Received: 15 March 2016
Revised: 26 September 2016
Accepted: 17 November 2016
DOI 10.1002/dac.3264
RESEARCH ARTICLE
Multihop data gathering in wireless sensor networks with a mobile sink Farzad Tashtarian1
Khosrow Sohraby2
Amir Varasteh1
1 Department
of Computer Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran 2 General Motors, Correspondence Farzad Tashtarian, Department of Computer Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran. Email:
[email protected]
Summary Network performance can be improved by using a mobile sink (MS) to collect sensed data in a wireless sensor network. In this paper, we design an efficient trajectory for MS, collecting data from sensor nodes in a multihop fashion, with the aim of prolonging the network lifetime. Considering event-driven applications, we present an approach to jointly determine the optimal trajectory for MS and data paths and transmission rates from source nodes to MS, without considering any rendezvous points. In these applications, an MS is supposed to harvest the data from source nodes in a given time-slot. We first show that this problem is in form of a mixed integer nonlinear programming model, which is NP-hard. Then, to achieve an approximate solution, we divide the mentioned problem into 2 simple subproblems. In fact, after determining an approximate zone for the trajectory of MS, the optimal data paths and transmission rates from source nodes to the MS are obtained through a mathematical optimization model. Finally, to illustrate the efficiency of the proposed approach, we compare the performance of our algorithm to an rendezvous point–based and also the state-of-the-art approach in different scenarios. KEYWORDS
wireless sensor network, multihop data gathering, mobile sink, mathematical optimization
1
INTRODUCTION
Recently, the idea of boosting network performance using mobile elements in wireless sensor networks (WSNs) has been profoundly investigated by many researchers.1–9 For instance, using a mobile sink (MS) to gather sensory data has been explored in many studies. In addition, a number of studies have addressed the challenge of increasing WSN lifetime by determining the MS trajectory.10–22 Studying the proposed strategies of MS movement, 2 main classes of sink mobility can be defined: random sink mobility (RSM)10,11 and controlled sink strategy (CSM).1,2,13,18,19,23–25 The MS in RSM class can move freely in the network and harvest the sensory data. It is obvious that uncontrolled behavior of MS in RSM decreases the performance of network in the network lifetime.16 In contrary, the CSM algorithm designs a trajectory for MS with respect to different vital parameters such as the velocity of MS, the Int J Commun Syst 2017; e3264; DOI 10.1002/dac.3264
priority of buffered data, the buffer size of sensor nodes, and so on. The CSM introduces challenges such as trajectory design, delivering data to MS, mobility scheduling, and controlling MS velocity.26 The majority of studies have shown that designing trajectory corresponding to the different mobility models is the main problematic issue in CSM that can significantly improve network performance in network lifetime and energy consumption.24,25 In this paper, we focus on the event-driven applications to jointly determine the optimal data paths and data transmission rates from source nodes to the MS and specify the optimal MS trajectory. Although the problem of maximizing the network lifetime in CSM has been investigated in the literature,1–7 there are still some issues that need to be investigated: (1) Single hop data transmission from source nodes to MS is not a practical strategy in all applications of WSN. This issue gets more challenging when the MS needs to harvest the data in
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TASHTARIAN ET AL.
a specific time-slot.24 Therefore, in this study, which is an extension on our previous work,25 we are going to scrutinize multihop data gathering by an MS in a WSN. (2) Furthermore, considering the MS travel time between any 2 different geographical locations in the network without any infrastructure such as virtual grid or rendezvous points (RPs) makes our work more reasonable and efficient on real deployment. The remainder of the current paper is organized as follows. A summary of the related work is provided in section 2. Section 3 presents the system model and in section 4, we describe and model the problem in different scenarios. In the next section, we propose a heuristic algorithm to provide an approximate solution to the addressed problem. The performance of the proposed approach is scrutinized via simulation in section 6, and we conclude the paper in section 7.
2
RELATED WORK
Compared with the uncontrollable sink mobility algorithms like Li et al27 and Vecchio et al,28 CSM algorithms would profoundly improve the system performance in data delivery, network lifetime, and energy efficiency. In this section, we study the merits and defects of various categories of CSM and then introduce our main contributions. The proposed models in CSM can be divided into 2 main classes: restricted (RCM) and unrestricted (UCM) controlled mobility classes. In the RCM, the mobility of MS is restricted among a set of predefined location points as RPs (RCM-RP)13,22 and29 or along a fixed path (RCM-path),4 and,30 eg, freeways and railways. The second class is characterized by the complete freedom movement of MS in the network field without considering any predefined paths or location points (see Figure 1). 2.1
Restricted controlled mobility–rendezvous point
The proposed algorithms in the RCM-RP modeled the trajectory of MS regarding predefined points in different applications.2,13–15,18,19,31 Gandham et al29 increased the performance of WSN by proposing an integer linear programming (ILP) model and controlling the movement of MS on the basis of predefined RPs. Yun et al13 considered delay tolerant applications and then proposed a linear programming (LP) model
FIGURE 1
with the objective function of increasing the network lifetime to determine the optimal trajectory for MS. They explored the trajectory on the basis of predefined RPs. However, ignoring the MS traveling time between any 2 RPs and producing trajectory based on RPs are the main drawbacks in 1 study.13 Xing et al in 1 study14 proposed an efficient approach to specify MS trajectory in the network by considering a set of nodes as RPs. Also, in 1 study,14 the rendezvous nodes were responsible for aggregating data from source nodes and sending to the MS at an appropriate time. The trajectory of MS was designed on the basis of constructed routing tree rooted at RPs. Although the objective function for determining the tree minimized the total length of the tree, it could not guarantee to reach the maximum network lifetime. The idea of using resource-limited mobile sinks to harvest sensory data from source nodes was presented by Wang et al in 1 study.15 The main challenge was how to distribute mobile sinks in the network to minimize the number of missed events. A mixed integer linear programming (MILP) model was offered by Basagni et al in 1 study18 to design an efficient trajectory for MS based on some predefined RPs in the network. Moreover, they determined the sojourn time at each RP to harvest data from sensor nodes. In fact, sojourn time could be different for specific applications. The authors ignored the MS traveling time between any 2 points RPs in the network. Authors in 1 study19 showed that the problem of finding an optimal location for MS to harvest data from sensor nodes is in form of NP-hard and then they proposed a (1 − 𝜖) optimal algorithm to determine an optimal location for MS. The authors divided the smallest disk that covers all sensor nodes into some subareas and then selected a subarea as an optimal location for MS based on linear mathematical optimization model. 2.2
Restricted controlled mobility–path
Many studies have been proposed on RCM-path to investigate the effects of utilizing MS on the network performance in terms of network lifetime and energy consumption.4,6,30,32–34 In this category, the velocity of MS on a predefined path can be easily determined through the amount of buffered data by nearby sensors along the path. In 1 study,4 the authors divided sensing area into subareas and used multiple controllable MULEs (mobile ubiquitous LAN extensions) as MS to gather data while moving on linear paths. The main defect of 1 study4 was the fact that the algorithm worked well only
Different types of control sink mobility. MS, mobile sink; RCM, restricted controlled mobility ; UCM, unrestricted controlled mobility
TASHTARIAN ET AL.
