Distributed Scheduling of a Network of AdjustableRange Sensors for Coverage Problems

Akshaye Dhawan

Department of Mathematics and Computer ScienceUrsinus College

Collegeville, PA 19426

Aung Aung and Sushil K. Prasad

Department of Computer ScienceGeorgia State University

Atlanta, GA 30030

Abstract

In this paper, we present two distributed algorithms to maximize the lifetime ofWireless Sensor Networks for target coverage when the sensors have the ability toadjust their sensing and communication ranges. These algorithms are based on the en-hancement of distributed algorithms for fixed range sensors proposed in the literature.We outline the algorithms for the adjustable range model, prove their correctness andanalyze the time and message complexities. We also conduct simulations demonstrating20% improvement in network lifetime when compared with the previous approaches.Thus, in addition to sleep-sense scheduling techniques, further improvements in net-work lifetime can be derived by designing algorithms that make use of the adjustablerange model.

1 Introduction

Wireless Sensor Networks (WSNs) are networks of low-cost sensing devices equipped with aradio. These sensors are deployed over an area of interest usually in large numbers, resultingin dense networks with significant overlaps. Once deployed, the sensor nodes monitor theirenvironment and send this information to a base station. A critical constraint of WSNs istheir limited energy supply. This is because sensors are powered by a battery that is non-replenishable. This limitation has given rise to various power saving techniques for WSNs.The most common of these is that of scheduling sensors into sleep-sense cycle [1, 2, 3, 4,

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5, 6, 7]. The objective of these algorithms is to select a minimal subset of sensors thatcompletely cover the region/targets of interest to switch on while the remaing sensors canenter a low-power sleep state.

In addition to switching off redundant sensors, another means of prolonging the networklifetime is proposed via the adjustable sensing and communication range model. To thebest of our knowledge, this model was introduced in [8] and [9]. Algorithms based on thismodel were presented in [10, 11]. In this model, a sensor has the capability to adjust itssensing as well as communication range. Each sensor has the option of different rangesr1, r2, ..., rp and corresponding energy consumptions of e1, e2, ..., ep. Instead of operating ata fixed range, a sensor can now reduce its range to the needed amount and thus reduce itsenergy consumption.

A slightly different model was proposed by us in [12]. In this model, instead of selecting arange from one of p fixed values, a sensor has the ability to smoothly vary its range between0 and rmax where, rmax is its maximum range. This has the advantage of allowing a sensorto precisely adjust its range to that needed to cover a target t, thus resulting in additionalenergy gains. In this paper, we utilize the smoothly varying range model.

Our Contributions: In this paper, we present two distributed algorithms for the ad-justable range model - ALBP which is an extension of the fixed range protocol LBP [13]and ADEEPS which is an extension of DEEPS [14] to the adjustable range model. We com-pare the performance of these algorithms to their fixed range counterparts through extensivesimulations that show lifetime improvements of 20% or higher. This shows that in general,the adjustable range model is an effective energy-efficient approach that adds to the lifetimeimprovements of existing sleep-sense scheduling algorithms. Part of this work was completedfor meeting thesis requirements [15].

The remainder of this paper is as follows. In Section 2, we briefly survey existing workon the adjustable range model and then discuss the fixed range algorithms LBP [13] andDEEPS [14]. In Section 3, we present the adjustable range variations, ALBP and ADEEPS.We outline the operational details of these algorithms along with their time and messagecomplexities. In Section 4, we present our simulation results, where we compare the ad-justable range algorithms with their fixed range versions. Finally, we conclude in Section5.

2 Background

The problem of scheduling sensors into minimal sensor covers that are activated successivelyso as to maximize the network lifetime has been shown to be NP-complete [3, 5].

Existing work on this problem has looked at both centralized and distributed algorithmsto come up with such a schedule. A common approach taken with centralized algorithms isthat of formulating the problem as an optimization problem and using linear programming(LP) to solve it [16, 5, 17, 12]. The distributed algorithms typically operate in rounds. Atthe beginning of each round, a sensor exchanges information with its neighbors, and makesa decision to either switch on or go to sleep. In most greedy algorithms [1, 2, 3, 4, 13, 14],the sensor with some simple greedy criteria like the largest uncovered area [2], maximumuncovered targets [13], etc. is selected to be on.

