On Maximizing the Lifetime of Wireless Sensor Networks ... - IEEE Xplore

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2014.2354338, IEEE Transactions on Vehicular Technology

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT)
REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < periodically to the application server at the specific reporting time slots. The application server can adjust each reporting time slot according to the locations of the ASN, critical level of the application and previously received data. Hence, regarding the prescribed reporting time slot, the data of the sensor nodes are harvested in a hybrid manner. This implies that some sensor nodes send their data separately or simultaneously to MS, while MS uninterruptedly moves through the network. By proposing a convex mathematical model inspired by the Support Vector Regression (SVR) technique [33], MS determines its COT. Through comprehensive simulation, we show that our proposed model yields a substantial gain in the lifetime of event-driven applications with single hop data delivery. The remainder of this paper is organized as follows: we review some related works in section II. In section III, we describe the system model and the problem formulation. Section IV describes the proposed algorithm and the Continuous and Optimal Trajectory. The performance of the proposed algorithm is examined via simulation in section V, and finally, section VI concludes the paper. II. RELATED WORKS Recently, the issue of prolonging the network lifetime of WSN has been targeted through the mobility management of the Mobile Sink (MS) [1]-[7]. We can classify sink mobility into two categories: random mobility [8], [9] and controlled mobility [1], [2], [11], [16], and [18]. In [8], the authors proposed an approach based on the random mobility of mobile agents, called data MULEs (Mobile Ubiquitous LAN Extensions), to collect buffered data of sensor nodes in sparsely deployed networks. In a similar study [9], the data of sensor nodes are harvested by a mobile agent that is flying above the sensor field. The main advantage of the proposed algorithm in this category is simplicity in implementation. However, the random mobility algorithm brings some restrictions like the buffer overflow in sensor nodes and the delay of data delivery. The majority of existing works in sink mobility is proposed for controlled mobility. Shi and Hou in [18] addressed the sink mobility through the determination of the optimal location for MS in the network. Since this problem is NP-hard, they proposed a heuristic model and extended their algorithm to support multiple MS. Basagni et al. in [16] offered a Mixed Integer Linear Programming (MILP) problem formulation to obtain an optimal trajectory of MS and the sojourn time at RPs for maximizing the lifetime of the network. However, they assumed that the routes were predetermined, and they also ignored the traveling time of MS on its trajectory. Yun et al. in [11] proposed a new model for increasing the network lifetime in delay tolerant applications known as the Delay Tolerant Mobile Sink Model (DT-MSM). They determined the trajectory of MS according to the predefined RPs as the special location for data harvesting. Moreover, they assumed that the travelling time of MS between any two RPs is negligible. Gandham et al. in [2] controlled the movement of MS among some RPs through proposing an Integer Linear

2

Programming model. Wang et al. in [13] proposed an approach to control multiple mobile sensors to travel among event locations and harvest data. They assumed that each mobile sensor had limited residual energy. Hence, the main problem was how to dispatch the mobile sensors among the event locations to maximize the number of rounds until some event locations could not be reached. Konstantopoulos. C et al. in [21] addressed the sink mobility by considering the constrained path. Their proposed algorithm focused on a clustering network and routing of the captured data. The targeted application in [21] is environmental monitoring, i.e., urban park, building block. Xing. G et al. in [12] offered an efficient rendezvous algorithm to determine the trajectory for MS. They considered a subset of sensor nodes to serve as rendezvous points that buffer and aggregate data that originated from sources and they transferred them to the MS when it arrived. They determined the trajectory of MS by considering a routing tree that rooted at RPs. The objective function was to minimize the total edge length of the tree. Although this criterion has shown significant performance on network energy consumption, it cannot guarantee the maximization of the network lifetime. In the majority of the proposed algorithms in sink mobility, the predetermined structure (like a virtual grid or RPs) was considered [14], [35]. In [11], the authors showed that the number of RPs had a significant influence on the performance of the algorithm and the quality of the solution. However, considering the infinite number of RPs to determine at least one optimal location of a sink is an NP-hard problem [10]. In this work, we determine the optimal trajectory for MS in onehop data delivery event-driven applications without considering any infrastructure or predetermined RPs. We show that the problem is NP-hard and, we then propose an approximation solution for the optimal trajectory for MS. The other aspects of using an MS in WSN and wireless Ad hoc networks have been pursued in [34]-[40]. F. Bia et al. [36] proposed an estimation technique based on a marginal value theorem used by a mobile-data-harvesting agent. Sharifkhani and Beaulieu [34] proposed a transmission-scheduling algorithm to optimize a metric which was a tradeoff between transmission power and reliability. G.Shi et al. in [38] studied the double-blind data discovery problem in a large-scale WSN with MS(s), where sensor nodes and MS(s) do not know the locations of each other a priori. G. Shi et al. proposed an efficient data discovery mechanism to exploit a simple geometric property of a planar. III. SYSTEM MODEL AND PROBLEM FORMULATION In this study, mobility management of an MS in eventdriven applications is investigated. In these applications, each sensor node has two operation modes: monitoring and sending. By capturing an event, a sensor node changes its status from the monitoring to sending mode. The group of sensor nodes in the sending mode called the Active Sensor Node (ASN) group, sends their data to MS in a one-hop manner. We define a reporting time slot denoted by as a

0018-9545 (c) 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.


> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < deadline for MS to harvest the data from ASN over the network determined by the application server and depend on the locations of ASN, criticalness level of application, and previous received data. Due to the limited velocity of MS and transmission range of ASN, harvesting data from ASN in a large-scale WSN during is impractical, especially when there are some obstacles in the network field. Hence, we can divide the WSN field into autonomous zones where each zone consists of randomly deployed wireless sensor nodes ( ) , = 1: , and a sink mounted on a mobile robot that moves with a finite constant velocity (see Fig. 1). By assuming one-hop data delivery between members of ASN, which are denoted by ̅ ( ) , = 1: , ( ≤ ), and MS, the size of each zone is determined1. Moreover, we assume that the member of ASN can adjust their transmission range by receiving a control packet from MS. In this system, without considering any RPs or virtual structures, we determine an optimal trajectory for MS in each zone called Continuous and Optimal Trajectory (COT). Without loss of generality, we can consider a network with one zone and focus on obtaining a COT. Here, we emphasize that we take into account an existing ASN and the reporting time slot. Similar to [18] and [43], ̅ ( ) consumes the minimal transmission power to transmit a composite bit-stream to MS at a constant transmission bit rate (in bps) [19] and over the Euclid distance between ̅ ( ) and MS, as follows: ( + ) , → < → = (1) ( + ) , → ≥ → ⁄ where if → is less than a threshold (≈ ), the free space model with amplifier energy equal to is → used ( → power loss); otherwise, the multipath model with amplifier energy equal to is used → ( → power loss), where dimensions of and are joule/bit/m2 and joule/bit/m4, respectively. The nonnegative constant term depends on the electronics energy and is in joule/bit [43]. We note that both free space and multipath models are applicable in our algorithm, however due to the limited area of zone, the free space model is considered (usually denoted by ). The network lifetime is defined as the elapsed time since the beginning operation of the network till the first active sensor node dies. Since predicting the location and occurrence time of an event in the network is out of the scope of this zone 2

3

paper and moreover the transmission time of an active sensor node is limited, we focus on the occurred events, the existing ASN and reporting time slot to determine COT for MS so that the lifetime of the existing ASN is maximized. We introduce the fitness value as a criterion to evaluate the suitability of COT in terms of increasing the lifetime of ASN. In fact, by increasing in the following general optimization model (without any restriction on the length of COT), the optimal value of → will be determined by considering the actual remaining energy and the specified visiting time of the each member of ASN: (2) maximize () ∀ (I) ̅ ( ) subject to: + ≤ , → (̅ ) ∀ (II) = , > 0,

variables:

→

≥ 0 , = 1: ,

() () where ̅ and ̅ are the amount of the residual energy (in joule) and captured data (in bit) of ̅ ( ) , respectively. According to this description, the maximum of =

{

̅( )

, = 1: }.