when the paths were remained fixed. In 1 study,22 the authors offered an efficient strategy to control the mobility of MS on a constrained path. Dividing the network into some clusters and routing the data from clusters to the MS was its main idea. The algorithm was designed to be used in environmental monitoring, eg, urban parks. Gao et al30 proposed an RCM-path algorithm in which data were delivered to MS according to an efficient routing path. They formulated routing algorithm as a binary linear programming (BLP) and the objective function was designed to minimize the sum of the shortest hops. They solved the BLP model by presenting a genetic algorithm. 2.3
Unrestricted controlled mobility
Because in RCM a predetermined structure such as virtual grid, RPs or fixed-path trajectory is taken into account, some defects and challenges are emerged, the main ones of which are the following: (1) Although considering a large number of RPs increases the quality of solutions, the time complexity of algorithm elevates dramatically12 ; (2) The majority of RCM algorithms are not scalable and applying them in a larger WSN may result in the significant increment of data latency. To mitigate this challenge, some studies have assumed that MS has very high velocity for traveling between RPs; however, there is still some space for further improvements. In UCM algorithms, MS has complete freedom to move and gather the sensed data.35 In this category, there are numerous studies that do control the speed of MS.35–37 To reduce data latency, the authors in 1 study24 divided the network into some smaller areas and then used multiple MSs to gather data. Unrestricted controlled sink strategy considers 2 traffic models: static and dynamic. In the static model, each sensor node generates data with constant rate during the network operation.37 On the other hand, many schemes consider dynamic traffic model when the traffic in the network is time-varying.35,36,38 In 1 study,36 authors introduced SenCar as a mobile robot to gather data from static sensor nodes in a multihop manner. They proposed an energy efficient heuristic algorithm to explore the trajectory of SenCar. Luo and Hubaux35 proposed an MILP to determine the trajectory of mobile collectors in the network. All sensor nodes and sink were supposed to be static. In 1 study,35 while a collector node was moving, it polled the nearby sensor nodes one by one to gather data. However, in both studies Ma and Yang36 and Luo and Hubaux,35 the routing issue has not been fully investigated,
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which is the main drawback. Furthermore, regarding the position of SenCar and given destination point, the authors have proposed a simple method for determining the returning trajectory points. The proposed strategy considers neither the routing issue in the procedure of specifying trajectory nor the optimal destination point. We have addressed the problem of determining the optimal trajectory of MS without considering any predefined structure such as RPs in 1 study24 and in another study.25 In Tashtarian et al,24 we proposed a convex optimization model inspired by the support vector regression technique to determine a continuous and optimal trajectory (COT) for the MS. In another study,25 we addressed the problem of ignoring the traveling time of the MS between the ending point of the previous COT and the starting point of the next COT (that was assumed in 1 study24 ) through proposing an algorithm based on a decision tree and dynamic programming. In our previous studies such as Tashtarian et al24 and Tashtarian et al,25 the network was divided into some autonomous areas to enable the MS in each area to harvest data in a single-hop manner. However, it is not a cost effective solution, specially in nodes energy consumption and the number of employed MSs. Hence, we are going to extent our proposed algorithm in 1 study25 by jointly determining the optimal data paths and transmission data rates from the source node to the MS and MS trajectory.
3
SYSTEM MODEL
First, the general assumptions about the WSN model are described. Suppose n sensor nodes Si ,i = 1:n, are uniformly and randomly deployed in a circular area with radius h. The initial energy and coordinate location of Si are denoted by ei and (xi ,yi ), respectively. Furthermore, suppose a mobile robot with a constant velocity v as the MS. Considering event-driven applications, let 𝒩 be a set of source nodes that captures events in the network. In these applications, upon detecting an event by Si , it becomes the member of 𝒩 and begins to send data, which are generated at constant rate gi , to the MS via multihop data transmission (see Figure 2). Notably, we have not addressed the -node/link failure that could be an issue in multihop data gathering.39 In our system model, the deadline 𝜃 is defined as the maximum time-slot in which the MS must harvest a segment of data from 𝒩 , which ∑ ∑ equals to i∈𝒩 gi ti , where i∈𝒩 ti ⩽ 𝜃. The value of ti is predefined by the underlying applications based on the priority of
A, Multihop data transmission from the source nodes to MS when it has less time 𝜃 . B, Multihop data transmission from the source nodes to MS when it has more time 𝜃 . MS, mobile sink
FIGURE 2
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TABLE 1
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Notations
Notation
Description
increasing 𝜃, MS has more time to gather data from source nodes, even separately in fewer multihop data transmission. As shown in Figure 2, by increasing 𝜃, the territory of MS movement can be broadened and, then, data can be transmitted with less energy consumption. Our other assumptions are as follows:
n
The total number of sensor nodes in the network
Si
The ith sensor node in the network
ei
The initial energy of 𝒩 i
(xi ,yi )
The coordinate location of Si
v
The constant velocity of MS
𝒩
The group of sensor nodes that captures an event (source nodes)
gi
The constant data generation rate by Si , i ∈ 𝒩
𝜃
The maximum time-slot in which the MS must harvest a segment of data from 𝒩
ti
The data gathering time from Si , i ∈ 𝒩 , by MS in each time-slot 𝜃
•
The network lifetime that is defined as the total operational time of the network
•
T
• •
till one sensor node drains its entire battery
F
The fitness value which is defined as the number of round trip movement of MS on its trajectory until one sensor node drains its entire battery
• •
It is assumed that sensor nodes can adjust their transmission power level to support different transmission ranges. The data transmission path from 𝒩 to MS is fixed until at least 1 node joins or leaves 𝒩 . The location, event occurrence time, and amount of data produced by sensor nodes are unknown. The determined link rate by the MAC-layer protocol is large enough, so that constraints on the data rates or the MAC-layer contention are not considered.13 We assumed that sensor nodes have identical buffer sizes. Although there are algorithms to determine the position of sensor nodes,40–42 we assume that the MS already knows the sensor nodes position in the network.
