS’04)
(2004) - et al.
On k-coverage in a mostly sleeping sensor network
Proceedings of the 10th Annual International Conference on Mobile Computing and Networking (MobiCom ’04), Philadelphia, PA, USA
(2004)
Cited by (0)
Luiz F. M. Vieira is an Assistant Professor at the Computer Science Department, Universidade Federal de Minas Gerais (UFMG). He received his undergraduate and M.S. degree at the Universidade Federal de Minas Gerais in Belo Horizonte, and Ph.D. degree in Computer Science from the University of California Los Angeles (UCLA). His research interest is in Computer Networking.
Marcelo G. Almiron is a post-doc. He received his Ph.D. degree in Computer Science from the Universidade Federal de Minas Gerais.
Antonio A. F. Loureiro is a Full Professor at the Computer Science Department, Universidade Federal de Minas Gerais (UFMG). He received his undergraduate degree at the Universidade Federal de Minas Gerais in Belo Horizonte, and Ph.D. degree in Computer Science from the University of Britich Columbia (UVC). His research interest is in Computer Networking.