The dynamic routing algorithm for renewable wireless sensor networks with wireless power transfer

Computer Networks 74 (2014) 34–52

Contents lists available at ScienceDirect

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

The dynamic routing algorithm for renewable wireless sensor networks with wireless power transfer Lei Shi a,⇑, Jianghong Han a, Dong Han b, Xu Ding a, Zhenchun Wei a a b

School of Computer and Information, Hefei University of Technology, Hefei, Anhui, China Department of Computer Science, University of Houston, Houston, TX, USA

a r t i c l e

i n f o

Article history: Received 6 January 2014 Received in revised form 30 July 2014 Accepted 31 August 2014 Available online 15 September 2014 Keywords: Wireless sensor network Wireless power transfer Optimization

a b s t r a c t Wireless power transfer is recently considered as a potential approach to remove the lifetime performance bottleneck for wireless sensor networks. By using a wireless charging vehicle (WCV) to periodically recharge each sensor node’s battery, a wireless sensor network may remain operational forever. In this paper, we aim to jointly optimize a dynamic multi-hop data routing, a traveling path (for the WCV to visit all the sensor nodes in a cycle), and a charging schedule (charging time for each sensor node) such that the ratio of the WCV’s vacation time over the cycle time can be maximized. The key challenge of this problem (caused by time-varying data routing) is the integration and differentiation terms in problem formulation, which yields a very challenging non-polynomial program. To remove these non-polynomial terms, we introduce the concept of ðN þ 1Þ-phase solution, which adopt a special dynamic routing scheme. We prove that an optimal ðN þ 1Þ-phase solution can achieve the same objective value as that by an optimal time-varying solution. We further prove that the optimal traveling path must follow the shortest Hamiltonian cycle. Finally, we linearize the problem for data routing and charging schedule and thus obtain an optimal solution in polynomial-time. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction Due to limited battery energy at each sensor node, how to prolong the network lifetime of a wireless sensor network (WSN) is a hot research field and has been extensively studied over the last decade. A simple approach is to replace batteries when they are used up. But this approach could be too expensive for a WSN with many nodes. Some efforts are based on designing energy efficient schemes (e.g., [1–5]). However, even with the most energy efficient scheme, batteries will eventually be used up. Another approach is applying energy-harvesting techniques such as solar cell (e.g., [6,7, Chapter 9] [8–11]). By

⇑ Corresponding author. Tel.: +86 551 62901397. E-mail address: [email protected] (L. Shi). http://dx.doi.org/10.1016/j.comnet.2014.08.020 1389-1286/Ó 2014 Elsevier B.V. All rights reserved.

energy-harvesting, batteries are possible to have unlimited power and thus the WSN may have the potential of unlimited network lifetime. However, the harvested energy depends on the environment, and thus the environment may not provide enough energy during certain period of time. Moreover, the size of an energy-harvesting device may be too large to fit in a small sensor node. Recently, a new technique named wireless power transfer (WPT) provides another approach to remove the lifetime bottleneck for WSNs. This technique is based on a recent breakthrough by Kurs et al. [12], where a practical magnetic resonance-based WPT technique was designed. Kurs et al. showed that a 60 W light bulb at a distance of two meters away can be fully powered by their technique. They have launched a start-up company called Witricity Corp., and enhanced this technique to transfer energy to multiple receiving coils at the same time [13].