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We now focus on the various algorithms that make use of the adjustable range model.

2.1 The adjustable range model

The adjustable range model was proposed independently by [8] and [9]. In this subsection,we briefly survey the literature on extending the lifetime of WSNs using the adjustable rangemodel.

In [8], the authors present two different coverage algorithms based on an adjustable modelwhere the sensor can chose between one of two different ranges and one of three (maximum,medium and small) different ranges. Their objective was to minimize the overlapped areabetween sensor nodes, thereby resulting in energy savings.

[9] also address the problem of selecting a minimum energy connected sensor cover whennodes have the ability to vary their sensing and transmission radius. The authors present anumber of centralized and distributed heuristics for this problem including greedy formula-tions, Steiner Tree and Vornoi based approaches. The Vornoi based approach was shown toresult in the best gains.

In [10], the authors utilize a sensing model that allows a sensor to adjust its rangefrom one of several different fixed values. The authors address the problem of finding themaximum number of sensor covers and they present a linear programming based formulation,a linear programming based heuristic and also greedy formulations for this problem. A moreconstrained case of the same problem is studied in [11]. The authors add the requirementof connectivity to the sensor covers and present distributed heuristics to maximize the totalnumber of rounds.

We present a different model for adjusting the range in [12]. Instead of allowing a sensorto adjust its range between a number of fixed options, we allow the nodes to vary their rangesmoothly between 0 and rmax where, rmax is the maximum value for the range. We givea mathematical model of this problem using a linear program with exponential number ofvariables and solve this linear program using the approximation algorithm of [18], to providea (1 + ε)(1 + 2lnn) approximation of the problem.

More recently, [19] uses the adjustable range model to give two localized range optimiza-tion schemes based on a one-hop approximation of the Delaunay Triangulation. They alsoshow that their approximation scheme achieves the same results as the original DelaunayTriangulation.

Next, we look at the two fixed range protocols for which we propose adjustable rangevariations. The main idea of the load balancing protocol (LBP) is to keep the maximumnumber of sensors alive for as long as possible by means of load balancing [13]. LBP isgreedy on battery life. The Deterministic Energy-Efficient Protocol for Sensing (DEEPS)was presented by the same authors in [14]. The main intuition behind DEEPS is that theytry to minimize the energy consumption rate for low energy targets while allowing higherenergy consumption for sensors with higher total supply.

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3 Distributed Algorithms using Adjustable Range

3.1 Adjustable Range Load Balancing Protocol (ALBP)

In this section, we present a distributed load balancing protocol for sensors with an adjustablerange called ALBP. This protocol extends the ideas of LBP [13] to the adjustable rangemodel. As with LBP, the objective of the protocol is to maximize the time for which alltargets in the network are covered. The intuition behind the protocol is also similar to LBP,the aim is to keep as many sensors alive by balancing their load so as to let them exhausttheir batteries simultaneously. ALBP, however, differs from LBP in that while making adecision on switching a sensor to an active state, it also needs to decide what range thissensor should have.

We begin by defining the different states a sensor can be in at any point of time:

• Active: The sensor is monitoring a target(’s).

• Idle: The sensor is listening to other neighboring sensors, but does not monitor targets.

• Deciding: The sensor is presently monitoring a target, but will change its state toeither active or idle soon.

Setup: The setup phase starts as before. At the beginning of a round, each sensorexchanges information on its battery level and the set of targets it covers with its neighbors.However, after broadcasting this information, a sensor enters the deciding state with itsmaximum range.

Figure 1: State transitions for the ALBP protocol

Transitions : Figure 1 shows the state transitions for ALBP. At the end of the setupphase, each sensor is in the deciding state with its maximum range. A sensor then changesits state according to the following transition rules:

• Active state with a range r: A sensor transitions to the Active State with a ranger, if there is a target at range r which is not covered by any other active or decidingsensors.

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• Deciding state with lower range: A sensor in the deciding state with some range rcan decrease its range to the next closest target if all its targets at range r are coveredby another sensor in the active state or by a sensor that is in the deciding state andhas a higher battery life.

