School of Information Science and Engineering, Southeast University, Nanjing, China

* The corresponding author, email: zzhlixia@seu.edu.cn

I. INTRODUCTION

Underwater wireless sensor networks(UWSNs) with high civil and military value,can be widely used in ocean information collection, resource exploration, environmental monitoring, disaster forecast, auxiliary navigation, distributed tactical observation and so on[1]. Network capacity is an important performance index of wireless sensor network.The research of relations between network capacity and the various parameters has very important research value and significance, because it is conducive to a better understanding and the design of network

The capacity research in terrestrial wireless sensor network started relatively early.As early as 2000, Gupta and Kumar began to set about the theoretical analysis of network capacity [2], then many scholars successively into it, and some useful research results were obtained [3-5]. In recent years, with increasingly high data transmission rate demanding in UWSNs, scholars began to focus on the issue of capacity performance in the underwater networks [6-9]. However, the limited bandwidth, long propagation delay and severe multipath bring huge challenges to the capacity research of UWSNs. Thus the literatures on this subject are rare, either focusing on pure theoretical analysis of capacity or applying for only specific fixed networks. Basing on stochastic geometry theory and structure features of network, we propose a feasible network capacity analytical model with some insightful simulation results for the increasingly popular cluster-based UWSNs.

Based on stochastic geometry, a network capacity analysis model applied to the cluster-based UWSNs is presented in this paper.

II. COMMUNICATION MODEL

2.1 Cluster-based network model

For marine environmental monitoring, a typical UWSNs is based on hierarchical topology,and its structure is shown in Figure 1. Nodes are divided into clusters, each cluster consists of one cluster head and several cluster members. Information will be collected by cluster members and then distributed to the cluster heads. After a certain degree of data convergence, the cluster heads will forward it to the surface SINK node.

2.2 Channel transmission model

The high electrical conductivity of the seawater makes the radio wave attenuate seriously,and the light wave is easy to be chromatic dispersion because of the floating object in the sea. Thus the above two carrier forms are only used for short-range underwater communication. So far, acoustic wave is the most widely used carrier in the UWSNs. Compared with the traditional wireless channel, underwater acoustic channel has its own unique properties: the propagation speed is only 1500 m/s,thus it has a large propagation delay; for one thing the path loss increases with the increase of frequency, for another the power of ocean ambient noise is relatively high in the low frequency, therefore underwater acoustic channel has relatively narrow available bandwidth;the wave refraction caused by the water stratification and reflection of sea surface and sea bottom to the acoustic wave cause strong multipath effect for underwater acoustic channel.

Fig. 1 The sketch map of Cluster-based underwater acoustic sensor networks

(1) Pass loss

Path loss consists of two parts: spreading loss and absorption loss [10]. The path loss that occurs in an underwater acoustic channel over a distancel(km)for a signal of frequencyf(kHz)is given by

Where k is the spreading factor describing the geometric characteristics of the propagation andis the absorption coefficient.Andfor f in kHz can be expressed empirically by the Thorp’s formula [11] as:

All of the independent transmitters (TXs) are distributed on the 2D plane according to a homogeneous Poisson Point Process (PPP)[12], expressed asof densityλ, wheredenotes the location of the typical nodeConsidering a single cluster-based UWSNs, the cluster head of the given cluster as the common receiver (RX) is placed at the origin, then those TXs around RX form the corresponding cluster members. According to the Slivnyak’s Theorem [13], we can casually pick one of the cluster membersas the reference transmitter. Letdenotes the distance betweenandRX. The distribution of interference nodes is just shown in Figure 2.

(2) Multipath fading

In the underwater communication, acoustic wave affected by refraction and reflection will form a number of different paths to reach the receiver. After passing through the underwater acoustic channel, the received signal is the sum of a large number of statistically independent random variables, whose amplitude is random.Rayleigh fading model is a common statistical model for underwater acoustic signal propagation environment, the envelope of the received signal will obey a Rayleigh distribution.

2.3 Network interference model

In the remainder of this paper, we consider narrowband transmission, i.e., for nodetransmission that takes place in a “small”bandwidtharound the carrier frequencyWithinis given by Eq.(2). For node, the interference seen at the cluster head is

III. CAPACITY DEFINITION

3.1 SIR and outage probability

Fig. 2 The sketch map of the distribution of interference nodes

3.2 Transmission capacity

Assuming that the network can withstand a maximum outage probability of ε, the transmission capacity is defined as the product of the maximum transmission density in per unit area, the success probability and the data rate[14], expressed as

