ODT: Optimal deadline-based trajectory for mobile sinks in WSN: A decision tree and dynamic programming approach

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Contents lists available at ScienceDirect

Computer Networks journal homepage: www.elsevier.com/locate/comnet 5 6

ODT: Optimal deadline-based trajectory for mobile sinks in WSN: A decision tree and dynamic programming approach

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Farzad Tashtarian a, M.H. Yaghmaee Moghaddam a, Khosrow Sohraby b,⇑, Sohrab Effati c a

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Department of Computer Engineering, Ferdowsi University of Mashhad, Mashhad, Iran The School of Computing and Engineering, University of Missouri-Kansas City (UMKC), United States c Department of Mathematics and the Center of Excellence on Soft Computing and Intelligent Information Processing (SCIIP), Ferdowsi University of Mashhad, Mashhad, Iran b

a r t i c l e

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i n f o

Article history: Received 13 February 2014 Received in revised form 18 October 2014 Accepted 1 December 2014 Available online xxxx

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Keywords: Wireless sensor network Mobile sink Optimal deadline-based trajectory Dynamic programming Decision tree

a b s t r a c t Recent studies have shown that utilizing a mobile sink (MS) to harvest and carry data from a wireless sensor network (WSN) can enhance network operations and increase the network lifetime. Since a significant portion of sensor nodes’ energy is consumed for data transmission to MS, the specific trajectory has a profound influence on the lifetime of WSN. In this paper, we study the problem of controlling sink mobility in deadline-based and event-driven applications to achieve maximum network lifetime. In these applications, when a sensor node captures an event, it should determine a visiting time and a deadline with respect to the amount of captured data and the type of event. MS then has to determine its trajectory to harvest data from active sensor nodes in single hop transmission so that the network lifetime is increased. We show that this problem is NP-hard when there are no predefined structures like a virtual grid or rendezvous points in the network. We propose an algorithm based on a decision tree and dynamic programming to approximately determine an optimal deadline-based trajectory (ODT). ODT is obtained by considering the geographical positions of active sensor nodes and the properties of captured events. The effectiveness of our approach is validated via the extensive number of simulation runs and comparison with other algorithms.  2014 Published by Elsevier B.V.

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1. Introduction

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In recent years, a number of techniques have been proposed to prolong the lifetime of wireless sensor networks (WSN) by utilizing a mobile sink (MS) in networks [1–7]. In many applications of WSN, the sink that is mounted on a mobile robot regularly moves across the monitored area to collect data from all sensor nodes. A significant gain in the network lifetime could be achieved by the optimum control of the trajectory of MS, see [8–16,18– 21].

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Q3

⇑ Corresponding author.

The sink mobility in WSN can be categorized into two main groups: random mobility based [8,9] and controlled mobility based [1,2,11,16,18,32]. For the first group, MS can move freely and randomly over the network and harvests the buffered data [8]. Although the random mobility schemes are simple and easy to implement, they suffer from shortcomings like uncontrolled behaviors and poor performance [14]. The majority of existing works is allocated to the controlled mobility. The main challenge in this category is the determination of the optimal trajectory of MS depending on the status of sensor nodes and the WSN’s application. It has been shown that by designing the proper trajectory, the

http://dx.doi.org/10.1016/j.comnet.2014.12.003 1389-1286/ 2014 Published by Elsevier B.V.

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harvesting data via MS would significantly improve the network lifetime [1,11,14]. There are two main problems in the majority of the existing work in this category. The first and foremost is that the trajectory of MS has been assumed to be determined through some predefined special locations [11,16] or special nodes in hardware or resource [1,20,30,31], called rendezvous points (RPs). In most of the studies, RPs are predefined and independent of other sensor nodes locations. It is clear that the locations and the number of RPs have a great influence on the performance and the quality of the solution. Considering the fast mobility of MS during its movement on a trajectory, especially among RPs, is the second problem. Some previous proposals showed that the optimal trajectory of MS can be designed through mathematical optimization models based on RPs, but the majority of them avoid addressing the traveling time of MS on the selected trajectory [10,11]. In this study, we are interested in determining the optimal trajectory referred to as optimal deadline-based trajectory (ODT) for MS in deadline-based and event-driven applications by assuming single-hop data delivery. We do not assume pre-existing RPs or any other predefined structure. The process of designing ODT considers three main parameters: (1) the group of sensors that capture events (active nodes (AN) group), (2) the velocity of MS for traveling on ODT, and (3) the properties of captured data by AN: a visiting time and a deadline that are determined regarding the amount of captured data and type of event, respectively. Since the problem of determining ODT is NP-hard, as it will be proved, we divide the trajectory of MS into a limited number of steps, and then a convex programming model to obtain an optimal line segment for each step is proposed. We introduce a decision tree structure to cover the feasible solution space of ODT, and then through a dynamic programming approach, an optimal deadlinebased trajectory can be achieved. 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. Summary of related works are reviewed in Section 2. In Section 3, we describe the system model. The problem formulation is discussed in Section 4. In the next section, we propose an approximate algorithm to determine ODT. The performance of the proposed algorithm is examined via simulation in Section 6, and finally, Section 7 concludes the paper.

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2. Related works

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Prolonging the network lifetime of WSN has been targeted through the mobility management of the mobile sink (MS) recently [1–7,24]. We can classify sink mobility into two categories: random mobility [8,9] and controlled mobility [1,2,11,16,18]. 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

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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 introduces some problems such as 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 an approximate 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 and Xia 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 traveling time of MS between any two RPs is negligible. Gandham et al. in [2] Q4 controlled the movement of MS among some RPs through proposing an Integer Linear 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 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. Xing 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]. 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 one-hop data delivery event-driven applications without considering any infrastructure or predetermined RPs. We show that the problem is NP-hard and, we then

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propose an approximation solution for the optimal trajectory for MS.

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3. System model

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F. Tashtarian et al. / Computer Networks xxx (2014) xxx–xxx ðiÞ

ðiÞ

where Se and Sd are the amount of the residual energy (in joule) and captured data (in bit) of SðiÞ , respectively. Ei is the required energy per unit of time to transmit data from ith node to MS as follows [11,18]: l

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In deadline-based and event-driven 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 is called the active nodes (AN) group and each member of AN sends its data to MS in a single-hop manner. According to the different amount of captured data and types of events, different visiting time and a deadline are determined by AN, respectively. Due to limited velocity V of MS (0 6 V 6 Vmax) and transmission range of AN, harvesting data in a large-scale WSN from AN is impractical, especially when there are some obstacles in the network field. Hence, we can divide the WSN field into z autonomous zones. By assuming single-hop data delivery between sensor nodes and MS, the size of each zone is determined (see Fig. 1). Consider a zone consists of n randomly deployed wireless sensor nodes S(i), i = 1, . . . , n, and a sink mounted on a mobile robot that must harvest data from all members of AN (which are denoted by SðiÞ ; i ¼ 1; . . . ; m; ðm 6 nÞÞ in its zone with respect to the different values of visiting time ti (i = 1, . . . , m) and deadline si (i = 1:m). Moreover, we assume that each sensor node can adjust its transmission ranges by receiving a control packet from MS that includes the details of ODT. We denote T as the network lifetime defined as the elapsed time since the beginning operation of the network until 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 paper, we focus on the existing AN to determine ODT for MS so that the lifetime of the existing AN is maximized. The process of achieving ODT should be performed upon joining or leaving any sensor node to/from AN. Without loss of generality, we can consider a network with one zone and focus on obtaining ODT. We define F as the fitness value of the obtained ODT be as follows:

( F ¼ min

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Ei ¼ ða þ bðdi!MS Þ Þf ;



ðiÞ Se 

)

ðiÞ Sd

; i ¼ 1; . . . ; m ; t i ¼ t i Ei  f

ð1Þ

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4. Problem formulation

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In this section, we focus on the mathematical model of ODT for AN with size of m P 2 and show that the problem of finding ODT for m P 2 is NP-hard. In this study, the traveling time of MS between any two location points in the network field is taken into account. By capturing an event, SðiÞ selects si as a maximum deadline for sending its sensed data to MS. si implicitly specifies the priority of captured

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zone 3

MS

ODT

AN zone 6 zone 1

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where di?MS is the physical distance between ith node and MS, a (in joule/bit) and b (in joule/bit/m2) are nonnegative constant terms, l is the path loss index, which is typically in the range of 2–6 depending on the environment [11] (we set l = 2 in this study), and f is considered as a constant transmission bit rate (in bps) [15,19]. It is clear that by increasing F, the lifetime of the existing AN is increased. In fact, the process of determining ODT can be performed in MS by considering the properties of existing AN such ðiÞ as: geographical locations, Se , ti, and si which are sent through a small beacon packet from AN to MS in single hop upon capturing an event, and furthermore the limited velocity of MS. We note here that the paper does not consider the MAC-layer contention [11] and future work may 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 deadline-based and event-driven applications of WSNs: (1) investigating the problem of obtaining ODT for any size of AN analytically including mathematical modeling of ODT, and showing that the problem of determining ODT is NP-hard; (2) proposing an approximate algorithm to obtain ODT based on a decision tree and dynamic programming model; (3) considering different objective functions in the process of determining ODT; and (4) conducting simulation experiments to verify our theoretical results. Table 1 lists the notations and symbols used in this paper.

zone 2 Application Server

ð2Þ

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obstacle zone 4

zone 5

Fig. 1. The large-scale network field with six autonomous zones.

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Table 1 Notations. Symbol

Description

S(i)

The ith sensor node The ith active sensor node

SðiÞ n m  m

The number of sensor nodes The number of active sensor nodes The maximum number of nodes that MS can receive data from them simultaneously

ðiÞ

Sd ti

The volume of captured data by SðiÞ (in bits)

si

The specified deadline for sending data from SðiÞ to MS

Ei

The required amount of energy for sending data from SðiÞ to MS The constant transmission bit rate (in bps)

The required visiting time for sending data from SðiÞ to MS

f di?MS

The Euclidian distance between SðiÞ and MS The obtained fitness of ODT The network lifetime The number of steps of ODT The maximum velocity of MS The traveling time of MS on jth step of ODT The jth vertex of graph G The jth edge of graph G The ith decision-node in jth stage of decision tree A decision (arc) in jth stage of decision tree connecting two decision-nodes i and i0

F T k Vmax hj

vj Ej ni;j dj;ði;i0 Þ   ðqÞ ri;j ri;j   ðqÞ ei;j ei;j

The remaining visiting time of AN in ni;j (The qth element of ri;j ) The remaining power of AN in ni;j (The qth element of ei;j )

ðxÞ ðyÞ pi;j ðpi;j ; pi;j Þ ðqÞ

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  ðxÞ ðyÞ A candidate point for jth vertex in G with the coordinates of pi;j ; pi;j which is proposed by ni;j

C i;j

The average amount of data that should be sent by SðqÞ regarding all originated child decision-nodes from ni;j

b,a hq

The gradient and the y-intercept of L, respectively

nq

The horizontal distance between rotated SðiÞ and optimal location for MS on L

The vertical distance between rotated SðiÞ and rotated L

event regarding the type and occurrence time of event, and moreover the geographical location of SðiÞ . Since ti is the ðiÞ required time to transmit the volume of captured data Sd with constant bit rate f, si must be greater than or equal to ti. Definition 1. Let V and E be the two sets of vertices and edges respectively, where jVj ¼ k þ 1 and jEj ¼ k P 1. From the graph theory point of view, an ODT can be defined as a graph G ¼ ðV; EÞ where "j, degðV j Þ 6 2 and E j is a straight line segment with length of hjVj. The variables of hj and Vj are the traveling time and velocity of MS on E j , respectively; and V 1 shows the initial location of MS (see Fig. 2). In the above definition, k refers to the number of MS movements called steps. Since in the theoretical point of view, the optimal value of d1?MS should be decreased by approaching MS; however, it is impractical to do so due to

the hardware limitation of sensor nodes. Therefore, it should be assumed that when MS is moving on E j ; 8i; di!MS ¼ max fDfSðiÞ ; V j g; DfSðiÞ ; V jþ1 gg where Dfa; bg is the Euclidian distance between the two points of a and b, (see Fig. 2). Before introducing the mathematical model of ODT for AN with m P 2, we determine ODT for m = 1. In this simplest case, the best trajectory for MS is the straight line connecting initial location of MS and Sð1Þ . To harvest the data of Sð1Þ with the minimum energy consumption, MS has s1  t1 unit of time to move toward Sð1Þ to reduce the transmission range d1?MS. Thus, the optimal transmission range d1!MS ¼ DfSð1Þ ; MSg  ðs1  t1 ÞV max .

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Theorem 1. Determining ODT for AN with m P 2 is an NP-hard problem.

302

Proof. Without loss of generality, suppose that MS must harvest the data from AN through passing k edges of G. For achieving ODT in a topology with m P 2, a number of constraints must be satisfied. First, we define variables T and r. T is a two-dimensional binary array of size m  k to indicate valid edge(s) for AN to send their data to MS. T i;j ¼ 1 means that SðiÞ is permitted to send a portion of its data to MS when MS is moving on E j . The second variable r is defined as a two-dimensional binary array with the same size of T . If ri;j ¼ 1; SðiÞ will send a portion of its data to MS when MS is moving on E j .

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Fig. 2. A sample of ODT for AN with m = 4.