ℱ
1∕F
dij
The Euclidean distance between nodes i and j
diMS
The Euclidean distance between node i and MS
Eijk
The consumed energy by Si to send a composite bit-stream generated by source node k to Sj at rate fij(k) over dij
Ē jik
The consumed energy by Si per unit of time for receiving the data generated
This present study uses the simple model proposed in the previous studies12,13,24 for radio hardware energy dissipation. If the Euclidean distance between nodes i and j, which is denoted by dij , is less than threshold d0 , the free space (fs) model is applied using dij2 ; otherwise, the multipath (mp)
by source node Sk from Sj
model is used using dij4 . Thus, Si consumes Eij(k) to send a com-
𝒢i
The optimal subset of sensor nodes to send data of Si to the MS in a single-hop manner
ℛi
The optimal sub-set of sensor nodes that delivers the data from Si , i ∈ 𝒩 , to 𝒢 i
Oi (Oci )
The visiting area that surrounds all the members of 𝒢 i (the center of Oi )
𝒵
A circular boundary that surrounds all visiting areas
ℒ0
The current location of the MS
ℒ1
The optimal location of the MS in sub-problem I that is determined by12
events and also the buffer size of Si , Si ∈ 𝒩 . For instance, ti should be a small value for temperature sensing and fire detection and relatively larger for other applications like humidity sensing. For the sake of convenience, all symbols in the paper are listed in Table 1. Definition 1. The network lifetime T is defined as the time duration since the launch of network operation till the first sensor node drains its entire battery. Definition 2. Because it is assumed that the MS has to har∑ vest i∈𝒩 gi ti amount of data from 𝒩 in each round trip movement on its trajectory, we define the fitness value F as a number of round trip movement of MS on its trajectory till the first sensor node drains its entire battery. In fact, we introduce F to measure the quality of a solution in network lifetime T. Assuming identical buffer sizes, minimum 𝜃 can be equal to max{ti |i ∈ 𝒩 }. In this case, MS has to collect data from 𝒩 simultaneously. However, by
posite bit-stream generated by source node k to Sj at rate fij(k) over dij , as follows: { 𝛼 + 𝛽fs dij2 , dij < d0 (k) (k) , (1) Eij = cij fij , where cij = 𝛼 + 𝛽mp dij4 , dij ⩾ d0 √ where d0 ≈ 𝛽fs ∕𝛽mp . The constant term 𝛼 (joule/bit) is selected on the basis of electronic energy, and the amplifier energy, 𝛽 f s or 𝛽 mp , is determined with respect to the distance to the receiver and the acceptable bit-error rate. The dimensions of 𝛽 f s and 𝛽 mp are joule/bit/m2 and joule/bit/m4 , respectively. The consumed power by node Si per unit of time for receiving data, generated by source node k, from Sj is expressed by: Ē ji(k) = 𝛼fji(k) . (2)
4
PROBLEM DESCRIPTION
In 1 study24 and another study,25 we have addressed the challenges of controlling MS to collect sensory data from source nodes in a single-hop manner. To enhance power-efficiency, however, the data should be delivered from 𝒩 to MS in a multihop manner. Therefore, in addition to determining the optimal trajectory for MS, the data paths from source nodes to MS and data transmission rates should be efficiently specified. Before studying the problem of determining an optimal trajectory for |𝒩 | ⩾ 2, the optimal trajectory for a single source nod (|𝒩 | = 1) is proposed.
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4.1
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|𝒩 | = 1
Let us consider the MS trajectory as a 𝜇-connected line segments denoted by graph G = (M, N) with M and N as vertices and edges, where |M| = 𝜇 + 1,|N| = 𝜇⩾1,∀ideg(M i )⩽2, and ∑ i=1∶𝜇 length(Ni ) ⩽ 𝜃 × v. It is clear that the optimal MS trajectory for |𝒩 | = 1 is the straight line connecting MS and the source node. To efficiently gather the data from the source node, MS continuously must move toward the source node to reduce the distance between them. In fact, the number of participating sensor nodes in delivering data from the source node to MS will be diminished by moving on the straight line. Because the MS has to gather gi ti amount of data from Si , i ∈ 𝒩 , in each time-slot 𝜃, it must move toward Si
In other words, the sum of total incoming flow rates plus self-generated data rate is equal to the sum of total outgo(i) ing flow rates to MS, fiMS , and other nodes, fik(i) . The energy constraint for all nodes can be formulated as follows: ( ) ∑ (i) (i) Eik ⩽ ei , ∀i ∈ 𝒩 ti EiMS + ( ti
k
∑ k(k≠i)
Ē kj(i)
+
(i) EjMS
+
∑
) Ejk(i)
, ⩽ ej , ∀j ∉ 𝒩
k(k≠i)
(4) ie, due to receiving and transmitting data, total consumed energy cannot exceed its initial energy, ei . Now, the fitness value F can be achieved through the following equation25 :
⎧ ⎫ ⎪ ⎪ ej ⎪ ⎪ ei F = min ⎨ ( ), ( ) |∀i ∈ 𝒩 , j ∉ 𝒩 ⎬ . ∑ (i) ∑ ̄ (i) ⎪ ti E(i) + ∑ E(i) ⎪ (i) ti Ekj + EjMS + Ejk iMS ik ⎪ ⎪ k k(k≠i) k(k≠i) ⎩ ⎭
in 𝜃 − ti units of time, and then stop for ti to harvest the data in the multihop. The MS repeats this procedure until Si , i ∈ 𝒩 and transmits all the gathered data. Regarding the defined graph G, Mi is the location of MS in the lth movement, where dMl−1 Ml = (𝜃 − ti ) × v (see Figure 3A for source node S1 ). The optimal data paths and transmission rates for the sensor nodes participating in delivering data from Si , i ∈ 𝒩 to MS located at M l can be obtained through satisfying the following constraints: The first constraint guarantees that all the generated data by Si , i ∈ 𝒩 are transmitted to the MS. Thus, we have the following:
(i) + fiMS
∑
fik(i) = gi ,
k
(i) fjMS +
∑ k(k≠i)
FIGURE 3
fjk(i) =
∑ k(k≠i)
∀i ∈ 𝒩 fkj(i) , ∀j ∉ 𝒩
.
(3)
(5)
In fact, for the small number of source nodes, maximizing F cannot guarantee to prolong the network lifetime. It is possible that some sensor nodes Sj , j ∉ 𝒩 inefficiently participate in delivering data from Si , i ∈ 𝒩 to MS, because of the volume of the delivered data, and consequently, their amount of energy consumption may not influence F. Therefore, this inefficient energy depletion may result in reduction of F. Hence, we consider the following multiobjective function as follows: maximize 𝜔1 (fitness valueF) −𝜔2 (total energy consumption),
(6)
where weights 𝜔1 and 𝜔2 are determined regarding the desired data harvesting mechanisms in network lifetime and energy efficiency, respectively. Therefore, the LP model for |𝒩 | = 1 can be formulated as follows:
A, Optimal data harvesting for |𝒩 | = 1. B, Optimal data harvesting for |𝒩 | ⩾ 2
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A, Measuring the maximum fitness value F and the total energy consumption for different values of 𝜔1 . B, Data paths from source nodes to the sink 𝜔1 = 0.05. C, Data paths from source nodes to the sink 𝜔1 = 0.5. D, Data paths from source nodes to the sink 𝜔1 = 0.9.gi = 300 bps and ti = 3 s. The other simulation parameters are listed in Table 2
FIGURE 4
∑
𝛿i
i=1∶n
𝜔1 ℱ + 𝜔2 ∑
minimize
ei
i=1∶n
s.t.
constraint (3) ( ) ∑ (i) (i) Eik = 𝛿i , ti EiMS + ( ti
∑
(i) Ē kj(i) + EjMS +
k(k≠i)
∑
) Ejk(i)
,
l
∑
= 𝛿j , ∀j ∉ 𝒩
k(k≠i)
𝛿i ⩽ ℱ ei , vars.