• Idle state: A sensor is in the idle state if it has reduced its range to zero (i.e., all itstargets are covered by active sensors or higher energy sensors in the deciding state)

Every sensor uses these rules to make a decision on whether to enter the active or idlestate. The sensors will stay in this state until the end of a round, upon which theprocess will repeat itself. When the network reaches a point where a target cannot becovered by any sensor, the network is considered dead.

Correctness : In ALBP, a sensor can enter the idle state only when its range reacheszero. To achieve this, all its targets had to have been covered by an active or decidingsensor that had a higher energy than it. Hence, all targets are always covered.

Time and Message Complexity: The time complexity of ALBP is O(∆2) and themessage complexity is O(∆) where, ∆ is the maximum degree of the sensor graph.

At the start of a round, each sensor receives from every neighbor a message containingthe targets that neighbor covers, and its battery life. If ∆ is the maximum degree ofthe graph, a sensor can have no more than ∆ neighbors. This means that a sensor canreceive no more than ∆ messages, which it can process in O(∆) time.

In the worst case, a sensor may have to wait for all its neighbors to decide their statebefore it can make a decision. Thus, the waiting time accumulates as O(∆2), hencethe time complexity.

Since each sensor has at most ∆ neighbors and during a round a sensor sends atmost two messages to its neighbors (its battery and targets covered information, andits status - on/off), at most O(∆) messages are sent in the setup phase. Hence, themessage complexity is O(∆).

3.2 Adjustable Range Deterministic Energy EfficientProtocol (ADEEPS)

In this subsection, we present an adjustable range version of the DEEPS protocol [14]. Eachsensor can once again be in one of three states - active, deciding or idle. The definition ofthese states remains as before.

We make use of similar concepts of sink and hill targets as DEEPS. However, instead ofdefining these using the total battery of the sensors covering a target (as DEEPS does), wedefine them with respect to the maximum lifetime of the target. Let the lifetime of a sensorwith battery b, range r and using an energy model e be denoted as Lt(b, r, e). Then, themaximum lifetime of a target would be Lt(b1, r1, e)+Lt(b2, r2, e)+Lt(b3, r3, e3)+ ... assumingthat it can be covered by some sensor with battery bi at distance ri for i = 1, 2, ....

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Now, we can define the sink and hill targets as follows. A target t is a sink if it is thesmallest maximum lifetime target for at least one sensor covering t. A hill is a target whichis not a sink for any of the sensors covering it. We define the in-charge sensors for a targett as follows:

• If the target t is a sink, then the sensor s covering t with the highest lifetime Lt(b, r, e)for which t is the poorest is placed in-charge of t.

• If target t is a hill then out of the sensors covering t, the sensor s whose poorest targethas the highest lifetime is placed in-charge of t. If there are several such sensors, thenthe richest among them is placed in-charge of t.

Setup: Each sensor initially broadcast its lifetime and covered targets to all neighbors ofneighbors. This is similar to [14]. After this, it stays in the deciding state with its maximumrange.

Transitions : A sensor that is in the deciding state with range r changes its state accordingto the following rules:

• Active state with a range r: If there is a target at range r which is not covered byany other active or deciding sensors, the sensor enters the Active state with range r.

• Deciding state with lower range: A sensor in the deciding state with some ranger can decrease its range to the next closest in-charge target if all its in-charge targetsat range r are covered by another sensor in the active state or by a sensor that is inthe deciding state and has a higher battery life.

• Idle state: When a sensor s is not in-charge of any target except those already coveredby on-sensors, s switches itself to the idle state.

The algorithm again operates in rounds. When there exists a target that cannot becovered by any sensor, the network is considered to be dead.

Correctness : The correctness of ADEEPS can be proved from the fact that each targethas a sensor which is in-charge of that target and the transition rule to active state assuresthat the resultant sensor cover is minimal in which each sensor s has a target covered onlyby s.

Time and Message Complexity : In each round, the time complexity of ADEEPS isO(∆2) and the message complexity is O(∆2) where, ∆ is the maximum degree of the graph.