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Considering the deadlines of AN, the first constraint selects the optimal number of edges for harvesting the data P from AN. Thus, kj¼1 T i;j hj 6 si ; 8i and T i;j P T i;jþ1 . In the next constraint, by selecting the optimal binary variable ri,j, the appropriate visiting time ti;j for SðiÞ on E j is determined; therefore, we have 0 6 t i;j 6 ri;j hj T i;j , "i, Pk   where j¼1 t i;j ¼ t i . By having t i;j , the maximum distance ðjÞ between SðiÞ and MS on E j , which is denoted by di!MS , is obtained through the following constraint:

ðjÞ

6 di!MS ;

where Hi,j is referred to as the closest point located on E j to     ðiÞ ðiÞ SðiÞ with the coordinates Hxi;j ¼ Sx  qj Sy  qj cj = 1 þ q2j     ðiÞ ðiÞ and Hyi;j ¼ qj Sx  q2j Sy þ cj = 1 þ q2j , where qj ¼       V yjþ1  V yj = V xjþ1  V xj and cj ¼ V yj  qj V xj (see Fig. 2). ðjÞ di!MS

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V xj 6 Hxi;j þ rxi;j 6 V xjþ1 ; and V yj 6 Hyi;j þ ryi;j 6 V yjþ1 ;

341

342

ð4Þ

where ryi;j ¼ qj rxi;j þ cj . Moreover, using variable Di,j, li,j is placed between V j and V jþ1 :

V j t i;j 6 DfHi;j ; V j g þ Di;j ; and V j ti;j 344 345 346 347 348 349

06

t i;j

8i; j

6 ri;j hj T i;j

k X ti;j ¼ ti

8i; j

8i

k X ðjÞ F t i;j f ða þ b:di!MS Þ 6 SeðiÞ

8i

j¼1

Constraints ð3Þ  ð5Þ variables : T i;j ; ri;j 2 f0; 1g; t i;j ; Di;j ; V j ; hj P 0; F > 0 Hxi;j ; Hyi;j ; V xj ; V yj ; rxi;j ; ryi;j are free By maximizing the fitness value F, the lifetime of AN is increased. It is obvious that the proposed model is in the form of the mixed integer non-linear programming (MINLP), which is NP-hard in general [28,29]. h

6 DfHi;j ; V jþ1 g þ Di;j :

ð5Þ

Furthermore, the length of li,j must be less than or equal to the Euclidian distance between V j and V jþ1 ; thus, the constraint V j t i;j 6 DfV j ; V jþ1 g should be satisfied. Finally, the optimization problem for obtaining ODT for AN with m P 2 can be written as follows:

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ðiÞ

during receiving data from S (we assumed that the velocity of MS on each step is constant). To place Hi,j between V j and V jþ1 , we define two constraints as follows:

340

T i;j P T i;jþ1

ð3Þ

line segment li,j on E j that MS traversed with velocity Vj

337

8i

j¼1

V j ti;j 6 DfV j ; V jþ1 g 8i; j

333

335

k X T i;j hj 6 si

j¼1

332

334

subject to :

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 2 1  2 2 V j t i;j þ Di;j þ ðrxi;j Þ þ ðryi;j Þ DfHi;j ; SðiÞ g þ 2

In the best case, the is equal to rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 2 DfHi;j ; SðiÞ g þ 12 V j ti;j , where V j t i;j is the length of a

331

maximize Fð6Þ

To extend our analytical analysis, we propose a mixed integer linear programming (MILP) model to determine the ODT by considering some rendezvous points (RPs) in the network which is called R-ODT. We’ll evaluate the performances of R-ODT and ODT in the simulation part to show the impact of considering RPs in the process of determination of MS’s trajectory. The MILP model must consider some pre-determined virtual RPs in the network and R-ODT can be defined as a set of RPs and a scheduling table of transmission that maximizes the lifetime of AN; for example in Fig. 3(a) and (b), an optimal set of RPs and a scheduling table are shown, respectively. In MILP model, we assume that MS has an equal time h in each movement; therefore, the maximum length in each movement of MS is limited to hVmax. The

Fig. 3. R-ODT for MS in the network with m = 5 active sensor nodes and 33 rendezvous points located in the convex area of AN (a), and the schedule table of sending data (b).

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Fig. 4. Two sample pairs of the optimal scheduling tables of data harvesting (a) and (c) according the selected ODTs (b) and (d), respectively.

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maximum number of steps is bounded to q ¼

l

maxfsi ji¼1:mg h

m ,

where m is the size of AN. Since R-ODT is located in the convex area by AN [10,23], the solution space of MILP model can be narrowed by considering RPs located in the convex area that yields the significant gain in the time complexity of the MILP model. Before presenting the MILP model, let us define sets and variables as follows: Sets:

402 ðrÞ

 ai;j ðr ¼ 1 : q; i ¼ 1 : r; j ¼ 1 : rÞ: Auxiliary binary variable used to determine the amount of consumed enerðrÞ gies of AN. ai;j ¼ 1 means that SðiÞ sends its data to MS located at jth RPs at its rth movement.  l: l ¼ 1=F, where F is the fitness value of R-ODT. MILP formulation:

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393 394 395 396 397 398 399 400 401

 Hi,j (i = 1:m,j = 1:r): Euclidean distance (in meters) between SðiÞ and jth RP where r is the number of considered RPs in the convex area of ASN.  Hi;j (i = 1:r,j = 1:r): Euclidean distance (in meters) between ith and jth RPs.  A⁄: The two-dimensional binary array with the size of m  q limits the number of steps that ith active sensor node is allowed to send its data. A⁄ is configured based on the deadline of AN; thus, Ai;j ¼ 1 if j 6 si else Ai;j ¼ 0.

405 406 407

409

ð7Þ

subject to : M1;lMS ¼ 1 ðIÞ X Mi;j ¼ 1 8j ðIIÞ i¼1:r

ðMi;r þ M j;rþ1  1ÞHi;j 6 hV max X Ai;j ¼ dti =he 8i ðIVÞ

8i;j;ðr ¼ 1 : qÞ ðIIIÞ

j¼1:q 

8i;j ðVÞ

Ai;j 6 Ai;j

Variables:

ðrÞ

8i;j;r ðVIÞ

Hi;j ðMj;r þ Ai;r  1Þ 6 di;j

 M: The two-dimensional binary array with the size of r  (q + 1) provides optimal list of RPs. Mi,j = 1 implies that MS at jth movement is at ith RP.  A: The two-dimensional binary array with the size of m  q is the scheduling table of data transmission from AN to MS. Ai,j = 1 implies that ith active sensor node sends it data to MS at jth movement. ðrÞ  di;j ðr ¼ 1 : q; i ¼ 1 : m; j ¼ 1 : rÞ: shows the transmission range of ith active sensor node for sending data to MS located at jth RPs at its rth movement.

404

408

l

minimize

403

ðrÞ

Hi;j ðMj;rþ1 þ Ai;r  1Þ 6 di;j 8i;j;r X ðrÞ ðrÞ ai;k Hi;k ¼ di;j 8i;j;r ðVIIIÞ

ðVIIÞ

k¼1:r

X

ðrÞ

ai;j 6 1 8i;r

ðIXÞ

j¼1:r

XX

ðrÞ

ai;j f hða þ bH2i;j Þ 6 SeðiÞ l 8i ðXÞ

r¼1:qj¼1:r

variables : ðrÞ

ðrÞ

Mi;j ;Ai;j ;ai;k 2 f0;1g;di;j P 0; and l > 0;

8i;j;r;k

Fig. 5. The proposed structure of the decision tree.

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The objective function (7) maximizes the fitness value of R-ODT by decreasing l ¼ 1F. The first constraint sets the current location of MS to the nearest RP denoted by lMS in the matrix M. The two next constraints enforce MS to select one RP at each movement so that the maximum distance between two selected successive RPs is less than or equal to Vmax. The constraint (IV) guarantees SðiÞ ; i ¼ 1 : m, to visit MS for dti/he times to send its data (where dxe is the ceiling function). The valid steps for SðiÞ to transmit its buffered data to MS is distinguished by the constraint (V) with respect to si and h. The minimum transmission range of SðiÞ in jth step of MS movement is obtained regarding jth and (j + 1)th selected RPs by constraints (VI) and (VII). Through constraints (VIII) and (IX), ðrÞ the value of ai;j is determined so that SðiÞ consumes amount of its power which is proportional to the appropriate transmission range in each step. Minimizing l in the last constraint decreases the maximum total amount of consumed energy in AN during MS movement on R-ODT. Since the proposed MILP model is in the form of NPcomplete [28], we will present a tabu search based (TSbased) algorithm [25] as a meta-heuristic solution in the simulation part.