∀i ∈ 𝒩
k
k
(i) ℱ , EiMS , Eik(i) , Ē kj(i) , and𝛿i ⩾ 0
where 𝜔1 + 𝜔2 = 1 and ℱ = F1 . Let us take an example to manifest the impact of 𝜔1 on the fitness value and total energy consumption in a network with a static sink. In this experiment, we consider a topology with 20 sensor nodes and a singe source node (see Figure 4). We measure the maximum fitness value F and the total energy consumption for different values of 𝜔1 from 0.01 to 0.99. As can be observed, by increasing 𝜔1 , the proposed model (Equation 7) is more focused on ℱ and provides solutions with higher performance in network lifetime. The impact of 𝜔1 on the solutions, especially on data paths from source node to the sink, is shown in Figure 4B-D. As it is conceived, when 𝜔1 increases, some unsolicited sensor nodes like {Si , Sj }, i, j ∉ 𝒩 participate in data delivery, because their amount of consumed energy cannot affect the maximum ℱ . |𝒩 | ⩾ 2
Theorem 1. The problem of jointly determining the optimal data paths and transmission rates, and optimal MS trajectory is an NP-hard when |𝒩 | ⩾ 2. Proof. Without loss of generality, assume that the MS must collect the sensory data from 𝒩 through traveling over 𝜇 edges of G (see Figure 3B for 𝜇 = 3). To achieve an optimal trajectory for |𝒩 | ⩾ 2, we must satisfy the following
∑
(il) f̄jMS +
l
i=1∶n (7)
4.2
constraints. First, let us define binary array 𝒯 to indicate valid edges for sensor nodes to send their data to MS. If 𝒯 (i,j,l) = 1, Si delivers a portion data of source node Sj , j ∈ 𝒩 to MS moving on N l (the lth edge of graph G). Similar to the proposed model (Equation 7), we define the data flow constraint as follows: ∑ ∑ ( j) ∑ (il) ∑ (i) fik = gi + fki , ∀i ∈ 𝒩 f̄iMS + fjk(i)
∑
=
k(k≠i)
j≠i k(k≠i
fkj(i) ,
, (8)
∀i ∈ 𝒩 , j ∉ 𝒩
k(k≠i)
(il) where f̄jMS indicates the transmission rate from Sj to MS (carrying data from source node Si , i ∈ 𝒩 ) when MS is moving on N l . In the next constraint, we calculate the total energy consumption of sensor nodes: ( ( ( )2 ) ∑ ∑ (l) ∑ (i) ̄t 𝛼f + 𝛼 + 𝛽fs d̄ (l) f̄ (il) i
l
i
kj
jMS
k
+
∑
jMS
) cjk fjk(i)
k
∑
= 𝛿j , ∀j
̄ti(l) = ti ,
.
(9)
∀i
l
The Euclidean distance between Si and MS moving on N l (l) which is denoted by d̄ iMS can be obtained by the following constraints: (l) , 𝒯 i, j,l × dSi Ml ⩽ d̄ iMS
(10)
(l) 𝒯 i, j,l × dSi Ml+1 ⩽ d̄ iMS .
(11)
To allocate sufficient time to the MS for gathering data of Si , i ∈ 𝒩 , when it is moving on N l , the following constraint must be satisfied: ̄tj(l) ×𝒯 i, j,l × v ⩽ dMl Ml+1 ,
∀l, and ∀j ∈ 𝒩 .
(12)
The next constraint guarantees that MS traveling time must be less than or equal to 𝜃 units of time: ∑ dMl Ml+1 ⩽ v × 𝜃. (13) l=1∶𝜇
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FIGURE 5
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A, Sets of 𝒢 i and ℛ i . B, Visiting area for MS to gather data from 𝒢 i . C, A sample of three visiting areas in the network field. MS, mobile sink
Finally, the optimization problem for |𝒩 | ⩾ 2 can be modeled as follows: ∑ 𝛿i i=1∶n minimize 𝜔1 ℱ + 𝜔2 ∑ ei i=1∶n
s.t.
𝛿i ⩽ ℱ ei ,
i=1∶n
.
(14)
constraints (8 − 13) vars.
(il) ̄ ℱ , fjk(i) , f̄jMS , dMl Ml+1 , dSi Ml ⩾ 0
Ml is free, 𝒯 i,j,l ∈ {0, 1} The proposed model (Equation 14) is in the form of MINLP which is NP-hard.43,44
5
PROPOSED HEURISTIC ALGORITHM
In this section, a heuristic algorithm to determine an approximate solution for the proposed model (Equation 14) is presented. Our main contribution is to specify the territory of MS movement by considering the locations of source nodes, 𝜃 and v, and then determine the optimal data paths and transmission rates from the source nodes to the MS moving on an achieved trajectory. Let us divide the addressed problem into 2 subproblems I and II. In subproblem I, we propose a heuristic algorithm to achieve a trajectory for the MS. Moreover, an optimization model to determine the optimal data paths and transmission rates from 𝒩 to MS is proposed. Since in subproblem I, we address the problem by assuming that MS is located at optimal location ℒ 1 , in the second subproblem, we relax this assumption and propose an efficient strategy for the MS to move from its current location, denoted by ℒ 0 , to the optimal location ℒ 1 . 5.1
Subproblem I
By considering ℒ 1 as an optimal and fixed location for a sink obtained by the proposed algorithm in 1 study,12 the optimal data paths and data transmission rates can be easily achieved by the proposed model (Equation 7). In this subproblem, by assuming the MS located at ℒ 1 , we propose a heuristic approach to obtain a trajectory for the MS and data paths from source nodes to it as well. In our heuristic algorithm, we categorize the participating sensor nodes in delivering data from
𝒩 to the MS into 2 groups: gateway nodes 𝒢 i and relay nodes ℛ i that are defined as an optimal subset of sensor nodes, which sends the data of Si to MS in single-hop and an optimal subset of sensor nodes and delivers data from Si , i ∈ 𝒩 to 𝒢 i , respectively (see Figure 5A). To determine 𝒢 i and ℛ i , we first need to specify the territory of MS movement, because it is assumed that only the members of 𝒢 visit MS and transmit the buffered data in a single-hop manner. Therefore, the locations and distance between the members of 𝒢 and MS must be determined. Let us define Oi as the visiting area that surrounds all the members of 𝒢 i . Since MS has to visit 𝒢 i for ti units of time, radius of Oi should be equal to v × ti /2 by assuming that MS moves across the center of Oi , denoted by Oci (see Figure 5B). By having Oi , i = 1 ∶ |𝒩 |, the problem of finding an optimal trajectory for MS passing all centers of all Oi in 𝜃 units of time is the same as the travelling salesman problem, which is in form of NP-hard45 (see Figure 5C). As it is discussed earlier, the MS has 𝜃 units of time to visit all source nodes in each Oi , i = 1 ∶ |𝒩 |. Thus, to simplify the proposed approach, we assume that the center of all Oi s forms a regular |𝒩 |-sided polygon as the appropriate trajectory for the MS by using ℒ 1 as the center of |𝒩 |-sided polygon. Also, the length of each side and circumradius of the |𝒩 |-sided polygon, denoted by l and r, respectively, are calculated as follows (see Figure 6): l=v×
r= 2 sin
𝜃 , |𝒩 |
(15)
l (
(16)
𝜋 |𝒩 |
).