We

note

that

the

other

→

constraints in producing COT will be added into the general model (2). The process of determining COT can be performed in MS by considering three main parameters: (1) remaining energies, the volumes of captured data, and the geographical locations of ASN which are sent through a 'beacon' packet from ASN to MS upon capturing an event, (2) the reporting time slot , which is prescribed by the application server, and (3) the limited velocity of MS that forces the length of COT less than or equal to . Since MS has to visit ASN during , we introduce the visiting area for ̅ ( ) as a circular region ( ) . () MS must travel across the center point of ( ) denoted by , located on COT, to receive captured data during (see Fig. 2). Thus, we can calculate the radius length of ( ) , denoted by () ( ) ( ) = , where is the velocity of MS. Let and be the two intersection points of ( ) and COT. Hence, the transmission radius of ̅ ( ) denoted → is equal to the ̅( ) , ( ) , = 1,2 , where ( , ) is the maximum of Euclidian distance between the two points of and . We assume that ̅ ( ) start sending captured data when MS arrives () ( ) at and continues until MS leaves ( ) from . The traveling time of MS between the ending point of the previous COT and the starting point of the new COT is negligible (and hence, ignored) for the application server. This is a usual

zone 3 Application Server

MS

COT

S

obstacle

S

(2 )

(1)

d1 MS

S

COT

S

(1)

v S

zone 1

(4)

( 3)

(1) c

p2(1) zone 4

Fig. 1. The large-scale network field with four autonomous zones. 1 In this study, we set the diameter of a zone equal to the maximum transmission range which is supported by sensor nodes.

p1(1)

vr(1) v (1)

COT Fig. 2. COT for MS to harvest data from ASN in the sensor field with one zone (a), and visiting area ( ) (b).



> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < assumption in most recent works reported in the open literature [11], [18], and [32]. Moreover, the paper does not consider the MAC-layer contention [11]. Future work may try to relax these assumptions. Our analytical analysis in this paper addresses the following fundamental issues that relate to the key characteristics of using MS in the event-driven applications of WSNs: (1) Investigating the problem of obtaining COT for any size of ASN analytically: (a) Solving the problem of determining COT for = 2, and (b) Modeling COT for > 2 and proving it as an NP-hard problem; (2) Proposing a heuristic algorithm to determine COT to achieve maximum network lifetime; (3) Extending the algorithm to provide COT according to the different values of τ; and (4) Conducting simulation experiments to verify our theoretical results.

COT

d 2MS r*

S

*

→

=

∗

=

→ ∗

∗

,

(4) ∗

∗

where = { − + 4 , 2 } and = . Proof: In the symmetric case, both active sensor nodes have the same visiting areas ∗ with radius of ∗ . When MS has enough time to harvest data ( ≥ ( + 2 ∗ )/ ), the two active sensor nodes should consume the minimum energy for ( ) ( ) transmitting the capture data. In this case, and should ( ) ( ) ̅ ̅ be located at the coordinates of and , respectively (Fig. 3 (a)). For < ( + 2 ∗ )/ , we have (Fig. 3 (b)): + → = − + 4 ∗, (5) → Therefore, by having (2):

r

(b) ( )

Fig. 3. COT for τ ≥( +

+

( )

{

=

)/ (a), and COT for τ2 In this part, we investigate how to obtain COT for ASN with more than two members. At first, we show that obtaining COT in the simplest pattern, as a straight line, is an NP-hard problem, which includes some non-linear constraints [17]. Moreover, we show that COT as a non-straight line can be designed in the form of a Mixed Integer Non-Linear Programming (MINLP) problem. Let us consider a topology with three active sensor nodes in Fig. 5. For achieving COT in the simplest pattern for active sensor nodes (Fig. 5 (a)), the following four constraints must be satisfied. The first

The network lifetime (seconds), In logarithmic scale

( ) ( ) ̅ ( ) − ̅ ( ) and = ̅ ( ) − ̅( ). where = ̅ − ̅ Theorem 1: To achieve the maximum of the fitness value of COT in the symmetric situation, when two active sensor nodes have equal visiting time ∗ and residual energy ∗ , the optimal transmission range of ̅ ( ) and ̅ ( ) is: ∗

S

V l

*

( 2)

(a)

∗

(1)

r

→

A. COT for m=2 This is the simplest non-trivial case. Consider ASN with two members of ̅ ( ) and ̅ ( ) with arbitrary locations in the network. Let be the distance between the two members of ASN. MS must visit ̅ ( )and ̅ ( ) during visiting times and , respectively. If the application server lets MS harvest the data of ASN during τ that is greater than or equal to ( + ( ) ( ) + )/ , COT will be the line L connecting ̅ ( ) and ̅ ( ) . This is because the centers of the visiting areas are located on the coordinates of ASN, and in this case, ASN consumes the minimum energy to transmit data to MS () ( → = , = 1,2) (see Fig. 3 (a)). The line L is defined as: : = + , (3)

S

(1)

S

l

where

L

d 1MS

COT

(2 )

IV. THE CONTINUOUS AND OPTIMAL TRAJECTORY In this section, we focus on the mathematical model of COT for different sizes of ASN ( ). First, we provide an analytical solution for the simplest case of = 2. Then, we extend our investigation for any m and show that the problem of finding COT for > 2 is NP-hard.

d2MS

L

4

T*

1 *

T2 (t=2) 10

T* (t=3)

2

2 *

T2 (t=4) *

T2 (t=5) T* (t=6) 2 *

T2 (t=7) 10

1

T* (t=8) 2

T* (t=9) 2 *

T2 (t=10) T* (t=11) 2 *

10

T2 (t=12)

0

2

4 6 8 The length of transmission range (S(1)) (m)

10

r

Fig. 4. The measured maximum of fitness value of COT according to the ( ) ( ) different values of τ and . The other parameters are: ̅ = ̅ =1 → (joule), = = 1 (sec), d=10 (m), f=2500 bps, V=1m/s, α=0.005 joule/bit, β=0.0025 joule/bit/m2.



> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < constraint states that the total length of COT must be less than or equal to the distance that MS can traverse during seconds with constant velocity . Therefore, we have ( , ) ≤ , where A and B are the starting and ending points of COT with coordinates of ( , ) and ( , ), respectively. Let (

() () . , . ) be ()

()

the coordinates of

function must be convex; (2) The inequality constraint functions , … , must be convex; and (3) The equality constraint functions ℎ ( ) must be affine. Let = > 0, so the problem (11) can be presented as follows:

. The center of all visiting subject to:

areas ( , = 1: ) must be located on COT. Therefore, the second constraint is written as follows: () () − − . − . = 0, ∀ , (8) where w =

minimize ( , )≤ − ( ≤

()

that the intersection points between , = 1: , and COT will be located between and . Thus, we introduce the next () () constraint to restrict the location of and , ( = 1: ) between A and B as follows: () () ≤ . ≤ , ≤ . ≤ , ∀ , () () (9) () () ( ) () . = . − , . = . + ,∀ .