IV. THEORETICAL ANALYSIS AND SIMULATION

4.1 Theoretical analysis

There is a certain number of transmitting nodes in the network, obeying a distribution of Poisson Point Process with densityλ. The cluster members in the given cluster are numbered in turn from 1 to N, resulting in a set of transmitters, i.e.The number of sub-channels is n. Selecting a node(idenotes the ID.) casually, the absorption coefficient in the responding sub-channel iswhereWe useto denote the set of other transmitters in the same sub-channel with nodebutis not included. According to the transmission model of underwater acoustic channel, takeas the reference transmitter, then from Eq. (4),the interference signal at the cluster head can be expressed as:

In accordance with the definition of outage probability, substitute Eq. (5) into Eq. (6).Owning to, we can know that

Substituting Eq. (13) into Eq. (12), we get

Then according to Eq. (14) and Eq. (11),we can obtain

The integral of exponent part does not have closed analytical solution in the above equation, so the numerical integration method is used to obtain the theoretical approximate value in the subsequent content. Due to the condition of a given 2-dimensional planar network, so we haveAnd θ is a constant,so the expression can be simplified as

Hence, Eq. (15) can be just simplified as

On the basis of the above formula, the outage probability of sub-channelcan be expressed as

Considering the ideal channel allocation method, we can assume that the number of nodes on each sub-channel is approximately equal, so that the outage probability of the whole network can be obtained by simply summing and averaging as follows:

The following task is of course to solve the expectationIn order to get the expression on the node densityλ, do as the follows：

As the cluster members are independent identically uniform distributed in the cluster’s coverage area, suppose R as radius of a single cluster,then the distance of cluster members to cluster head has the following distribution function:

Then the corresponding probability density function is:

c) Substitute Eq. (21) into the expectation.

d) According to the definition of definite integral, we can get

According to the definition of expectations and the solving method of double integral, we obtain the expression ofas follows:

Substituting Eq. (24) into Eq. (22), we can getAnd the theoretical value of outage probability can be obtained by substituting the Eq. (22) into Eq. (19), then we can get the corresponding theoretical value of the transmission capacity according to the definition. The changing curves of outage probabilityand transmission capacityon node densityare studied by simulation experiments. The comparison and analysis between simulation results and theoretical value are also given in the following section.

4.2 Simulation experiment

In this section, we try to simulate the outage probability and transmission capacity under different sizes of simulation regions using MATLAB, then compare with the theoretical value.

Assume that all nodes use the same transmission power in an interference-limited network, and the transmitters assign sub-channels in a random competition way. In the theory analysis, interference nodes are assumed to distribute in an infinite region. However, the infinite region can’t be achieved in simulation experiment. Besides, considering the limited transmission power and transmission radius in actual networks, the radius of limited region that interference nodes exist is defined as interference radius ρ. The simulation parameters settings are listed in Table I.

The outage probability and transmission capacity changing with node density are shown in Figure 3 and Figure 4 respectively.

According to the figures we can see that whether in simulation results or theoretical value, with the increase of node density, the outage probability of the system is alwaysincreasing, and the transmission capacity will increase at first and then decrease. That is to say that there is an optimal node density which can make the network’s transmission capacity reach a maximum value. For the outage probability, the simulation results are always less than the theoretical value, and for the transmission capacity, the former is also better than the latter. It is known that the results of“theory inf” corresponds to an infinite interference region, then the number of interference nodes considered will far exceed that of the simulation experiment with limited interference radius ρ. Therefore compared with the theoretical value, the outage probability will be relatively low and the transmission capacity will be slightly high accordingly in the case of simulation experiment.

Table I Simulation parameters settings

Fig. 3 The comparison of simulation results and theoretical value about the outage probability I

Fig. 4 The comparison of simulation results and theoretical value about the transmission capacity I

Simulation results are gradually approaching the theoretical value as the radius ρ increases from 5 to 12. It is easy to speculate that, as long as the value of ρ continues to increase, the simulation results must be able to approximate the theoretical value within the range of given error accuracy. In order to measure the error between theoretical value and simulation results, the upper limit of integral in Eq. (15) is set as the value of limited interference radius ρ, obtaining theoretical value of finite integral upper limit, finally comparing with the corresponding simulation results. The comparisons between simulation results and theoretical value about outage probability and transmission capacity are shown in Figure 5 and Figure 6 respectively.

The following two figures show that the theoretical value with finite interference radius will quickly converge to the theoretical upper bound with infinite interference radius as ρ increases from 5 to 12; at the same time, the simulation results with finite interference radius are also heading towards the “theory inf” with the increase of ρ, but its speed is far less than the corresponding part of theoretical value. Therefore, with the increase of ρ, the error with finite interference radius between the theoretical value and simulation results have a tendency to expand. In a word, the changing tends of outage probability and transmission capacity with node density are consistent, and the optimal node density corresponding to the maximum transmission capacity between the theoretical value and simulation results is consistent, thus the validity of the theoretical analysis is verified.

