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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Transmission range plays an important role in the deployment of a practical underwater acoustic sensor network (UWSN), where sensor nodes equipping with only basic functions are deployed at random locations with no particular geometrical arrangements. The selection of the transmission range directly influences the energy efficiency and the network connectivity of such a random network. In this paper, we seek analytical modeling to investigate the tradeoff between the energy efficiency and the network connectivity through the selection of the transmission range. Our formulation offers a design guideline for energy-efficient packet transmission operation given a certain network connectivity requirement.

Typically, underwater acoustic sensor networks (UWSNs) consist of sensors that are deployed to perform collaborative monitoring tasks over a given region, such as oceanographic data collection, marine pollution monitoring, offshore exploration and disaster prevention and tactical surveillance [

Sensor nodes are prone to failures due to fouling and corrosion in the underwater environment. They are battery powered, which implies a limited operational lifetime. Due to the deployment remoteness of UWSNs, replacing faulty or flat sensor nodes incurs high cost. Thus, the deployment of UWSNs plays an important role in the

There are broadly two strategies in sensor node deployments. If a certain precision of location can be achieved in sensor node deployment, a precise planning of sensor node location can be sought to ensure full functions of complete sensing coverage and network connectivity based on a certain geometrical arrangement with the least number of sensor nodes [

In this paper, we consider an UWSN with random sensor node deployment. As opposed to the high-precision deployment of sensor nodes, here sensor nodes are deployed at random locations with no particular geometrical arrangements, which forms a random network. As a result, the function and the efficiency of such UWSN can no longer be guaranteed. Full sensing coverage and network-wide connectivity may not be reached, and operation may not be optimized for energy efficiency

In the aspect of energy efficiency, one key influencing factor is the transmission power of each sensor node. Intuitively, when a higher transmission power is used in a packet transmission, the transmission can reach a longer distance, hence a fewer number of transmission relays is involved in delivering a packet to the sink. However, this fewer involvement in transmission relays is achieved at the expense of high energy consumption per transmission. Additionally, a larger transmission radius also introduces interference which may eventually translate into a higher overhead for each successful packet transmission. On the other hand, when a lower transmission power is used in a packet transmission, less energy is used for each packet transmission or relay. However, a higher number of transmission relays is required, which may result in a higher energy consumption for an end-to-end packet transmission. Thus, there exists an optimum transmission range that maximizes the energy efficiency or minimizes the energy consumption.

On the other hand, in the aspect of function, with the randomness in sensor node locations, the full coverage of sensing and communication may not be fulfilled. The effectiveness of sensor data collection is dictated by the network connectivity from a sensor to the sink. A particular transmission range setting leads to a certain probability of network connectivity where a longer range gives a higher probability of full network connectivity.

In [

As opposed to the study in [

We would like to point out that there have been a number of studies focusing on the optimal transmission range in the literature for both terrestrial networks and underwater networks. Most of the studies primarily focus either on how to enhance throughput by adjusting transmission range [

The remainder of this paper is organized as follows. In Section 2, we describe our considered UWSN model, followed by the derivations of the energy efficiency and the network connectivity in Section 3. In Section 4, we present numerical and simulation results and illustrate the optimal transmission range and the tradeoff between energy efficiency and network connectivity Finally we conclude this paper in Section 5.

A reference architecture for two-dimensional underwater sensor networks is shown in

To facilitate the analysis of an UWSN, we model it as follows. Suppose that the sink (denoted by

As pointed out in [

In this section, we first describe the physical-layer underwater energy consumption model, which tells how large the energy consumption is in one transmission given a transmission range. Then, we analyze the average energy consumption w.r.t. an end-to-end packet transmission,

The attenuation or path loss that occurs in an underwater acoustic channel over a distance

The above formula is generally valid for frequencies above a few hundred Hz. For lower frequencies, it is suggested to use the following formula:

The power consumption (denoted as _{t}_{t}^{−17.2} is the conversion factor and

In practice, a certain non-zero minimum level of power is always radiated for a transmission regardless of how short the distance is [_{min}_{r}_{d}

Here, we study the

Let

Denote by

Like [

It should be pointed out that, the transmission strategy adopted in our study is

Denote by

As a result, the one-hop energy-distance ratio or the average energy consumption (denoted by