Therefore, the mathematical model for obtaining COT for ASN with > 2 can be written as follows: (11) maximize subject to:

( , )≤ Constraints (8-10) ()

variables:

()

,

,

() . (

= 1:

), , , are free , > 0.

Theorem 2: The above proposed non-linear programming model is NP-hard. Proof: Since if an optimization problem is non-convex, it is NP-hard (see Theorem 22 in [44]), we need to show that the problem (11) is non-convex. A convex optimization problem is presented as follows: (12) minimize ( ) ( ) ≤ 0, = 1: subject to: ℎ ( ) = 0, = 1: , where , , … , are convex functions [45]. The convex problem has three main requirements: (1) The objective S

v (1)

S

d 2 MS

d 3 MS

v (3)

S

A

S

(1)

d1 MS

S (a)

(1)

v (3)

( 2)

X3

v ( 2)

v(2)

A

( 3)

X2

B

() . () .

S

( 2)

= =

() . ≤ () . ≤ () . − () . +

() .

()

− . = 0, − ,

variables:

()

,

()

,

() . ,

,

(13) (I) ∀ (II)

,

(III) ∀ (IV)

,

∀ (V)

()

(1 +

) ,

∀ (VI)

()

(1 +

) ,

∀ (VII)

( + → ) ≤ ̅( ), ̅( ), ( ) ≤ → , ̅( ), ( ) ≤ → , ,

,

∀ (VIII) ∀ (IX) ∀ (X)

, are free, > 0.

Although the objective function and constraints (IV), (V), (VIII) are in convex forms, the other constraints cannot meet the requirements (2) and (3) of the convex optimization model (12). Therefore, we can conclude that the problem (13) is non-convex. ■ Theorem 3: The problem of determining COT as a nonstraight line is in the form of a Mixed Integer Non-Linear Programming (MINLP), which is NP-hard. Proof: A non-straight line COT for ASN with a size of can be designed through determining the maximum point(s) of , = 1: , where each can be considered as a ( ) candidate location point for , ∈ {1, … , } (see Fig. 5 (b)). Let be a two-dimensional binary array with a size of × . If , = 1, it means that ̅ ( ) sends its data to . We can define the MILP model as follows: maximize (14) subject to: ∑ : , = 1, ∀i (I) ( , )+∑ : ( , )+ ( , ) ≤ (II) () ∑ ( ) ( ) ≤∑ : , + , , , : ∀i (III) () ( , ) + ( , ) , ≤∑ : , ∑ : ∀i (IV) ∑

( 3)

− )=

− ≤

. The second constraint cannot guarantee

As discussed earlier, MS should determine its COT with the maximum fitness value . Thus, we can model the last constraint as follows: ( + → ) ≤ ̅ ( ) , ∀ , where (10) ̅ ( ) , ( ) ≤ → , and ̅( ), ( ) ≤ → .

5

:

,

̅( ),

( + → )≤ variables: > 0, , ,

()

+ ̅( )

≤

,

→

∀i (V)

, ∀i (VI) are free,

,

∈ {0,1} , , = 1: .

B X1

v (1) (b)

Fig. 5. The sample of COT in the network with three active sensor nodes; simplest pattern of COT as a straight line (a), and a non-straight COT (b).

The first constraint guarantees that each member of ASN just send its data to the one location point of . The second constraint states that the total length of COT must be less than or equal to the distance that MS can traverse during seconds with constant velocity . Through the next two constraints, COT must be stretched in all visiting areas of ASN (see Fig. 5



> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < (b)). The fifth constraint determines the approximate transmission range of ̅ ( ) based on its distance to the selected () and the average of . To obtain an optimal COT, the last constraint increases the fitness value . It is obvious that the problem is in the form of MINLP, which is NP-hard in general [28] and [29]. ■ C. Proposed Heuristic Algorithm In this part, we propose a heuristic algorithm to provide COT for general and that consists of two simpler subproblems. Regarding the value of , the achieved COT can be in the form of a straight or non-straight line, e.g., zigzag pattern. In the first sub-problem, we present a convex optimization model inspired by a Support Vector Regression [33] to attain COT as a straight-line segment without considering the value of . The proposed model achieves the optimal straight-line segment by taking into account the amount of residual energies, locations, and the amounts of captured data of ASN. Regarding the achieved COT, we calculate the traveling time of MS on COT, which is named as Unsolicited Traveling Time (UTT). In the second subproblem, we consider the reporting time and propose two algorithms. The first algorithm obtains Shorter-COT (S-COT) as a straight line for < UTT through a convex optimization model. In the second algorithm, we present a heuristic approach to determine a non-straight COT for > UTT in polynomial time called the Longer-COT (L-COT). We note that both algorithms in the second sub-problem are based on the achieved COT from the first sub-problem. 1) Support Vector Regression Support vector regression (SVR) is gaining popularity due to many attractive features and promising experimental performance in widely used applications. Given a set of training instance-target pairs {( , )}, ∈ , ∈ , = 1: , a linear SVR finds and for the linear function ( ) = + that has at most deviation from the actually obtained targets for all the training data and at the same time as flat as possible [33]. Therefore, it is required to minimize the Euclidean norm i.e.‖ ‖ , which can be written as the convex optimization problem: (15) minimize ‖φ‖ subject to: variables:

−

− ≤ , + − ≤ , φ, are free.

∀ ∀

Since the above convex optimization problem is not feasible in all cases, some errors are allowed through introducing slack variables and ∗ .Thus, the convex problem (15) becomes: minimize ‖ ‖ + ∑ : ( + ∗ ) (16) subject to: variables:

−

− ≤ + , + − ≤ + ∗, φ, are free, and ∗ ≥ 0.

∀ ∀

The constant > 0 makes the tradeoff between the flatness of function F and the amount up to which deviations larger than ε are tolerated.

6

2) The First Sub-problem Since active sensor nodes transmit their data to MS in a one-hop manner, the distance from ASN to COT has a significant influence on the energy consumption of ASN. Let () () ( ̅ , ̅ ) be the coordinates of ̅ ( ). We can estimate a linear function (i.e., ℒ) for COT as follows: () () ℒ ̅ = ̅ + , (17) where φ and θ are the gradient and the y-intercept of ℒ. Since in this part, the reporting time slot has been ignored, the optimal transmission range of ̅ ( ) for covering MS on COT can be estimated to be ()

()

→

equal to

̅( )

+

()

,

() ≤ ̅ (|.| denotes the absolute () () value). By decreasing the value of ̅ − ̅ − , ∀ = {1: }, the length of the orthogonal lines and consequently the transmission ranges of ASN will be decreased. In fact, if the length of the orthogonal line from ̅ ( ) to COT is considered, then the problem will not be formulated as a convex optimization model. Therefore, we propose the following optimization model inspired by SVR technique to achieve the optimal values of and : (18) minimize + ( ) ( ) ( ) ∀ (I) subject to: ̅ − ̅ − ≤ ̅ ,

where

̅

−

̅

−

̅( )