If r is greater than r̄ which is defined as the average distance between the MS at ℒ 1 and source nodes, we set r = r̄ and update l accordingly. It is obvious that MS can travel the |𝒩 |-side polygon in 𝜃 units of time. Because the locations of Oci are unknown, we define boundary 𝒵 to surround all visiting areas. The boundary 𝒵 is equal to the difference between 2 circles C1 (ℒ 1 , r + r∗ ) and C2 (ℒ 1 , r − r∗ ), where 1 ∑ r∗ = 2|𝒩 v × ti (see Figure 6). In fact, we assume | i=1∶|𝒩 | that all covered nodes by 𝒵 are candidates for 𝒢 . Now by having this assumption, the following LP model determines the optimal sets 𝒢 i , and ℛ i for Si and the data path from the 𝒩 to the MS:
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Boundary 𝒵 and MS trajectory for different numbers of source nodes, |𝒩 |. A, Boundary 𝒵 and MS trajectory for |𝒩 | = 2. B, Boundary 𝒵 and MS trajectory for |𝒩 | = 3. C, Boundary 𝒵 and MS trajectory for |𝒩 | = 4. D, Boundary 𝒵 and MS trajectory for |𝒩 | = 5. MS, mobile sink FIGURE 6
∑ 𝜔1 ℱ + 𝜔2
minimize s.t.
∑∑ k
∑
∑∑
k
∑ ∑
k
fij(k)
j
( tj
𝛿i ei
fji(k) ,
j
fij(k) = gi +
∑∑ k
k
fji(k)
j
fij(k) =
j
∑∑ k
if i ∈ 𝒩
if i ∉ 𝒩 fji(k) ,
j
∑
fji(k) ,
( j)
and
diMS ⩾ r + r∗
diMS < r + r∗ .
Si is not located in𝒵
and
(17)
Si is located in𝒵
) ( j) Eik
= 𝛿i ,
∀i
k(k≠j)
∀i ( j)
ℱ , fik , fiMS , 𝛿i ⩾ 0
We note that the transmission radius for the members of 𝒢 i is set to the diameter of Oi . 5.2
and
if i ∉ 𝒩
𝛿i ⩽ ℱ ei , vars.
and
j
= 0,
( j) ( j) Ē ki + EiMS +
k(k≠j)
if i ∈ 𝒩
∑∑
j
−
∑
k
∑∑ k
(k) fiMS +
k
∑∑
fij(k) = gi +
(k) fiMS +
k
∑
i=1∶|𝒩 |
j
∑∑
j
i=1∶|𝒩 |
Subproblem II
In subsection 5.1, it is assumed that the MS is located at the optimal point ℒ 1 , which is determined by the proposed method in 1 study.12 Relaxing this assumption poses an interesting challenge, which makes up the second part of the proposed heuristic algorithm in this paper. Suppose the MS is located at ℒ 0 as its current position. Because ℒ 1 is selected as the optimal starting point for MS, it should move toward ℒ 1 with the constant velocity v. During its movement, the MS collects the sensed data of 𝒩 using 𝒢 . Thus, the optimal trajectory is the straight line connecting ℒ 0 and ℒ 1 denoted by L0 (see Figure 7). In this case, the sets of 𝒢 and ℛ must be updated based on the location of MS on L0 . As it is mentioned earlier, data gathering is performed in each time-slot. During each time-slot, MS has to harvest data ̄ with length from 𝒢 i regarding a defined rectangular area 𝒵 * th min{v × 𝜃, dℒ̄ k ℒ 1 } and width 2r in the k time-slot, where ̄ k is the location of MS at the end of the kth time-slot. ℒ
Therefore, we can simply apply our proposed model (Equation 17) to determine the optimal data paths from Si , i ∈ 𝒩 , to the optimal set 𝒢 i by considering v × ti /2 as the transmission radius for the members of 𝒢 i in the mathematical model (Equation 17). 6
PERFORMANCE EVALUATION
In this section, we compare the performance of the proposed algorithm with other approaches, such as the rendezvous-based algorithm. Firstly, to obtain a
FIGURE 7
̄ of the first round with t1 = t2 Temporary rectangular area 𝒵
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A, Multihop data transmission from source nodes to MS moving on ROT. B, Strategy of the MS movement. MS, mobile sink; ROT, rendezvous-based optimal trajectory FIGURE 8
rendezvous-based optimal trajectory (ROT), we introduce a mixed integer nonlinear programming (MINLP) model by considering our addressed problem in section 4. We note that, by using rendezvous points (RPs), we just determine the ROT and it does not mean that MS must stay at these points to gather data from source nodes. 6.1
Rendezvous-based optimal trajectory
x
Rendezvous-based optimal trajectory is introduced as the order of RPs for MS to collect data from source nodes (see Figure 8). Because ROT is placed in the convex area formed by source nodes in 1 study12 and in another study,13 we can drastically decrease the execution time of finding a solution by considering only RPs enclosed by this area. Let us define the 4 parameters 𝜎,𝜌,qij , and pij as follows: • • • •
𝜎: The number of RPs in the convex area formed by source nodes, 𝜌: The desired number of line segments of ROT where 𝜌 ≪ 𝜎, qij : The Euclidean distance (in m) between RPs i and j,and pij : The Euclidean distance (in m) between node i and RP j.
Now, ROT can be achieved through satisfying the following constraints. Let us define a 2-dimensional binary array Y where if Y ij = 1, the MS in its jth movement must be at the ith RP. The first constraint limits MS to visit, at most, 1 RP at each movement and the second constraint guarantees the length of ROT to be less than and equal to 𝜃 × v/2. Thus, the first 2 constraints are presented as follows: ∑
Yij ⩽ 1,
∀j,
(18)
i=1∶𝜎
∑
∑
Yik Yj(k+1) qij ⩽ 𝜃 × v∕2.
(19)
k=1∶𝜌−1 i=1∶𝜎
Through the third constraint, the continuity of ROT must be guaranteed. It means that 2 successive line segments of ROT must have a common RP: ∑ ∑ Yi( j+1) ⩽ Yij , ∀j. (20) i=1∶𝜎
i=1∶𝜎
The next constraints are related to the determination of data paths and energy consumption that are investigated in the (k) Equations 8 and (9), respectively, in which d̄ iMS is defined as the minimum distance between Si , i ∈ 𝒩 , and the MS mov(k) ing on the kth line segment. The value of d̄ iMS should satisfy the following two constraints: ∑ (k) pix Yxj Hijk ⩽ d̄ iMS , ∀j, (21) ∑
(k) pix Yx( j+1) Hijk ⩽ d̄ iMS ,
∀j,
(22)
x
where H ijk is defined as a 2-dimensional binary array that determines at which movement of MS, the member of 𝒢 must send its data. H ijk = 1 shows that node i sends the data generated by Sk , k ∈ 𝒩 , to MS in the jth movement of MS. Now, ROT can be obtained through the following mathematical model: ∑ 𝛿i minimize
𝜔1 ℱ + 𝜔2
i=1∶|𝒩 |
∑
i=1∶|𝒩 |
s.t.
𝛿i ⩽ ℱ ei ,
ei .
∀i
constraints(8), (9),
and
(19 − 25)
(il) ( j) ̄ (k) Yij , Hijk ∈ {0, 1} and f̄jMS , fki , diMS , 𝛿i , ℱ ⩾ 0 (23) It is obvious that the above proposed model is in form of MINLP model that is NP-hard43,44 ; therefore, we present a tabu search algorithm46 as a meta-heuristic solution. Tabu search is a meta-heuristic method that finds a near-optimal solution through iterative local search strategy. The process of tabu search can be realized in 4 parts26,47 : (1) designing an initial solution, (2) finding a neighborhood for a solution 𝒮 , (3) determining the fitness function, and (4) defining the tabu list and aspiration criteria. The number of algorithm iterations is restricted by thresholds, q1 or q2 , where q1 is the upper bound of the iterations and q2 is the maximal number of passed iterations that the last best solution is not boosted. The present work assumes that MS can move in 8 directions from RP i (see Figure 8B). Moreover, it is assumed that each line segment of ROT is obtained by 2 adjacent RPs according to the defined directions. Let 𝜏
vars.