Each sensor has no more than ∆ neighbors. At the start of each round, every sensorreceives from its neighbors and their neighbors (2-hops) information about their lifetime andthe targets covered. Thus a sensor can receive at most ∆2 messages. Once this informationhas been received, all decisions on sink/hill targets, in-charge sensor and the active/idle statecan be made locally. Thus the time complexity is O(∆2).

Also, since a sensor has at most ∆ neighbors and it needs to communicate the setupinformation to two-hops, each sensor sends O(∆2) messages in the setup phase. This meansthat the message complexity is O(∆2).

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4 Simulations

To evaluate the performance of the new algorithms and to make comparison with the al-gorithms in [13, 14], the new algorithms are implemented by using C++. We built on thesource code for [13, 14]. For the simulation environments, a static wireless network of sensorsand targets which are scattered randomly, while ensuring that all targets can be covered, in100m×100m area is considered. The location of the sensor nodes can be randomly generatedand the targets can also be placed randomly. We assume that the communication range ofeach sensor is two times the sensing range. Simulations are carried out by varying the numberof sensors and the lifetime is measured. We also vary the maximum range, energy models,and numbers of targets with various combinations. For these simulations, we use the linearenergy model wherein the power required to sense a target at distance d is proportional tod. We also experiment with the quadratic energy model (power proportional to d2). Notethat to facilitate comparison, we follow the simulation setup of [13, 14].

Figure 2: Variation in network lifetime with the number of sensors with 25 targets, linearenergy model, 30m range

In the first simulation shown in Figure 2, we limit the maximum range to 30m. Thismeans that a sensor can smoothly vary its range from 0 to 30m. The simulation is conductedwith 25 randomly deployed targets, 40 to 200 sensors with an increment of 20 and a linearenergy model. As is expected, increasing the number of sensors while keeping the number oftargets fixed causes the lifetime to increase for all the protocols. Also, using the adjustablerange model shows performance improvements when compared to the fixed range model.As can be seen from the figure, ALBP outperforms LBP by at least 10% and ADEEPSoutperforms DEEPS by around 20%.

In the second simulation shown in Figure 3, we study the network lifetime while increasingthe number of targets to 50 and keeping maximum range at 30m. The numbers of sensors arevaried from 40 to 200 with an increment of 20 and the energy model is linear. The results ofsimulations are consistent and showed that the network lifetime increases with the number

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Figure 3: Variation in network lifetime with the number of sensors with 50 targets, linearenergy model, 60m range

of sensors. When compared with the results of Figure 2, the network lifetime decreases asmore targets are monitored. This is also a logical conclusion, since a larger number of targetsimplies that there is more work to be done by the network as a whole.

In Figure 4, we change the energy model to the quadratic model. We use the same numberof sensors (40 to 200 with increment of 20), the maximum range is 30m and the energy modelis quadratic. As in Figure 2, for both energy models, the result indicates that the networklifetime increases with the number of sensors. As is expected, the quadratic model causes allprotocols to reduce in total lifetime when compared to the linear model of Figure 2. It canalso be seen that the network lifetime is significantly improved with ALBP and ADEEPSin the quadratic model. This phenomenon is quite logical since in the fixed sensing model,each sensor consumes more energy than the adjustable range model. Improvements here arein the range of 35-40% when compared to their fixed range counterparts.

From the results, the overall improvement in network lifetime of ALBP over LBP isaround 10% and ADEEPS over DEEPS is about 20% for linear energy model. For quadraticenergy model, the improvements are even higher.

5 Conclusion

In this paper, we present two distributed algorithms for maximizing the lifetime of wirelesssensor networks when the sensors have adjustable ranges. These algorithms are built as anextension of earlier algorithms that were devised for the fixed range model. Simulations showthat significant improvements in network lifetime can be obtained for comparable algorithmsusing the adjustable range model. Overall the new heuristics exhibit a 10-20% improvementover their fixed range versions for the linear energy model and 35-40% improvement for thequadratic energy model.

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Figure 4: Variation in network lifetime with the number of sensors, with 25 targets, quadraticenergy model and 30m maximum range

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