+ variables:

( ) > 0, ̅ , and

→

+

()

() ≤ ̅ ,∀ (II)

≥ 0, and

are free,

where C is assigned a large positive value. We convert the problem shown above into a convex model by introducing the new variable µ = > 0. Thus, the objective function (18) and the second constraint can be reformulated as "minimize

+

" and

"

+

̅( )

+

()

() ≤ ̅ ,

∀ ", respectively. To attain the minimum value of the objective function, the value of should be decreased. In fact, by decreasing the value of , the lifetime of ASN will be increased through the second constraint. Thus, convex model obtains COT in polynomial time [41], and [42]. We note that ̅ ( ) in the second constraint is equal to the actual residual energy of ̅ ( ) after sending beacon packet to MS. By considering two arbitrary points and on the obtained ( ) () ( ) ̅ COT, the coordinates of and for can be calculated easily. Therefore, UTT can be obtained as: UTT = max

(

()

,

( )

) ∀ , ∈ {1: } / . (19)

3) The Second Sub-problem In the second sub-problem, the value of the reporting time slot is taken into account. We propose a convex programming model and a heuristic algorithm to find COT for τ < UTT(SCOT) and τ > UTT (L-COT), respectively. Both algorithms are based on achieved COT from the first sub-problem.



> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < S-COT: 0. Thus, the objective function (20) and the fifth constraint are presented as "minimize "

"

and

+

(ℎ ) + (

+

()

)

≤

̅ ( ) , ∀ ", respectively. The convex model maximizes the fitness value of S-COT through decreasing =1/ . Constraint (I) guarantees that the length of S-COT is less than or equal to . The two next constraints force the circular visiting are ( ) to be located between starting and ending points of S-COT ( and , () respectively). The total length of relocation of is () determined by (constraint (IV)). Regarding ℎ , , and , the transmission range of ̅ ( ) is obtained to be equal to (ℎ ) + (

+

()

) (see Fig. 6 (b)). Therefore, the last

constraint decreases the total energy consumption of the active sensor nodes for sending their captured data to MS by decreasing the value of . The proposed convex model

h1

v(2)

A

S (3 )

h3

rm sfo

T CO d1MS d 3 MS S

n Tra

Transformed ASN (1)

v (3) B

v

(1)

d2MS h2 (a )

S (2)

d1MS d 3 MS

A

Transformed COT

b

a d2MS

Z1

(b )

B

S  COT

a

b

d2MS (c )

Fig. 6. Transformed COT and ASN (a); S-COT as a line segment of COT (b); the result of the post-processing algorithm on S-COT (c).

7

guarantees to obtain S-COT with the maximum fitness value in polynomial time however it is possible that the new () positions of some v impose more energy consumption. This ( ) is because the new location of does not influence the obtained maximum fitness value. Therefore, we propose a simple post-processing algorithm to enhance the already obtained S-COT. Indeed, the post-processing algorithm () determines the best locations for ( = 1: ) between the achieved starting and ending points of S-COT ( and , respectively) (see Fig. 6 (b) and (c)). The pseudo-code of the proposed algorithm is given in Fig. 7. b) L-COT: >UTT In this case, L-COT should be longer than the achieved COT from the first sub-problem because MS has more time to approach ASN and harvests their data. Therefore, L-COT can be in the form of non-straight line, e.g., zigzag pattern. As it was shown earlier, the problem of obtaining a non-straight COT for ASN with size of m > 2 is MINLP, which is NPhard. In this part, we propose a heuristic method to determine L-COT in polynomial time based on the achieved COT from the first sub-problem. The main idea of the algorithm is to () determine the optimal locations for , = 1: . By considering the transformed ASN and COT, we divide the vertical line from ̅ ( ) to COT into the set of segments = 1: ℎ / , where is the length of the segment (see , , Fig. 8 (a)). Now, we can define L-COT as an optimal trajectory with length less than or equal to regarding the () optimal location of ∈ , . The pseudo-code of determining L-COT is given in Fig. 9. Before describing the algorithm, we note that although L-COT is obtained based on some certain points on the vertical lines between transformed ASN and COT, the proposed algorithm belongs to the group of Continuous and Optimal Trajectory because the points are achieved through the first sub-problem. Let VcLocation be an array with a size of . At the end of the process, VcLocation(i) represents the optimal segment () number for . The closest segment to COT is numbered by one and the maximum segment number for ̅ ( ) is held in a maxIndex array. A while loop constructs the main body of the heuristic algorithm (line (2)) with the Boolean condition of ((Len)< ) AND (maxIndex≠ VcLocation)). In the while () loop, we consider the new positions for , = 1: , by increasing the value of VcLocation by one (line (3)). It is clear that the maximum number of new positions should be limited by maxIndex (line (5)). In each round of while loops, the Function Post-Processing Algorithm () () 1. For all () () 2. If . < ̅ then (̅ ) () () ̅( ) else 3. If + < then . = 4. else () () () ̅( ) else 5. If ̅ − > then . = Fig. 7. The pseudo-code of post-processing algorithm.


() .

= −

()

() .

= +

()


> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < ()

function UpdateSegment selects the best replacement for , ∈ {1, … , }. To do this, the UpdateSegment calculates ∗ of ̅ ( ) as follows: ∗

̅( )

=

̅( )

×∇ ,

()

()

,

(21)

and then updates VcLocation so that if ̅ ( ) has the minimum ∗ in ASN then ()= ( ) else ()= ( ) − 1. In line (8), the function CalculateNonStraight determines the L-COT regarding the updated VcLocation. Suppose a complete graph = { , E}, where is the set of new location points of VcLocation and = , ∀ , ∈ {1, … , } . Let ∆ , be the degree of , which is equal to the number of the connected edge to . Now, for determining L-COT, we should find a minimum spanning tree where ∆ ≤ 2, ∀ . This problem is known as the Minimum Cost Hamiltonian Path problem, and thus, is NPhard [22]. Since the size of ASN in comparison with the number of all sensor nodes is small, we propose a greedy heuristic algorithm to determine L-COT in Fig. 10. V. SIMULATION AND PERFORMANCE E VALUATION In this section, we evaluate the performance of the proposed models and verify our theoretical analysis with other approaches like a rendezvous-based algorithm. To do this, we first propose a Mixed Integer Linear Programming (MILP) optimization model regarding the defined problem in IV-B to obtain a Discrete and Optimal Trajectory (DOT) based on predetermined virtual Rendezvous Points (RPs) in the network. A. Rendezvous-based Algorithm The MILP model considers some pre-determined virtual RPs in the network so that DOT can be defined equal to a set of RPs (see Fig. 11). It is clear that the maximum length of RPs which is denoted by is equal to . Since DOT will be S

h1

(1)

segment

d1MS

S

S ( 3)

h3

v( 2 )

v (1) B

h2 (a)

S (2)

S ( 3)

d1MS

v (3) d 3 MS

v (3 )

A

v (1)

(1)

B

A d 2MS

COT L  COT

v( 2 )

S (2)

(b)

Fig. 8. The vertical line from ̅ ( ) to COT is divided into some segments (a), and the sample of L-COT (b).