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be the constant MS travel time on each line segment; therefore, by considering MS at the closest RP to ℒ 1 , the initial solution can be defined as an integer array 𝒮 with the size of ⌊𝜃/2𝜏⌋ and arbitrary values of 1 ⩽ 𝒮 (i) ⩽ 8, i = 1⌊𝜃∕2𝜏⌋. Because the performance of tabu search in execution time can be influenced by the process of exploring the neighborhood of solution 𝒮 ,47 a simple and efficient strategy to find a neighbor of 𝒮 is proposed. In our strategy, a neighbor of 𝒮 can be easily obtained by changing m last directions at 𝒮 , where m is a random number between 1 and ⌊𝜃/2𝜏⌋. Now, by having the solution 𝒮 and the distance between nodes and the MS at each line segment of its trajectory, we can determine ROT by running the following LP model: ∑ 𝛿i minimize
𝜔1 ℱ + 𝜔 2
Simulation Parameters
Parameter
Value
n
100 and 200
h
50, 100, 150, and 200 m
ei
2J
v |𝒩 |
10 m/s 4, 6, 8, and 10
gi
3 packets per second (3 × 1024 bps)
𝜃
4,6,8, and 10
ti
2 and 4 s
𝛼
50 nJ/bit
𝛽
10 pJ/bit/m2
q1
200
q2
1000
i=1∶|𝒩 |
∑
i=1∶|𝒩 |
𝛿i ⩽ ℱ ei ,
s.t.
TABLE 2
ei
∀i
.
constraints(8) and (9) vars.
(j)
ℱ , fik , 𝛿i ⩾ 0
(24) In the proposed algorithm, let 𝒟 be a tabu list used as a 2-dimensional array, the size of (𝜎 × 𝜎) to record forbidden moves. The value of 𝒟 ij will grow if the line segment with 2 endpoints (RPs), i and j, is selected. The key goal of the tabu list is to keep track of the solutions that have been evaluated in the past.46,47 Therefore, the 𝒟 will be updated by the manipulated cells of the best neighbor of 𝒮 in each execution round. Although the tabu list improves the tabu search, it could limit the neighborhood of solution 𝒮 and overlook some appropriate solutions. To cope with this problem, we find that the aspiration criterion allows tabu search to invalidate the limitations of 𝒟 . Therefore, it is possible to accept the superior solution 𝒮 , despite all the columns of 𝒮 that are altered at 𝒟 . This phenomenon is known as intensification.46,47 Tabu tenure adjusts how long a record is kept on the tabu list.46,47 Because tabu tenure specifies the boundary of the search area, long tabu tenure permits tabu search to discover the unvisited search space (which is called diversification). 6.2
Simulation
In this subsection, we present the results from the numerical experiments and simulations. We evaluate the performance of the proposed approximate algorithm in comparison with MINLP model (Equation 23) using 3 metrics: maximum round-trip movement of the MS (F), time complexity, and energy consumption. We consider different scenarios, which are written in MATLAB,48 MOSEK optimization package,49 and OMNeT++.50 The simulations are run on a computer with Intel core i7-3540 M 3.00 GHz CPU and 8GB memory. The other simulation parameters are listed in Table 2. Because the size of tabu tenure has a great influence on the quality of the solutions, we first run tabu search in different scenarios to estimate the appropriate size of this parameter.
Impact of tabu tenure on the average multiobjective function (Equation 24) FIGURE 9
The tabu search method is run for different number of sensor nodes n = [5,10] (randomly scattered in the network) and different values of 𝜃 = [8,10,12] sec. The thresholds q1 and q2 are set to 200 and 1000, respectively. We run the simulation 20 times and measure the average of minimum of the multiobjective function (Equation 24) with respect to different sizes of the tabu tenure (we set 𝜔1 = 0.5). Figure 9 shows that if the value of tabu tenure is set to 15, the minimum average of the objective function (Equation 24) can be achieved in almost all different settings of |𝒩 | and 𝜃. In the next experiment, we investigate the influence of 𝜔1 on the fitness value and the energy consumption with respect to different setting of n, |𝒩 |, and h in Figures 10 and 11, respectively. The measured values are the average of 20 executions. As shown in Figure 10, by increasing 𝜔1 in the objective function of the proposed model (Equation 17), the fitness values are boosted in all the scenarios, because, by increasing 𝜔1 , any variations of F (or ℱ ) have a greater influence on the overall value of multiobjective function (Equation 17)
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Impacts of different values of n, |𝒩 |, and h on the average of
FIGURE 12
FIGURE 11 Impacts of different values of n, |𝒩 |, and h on the average of energy consumption
FIGURE 13
compared with the term of energy consumption (see Figure 11 for the energy consumption of sensor node regarding different values of 𝜔1 ). Moreover, Figure 10 shows that the impact of increasing 𝜔1 on fitness value is independent from the number of sensor nodes, source nodes, and even the network size. However, the trend of fitness value for the fixed 𝜔1 is gradually decreased by increasing the network size or the number of nodes. For instance, by considering the first 2 scenarios, increasing the network size leads to increasing more energy consumption and, consequently, less fitness values (see Figure 11). In addition, for a specific network size, increasing the number of source nodes causes more sensor nodes to participate in delivering data to MS and consumes more energy accordingly. As the third experiment, we are going to scrutinize the impact of 𝜃 and ti on the fitness value and energy consumption with regard to different network sizes. In this simulation, by considering various values for 𝜃 and ti , the trends of the fitness value and energy consumption are illustrated in Figures 12
and 13, respectively. As it is conceived, by increasing the network size, the obtained fitness values for all the pairs of 𝜃 and ti are decreased significantly (see Figure 12) and the energy consumption of sensor nodes is increased drastically (see Figure 13). Regarding a constant number of sensor nodes (n = 100 and |𝒩 | = 5), these 2 phenomena are related to the distances between sensor nodes that are increased by growing the network size h. The results of increasing time-slot 𝜃 on the fitness value and energy consumption show that, when MS has more time to collect sensory data, it can form an |𝒩 |-sided polygon with greater circumradius that results in more fitness value and less energy consumption. In addition, by considering any 2 pairs of 𝜃 and ti with the same values of 𝜃, we can find that increasing ti causes more energy consumption by the sensor nodes on the data paths from 𝒩 to MS and, consequently, achieves less fitness value. However, because the length of MS trajectory is not affected by ti , the differences between the obtained fitness values and energy consumptions for the 2 pairs of the same and different ti are not dramatic.