435

5. Proposed approximate algorithm

436

456

In this part, we propose an algorithm based on a decision tree and dynamic programming approach to approximately determine ODT for AN with m P 2. Before describing the structure of the decision tree, we consider an example of ODT from a different point of view. Consider two sets t = [2, 2, 2, 1] and s = [5, 4, 4, 2] (in seconds) for AN with m = 4 and assume that "j, hj = 1 s (see Fig. 4). Therefore, ODT can be achieved by solving two related sub-problems: (1) determining the optimal location points for the vertices of G (see Fig. 4(b) and (d)), and (2) specifying an optimal scheduling of data harvesting regarding the first sub-problem (see Fig. 4(a) and (c)). Among all candidate solutions for ODT, we should select a solution that maximizes the lifetime of AN. Using a decision tree structure we are going to explore an optimal solution for this decision: ‘‘whether ith member of AN sends its data to MS in the jth step or not?’’. However, the main challenge is how to construct the graph G when there are infinite candidate points for the vertices of G. Before finding an optimal solution for the main challenge, let us introduce the structure of the decision tree.

457

5.1. The structure of the proposed decision tree

458

A decision tree is a pictorial description of a welldefined decision problem [35]. To cast the problem of determining ODT into a decision tree, two assumptions should be taken into account1: (1) MS has an identical traveling time h at each step; and (2) throughout each step, MS can harvest data from i active sensor node(s) simulta The actual value of m  depends neously, where 0 6 i 6 m.  ¼ m in this on the hardware restrictions of MS but we set m

412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433

437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455

459 460 461 462 463 464 465

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F. Tashtarian et al. / Computer Networks xxx (2014) xxx–xxx

1

We note here that the described problem in the Theorem 1 remains in the form of NP-hard even by considering these two assumptions.

 may impact the maxwork. It is possible that the value of m imum time of data harvesting by MS. We note here that our  algorithm can work with any value of m. Consider a multi-stage decision tree with the limited number of arcs and decision-nodes in each stage (see Fig. 5). An arc connects two decision-nodes located at two consecutive stages. In fact, each arc represents a decision that creates a new decision-node (child decisionnode) which is originated from the current decision-node (parent decision-node). The child decision-node contains the updated attributes regarding the taken decision. The process of making decision starts from the root of the decision-node in the first (initial) stage of the system. The stages are indexed by j 2 {1, . . . ,k + 1} where k is the number of steps. Therefore, the jth stage in decision tree (j – k + 1) refers to the jth step of MS movement on ODT. We denote the ith decision-node in jth stage by ni;j which contains two sets ri;j and ei;j . The qth element of ri;j ðqÞ

ðqÞ

466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483

and ei;j , are represented by ri;j and ei;j , respectively, show

484

the remaining visiting time and consumed energy of SðqÞ . Furthermore, ni;j holds a location point pi;j as a candidate point for the jth vertex in G with the coordinates of   ðxÞ ðyÞ pi;j ; pi;j . Hence, the attributes of the root decision-node

485

ðqÞ

ðqÞ

n1;1 are r1;1 ¼ tq =h; e1;1 ¼ 0,"q and p1;1 ¼ V 1 (the initial location of MS). dj;ði;i0 Þ is defined as the decision connecting ni;j and ni0 ;jþ1 which is taken based on the attributes of ni;j and the deadlines of AN (see Fig. 5). The qth element of ðqÞ

dj;ði;i0 Þ , denoted by dj;ði;i0 Þ , is a binary value that shows the ðqÞ

transmission status of SðqÞ in jth step. dj;ði;i0 Þ can be assigned to zero if dsq =he 

ðqÞ drði;jÞ e

486 487 488 489 490 491 492 493 494

> 1 otherwise it should be set to 1.

495

If dj;ði;i0 Þ ¼ 1, SðqÞ has to send a portion of its data to MS

496

ðqÞ

which is equal to fh bits, otherwise, it postpones its transmission to the next step. By having dj;ði;i0 Þ , the two attributes ðqÞ

ðqÞ

ðqÞ

of ni0 ;jþ1 are obtained equal to ri0 ;jþ1 ¼ ri;j  dj;ði;i0 Þ ; 8q and   ðqÞ ðqÞ ðqÞ ei0 ;jþ1 ¼ dj;ði;i0 Þ Eq ni;j ; dj;ði;i0 Þ ; 8q. The function Eq calculates

497 498 499 500

the amount of energy consumption for sending data from qth AN to MS when MS is moving on E j from V j ðpi;j Þ to V jþ1 ðpi0 ;jþ1 Þ. As mentioned earlier, the main challenge in the proposed decision tree is how to determine the vertices of G so that the lifetime of AN is maximized. In fact, for   ðqÞ obtaining the appropriate value of Eq ni;j ; dj;ði;i0 Þ , the opti-

501

mal location point pi0 ;jþ1 is needed to be determined first. We note that the initial location of MS ðV 1 ¼ p1;1 Þ is known.

507

5.1.1. Determining the optimal vertices for G Our proposed strategy for solving this challenge is based on a convex mathematical model that by having pi0 ;j1 (j P 2) determines an optimal location pi;j for the child decision-node ni;j . Since producing child decisionnodes of ni;j are influenced by the single location point pi;j , so the procedure of determining pi;j should take into account all originated decisions from ni;j . In fact, the originated decisions from ni;j specify the average amount of data that should be sent by AN to MS. Therefore, the problem of finding pi;j can be defined as follows:

509

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522

‘‘Determine pi;j so that by moving MS on the straight line segment L connecting two points (pi0 ;j1 and pi;j ) and harvesting data, the fitness value of AN is maximized’’.

523

Theorem 2. The problem of finding pi;j is NP-hard.

520 521



ðxÞ pi;j ;

ðyÞ pi;j



525

as the coordinates of pi;j . A   ðxÞ ðyÞ mathematical model for determining pi;j ; pi;j to maxi-

526

mize the lifetime of AN can be written as follows:

524

527

Proof. Suppose

wise rotation of both L and AN around the coordinates of (0, a) with the angle of arctan(b) (see Fig. 6(b)), the optimal location point pi;j can be easily achievable in the second phase. In fact in this phase, we transform the proposed non-convex problem in Eq. (8) to a convex optimization model through redefining the constraints with respect to the rotated L and AN. The convex mathematical model is proposed as follows:

minimize

ð8Þ

ðqÞ

2 hq

532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547

548

R? ¼ 550 551 552 553 554

m X ðqÞ  2 C i;j d q ¼ q¼1

q¼1

a¼ 557 558

1þb

1

u

where u ¼ Pm

ðqÞ q¼1 C i;j

maximize F



and b ¼ B 

2

m X

ðqÞ

Sy

 /1

!2 ðqÞ ðqÞ

C i;j Sy

q¼1

2

1 /

m X q¼1

ðqÞ ðqÞ C i;j Sy



m m  2 X X ðqÞ ðqÞ ðqÞ ðqÞ C i;j Sx þ /1 C i;j Sx

q¼1

ðqÞ ðqÞ C i;j Sx

575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596

597

ð13Þ

ðqÞ

ðqÞ

F l ðC l;j hf ða þ bdl;q!MS ÞÞ 6 Sl;e 8q;l ðIÞ hl;q þ n2l;q 6 dl;q!MS 8q;l ðIIÞ     ðqÞ  X l  Rotated Sl;x  6 nl;q 8q;l ðIIIÞ n    o ðqÞ D Rotated Sl ;Rotated pi0 ;j1 6 dl;q!MS 8q;l ðIVÞ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi B2 þ 1, where

q¼1 m X

ðIVÞ

subject to :

ð10Þ

ðqÞ q¼1 C i;j

B¼



m X

!