Function LCOT () 1. VcLocation (1: ) = 0, Len= 0 2. While (Len< )AND (maxIndex≠ VcLocation) 3. temp()=VcLocation() + 1 4. For each item in temp () 5. If temp>maxIndex then Temp=maxIndex 6. End for 7. VcLocation()=UpdateSegment(VcLocation(),temp) 8. [LCOT, Len]=CalculateNonStraight (VcLocation()) 9. End while 10. Return LCOT Fig. 9. The Pseudo-code of determining L-COT.

8

located in the convex area by ASN [10], [23], the solution space of MILP model is narrowed by RPs located in this convex area that yields the significant gain in time complexity of MILP model. Before presenting MILP model, let us define sets and variables as follows: Sets:  , ( = 1: , = 1: ): Euclidean distance (in meters) between ̅ ( ) and RP where is the number of considered RPs in the convex area of ASN.  , ( = 1: , = 1: ): Euclidean distance (in meters) between and RPs. Variables:  :The two-dimensional binary array with the size of ( = 1: , = 1: ) provides optimal list of RPs in DOT. movement is at RP. , = 1 means that MS at  : The two-dimensional binary array with the size of ( = 1: , = 1: ) provides the sequence of sending captured data from ASN to MS. , = 1 means that active sensor node sends it data to MS at movement. ( ) ( )  , ( = 1: , = 1: , = 1: ): , shows the amount of data transmitted data from active sensor node to MS located at RPs at its movement. ( ) ( )  , ( = 1: − 1, = 1: , = 1: ): shows the , traveled distance of MS between and RPs at its movement.  : = 1/ , where is the fitness value of DOT. (22) minimize ∀ (I) subject to:∑ : , ≤ 1, ∑ ∑ : ≤ , ∀ (II) , : , ∀ (III) ∑ : , = 1, ∑ ∑ : ≤ × , ∀ (IV) , : , (̅ ) ( ) ∀ , , (V) , + , −1 ≤ , , CalculateNonStraight (VcLocation ()) 1. Create Graph = { , } 2. Sort the edges in the increasing order of length 3. =Select the least cost edge 4. , =Vertex of ( ) 5. TotalLen=Length ( ) 6. = + 1 7. LCOT ( )= 8. While ( ≠ ( − 1)) 9. = + 1 10. =Select the least cost edge connected to 11. =Select the least cost edge connected to 12. If Length ( ) ≤Length ( ) 13. = Vertex of ( ) not in { , } 14. TotalLen=TotalLen + Length ( ) 15. LCOT ( )= 16. Else 17. = Vertex of ( ) not in { , } 18. TotalLen=TotalLen + Length ( ) 19 LCOT ( )= 20. End if 21. End while 22. Return LCOT, TotalLen Fig. 10. The Pseudo-code of the greedy heuristic algorithm to determine LCOT based on VcLocation.



> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < ∑

( ) , ( ) ,

≤ ≤

,

( ) ,

+

−1

,

,

×

,

,

×

,

∑

:

,

:

+

,

∑ variables: , and

∑

:

:

∈ {0,1} ,

≤

,

̅( ),

∀ (VI)

( ) , ,

≤

∀ , , = 1: − 1 (VII) , ∀ , , = 1: − 1 (VIII) ,

∑ ( ) ,

:

≥ 0,

∀ , , = 1: − 1 (IX) (X) ≤ ,

( ) ,

( ) ,

≥ 0,

> 0, ∀ , , , .

The objective function (22) maximizes the fitness value of DOT by decreasing the value of = . The first constraint limits MS to select (at most one) RP at each movement and the second constraint lets MS have a DOT with the length between 1 and . The constraint (III) indicates that at which step of MS movement, an active sensor node must send its data. It is assumed that MS can receive the data from ASN simultaneously; hence, the constraint (IV) limits the maximum size of ASN to in each movement of MS. When MS at movement moves to RP then ∀ ∈ {1, … , } if , = 1, ̅ ( ) has to send its buffered data to MS (constraint (V)). Constraint (VI) decreases the total energy consumption of the active sensor nodes for sending their captured data to MS by increasing the fitness value of DOT (decreasing µ). Finally, the four last constraints guarantee that the total length of COT should be less than or equal to . Since the proposed MILP model is in the form of NP-complete [28], we present a tabu search based (TS-based) algorithm as a meta-heuristic solution [24], and [25]. 1) Solving MILP Model: A TS-based Approach A tabu search is a local search strategy with a flexible memory structure and consists of four steps [26]: (1) generating an initial solution, (2) exploring the neighborhood ( ) for a solution , (3) designing the fitness function and appropriate move, and (4) defining the tabu list and aspiration criteria. The algorithm ends when one of two thresholds or has been reached, where is defined as a maximal number of allowed iteration and is defined as a maximal number of iterations that can occur while the best solution is not promoted. Let be a two-dimensional integer array with the size of 2 ( =size of ASN), considered as a solution's structure. The first row of shows the order of data harvesting from ASN and the second row includes the set of visited RPs;

RP

S (1)

S ( 3)

S (5)

S (2 )

DOT

S ( 4)

Fig. 11. DOT for MS in the network with five active sensor nodes and 33 rendezvous points located in the convex area of ASN.

9

for example, Fig.11 shows that MS visits ̅ ( ) , ̅ ( ) , ̅ ( ) , ̅ ( ) , and ̅ ( ) at RPs 7,24,20,20, and 10 and harvests their data, respectively. An arbitrary order of ASN can be used in the first row; however, for the second row, a set of randomly selected RPs will be admissible if the length of DOT is less than or equal to . We note that by having the matrix of and random starting RP, a valid set of RPs can be constructed with a time complexity of ( ). Proposing an algorithm to explore the neighborhood ( ) of a solution has a direct impact on the execution time. We propose a method that includes two simple strategies to obtain a neighbor of ( ). The selection between strategies is based on the threshold value of 0 < < 1. A neighbor of can be obtained through swapping , and , where 1 and 2 are random integer numbers between 1 and (strategy I) or by changing last RPs in the second row of where is a random number between 1 and (strategy II). The fitness value of each solution can be easily obtained by (2). Let be a tabu list used in the proposed algorithm as a three-dimensional data structure array to record forbidden moves that is a salient feature of TS. The value of , , , ( = 1: , = 1: , = 1: ) will be increased if active sensor node or RPs at movement of MS is changed ( = , , = , ). The main concept of the tabu list is to keep track of the solutions that have been considered in the past [24]-[26]. Therefore, will be updated by the changed columns of the best neighbor of in each execution round of algorithm. Although the tabu list helps tabu search to prevent the search from the local optima, it could drastically restrict the neighborhood ( ). Moreover, it is possible that some attractive solutions will be missed. The aspiration criterion lets tabu search override the tabu list restrictions. Consequently, it allows the superior solution ∗ to be accepted despite the all changed columns of ∗ that have been changed in . The effect of this criterion on tabu search is known as intensification [24]-[26]. The duration of keeping a record in the tabu list is adjusted by tabu tenure [24]-[26]. In fact, tabu tenure determines the boundary of the search area, thus long tabu tenure lets tabu search explore unvisited territory. This phenomenon is named by diversification. B. Performance Evaluation We categorize the results of performance evaluations into two phases. In the first phase, we are going to compare the performance of the proposed heuristic algorithm with the proposed MILP model. Two groups of sink mobility (DOT and COT) will be evaluated in this phase. Since in the first sub-problem, COT is determined through maximizing the fitness value , we evaluate the quality of solutions for different objective functions in the second phase. The second phase is based on a dynamic network; in fact, in each round of network operations, some events occur at arbitrary locations in the network and MS must harvest the captured data from ASN during the reporting time slot, which is selected arbitrarily in each round. Regarding this setting, for different sizes of ASN, we measure the round number that the data of at least one event is missed since there is no live node to cover it. The



> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 100 Proposed heuristic algorithm TS-based algorithm (RP=200) TS-based algorithm (RP=600)

90 The average of fitness value

evaluation metrics in two phases are the fitness value, network lifetime, energy consumption, and computation time. Our simulations are written in MATLAB [27] and are run on a computer with Intel core i5-3210M 2.5 GHz CPU and 6GB memory. The other simulation parameters are =15 m/s, n=100, (the initial energy of sensor nodes)=1 Joule, (the radius of zone) =100 m, =50 nJoule/bit, =10 pJoule/bit/m2, =250 Kbps, and γ (the number of zones)=1. 1) The First Phase Before comparing the performance of the proposed heuristic algorithm with the TS- based method, we are going to estimate the appropriate size of the tabu tenure because it has a direct impact on the quality of the solutions. We run the TS-based method for ASN with sizes 5 and 10 (randomly scattered in the network) and reporting time slots 8, 10, and 12. Moreover, the visiting time values of ASN are selected randomly between 1 and 4 seconds. The thresholds and are set to 200 and 1000, respectively. However, to obtain the optimal value of , we consider the different values of q = 0.1,0.2, … ,0.9. For each value of , we run the simulation 20 times and measure the average of the maximum fitness value with respect to the different sizes of tabu tenure. The optimal solutions for =5, 10 and =8, 10, 12 are achieved when = 0.5 (Fig. 12). As shown in Fig.12, by setting the size of the tabu tenure equal to 25, the TS-based method can obtain the maximum fitness value in almost all considered values of and . Since the obtained confidence band is narrow, it is not reported. In the next experiment, we evaluate the quality of solutions obtained from the TS-based method and the proposed heuristic algorithm in terms of fitness value. The reported results are the average of 20 times simulation runs. Fig. 13 illustrates the measured average of the fitness value in all the simulation runs. It is shown that by increasing the reporting time slot, the achieved fitness value in both algorithms is increased. This is because by increasing the value of the reporting time slot, MS has enough time to reach ASN and receive their data in shorter range. However, the curve of the proposed heuristic algorithm looks steady after = 14 because MS can visit ASN in the closest location. It is obvious that the performance gap between the TS-based and heuristic algorithm grows larger by increasing the reporting time slot, especially when > UTT

10

TS-based algorithm (RP=1200) UTT 80

70

60

50

40 0

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20 Reporting time slot (second)

Fig. 13. The difference of average of fitness value between the heuristic algorithm and TS-based method.

(the average of UTT is obtained 11.7 seconds). Although the proposed simple algorithm for > UTT determines L-COT through the greedy algorithm, it shows a better performance than the proposed TS-based method. As shown in Fig. 13, by increasing the number of RPs, the fitness values of solutions of the TS-based are increased slightly. The averages of energy consumption for both algorithms are shown in Fig. 14. The proposed heuristic approach confirms its better performance in comparison with the TS-based algorithm. Furthermore, Fig. 15 depicts the difference between the measured values of average computation time of the heuristic algorithm and the TS-based approach. The comparison between the two algorithms reveals that the time complexity of the heuristic algorithm for achieving COT is considerably lower than the TS-based approach. Although the time complexity of the heuristic model remains almost steady by increasing the reporting time slot, the TS-based approach decreases gradually, because the search space of the TS-based algorithm for a solution with the same starting RP is restricted by considering a shorter reporting time slot. 2) The Second Phase In the first sub-problem, the objective function Maximizes the Fitness Value of COT (MFV); in other words, it increases the lifetime of ASN. In this phase, we are interested in investigating the impact of different metrics on the first sub0.38

65

60

55

50

0.34 0.32 0.3 0.28 0.26 0.24 0.22 0.2

45

0.18 40

Proposed heuristic algorithm TS-based algorithm (RP=200) TS-based algorithm (RP=600) TS-based algorithm (RP=1200)

0.36

m=5, reporting time slot=8 m=5, reporting time slot=10 m=5, reporting time slot=12 m=10, reporting time slot=8 m=10, reporting time slot=10 m=10, reporting time slot=12

The average of energy consumption (Joule)

The average of maximum fitness value

70

5

10

15

20 25 30 The size of tabu tenure

35

40

Fig. 12. The impact of tabu tenure on the maximum of fitness value.

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20 Reporting time slot (second)

Fig. 14. The difference of average energy consumption between the heuristic algorithm and TS-based algorithm.



> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < problem such as Maximizing the Minimum of Residual energy (MMR), Minimizing the Maximum of Energy consumption (MME), and Minimizing the Sum of Energy consumption (MSE). Fig. 16 shows the proposed convex models of MMR, MME, and MSE. We run the simulation with different sizes of ASN and measure time until the first active sensor node dies. In other words, we measure the round number in which an event occurs in the network, but is not captured. The measured values are the average of 20 simulation runs (since the obtained confidence band is narrow, it is not reported). In addition, for each size of ASN, the number and locations of events, the location and visiting time of each active sensor node, and the requested reporting time slots are the same for all models. As Fig. 17 shows, the proposed model in the first sub-problem confirms its better performance in comparison with the other models. The MMR model focuses on the residual energy of ASN. In fact, the residual energy of ASN has a profound influence on the process of producing COT; however, it cannot guarantee that MMR obtains the maximum network lifetime. This is because increasing the minimum amount of residual energy might have an effect on the energy consumption of the other nodes with more energy. Since MME produces COT as it regards the maximum energy consumption, it cannot maximize the lifetime of the network, because it is conceivable that an active sensor node with less energy consumption dies earlier than another node with more energy consumption. In addition, the last model (MSE) focuses on the sum of energy consumption of ASN and

it is possible that the energy consumption of the sensor nodes with low energy would be ignored. As shown in Fig. 17 , in all sizes of ASN, the performance of the TS-based method is lower than other models and it is obvious that by increasing the number of ASN, the maximum network lifetime decreases. Fig. 17 shows that increasing the fitness value of COT impacts the network lifetime of WSN. Moreover, in Fig. 18, we illustrate the average energy consumption for different size of ASN. Interestingly, the average of energy consumption for the heuristic model and other convex models (MMR,MME, and MSE) are almost the same while the TS-based algorithm shows more energy consumption. Moreover, considering the average of computation time in Table 1, the time complexity of the TS-based algorithm is considerably greater than other models. VI. CONCLUSION In this paper, we studied how to determine a trajectory for a mobile sink without considering any predefined rendezvous points or virtual structures. The trajectory, which is named the Continuous and Optimal Trajectory (COT), yields a significant gain in terms of the network lifetime. We considered the event-based application, where an application server specifies a reporting time slot for the mobile sink to harvest all the data from a group of Active Sensor Nodes (ASN). We proposed an exact solution to obtain COT for the topology with two sensor nodes and any required . Moreover, it was shown that the problem of finding COT for topologies

35

250

Proposed heuristic algorithm TS-based algorithm (RP=200) TS-based algorithm (RP=600) TS-based algorithm (RP=1200)

30

The average of first node die (round)

The average of computation time (second)

11

25

20

15

10

MFV(proposed heuristic algorithm) MMR MME MSE TS-based algorithm (RP=1200)

200

150

100

50

5

0

4

5

6

7

8

0

9 10 11 12 13 14 15 16 17 18 19 20 Reporting time slot (second)

Fig. 15. The difference of average computation time between the heuristic algorithm and TS-based algorithm.