FIGURE 10
fitness value
Impacts of different network sizes on the average of fitness value with n = 100, |𝒩 | = 5, and 𝜔1 = 0.5
Impacts of different network sizes on the average of energy consumption with n = 100, |𝒩 | = 5, and 𝜔1 = 0.5
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Impacts of different values of 𝜃 and |𝒩 | on the average of fitness value with n = 100 and 𝜔1 = 0.5. ROT, rendezvous-based optimal trajectory FIGURE 14
FIGURE 15 Impacts of different values of 𝜃 and |𝒩 | on the average of energy consumption with n = 100 and 𝜔1 = 0.5
The last experiment shows the comparison in the performance of the proposed model (Equation 17) and ROT model (Equation 24). Furthermore, we evaluate the influence of using different approaches for determining MS trajectory, eg,25 on the performance of the proposed algorithm in fitness value, energy consumption, and time complexity. By considering different numbers of RPs, Figure 14 shows the comparison of the performance of the tabu search method (ROT) and proposed model (Equation 17) in the obtained fitness value. For the proposed model (Equation 17), we consider 2 cases: the simple proposed strategy for determining MS trajectory in subproblem I and the proposed model in 1 study.25 Figure 14 shows that, by increasing 𝜃, the average fitness values of all algorithms are increased, because MS has more time to harvest data from gateway nodes in a shorter distance. It is obvious that, for a constant 𝜃, increasing |𝒩 | does not improve the average fitness value. Although by increasing the number of RP, the average of fitness value improves in the tabu search algorithm, the proposed model (Equation 17)
TASHTARIAN ET AL.
FIGURE 16 Impacts of different values of 𝜃 and |𝒩 | on the average of computational time complexity with n = 100 and 𝜔1 = 0.5. ROT, rendezvous-based optimal trajectory
shows better performance in all cases of 𝜃 and |𝒩 | in comparison with tabu search algorithm (see Figure 14). This is because, the proposed model (Equation 17) determines the trajectory of MS regardless of any RPs. As shown in Figures 14 and 15, by increasing |𝒩 |, the difference between the simple proposed algorithm in subproblem I and the proposed one in another study25 is gradually decreased, because MS does not waste more time on its trajectory and harvests data from gateway nodes in most of the traveling time. Although the proposed algorithm does not have superior fitness value and energy consumption compared with 1 study,25 it has the lowest computational time complexity (see Figure 16). The disadvantage of the proposed approximate algorithm25 is high time complexity for large values of 𝜃. As illustrated in Figure 16, by using 1 study25 and determining MS trajectory, the time complexity increases with a sharp slope, especially for 𝜃 = 8 and 10. Even though the time complexity of the tabu search algorithm increases gradually with an increase in the number of RPs, it is not greatly impacted by an increase in the number of source nodes and value of 𝜃, which is the advantage of the tabu search method.
7
CONCLUSION
In this paper, we focused on how to use a mobile sink in event-driven applications of WSNs to collect data from source nodes (a group of sensor nodes that capture events) in an energy-efficient way. Considering a time-slot 𝜃, predefined by the underlying application, MS must collect sensed data from source nodes. According to previous studies, designing and efficient trajectory for the MS, could prolong the network lifetime. On the other hand, it has been shown that how much multihop transmission between network nodes could reduce the amount of energy consumption. Therefore, we proposed an approximate solution to jointly determine the trajectory of the MS, and the data paths and transmission rates from source
TASHTARIAN ET AL.
nodes to the MS. We showed that this problem is in the form of MINLP model, which is NP-hard. After describing the problem and formulating it as a mathematical program model, we divided it into 2 subproblems. In the first subproblem, we proposed a heuristic algorithm to design a simple and efficient trajectory for the MS. By having the trajectory of MS, a mathematical optimization model to determine the optimal data paths and transmission rates from source nodes to the MS was presented. Since in the subproblem I, we assumed the MS located at the optimal point ℒ 1 , in the second subproblem, we proposed an optimal strategy for the MS to move from ℒ 0 as its current location to the optimal location ℒ 1 . For a comprehensive investigation, an MINLP model was proposed to specify a near-optimal rendezvous-based trajectory by considering a limited number of RPs in the network. We simulated and evaluated the performance of our algorithms in different scenarios using MATLAB and OMNeT++ tools. We compared our proposed algorithm with other approaches in network lifetime, time complexity, and energy consumption. The simulation results proved the effectiveness of the proposed algorithm over similar methods. Although many studies exist on mobile sink in WSN, there are still more interesting challenges to address, for instance, considering MAC-layer limitations, adjusting MS velocity in different applications, and proposing a distributed algorithm to design MS trajectory in 2D and 3D space.
REFERENCES 1. Luo J, Hubaux JP. Joint sink mobility and routing to maximize the lifetime of wireless sensor networks: the case of constrained mobility. IEEE/ACM Trans Netw. 2010;18(3):871–884. 2. Akkaya K, Younis M, Bangad M. Sink repositioning for enhanced performance in wireless sensor networks. Comp Net. 2005;49(4):512–534. 3. Jain S., Shah RC, Brunette W, Borriello G, Roy S. Exploiting mobility for energy efficient data collection in wireless sensor networks. Mobile Netw Appl. 2006;11(3):327–339. 4. Chakrabarti A, Sabharwal A, Aazhang B. Communication power optimization in a sensor network with a path-constrained mobile observer. ACM Trans Sensor Net. 2006;2(3):297–324. 5. Gu Y, Bozda˘g D, Brewer RW, Ekici E. Data harvesting with mobile elements in wireless sensor networks. Comp Net. 2006;50(17):3449–3465. 6. Jea D, Somasundara A, Srivastava M. Multiple controlled mobile elements (data mules) for data collection in sensor networks. International Conference on Distributed Computing in Sensor Systems. Springer, Berlin Heidelberg; 2005:244–257. 7. Akyildiz IF, Su W, Sankarasubramaniam Y, Cayirci E. Wireless sensor networks: a survey. Comp net. 2002;38(4):393–422. 8. Chen H, Shi Q, Tan R, Poor HV, Sezaki K. Mobile element assisted cooperative localization for wireless sensor networks with obstacles. IEEE Trans Wirel Commun. 2010;9(3):956–963. 9. Chen H, Liu B, Huang P, Liang J, Gu Y. Mobility-assisted node localization based on toa measurements without time synchronization in wireless sensor networks. Mobile Net Appl. 2012;17(1):90–99. 10. Sushant RCSSR. Data mules: Modeling a three-tier architecture for sparse sensor networks. IEEE Workshop on Sensor Network Protocols and Applications (SNPA). IEEE, Anchorage, AK, USA; 2003:30–41. 11. Tong L, Zhao Q, Adireddy S. Sensor networks with mobile agents. Military Communications Conference, 2003. Milcom’03. 2003 IEEE, vol. 1. IEEE, Boston, MA (USA); 2003:688–693.