!2

q¼1

F 6 F l 8l ðVÞ variables : :

ðqÞ ðqÞ ðqÞ C i;j Sy Sx

q¼1

561

ð11Þ

562

Fig. 6(a) shows a sample topology with six active sensor nodes and the obtained line L. Now, by having L and clock-

563

equal to l. Dfx; yg is the Euclidian distance between the two points of x and y. The location of X on L is determined so that the maximum required energy for transmitting ðqÞ C i;j hf amount of data is decreased (the first constraint). In other words, to achieve maximum of F, the maximum energy consumption of AN with respect to Ci,j should be decreased. Through the last three constraints, the optimal location of X and the transmission range of AN are interrelated. The last constraint restricts the minimum value of dq?MS to the distance between the rotated position of SðqÞ and the current location of MS which is equal to pi0 ;j1 . Finally, the anticlockwise rotation of the optimal value of X around the coordinates of (0, a) with the angle of arctan(b) is considered as pi;j . Fig. 6(c) and (d) shows two samðqÞ ples of determining pi;j for the different values of C i;j . We can promote the proposed convex model to obtain all optimal location points for all child decision-nodes of ni0 ;j1 . To do this, we define Fl as the fitness value of lth child decision-node of ni0 ;j1 . Hence, by increasing F in the following mathematical model, the minimum value of Fl is maximized through the last constraint:

q¼1

Pm

572

2

! m m X X ðqÞ ðqÞ ðqÞ ðqÞ C i;j Sy  b C i;j Sx ; q¼1

571

574

1

ð9Þ

:

2

8q

      ðqÞ where hq ¼ Rotated Sy  a and the fitness value F is

In the first phase, the optimal values of a and b are desired ? ? to minimize R\; therefore, by setting @R ¼ 0 and @R ¼0 @a @b and after a fair bit of algebra, the optimal values of a and b are:

555

559

 2 ðqÞ  a þ bSy

570

ðIIIÞ

l > 0; nq P 0; dq!MS P 0; and X is free

By setting F ¼ l1 , the above proposed model minimizes l ðqÞ with respect to hf ða þ bdq!MS Þ 6 Se l as the first constraint which is in the convex form. However, constraints II and III that are in the non-linear form, change the class of the problem to the NP-hard. h Since the above problem is NP-hard, we propose an approximate algorithm to determine pi;j which is comðqÞ prised of two phases. Let C i;j show the average amount of data that should be sent from SðqÞ to MS regarding all originated child decision-nodes from ni;j . Therefore, if sq/ ðqÞ ðqÞ ðqÞ h > (j  1) then C i;j ¼ rði;jÞ =ðdsq =he  ðj  1ÞÞ, else C i;j ¼ 0. Now in the first phase, by considering Ci,j, we determine a straight line L = a + bx that minimizes the sum of square perpendicular distance between AN and L. We then propose an optimization mathematical model to determine pi;j on L in the second phase. Let R\ be defined as the sum of square perpendicular distance between AN and L 2 . Thus, we have: denoted by d q ðqÞ Sy

569

variables :

ðyÞ



568

ðIÞ

DfRotatedðS Þ; Rotatedðpi0 ;j1 Þg 6 dq!MS

pi;j ;pi;j are free; dq!MS P 0; and F > 0

ðqÞ m C X i;j

8q

ðqÞ

variables :

531

567

n2q

þ 6 dq!MS 8q ðIIÞ     ðqÞ  X  Rotated Sx  6 nq 8q

Fhf ða þ bdq!MS Þ 6 SeðqÞ 8q ðIÞ  2  2 ðxÞ ðyÞ SðqÞ þ SyðqÞ  pi;j 6 dq!MS 8q ðIIÞ x  pi;j  2  2 ðxÞ ðyÞ SðqÞ þ SðqÞ 6 dq!MS 8q ðIIIÞ x  pi0 ;j1 y  pi0 ;j1

530

566

ð12Þ

C i;j hf ða þ bdq!MS Þ 6 lSeðqÞ

subject to :

ðxÞ

565

subject to :

maximize F

529

l

564

F;F l > 0;nl;q ;dl;q!MS P 0; and X l is free   ðqÞ and dl,q?MS are qth rotated active senwhere Rotated Sl sor and its transmission range in lth child decision-node, ðqÞ respectively. Moreover, Sl;e shows the residual energy of ðqÞ S in lth child decision-node of ni0 ;j1 . We convert the

Q1 Please cite this article in press as: F. Tashtarian et al., ODT: Optimal deadline-based trajectory for mobile sinks in WSN: A decision tree and dynamic programming approach, Comput. Netw. (2014), http://dx.doi.org/10.1016/j.comnet.2014.12.003

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644 604 605 606

607

mathematical model shown above into a convex model by introducing the new variables F ¼ l1 and F l ¼ @1 . Thus, we l have: minimize

l

ð14Þ

subject to : ðqÞ

ðqÞ

2 hl;q þ n2l;q

6 dl;q!MS 8q;l ðIIÞ     ðqÞ  X l  Rotated Sl;x  6 nl;q 8q;l ðIIIÞ   n  o ðqÞ D Rotated Sl ;Rotated pi0 ;j1 6 dl;q!MS 8q;l ðIVÞ

@l 6 l 8l ðVÞ variables :

610 611 612 613 614 615 616

617

@l ; l > 0;nl;q ;dl;q!MS P 0; and X l is free To extend our investigation, we can consider another metric to determine the optimal location point X in the proposed model (12) that has substantial effect on the time complexity of creating the decision tree. In this metric, the objective function Ë for determining pi;j is designed to minimize the sum of AN’s energy consumption through minimizing the sum of distances between rotated AN and X:    2  X ðqÞ  2 €¼ minimize E C i;j f h a þ b hi þ X  Rotated SðqÞ : x q¼1:m

619

ð15Þ

620

@E By setting h@X ¼ 0, the optimal value of X is obtained equal P ðqÞ ðqÞ P ðqÞ to q¼1:m C i;j RotatedðSx Þ= q¼1:m C i;j . The performances of

621 622

these two metrics will be evaluated in the simulation part.

623

5.2. Time complexity of creating the decision tree

624

The computational complexity of creating the decision tree depends on several parameters: the size of AN, the values of deadline and visiting time, and the time complexity of the proposed convex model in Eq. (14). In simulation part, the impact of m, si and ti on time complexity of creating the decision tree and consequently determining ODT will be investigated.