From figure 6, as the interference radius ρ increases from 5 to 12, the optimal node density successively to be 0.11, 0.07, 0.06 and 0.05 nodes/km2is gradually decreasing. In other words, the optimal node density will be affected by the interference radius ρ. In fact,there are many influence factors of network capacity to be researched in the future, such as network structure, frequency band, channel allocation method, channel fading and so on.

V. CONCLUSION

Fig. 5 The comparison of simulation results and theoretical value about the outage probability II

Fig. 6 The comparison of simulation results and theoretical value about the transmission capacity II

Based on stochastic geometry, a network capacity analysis model applied to the cluster-based UWSNs is presented in this paper. In combination with the unique characteristics of underwater acoustic channel in path loss and multipath fading, considering the interference signal generated by other transmitters, the network’s outage probability and transmission capacity are both defined. Then some theoretical analysis and simulation experiments based on stochastic geometry are carried on respectively. Comparing the simulation results with the theoretical value, the validity of the theoretical analysis is verified, and the cause of error between them is also clearly explained. The results show that under a given network condition like specific monitoring scope, frequency band, channel-allocation method and so on, there is always an optimal network node density which can result in the maximum transmission capacity. It is conceivable that when some specific network conditions change, the optimal network node density should be varied correspondingly. Then the influence factors analysis on network capacity will be one of the following research topics and directions. In conclusion, we believe that the work of capacity research in this paper can provide some reference for the future application and design of the UWSNs.

ACKNOWLEDGEMENTS

This work has been supported by National Natural Science Foundation of China (No.61101164).

Reference

[1] AKYILDIZ I F, POMPILI D, MELODIA T. Underwater acoustic sensor networks: research challenges [J]. Ad Hoc Networks, 2005, 3(3): 257-279.

[2] GUPTA P, KUMAR P R. The capacity of wireless networks [J]. IEEE Transactions on Information Theory, 2000, 46(2) : 388-404．

[3] WEBER S P, YANG X, ANDREWS J G, et al. Transmission Capacity of Wireless Ad Hoc Networks with Outage Constraints [J]. IEEE Transactions on Information Theory, 2005, 51(12): 4091-4102

[4] ANDREWS J G, WEBER S P, KOUNTOURIS M,et al. Random Access Transport Capacity [J].IEEE Transactions on Wireless Communications,2010, 9(6):849-856

[5] BUSSON A, CHELIUS G. Capacity and interference modeling of CSMA/CA networks using SSI point processes [J]. Telecommunication Systems, 2014, 57 (1): 25-39

[6] LUCANI D, M′EDARD M, STOJANOVIC M. Capacity scaling laws for underwater networks[A]. Proceedings of 42nd Asilomar Conference on Signals, Systems and Computers [C]. Pacific Grove, CA, Oct. 2008.

[7] STAMATIOU K, CASARI P, ZORZI M. Throughput and Transmission Capacity of Underwater Networks with Randomly Distributed Nodes [A]. Proceeding of 2011 IEEE Global Telecommunications Conference (GLOBECOM 2011) [C]. 2011.1-9

[8] SHIN W Y, LUCANI D E, M′EDARD M, et al. On the order optimality of large-scale underwater networks–Part I: Extended network model [J].Springer: Wireless Personal Communications,2013, 71(3):1683-1700.

[9] SHIN W Y, LUCANI D E, M′EDARD M, et al. On the Eff ects of Frequency Scaling over Capacity Scaling in Underwater Networks–Part II: Dense Network Model [J].Springer: Wireless Personal Communications, 2013, 71(3):1701-1719.

[10] STOJANOVIC M. On the relationship between capacity and distance in an underwater acoustic communication channel [J]. ACM Mobile Comput. and Commun. Review, 2007, 11(4): 34–43.

[11] BREKHOVSKIKH L M, LYSANOV Y P. Fundamentals of ocean acoustics [M]. Springer, 2003.

[12] HAENGGI M, ANDREWS J G, BACCELLI F, et al.Stochastic geometry and random graphs for the analysis and design of wireless networks [J].IEEE Journal on Selected Areas in Communications, 2009, 27(7):1029-1046,

[13] MARTIN H, RADHA K G. Interference in Large Wireless Networks [J]. Foundations and Trends? in Networking: 2009, 3(2): 127-248

[14] H Hu, H.B Zhu, Q Zhu. Transmission Capacity Analysis in Wireless Ad Hoc Networks Based on Stochastic Geometry Theory [J]. Journal of Nanjing University of Posts and Telecommunications (Natural Sciences), 2013,33(2):1-7