An exact approach for maximizing the lifetime of sensor networks with adjustable sensing ranges

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Abstract

This paper addresses the problem of target coverage for wireless sensor networks, where the sensing range of sensors can vary, thereby saving energy when only close targets need to be monitored. Two versions of this problem are addressed. In the first version, sensing ranges are supposed to be continuously adjustable (up to the maximum sensing range). In the second version, sensing ranges have to be chosen among a set of predefined values common to all sensors. An exact approach based on a column generation algorithm is proposed for solving these problems. The use of a genetic algorithm within the column generation scheme significantly decreases computation time, which results in an efficient exact approach.

Introduction

Advances in signal processing and embedded systems are at the origin of the growing popularity of Wireless Sensor Networks (WSN) in a wide range of applications [1]. While they were initially used in remote or hostile environments (for battlefield surveillance, or tsunami monitoring), WSN are also increasingly used for health care [12]. Although these applications rely on very different types of sensors, most of them share the following characteristics: sensors operate on a battery that cannot be recharged and a large number of sensors is deployed for improving fault tolerance and lifetime. This paper is concerned with lifetime maximization, that is achieved by making the best possible use of sensors redundancy. Moreover, the sensors are supposed to have adjustable sensing ranges. Such a feature saves energy in the situation where a sensor needs to cover close targets only, as power requirement is a non-decreasing function of the distance between the sensor and the farthest target it covers.

More formally, suppose n sensors are randomly deployed in order to cover a set of m targets {τ1,,τk,,τm}. Each sensor si has an initial energy bi. Sensors can either be active or inactive. An inactive sensor does not cover any target, and its power consumption is negligible. When a sensor is active, its power consumption depends on its sensing range. All the sensors have the same maximum sensing range, denoted by Rmax: a sensor can cover a target if its distance is less than or equal to Rmax. Lifetime maximization is reached by gathering sensors into non-disjoint subsets called covers (each cover being such that each target can be covered by at least one sensor in that cover), and by scheduling these covers, i.e., by determining the amount of time during which each cover is used. The sensors that are not part of the cover that is currently being used are not active. Moreover, covers are not necessarily disjoint, as this allows for reaching longer lifetimes [15]. The schedule must be such that the total amount of energy consumed by sensor si is at most equal to its initial energy bi. The network lifetime is the sum of these durations: when it is exceeded, the coverage of all targets is no longer possible.

This paper addresses two close versions of the lifetime maximization problem:

  • Lifetime Maximization with Ad-hoc Sensing Ranges (LM-ASR): Each sensor can adjust its sensing range so as to cover targets with the minimum amount of necessary power (for all targets which distance to the sensor is less than Rmax). In this model, continuous variations of the sensing range are allowed, this problem version is addressed in [6],

  • Lifetime Maximization with Predefined Sensing Ranges (LM-PSR): Sensors have MPSR nonzero and distinct predefined sensing ranges, so all the targets that are under such a predefined range are covered at a predefined power level. In this model, MPSR predefined sensing ranges are supposed to be given, this problem version is addressed in [3].

A mathematical formulation based on integer or linear programming is proposed in [3], [6], but is never used for solving the problem, as both formulations rely on an exponential number of variables. In [17], [18], [16], [11], heuristic methods are proposed for finding the best possible cover for LM-PSR. More precisely, [11] uses a NSGA-II approach, where a trade off is sought between the coverage rate of the sensors in the cover (breach is allowed), the financial cost of the cover (which is proportional to the number of sensors in the cover) and the total power of the sensors used in the cover. These approaches, however, are concerned with the generation of a single cover, and the problem of maximizing lifetime is not addressed.

The present paper generalizes both problems, and provides exact approaches hybridized with metaheuristics for both of them. These approaches are based on column generation, which has been successfully used to address lifetime maximization problems in the literature [2], [8], [9], [14].

The remainder of this paper is organized as follows. Section 2 describes in detail the terminology and notations used in this paper and illustrates them, wherever appropriate, with suitable examples. Section 3 describes the problem models and their resolution approaches. Computational results along with their analysis are presented in Section 4. Finally, Section 5 outlines some concluding remarks and ideas for future works.

Section snippets

Definitions and notations

All the notations introduced in this section can be found in Table 1.

A mixed integer linear programming model

LM-ASR can be modeled as the following mathematical formulation:Maxj=1ctjs.t.j=1cr=1|Di|xi,r,jpi,rtjbi,i{1,,n}r=1|Di|xi,r,j1i{1,,n},j{1,,c}i=1nr=Ck,i|Di|xi,r,j1k{1,,m},j{1,,c}i=1nr=1|Di|xi,r,jmj{1,,c}xi,r,j{0,1}i{1,,n},r{1,,|Di|},j{1,,c}tj0j{1,,c}

The objective function is to maximize the network lifetime. Eq. (5), which is non-linear, ensures that energy limitations are respected for each sensor. It is a combination of Eqs. (2), (3). Eq. (6) states

Instances and experimental setup

The instances that we have used in our computational experiments have been generated as follows. n sensors and m targets are generated at random in a 500×500 area. The maximum sensing range of sensors is set to Rmax=150. The instance name format is nXXXmYYY, where XXX{50,100,150,200,300,600} is the number of sensors, and YYY{15,30,45,60,90,120,180,360} is the number of targets. Five instances of the same size have been generated, leading to a grand total of 60 different instances.

In addition

Conclusions

In this paper we have proposed an exact approach for maximizing the lifetime of a wireless sensor network where sensing ranges of sensors can be adjusted. Two versions of the problem are considered. In the first version, sensing ranges can be continuously adjusted within the maximum sensing range, whereas in the second one, sensing ranges can be adjusted only within the set of predefined values. We have modelled these problems using a column generation scheme and devised a matheuristic (i.e., a

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