13 of 14
12. Shi Y, Hou YT. Optimal base station placement in wireless sensor networks. ACM Trans Sensor Net. 2009;5(4):32. 13. Yun YS, Xia Y. Maximizing the lifetime of wireless sensor networks with mobile sink in delay-tolerant applications. IEEE Trans mobile comput. 2010;9(9):1308–1318. 14. Xing G, Li M, Wang T, Jia W, Huang J. Efficient rendezvous algorithms for mobility-enabled wireless sensor networks. IEEE Trans Mobile Comp. 2012;11(1):47–60. 15. Wang YC, Peng WC, Tseng YC. Energy-balanced dispatch of mobile sensors in a hybrid wireless sensor network. IEEE T Parall Distr. 2010;21(12):1836–1850. 16. Gu Y, Ji Y, Li J, Zhao B. Eswc: efficient scheduling for the mobile sink in wireless sensor networks with delay constraint. IEEE T Parall Distr. 2013;24(7):1310–1320. 17. Zhao M, Ma M, Yang Y. Efficient data gathering with mobile collectors and space-division multiple access technique in wireless sensor networks. IEEE Trans Comp. 2011;60(3):400–417. 18. Basagni S, Carosi A, Melachrinoudis E, Petrioli C, Wang ZM. Controlled sink mobility for prolonging wireless sensor networks lifetime. Wirel Net. 2008;14(6):831–858. 19. Shi Y, Hou YT. Theoretical results on base station movement problem for sensor network. The 27th Conference on Computer Communications In (INFOCOM). IEEE, Phoenix AZ, USA; 2008. 20. Xing G, Wang T, Xie Z, Jia W. Rendezvous planning in wireless sensor networks with mobile elements. IEEE Trans Mobile Comp. 2008;7(12):1430–1443. 21. Liang W, Luo J, Xu X. Prolonging network lifetime via a controlled mobile sink in wireless sensor networks. Global Telecommunications Conference (GLOBECOM 2010), 2010 IEEE. IEEE, Miami, florida, USA; 2010:1–6. 22. Konstantopoulos C, Pantziou G, Gavalas D, Mpitziopoulos A, Mamalis B. A rendezvous-based approach enabling energy-efficient sensory data collection with mobile sinks. IEEE T Parall Distr. 2012;23(5):809–817. 23. Yun YS, Xia Y, Behdani B, Smith JC. Distributed algorithm for lifetime maximization in a delay-tolerant wireless sensor network with a mobile sink. IEEE Trans Mobile Comp. 2013;12(10):1920–1930. 24. Tashtarian F, Moghaddam MHY, Sohraby K, Effati S. On maximizing the lifetime of wireless sensor networks in event-driven applications with mobile sinks. IEEE T Veh Technol. 2015;64(7):3177–3189. 25. Tashtarian F, Moghaddam MHY, Sohraby K, Effati S. Odt: Optimal deadline-based trajectory for mobile sinks in wsn: A decision tree and dynamic programming approach. Comp Net. 2015;77:128–143. 26. Gu Y, Ren F, Ji Y, Li J. The evolution of sink mobility management in wireless sensor networks: A survey. IEEE Commun Surv Tut. 2015;18(1):507–524. 27. Li X, Yang J, Nayak A, Stojmenovic I. Localized geographic routing to a mobile sink with guaranteed delivery in sensor networks. IEEE Journal on Selected Areas in Communications. 2012;30(9):1719–1729. 28. Vecchio M, Viana AC, Ziviani A, Friedman R. Deep: Density-based proactive data dissemination protocol for wireless sensor networks with uncontrolled sink mobility. Comp Commun. 2010;33(8):929–939. 29. Gandham SR, Dawande M, Prakash R, Venkatesan S. Energy efficient schemes for wireless sensor networks with multiple mobile base stations. Global Telecommunications Conference, 2003. GLOBECOM’03. IEEE, vol. 1. IEEE, San Francisco, USA; 2003:377–381. 30. Gao S, Zhang H, Das SK. Efficient data collection in wireless sensor networks with path-constrained mobile sinks. IEEE Trans Mobile Comp. 2011;10(4):592–608. ˙ Ersoy C. Efficient integer programming for31. Güney E, Aras N, Altınel IK, mulations for optimum sink location and routing in heterogeneous wireless sensor networks. Comp Net. 2010;54(11):1805–1822. 32. Liang W, Luo J, Xu X. Network lifetime maximization for time-sensitive data gathering in wireless sensor networks with a mobile sink. Wirel Commun Mobile Comp. 2013;13(14):1263–1280. 33. Smeets H, Shih CY, Zuniga M, Hagemeier T, Marrón PJ. Trainsense: a novel infrastructure to support mobility in wireless sensor networks. European
14 of 14
TASHTARIAN ET AL.
Conference on Wireless Sensor Networks. Springer, Ghent, Belgium; 2013:18–33.
43. Vavasis SA. Nonlinear optimization: complexity issues. England: Oxford University Press, Inc.; 1991.
34. Somasundara AA, Kansal A, Jea DD, Estrin D, Srivastava MB. Controllably mobile infrastructure for low energy embedded networks. IEEE Trans Mobile Comp. 2006;5(8):958–973.
44. Lee J, Leyffer S. Mixed integer nonlinear programming, vol. 154. Berlin, Germany: Springer Science & Business Media; 2011.
35. Luo J, Hubaux JP. Joint mobility and routing for lifetime elongation in wireless sensor networks. Proceedings IEEE 24th Annual Joint Conference of the IEEE Computer and Communications Societies, vol. 3. IEEE, Miami, Florida, USA; 2005:1735–1746. 36. Ma M, Yang Y. Sencar: an energy-efficient data gathering mechanism for large-scale multihop sensor networks. IEEE T Parall Distr. 2007;18(10):1476–1488. 37. Ekici E, Gu Y, Bozdag D. Mobility-based communication in wireless sensor networks. IEEE Commun Mag. 2006;44(7):56. 38. Nakayama H, Fadlullah ZM, Ansari N, Kato N. A novel scheme for wsan sink mobility based on clustering and set packing techniques. IEEE Trans on Autom Cont. 2011;56(10):2381–2389.
45. Cormen TH, Leiserson C, Rivest RL, Stein C. Introduction to Algorithms. Cambridge, MA, USA: MIT Press; 2001. 46. Glover F. Tabu searchpart ii. ORSA Journal on Computing. 1990;2(1):4–32. 47. El Rhazi A, Pierre S. A tabu search algorithm for cluster building in wireless sensor networks. IEEE Trans Mobile Comp. 2009;8(4):433–444. 48. MATLAB and Statistics Toolbox Release R2014b, Natick, Massachusetts, United States: The MathWorks, Inc.. 49. MOSEK ApS, The MOSEK optimization toolbox for MATLAB manual. Version 7.1 (Revision 28), 2015, Available at http://docs.mosek.com/7.1/ toolbox/index.htm 50. The omnet++ discrete event simulation system v5.0, 2016 Available at http:// www.omnetpp.com
39. Huang P, Chen H, Xing G, Tan Y. Sgf: a state-free gradient-based forwarding protocol for wireless sensor networks. ACM Trans Sensor Net. 2009;5(2):14. 40. Chen H, Gao F, Martins M, Huang P, Liang J. Accurate and efficient node localization for mobile sensor networks. Mobile Net Appl. 2013;18(1):141–147. 41. Shi Q, He C, Chen H, Jiang L. Distributed wireless sensor network localization via sequential greedy optimization algorithm. IEEE Trans Sig Proc. 2010;58(6):3328–3340. 42. Wang G, Chen H, Li Y, Jin M. On received-signal-strength based localization with unknown transmit power and path loss exponent. IEEE Wirel Commun Lett. 2012;1(5):536–539.
How to cite this article: Tashtarian F, Sohraby K, Varasteh A. Multi-hop data gathering in wireless sensor networks with a mobile sink. Int J Commun Syst. 2017;e3264. doi:10.1002/dac.3264