625 626 627 628 629 630 631 632 633

634 635

636

Theorem 3. The proposed convex model in Eq. (14) is in the form of second-order cone programming (SOCP) model and can be solved in polynomial time. Proof. A second-order cone program (SOCP) is the convex optimization problem [34] of the form: T

minimize f x

ð16Þ

Subject to :

2

2

ð17Þ

646

Now, through squaring roots of both positive sides, the constraint (II) can be written in the form of SOCP’s constraint:

647

  ðqÞ X l  Rotated Sl;x 6 nl;q   ðqÞ Rotated Sl;x  X l 6 nl;q

f 2 Rn ;Ai 2 Rni n ;bi 2 Rni ;ci 2 Rn ;di 2 R;F 2 Rpn ; and g 2 Rp

639 641

Since constraints (I), (IV), and (V) in the proposed model (Eq. (14)) are linear equations, we need to prove that the constrains (II) and (III) are in the form SOCP constraint.

642

By multiply both sides of the second constraint with

2 4hl;q

643

and then adding by dl;q!MS , we have:

648 649

650 652 653 654

655

8q; l ð19Þ

8q; l 

657

Like linear programming (LP) and quadratic programming (QP) problems, SOCP problems can be solved with great efficiency in polynomial time by interior point methods [34]. The computational effort per iteration required by these methods to solve SOCP problems is greater than that required to solve LP and QP which is related to the size of problem (the number of variables) [34]. Therefore, in this part, we propose a method to decrease the computational complexity of proposed model through decreasing the number of decision-nodes. Because by decreasing number of decision-nodes, the size of problem and consequently the computational complexity of creating decision tree are decreased.

658

The two decision-nodes ni;j and ni0 ;j in jth stage can be    ðqÞ ðqÞ ðqÞ ðqÞ  i;j if 8q; ri;j merged and appear as n  ri0 ;j ¼ 0; ei;j  ei0 ;j 

671

6 e and Dfpi;j ; pi0 ;j g 6 p , where e is a thresholds for the difference of consumed energies between corresponding AN in two decision-nodes, and p is defined as a neighborhood radius between two optimal location points for data i;j are p  i;j ¼ ðpi;j þ pi0 ;j Þ=2; harvesting. The attributes of n   ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ i;j ¼ ei;j þ ei0 ;j =2 and ri;j ¼ ri;j . The values of e 8q; e

673

and p should be selected so that growing decision tree i;j would not influence from the emerged decision-node n substantially on the quality of results. It is clear that when the values of e and p are selected smaller, the quality of results and time complexity of the proposed convex model are increased.

679

5.3. Exploring ODT from decision tree: a dynamic programming approach

685

In this part, we propose a recursive formulation of forward dynamic programming strategy [33] to explore ODT from the created decision tree as follows:

687

8i

Variables :

ð18Þ

and the third constraint can be casted into two linear constraints as follows:

f ðni;j Þ ¼ min 0

638

2

2

(

kAi x þ bi k2 6 cTi x þ di i ¼ 1;...;m Fx ¼ g

640

2

h i

2

2hl;q nl;q ; dl;q!MS  2hl;q 6 dl;q!MS ;

C l;j hf ða þ bdl;q!MS Þ 6 Sl;e @l 8q;l ðIÞ

609

4

4hl;q þ 4hl;q n2l;q þ dl;q!MS  4hl;q dl;q!MS 6 dl;q!MS

max

8q;i0 s:t 9dj1;ði0 ;iÞ

660 661 662 663 664 665 666 667 668 669 670

672 674 675 676 677 678 680 681 682 683 684

686

) n o ðqÞ ðqÞ 0 ei;j þ f ðni ;j1 Þ ; 8j

P3

659

688 689

690

ð20Þ

692

where for j ¼ 2; f ðni;2 Þ ¼ ei;2 . A topology for AN with m = 2 and the created decision tree with all attributes in three stage are shown in Fig. 7. ODT is obtained through executing the recursive procedure in Eq. (20) for j = 2–4 (the coordinates of AN and obtained vertices are shown in Fig. 7(b)). The optimal path on decision tree, which is marked in red

693

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Fig. 6. A line L in a zone with six active sensor nodes (a); clockwise rotation of L around the coordinates of (0, a) with the angle of (arctan(b)) (b); two ðqÞ samples of determining pi;j with different values of C i;j : Ci,j = [.5, .75, 0, 0, 0, .25] in (c) and Ci,j = [.5, 0, 0, 0, .5, .25] in (d).

699 700 701 702 703 704

with dashed lines, determines the optimal set of decisions: d1;ð1;1Þ ¼ ð0; 0Þ; d2;ð1;1Þ ¼ ð1; 0Þ, and d3;ð1;1Þ ¼ ð1; 1Þ. Therefore, MS has to harvest the AN’s buffered data when it’s moving from V 2 to V 3 and also from V 3 to V 4 . Fig. 7(b) shows the obtained ODT based on the decision tree in Fig. 7(a).

705

6. Simulation and performance evaluation

706

712

In this section, we evaluate the performance of the proposed model and compare our work with other approaches like the proposed rendezvous-based algorithm (R-ODT proposed in Part IV) and random mobility algorithm. To do this, we first present a tabu search based (TS-based) algorithm [25] as a meta-heuristic solution for the proposed MILP model (7).

713

6.1. Solving MILP model (7): a TS-based approach

714

A tabu search (TS) is a local search strategy with a flexible memory structure and consists of four phases [26]: (1) generating an initial solution; (2) exploring the neighborhood N ðSÞ for a solution S; (3) designing the fitness function and appropriate move, and (4) defining a tabu list and an aspiration criteria. The algorithm ends when one of two thresholds q1 or q2 has been reached, where q1 is defined as a maximal number of allowed iteration and q2 is defined as a maximal number of iterations that can occur while the best solution is not promoted. Let S be a two-dimensional integer array with the size of (m + 1)  (q + 1) considered as a solution’s structure (m = size of AN). The first m rows of S determines in which steps, active sensor nodes send their data and the last row that includes the set of randomly selected RPs. Fig. 8 shows the pseudo code with time complexity of O(m + q) to initialize a solution. The function NexRP in line 9 opts the next RP based on selected direction from the last selected RP in line 8. If there is no valid RP (for example, there is no left RP for the leftmost RP in the convex area of AN), it will return the last selected RP again. The distance between two adjacent RPs on a vertical or horizontal line is set to h Vmax. Proposing an algorithm to explore the neighborhood N ðSÞ of a solution S has a direct impact on the execution time of the algorithm. We propose a method that includes

707 708 709 710 711

715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738

two simple strategies to obtain a neighbor of N ðSÞ and they are selected based on the third threshold value 0 < q3 < 1. In the first strategy, we replace values of two cells j and k of ith row where i,j, and k are selected randomly (1 6 i 6 m and 1 6 j,k 6 si/h). In the second strategy, the last l RPs in the last row of S will be changed where l is a random number between 2 and q + 1. The fitness value of each solution can be easily obtained by (1). Let D be a tabu list used in the proposed algorithm as a three-dimensional data structure array with size of (i = 1:r, j = 1:r, k = 1: m⁄q) to record forbidden moves that is a salient feature of TS. The main concept of the tabu list is to keep track of the solutions that have been considered in the past [26]. The tabu list D will be updated by the changed cells’ value of the best neighbor of S in each execution round of algorithm. Therefore, the changed items of the best neighborhood set Di;j;k to tabu tenure that is the duration of keeping a record in the tabu list [26]. In fact, tabu tenure determines the boundary of the search area, thus long tabu tenure lets TS explore unvisited territory. This phenomenon is named by diversification [26]. Although the tabu list helps TS to prevent the search from the local optima, it could drastically restrict the neighborhood N ðSÞ. Moreover, it is possible that some attractive solutions will be missed. The Aspiration criterion lets TS override the tabu list restrictions. Consequently, it allows the superior solution S  to be accepted despite the all changed columns of S  that have been recorded in D. The effect of this criterion on TS is known as intensification [26].