The MMR model: minimize − subject to: ̅ ( ) − ̅ ( ) − ≤ ̅ ( ) , ∀ ̅( ) −

+

̅( ) +

()

≥ ,∀

variables: ≥ 0,

̅( )

≥ 0, ( = 1: ), , .

5

̅( ) +

()

≤ , ∀ variables: () > 0, ̅ ≥ 0, ( = 1: ),

9 11 The size of ASN

13

15

Fig. 17. The difference of average network lifetime between the proposed convex models and the TS-based algorithm.

The MME model: minimize + subject to: ̅ ( ) − ̅ ( ) − ≤ ̅ ( ) , ∀ +

7

,

.

The MSE model: minimize + subject to: ̅ ( ) − ̅ ( ) − ≤ ̅ ( ) , ∀ ∑

( + (

̅( )

+

()

)) ≤

variables: () > 0, ̅ ≥ 0, ( = 1: ), , .

Fig. 16. Different metrics on obtaining COT in the first sub-problem: MMR, MME, and MSE.



> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < [3]

The average of energy consumption (Jole)

0.06

[4] 0.05

0.04

[5]

0.03

[6] 0.02 MFV(proposed heuristic algorithm) MMR MME MSE TS-based algorithm (RP=1200)

0.01

0

5

7

9 11 The size of ASN

13

[7]

15

Fig. 18. The difference of average energy consumption between the proposed convex models and the TS-based algorithm.

with more than two sensor nodes is NP-hard. Hence, we introduced a new heuristic approach for attaining COT. Our methodology determines COT for all topologies and various reporting time slots through solving two simpler subproblems. In the first sub-problem, COT was determined through a convex optimization model without considering the reporting time slot. The traveling time of the mobile sink on the obtained COT was denoted as Unsolicited Traveling Time (UTT). In the second sub-problem, we considered the reporting time and proposed two algorithms: the first algorithm obtained COT as a straight line for UTT in polynomial time. The effectiveness of our proposed approach was validated through the extensive number of simulation runs. Several interesting directions can be pointed out as suggestion for future work. Proposing an optimal multi-hop delivery scheme is the first direction. Focusing on a distributed algorithm to determine COT can be a second direction. The last one would be to extend the proposed algorithm for supporting multi-mobile sinks and various velocities.

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16] [17]

[18]

[19]

TABLE 1. THE DIFFERENCE OF COMPUTATION TIME BETWEEN THE PROPOSED CONVEX MODELS AND THE TS-BASED ALGORITHM

5 7 9 11 13 15

MFV 0.3509 0.3844 0.3878 0.3868 0.4791 0.5031

Computation time (second) MMR MME MSE 0.3129 0.3448 0.3441 0.3206 0.3324 0.3353 0.3255 0.3604 0.3509 0.3552 0.3856 0.3758 0.4808 0.4430 0.4444 0.4268 0.4500 0.3856

TS-based 25.0500 25.1961 29.5323 36.2330 42.1382 46.9904

[20]

[21]

[22]

[23] [24]

REFERENCES [1]

[2]

J. Luo, J.-P.Hubaux, “Joint sink mobility and routing to increase the lifetime of wireless sensor networks: The case of constrained mobility,” IEEE/ACM Trans. on Networking, vol. 18(5), pp.1387–1400, 2010. Gandham S.R, M. Dawande, et al. “Energy efﬁcient schemes for wireless sensor networks with multiple mobile base stations”, proceeding of the Globecom’03, PP 377-381, 2003.

[25] [26]

[27]

12

Jain S., Shah R.C., Brunette W., Borriello G., Roy S., “Exploiting Mobility for Energy Efficient Data Collection in Sensor Networks, Mobile Networks and Applicationfs”, Vol.11, No.3, PP327-339, 2006. Chakrabarti A., Sabharwal A., Aazhang B., “Communication Power Optimization in a Sensor Network with a Path-Constrained Mobile Observer”, ACM Transaction. Sensor Networks, Vol.2, NO. 3, Pages 297-324, 2006. Song, Liang, and Dimitrios Hatzinakos. "Architecture of wireless sensor networks with mobile sinks: Sparsely deployed sensors." Vehicular Technology, IEEE Transactions on 56.4 (2007): 1826-1836. Jea, David, Arun Somasundara, and Mani Srivastava. "Multiple controlled mobile elements (data mules) for data collection in sensor networks." Distributed Computing in Sensor Systems. Springer Berlin Heidelberg, 2005. 244-257. Z. Li, M. Li, J. Wang, and Z. Cao, “Ubiquitous data collection for mobile users in wireless sensor networks,” in Proc. of IEEEINFOCOM’11, Shanghai, China, 2011. Shah, Rahul C., et al. "Data mules: Modeling and analysis of a three-tier architecture for sparse sensor networks." Ad Hoc Networks 1.2 (2003): 215-233. Lang Tong, Qing Zhao, and Srihari Adireddy, "Sensor networks with mobile agents". In Proc., IEEE MILCOM 2003, volume 22, pages 688– 693, Boston, MA, USA. October 2003. Y. Shi and Y. T. Hou, “Optimal base station placement in wireless sensor networks,” ACM Transactions on Sensor Networks, vol. 5, no. 4, 2009. YoungSang Yun, Ye Xia, ”Maximizing the Lifetime of Wireless Sensor Networks with Mobile Sink in Delay-Tolerant Applications” IEEE Transaction on Mobile Computing, pp: 1308 – 1318, 2010. Xing, G., Li, M., Wang, T., Jia, W., & Huang, J, "Efficient rendezvous algorithms for mobility-enabled wireless sensor networks", IEEE Tran. on Mobile Computing, 47–60. 2012. Y.-C. Wang, W.-C.Peng, Y.-C.Tseng, "Energy-balanced dispatch of mobile sensors in a hybrid wireless sensor network", IEEE Trans. Parallel Distrib. (12) 1836–1850. 2010. Y Gu, et al., "ESWC: Efficient Scheduling for the Mobile Sink in Wireless Sensor Networks with Delay Constraint", , IEEE Trans. Parallel Distrib. Syst. VOL. 24, NO. 7, JULY 2013. M. Zhao, M. Ma, and Y. Yang, “Efﬁcient data gathering with mobile collectors and space-division multiple access technique in wireless sensor networks,” IEEE Trans. Comput., vol. 60, no. 3, pp. 400–417, Mar. 2010. Stefano Basagni, et al. "Controlled sink mobility for prolonging wireless sensor networks lifetime," Springer Science Wireless Netw, 2008. M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-completeness, W. H. Freeman and Company, New York, NY, 1979. Y. Shi and Y.T. Hou, “Theoretical Results on Base Station Movement Problem for Sensor Network”, The 27th Conference on Computer Communications. IEEE INFOCOM, 2008. G. Xing, T. Wang, Z. Xie, and W. Jia, “Rendezvous Planning in Wireless Sensor Networks with Mobile Elements,” IEEE Trans. Mobile Computing, vol. 7, no. 11, pp. 1-14, Nov. 2008. W. Liang, J. Luo, and X. Xu."Prolonging network lifetime via a controlled mobile sink in wireless sensor networks", Proc. of Globecom'10, IEEE, 2010. Konstantopoulos C., Pantziou G., Gavalas D, Aristides Mpitziopoulos, Basilis Mamalis, “A Rendezvous-Based Approach Enabling EnergyEfficient Sensory Data Collection with Mobile Sinks”, IEEE Transaction Parallel Distributed System, Vol.23, Pages 809-817, 2012. Singh, M. and Lau, L. C. "Approximating minimum bounded degree spanning trees to within one of optimal", In STOC’07—Proceedings of the 39th Annual ACM Symposium on Theory of Computing 661–670. ACM, New York, 2007. Welzl, E." Smallest enclosing disks". Lecture Notes in Computer Science, 359–370. 1991. Glover, F. "Tabu Search", Part I. ORSA Journal on computing, 1, 190206, 1989. Glover, F. "Tabu Search", Part II. ORSA Journal on computing, 2, 4-32, 1990. Rhazi, A. E., & Pierre. S, "A Tabu search algorithm for cluster building in wireless sensor networks", IEEE Transactions on Mobile Computing, 8(4), 433–444. 2009. MathWorks—MATLAB and Simulink for Technical computing, http://www.mathworks.com



> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < [28] S. A. Vavasis. "Nonlinear Optimization: Complexity Issues". Oxford University Press, 1991. [29] Lee, J. and Leyffer, S., editors, "Mixed Integer Nonlinear Programming, IMA Volume in Mathematics and its Applications", Springer, New York, 2011. [30] Pan, J.,Cai, L., Hou, Y.T., Shi, Y., Shen, S.X. “Optimal base-station locations in two-tiered wireless sensor networks”, IEEE Transaction on Mobile Computing, Vol. 4, No.5, 458-473. 2005. [31] Gao S., et al., “Efficient Data Collection in Wireless Sensor Networks with Path-Constrained Mobile Sinks”, IEEE Transaction on Mobile Computing, Vol.10, No. 5, PP 1-9, 2011. [32] Yun, Y., Xia, Y., Behdani, B., and Smith, J. C., "Distributed Algorithm for Lifetime Maximization in Delay-Tolerant Wireless Sensor Network with Mobile Sink," IEEE Transactions on Mobile Computing, 2012. [33] V. N. Vapnik, “the Nature of Statistical Learning Theory”. NewYork: Springer-Verlag, 1995. [34] A. Sharifkhani and N. Beaulieu, “A mobile-sink-based packet transmission scheduling algorithm for dense wireless sensor networks,” IEEE Transactions on Vehicular Technology, vol. 58, no. 5, pp. 25092518, Jun. 2009. [35] D. N. Alparslan, K.Sohraby, "Two-Dimensional Modeling and Analysis of Generalized Random Mobility Models for Wireless Ad Hoc Networks ", IEEE/ACM Transaction on networking, vol.15, no. 3. 2007. [36] F. Bai, K. S. Munasinghe, and A. Jamalipour, “A Novel Information Acquisition Technique for Mobile-Assisted Wireless Sensor Network”, IEEE Transactions on Vehicular Technology, vol. 61 no. 4, 2012. [37] Y. Wu, L.Zhang, Y.Wu, Z. Niu,"Motion-Indicated Interest Dissemination With Directional Antennas for Wireless Sensor Networks With Mobile Sinks", IEEE Transactions on Vehicular Technology, vol.58 no.2, 2009. [38] G. Shi, J. Zheng, J. Yang, Z. Zhao, "Double-Blind Data Discovery Using Double Cross for Large-Scale Wireless Sensor Networks with Mobile Sinks", IEEE Transactions on Vehicular Technology, vol.61 no.5, 2012. [39] W. Liu, et al., “Performance analysis of wireless sensor networks with mobile sinks”, IEEE Transactions on Vehicular Technology, vol. 61, no. 6, pp. 2777–2788, 2012. [40] D. N. Alparslan, K. Sohraby, "A Generalized Random Mobility Model for Wireless Ad Hoc Networks and Its Analysis: One-Dimensional Case", IEEE/ACM Transaction on networking, vol.15, no. 3. 2007. [41] Nesterov and Nemirovski,” Interior-Point Polynomial Algorithms in Convex Programming” Society for Industrial and Applied Mathematics, 1994. [42] MartinSkutella, ”Convex Quadratic and Semi definite Programming Relaxations in Scheduling” Journal of the ACM, Vol. 48, No. 2, pp. 206 –242, March 2001. [43] Heinzelman, Wendi B., et al. "An application-specific protocol architecture for wireless microsensor networks."Wireless Communications, IEEE Transactions on 1.4 (2002): 660-670. [44] Manyem, Prabhu, and Julien Ugon. "Computational Complexity, NP Completeness and Optimization Duality: A Survey." Electronic Colloquium on Computational Complexity (ECCC). Vol. 19. 2012. [45] Boyd, Stephen P., and Lieven Vandenberghe. Convex optimization. Cambridge university press, 2004

Farzad Tashtarian received the B.S. degree in Computer Engineering from Islamic Azad University Mashhad Branch, Iran in 2005, and the M.S. degree in Information Technology from Islamic Azad University Qazvin Branch, Iran in 2007. He is currently pursuing his Ph.D. degree in Computer Engineering at

13

Ferdowsi university of Mashhad, Mashhad, Iran. His research interests include wireless sensor networks, mobile communications, mathematical modeling, optimization, and distributed control. Mohammad Hossein Yaghmaee Moghaddam received his B.S. degree in communication engineering from Sharif University of Technology, Tehran, Iran in 1993, and M.S. degree in communication engineering from Tehran Polytechnic (Amirkabir) University of Technology in 1995. He received his Ph.D degree in communication engineering from Tehran Polytechnic (Amirkabir) University of Technology in 2000. He has been with Department of Computer Engineering, Ferdowsi University of Mashhad since 2000. His current research interests include computer and communication networks, wireless sensor networks, multimedia networking and smart power grid. Refer to http://profsite.um.ac.ir/~hyaghmae/ for a detailed biography.

Khosrow Sohraby (S’82–M’84–SM’89) received the B.Eng. and M.Eng. degrees from McGill University, Montreal, Canada, in 1979 and 1981, respectively, and the Ph.D. degree from the University of Toronto, Toronto, Canada, in 1985, all in electrical engineering. His current research interests include design, analysis and control of high-speed computer and communications networks, traffic management and analysis, multimedia networks, networking aspects of wireless and mobile communications, analysis of algorithms, parallel processing and large-scale computations. Refer to http://www.sce.umkc.edu/~sohrabyk/ for a detailed biography. Sohrab Effati received the B.S. degree in applied mathematics from Birjand University, Birjand, Iran, and the M.S. degree in applied mathematics from Tarbiat Moallem University of Tehran, Tehran, Iran, in 1992 and 1995, respectively, and the Ph.D. degree in control systems from Ferdowsi University of Mashhad, Mashhad, Iran, in April 2000. Since 2005, he has been an Associate Professor at the Department of Applied Mathematics at Ferdowsi University of Mashhad in Iran. His research interests are in the areas of control systems, optimization, ODE and PDE, and neural networks and its applications in optimization problems.