Note that

For the first unknown term, using

To compute the fourth unknown term, we denote by _{s}_{α}_{β}^{n}e^{−ρA}/n

According to _{α}

It is to note that the event {_{β}_{α}

Substituting

As aforementioned, minimizing the energy-distance ratio

Let _{k}, υ_{k}_{k}, k^{th}^{th}_{k}_{k}_{k}_{k}, υ_{k}_{k}_{k+}_{1} = _{k} − x_{k}

Based on the definitions and ^{k}^{th}_{s}, A_{α}

Further, we define
_{k}_{1}) be the conditional probability that the source node _{1}. By [_{k}_{1}) can be approximately computed as:

Thus, the connectivity of the network (denoted by _{c}

We conduct simulation experiments to validate our analytical framework. The network coverage area is assumed to be a circle with radius ranging from 5,000 m to 15,000 m, and the sink is fixed at the center. The central frequency _{min}_{r}_{d}^{2}). All the results obtained are the average over 500 randomly selected topologies.

From ^{−8}, 1 × 10^{−7}, 2 × 10^{−7} are 0.00331, 0.00321 and 0.00313, respectively. This can be attributed to the fact that a larger node density makes a larger one-hop progress under the same transmission range, rendering a lower energy consumption. The same conclusions still hold for

To validate our analytical result for the energy efficiency, in ^{−7}. Clearly, both results reach a good agreement indicating the accuracy of our analytical approach. Similar to that of

_{opt}

_{c}^{−7} or 2 × 10^{−7}. Again, it demonstrates that the analytic and simulation results match closely. Moreover, it can be seen that a larger

^{−7}, the optimum transmission range is 2,852 m, resulting an average energy consumption of 0.00317 J/m but a connectivity of 0.4317, which does not meet the practical connectivity requirement for UWSNs. Thus, the selection of the transmission range needs to take into consideration the tradeoff between the energy consumption and the connectivity. In particular, a targeted network connectivity requirement can be fulfilled by increasing either the node density or the transmission range, each of which incurs a cost. Increasing node density introduces additional hardware cost while increasing transmission range causes higher operational energy and transmission interference. Our formulation enables network designers to determine the tradeoff and thus derive an adequate setup to meet the requirements.

So far, we have investigated two relationships: one is between the transmission range and energy consumption, and the other is between the transmission range and the network connectivity. Given the network size, the node density and the threshold of the network connectivity, we may determine the optimal transmission range for sensor nodes to operate. In other words, we may adjust the transmission power of sensor nodes such that the energy consumption in transmissions is optimally set.

It is easy to see that, for a given transmission range, the energy efficiency and the network connectivity can be calculated using _{c}

First, for a particular threshold of the network connectivity, we can find the lowest transmission range, say _{1}, by using _{2}, based on _{1} and _{2} as the transmission range for operation.

In our earlier discussions, we suggested increasing either node density or transmission range to achieve a certain network connectivity requirement. In this subsection, we demonstrate the employment of a multiple sink setup as an alternative solution to meet the requirement.

Assuming that a number of sensors are randomly deployed over a square area with the side length of 5,000 m, we consider two scenarios here: one is the scenario where there is only one sink located at the center of the square area, and the other is that there are four sinks individually placed at four different vertexes of the square. The node density ^{−7} or 2 × 10^{−7}, and the transmission range varies from 1,000 m and 5,000 m. We evaluate the impact of multiple sinks on the connectivity, as shown in

In this paper, we developed an analytical framework which describes the relationship between the transmission range and the energy efficiency as well as the relationship between the transmission range and the connectivity in an UWSN scenario. We illustrated that the selection of the transmission range needs to consider the tradeoff between the energy efficiency and the connectivity. Meeting a certain level of network connectivity incurs either cost for additional node deployment or energy due to operational deviation from the optimal transmission range. Our analytical framework provides a means for network designers to plan the deployment of an UWSN. We further illustrated that employing multiple sinks helps to meet the connectivity requirement in a more cost-effective way.

Although we consider the underwater environment in this paper, the developed analytical framework is general and can be applied to other random network scenarios. In the future, we shall extend our investigation by considering a medium access control (MAC) protocol such as [

The network model for 2D UWSNs [

An illustration of the forwarding progress.

The numerical results of the average energy consumption

Comparisons of the numerical and simulation results of the average energy consumption under different transmission ranges with ^{−7}.

The optimal transmission range (

The results of the network connectivity with different transmission ranges (

The simulation results of the network connectivity