739

6.2. Performance evaluation

768

In this part, we evaluate the performance of the proposed algorithm in comparison with the proposed MILP model (R-ODT) and random mobility algorithm inspired by [8] where MS can move randomly in the network field with variable velocity during the given deadline h. In this simulation the three following metrics are taken to account: the sum and maximum of energy consumption that is related to the fitness value of ODT2 (see Eq. (1)) and time complexity of the algorithm which is defined as

769

2 Minimizing the maximum energy consumption increases the fitness value of ODT when AN has the same power.

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Fig. 7. ODT for AN with m = 2: the decision tree (a), the obtained ODT(b).

Fig. 8. The algorithm of initialize a solution (pseudo code).

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the amount of time taken by the algorithm to return a solution. Therefore, we evaluate the performances of the random mobility algorithm, the proposed MILP model (R-ODT), and the approximate algorithm with two objective functions: Maximizing LifeTime (MLT) of AN through the proposed model in Eq. (14) and Minimizing total Energy Consumption (MEC) (Eq. (15)) in different scenarios which are written in MATLAB [27] by using MOSEK optimization package [36]. The simulations are run on a computer with Intel core i7– 3540 M, 3.00 GHz CPU and 8 GB memory. The network field (with single zone) is considered as a circle area with radius of 100 m, the initial power of sensor nodes are set to 2 J, and the other parameters are a = 50 nJ/bit, b = 10 pJ/bit/m2, f = 250 Kbps, Vmax = 15 m/s, and size of beacon packet = 10 bytes. In our simulation, we considered the energy consumption of sending and receiving all control packets (e.g. beacon packets) between AN and MS. We note that sending beacon packets imposes less energy consumption in comparison with transmitting data packet due to small size of beacon packets. Before comparing the performance of the algorithms, 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 proposed tabu search for AN with sizes 2–10 (randomly scattered in the network). Moreover, the average visiting time and deadline values of AN are

considered 2.5 and 3.5 in second, respectively. The thresholds of the TS-based approach are: q1 = 200, q2 = 1000, and q3 = 0.5. By this setting, we run the simulation for 20 times and measure the average of the fitness value (see Fig. 9). Since the obtained confidence band is narrow, it is not reported. As shown in Fig. 9, 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 m. In the first experiment, we compare the time complexity, the maximum and the sum of energy consumption and show the results in Figs. 10–12, respectively. In this experiment, we consider AN with the variable size between [2, 10]; in addition, the average of visiting time and deadline values are set to 3 and 5 s, respectively. The measured values are the average of 20 times of execution for each size of AN. Fig. 10 compares the time complexity values of the proposed algorithms and demonstrates that by increasing the size of AN, the time complexities of MLT, TS-based are increased, drastically. As shown in Fig. 10, the time complexity values of TS-based method for the different number of rendezvous points (q) are the same and elevate gradually by increasing the size of AN. Although the time complexity of TS-based does not strictly depend on the number of rendezvous points, the impact of q on the quality of solutions can be seen obviously in Figs. 11 and 12. By comparing the results of R-ODT and ODT, we can conclude that considering rendezvous points or some predefined virtual structures cannot increase the quality of produced trajectory specially in terms of energy consumption. Although the time complexity of MEC is almost steadystate (Fig. 10), the trend of MLT is increased with a sharp slop specially for m > 4. However, by comparing the obtained results in Fig. 11, we can see that determining ODT with the objective function MLT achieves the maximum lifetime by minimizing the maximum of energy consumption in comparison with other algorithms (we note here that all active sensor nodes have the same initial

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Fig. 12. The impact of size of AN on the sum of energy consumption. Fig. 9. The impact of tabu tenure on the maximum of fitness value.

Time complexity (sec.)

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Fig. 10. Measuring the time complexity of the proposed algorithm for different values of m.

The maximum of energy consumption (joule)

1.8 MLT MEC TS-based (rho=350) TS-based (rho=250) TS-based (rho=100) Random mobility(#10) Random mobility(#8) Random mobility(#5)

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Fig. 11. The impact of size of AN on the maximum of energy consumption.

mobility algorithm, we split the trajectory of MS into a different number of hop counts (steps) denoted by D (D = 5, 8, 10 in simulation). Hence, MS moves toward a random direction at a random velocity V (0 6 V 6 Vmax) during h, where h is the traveling time of MS on each movement which is set to 1 s. As shown in Figs. 10–12, the only advantage of the random mobility algorithm is the lowest time complexity while it causes more energy consumption in comparison with three other algorithms. Since the small values of D decrease the flexibility of MS’s movement, it is possible that MS increases the energy consumption of AN (see Figs. 11 and 12). The results of the second experiment are shown in Figs. 13–15. In this simulation, we investigate the influence of increasing deadline on the three metrics in both objective functions: MLT and MEC. We consider three scenarios based on the different sizes of AN, visiting time and deadline: Scenario (1), m = 2, "i, ti = 2 and si 2 {2, 3, 4, 5}; scenario (2), m = 3, "i, ti = 3 and si 2 {3,4,5}; and scenario (3), m = 4, "i, ti = 4,si 2 {4, 5}. In general, by increasing deadline, the sum and the maximum of energy consumption are decreased gradually in three scenarios (see Figs. 13 and 14). This is because, by increasing si, MS has more time to close to AN and harvests their data. However, due to

The sum of energy consumption (Joule)

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0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 2

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power). Fig. 12 shows the average of sum of energy consumption. In this metric, as we expected, MEC has a better performance in almost all the cases of m. For the random

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Fig. 13. The impact of different values of s and m on the sum of energy consumption.

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Table 2 The impact of increasing deadline in a topology with m = 5 (MLT algorithm).

MLT (m=2) MLT (m=3) MLT (m=4) MEC (m=2) MEC (m=3) MEC (m=4)

0.14

0.12

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Sum en. (J)

Max en. (J)

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Decisionnodes#

2,2,2,2,2 3,2,2,2,2 3,3,2,2,2 3,3,3,2,2 3,3,3,3,2 3,3,3,3,3 4,3,3,3,3 4,4,3,3,3 4,4,4,3,3 4,4,4,4,3 4,4,4,4,4

.244 .264 .263 .233 .230 .222 .222 .206 .189 .179 .179

.075 .075 .075 .063 .063 .063 .063 .053 .045 .044 .038

.24 .25 1.18 2.21 4.11 7.71 7.86 22.3 36.9 57.4 84.03

3 6 10 22 40 80 112 191 280 350 530

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Deadline (sec.)

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The time complexity (sec.)

10

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MLT (m=2) MLT (m=3) MLT (m=4) MEC (m=2) MEC (m=3) MEC (m=4)

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Fig. 15. The impact of different values of s and m on the time complexity of the proposed algorithm.

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the time complexity of MLT for determining P in each decision-node, the overall time complexity is increased drastically, while it is increased gradually in MEC (Fig. 15). In fact, by selecting a greater deadline value for AN, the size of decision trees are increased in both algorithms. In the third experiment, we are going to scrutinize the performance of MLT when the deadline of just one active sensor node will be increased. Let us define another scenario with m = 5, ti = 2 s, "i. By considering a constant topology of five active sensor nodes, besides measuring the sum and the maximum of energy consumption and time complexity, we measure the number of decision-nodes in the constructed decision tree (see Table 2). Table 2 shows that by increasing deadline even in the sensor level (see the first column), the sum of energy consumption is gradually decreased. However, the maximum of energy consumption is remained steady when the deadlines of some active sensor nodes are increased (e.g., the first three rows and rows 4–8). The main reason of this phenomenon lies in the geographical locations of active sensor nodes and the given deadlines. Let us show the two scenarios of the third and

fourth rows of Table 2 in Fig. 16(a) and (b), respectively. As it is shown in Fig. 16, the farthest node to MS is Sð3Þ , and when s3 = 2 (Fig. 16(a)), MS at the first step, moves toward Sð3Þ to reduce the distance for the next transmission, but in the second step, MS does not move. This is because even by moving toward Sð3Þ , the minimum length of transmission range of Sð3Þ is obtained equal to the distance between MS in location V 2 and Sð3Þ . However, in Fig. 16(b) when s3 = 3, Sð3Þ can postpone it transmission to the last two steps of MS movement; therefore, MS at the first two steps, can move toward Sð3Þ to reduce d3?MS, efficiently. Since in the proposed structure of decision tree, the number of decision nodes originated from ni;j is related to the deadlines of AN and the number of remained steps, by increasing the deadline of at least one active sensor node the number of originated decision node and consequently the time complexity of determining ODT are increased (see the fourth and last columns of Table 2). As mentioned earlier, merging two decision-nodes with identical remaining visiting time decreases the number of decision-nodes in the next stage and consequently reduces the time complexity of creating the decision tree. Two decision-nodes ni;j and ni0 ;j will be merged if the length of distance between pi;j and pi0 ;j is less than or equal to the threshold p and the differences between all AN’s residual energies are less than or equal to the threshold e . In the next experiment, we focus on the impact of p on the quality of ODT and furthermore the time complexity and the number of generated decision-nodes while the value of e is remained constant. The results of simulation for a topology with five active sensor nodes (ti = 2, si = 4, "i) are shown in Table 3. As it is expected, by increasing the threshold p , the time complexity of MLT is decreased but the sum and the maximum of energy consumption are increased. Because, an optimal location P for merged decision node is estimated as a middle location between the two Ps of decision-nodes. For the considered topology, Table 3 shows that reducing 26% of time complexity decreases up to 10% of lifetime. In the last experiment, the impact of different values of h on the three algorithms are evaluated. We setup three scenarios based on different values of visiting time and

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Fig. 16. The two topologies of the third (a) and fourth (b) rows of Table 3.

Table 3 The impact of p in a topology with m = 5 (MLT algorithm).

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Sum en. (J)

Max en. (J)

Time com. (s)

Decision-nodes#

5 10 15 20 25 30 35 40 45

.1788 .1789 .1789 .1789 .1654 .1919 .1884 .1884 .1884

.0382 .0384 .0384 .0384 .0398 .0458 .0462 .0462 .0462

84.03 81.03 79.15 79.07 76.70 73.4 63.3 63.3 62.3

530 463 439 436 411 315 312 311 311

deadline in a topology with five active sensor nodes: scenario (1), ti = 1 and si = 2; scenario (2), ti = 2 and si = 4, and scenario (3), ti = 3 and si = 6, "i. Figs. 17 and 18 illustrate the influence of different values of h on the four

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Tabu search (sum of energy cons.) MEC ( sum of energy cons.) MLT ( sum of energy cons.) MEC (max energy cons.) MLT(max energy cons.) Tabu search (max energy cons.)

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Energy (Joule)

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metrics: the sum and maximum energy consumption (Fig. 17), the time complexity and the number of generated decision-nodes (Fig. 18). The results are separated by a rectangle for each scenario. As shown in the previous experiments, by increasing visiting time, the sum and the maximum energy consumption of AN are increased; moreover, increasing the value of h in each scenario results in increasing the amount of energy consumption (Fig. 17). Since it is assumed that AN cannot change their transmission range during h, considering larger value of h leads to the inefficient energy consumption of AN. However, the only point of increasing h can be observed in the time complexity of algorithms. As it is shown in Fig. 18, by increasing the value of h in each scenario the number of considered decision-nodes and consequently the computational complexity of algorithms are decreased especially in the third scenario.

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Fig. 17. The impact of different values of h and s on the sum and the maximum of energy consumption.

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In this paper, we studied how to determine a trajectory for a mobile sink without considering any predefined rendezvous points or virtual structures in the network. The trajectory, referred to as optimal deadline-based trajectory (ODT), yields a significant gain in the network lifetime. We addressed deadline based and event-driven applications, where MS must harvest data from a group of sensors called active nodes group (AN) that capture an event. ODT was defined as an undirected graph with k edges and k + 1 vertices with degree 6 2. We showed that the problem of determining ODT is NP-hard when the size of AN P 2. Hence, we introduced a new approximate approach to achieve ODT. Our solution methodology, which is based on a decision tree and dynamic programming model, determined ODT for any arbitrary topology with the various values of visiting time and deadline. In fact, the structure of the decision tree covered the feasible solution space and ODT then obtained by running a recursive formulation of the forward dynamic programming on the decision tree. The optimal achieved path of the decision tree determined the vertices of the ODT graph and moreover the all transmission statuses of AN at each step. It was showed that the main challenge in creating decision tree is the estimation of the optimal location point in each decision-node. Therefore, we proposed an approximate convex model which is in the form of second order cone program (SOCP) and can be solved in polynomial time. For comprehensive investigation, we proposed a mixed integer programming (MILP) model to determine an optimal rendezvous-based trajectory by considering a limited number of rendezvous points in the network. The effectiveness of our approach is validated via the extensive number of simulation runs and comparison with MILP model and a random mobility algorithm.

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Two interesting directions can be pointed out as suggestion for future work. Relaxing the assumptions of the MAC-layer and then proposing an optimal method to determine ODT for MS with multi-hop data harvesting is the first direction. The second direction is to design a distributed algorithm to achieve ODT.

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Farzad Tashtarian (M’14) 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 Ferdowsi university of Mashhad, Mashhad, Iran. His research interests include wireless sensor networks, mobile communications, mathematical modeling, optimization, and distributed control.

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Mohammad Hossien Yaghmaee (M’08, SM’12) 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.

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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.

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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.

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Q1 Please cite this article in press as: F. Tashtarian et al., ODT: Optimal deadline-based trajectory for mobile sinks in WSN: A decision tree and dynamic programming approach, Comput. Netw. (2014), http://dx.doi.org/10.1016/j.comnet.2014.12.003