35

L. Shi et al. / Computer Networks 74 (2014) 34–52

WPT has aroused many scientists’ interests and many recent advances have been made, because it has many good properties: high power transfer efficiency, no lineof-sight requirement between the power transmitting and receiving devices, no need to move to a sensor node’s location to charge its battery, no interference to wireless communication, immune to the neighboring environment, small size of energy transfer device which made it possible to be equipped on a sensor node, and it is safe for human, and so on. For example, WPT has been applied in energyconstrained devices, e.g., [14, Chapter 4] [15–18], and some portable devices such as cell phones [19]. WPT is also a good solution to overcome energy constraint in WSNs by charging sensor nodes wirelessly. The Witricity Corp. has suitable small sized and light weighted products [38] that can be equipped on a wireless charging vehicle (WCV) to perform WPT. Li et al. proposed a charging-aware routing protocol (J-RoC) by incorporating nodal dynamic energy consumption rates into the design of data routing protocols [20]. Shi et al. [21] conducted theoretical studies on efficient usage of WPT in WSNs, and show the lifetime performance bottleneck may be removed by a WCV. Although some works have been done in WSNs, there are still many other problems that can be studied. In this paper, we propose a dynamic multi-hop data routing and charging schedule way when a WCV is charging in a WSN. In particular, we consider how to maximize the ratio of the vacation time (at the WCV’s home service station) over the cycle time by jointly optimizing a dynamic (time-varying) multi-hop data routing, the WCV’s traveling path, and a charging schedule (charging time at each sensor node). Our main contributions can be summarized as follows.  The key challenge for dynamic routing is that many integration and differentiation terms appear in the problem formulation, which yield a very complicate non-polynomial program. To remove these non-polynomial terms, we introduce the concept of ðN þ 1Þ-phase solution, which has a special dynamic data routing scheme (see Definition 2) and prove that an optimal ðN þ 1Þ-phase solution has the same objective value as that by an optimal time-varying solution. Thus, we successfully remove all the integration and differentiation terms from the original optimization problem without any performance loss.  For the objective of maximizing the ratio of the vacation time over the cycle time, we prove that the optimal WCV’s traveling path must follow the shortest Hamiltonian cycle. For the remaining optimization problem of data routing and charging schedule, we apply linearization techniques to reformulate it as a linear programming (LP), which can be solved optimally in polynomial-time. Our numerical results demonstrate that the optimal objective value for the dynamic routing case is indeed significantly larger than that for the static routing case. The WCV should be distinguished from a data mule [24] or a data ferry [25]. A data mule/ferry visits each node to collect data and visits the base station to deliver data. It

reduces energy consumption at sensor nodes but causes large delay due to physical movement before data delivery. A data mule/ferry cannot charge sensor nodes’ batteries. The WCV considered in this paper is designed for wireless charging, which does not participate in data routing and thus does not add any delay. Note that we focus on applications with strict delay requirement, e.g., monitoring applications, and thus all relays should forward data once received. Then the WCV cannot be a data mule simultaneously. The remainder of this paper is organized as follows. Section 2 describes how to employ WPT in a WSN and its problem formulation. Section 3 proves that we only need to consider ðN þ 1Þ-phase solutions to achieve the same optimal objective value. In Section 4, we first identify the optimal traveling path for the WCV, then formulate the remaining optimization problem for data routing and charging schedule as an LP. Section 5 shows how to construct an initialization cycle. In Section 6, we use numerical results to demonstrate our optimal solutions and discuss the benefits of dynamic data routing. Section 7 reviews related work and Section 8 concludes this paper. 2. WSNs with A WCV 2.1. Problem description We consider a set N of sensor nodes distributed over a two-dimensional area (see Fig. 1). In reality, the environment may be a three-dimensional area. Our model can be extended to the three-dimensional area since it only requires an extra parameter to identify the coordinates of nodes. We assume that each sensor node i 2 N senses the environment and generates data with a rate of Ri . It

Service station

WCV

Base station

Sensor node

Fig. 1. A WSN with a WCV.

36

L. Shi et al. / Computer Networks 74 (2014) 34–52

Table 1 Notation. ai B C ij (or C iB ) Dij DP DTSP Emax Emin ei ðtÞ m m f ij (or f iB )

Arrival time of the WCV at sensor node i in the first renewable cycle The base station The energy consumption coefficient for transmitting data from sensor node i to sensor node j (or the base station B) Distance between nodes i and j The traveling distance for a path P The minimal traveling distance in a cycle Battery capacity of sensor nodes Minimal energy to keep a sensor node operational Sensor node i’s energy at time t Flow rate from sensor node i to sensor node j (or the base station B) in phase m

g ij ðtÞ (or g iB ðtÞ)

Flow rate from sensor node i to sensor node j (or the base station B) at time t The set of sensor nodes in a WSN The number of sensor nodes in a WSN The traveling path of the WCV The set of all possible traveling paths of the WCV Energy consumption rate of sensor node i at time t Energy consumption rate of sensor node i in phase m The total data volume in phase m from sensor node i to sensor node j (or the base station B)

N N P P pi ðtÞ pm i m qm ij (or qiB ) Ri S Tm U ui V

Data rate generated at sensor node i The service station The set of all time instances in phase m Full charging rate of the WCV Charging rate for sensor node i in the initialization cycle Moving speed of the WCV The energy consumption coefficient for receiving data The ratio of the time length in phase m to the entire cycle time The ratio of the vacation time to the entire cycle time The i-th sensor node visited by the WCV along a path P Overall time spent in a cycle Time spent by the WCV to charge sensor node i’s battery Time length when the WCV does not charge any sensor node’s battery Vacation time in a cycle Traveling time of the WCV for a path P Minimal traveling time of the WCV in a cycle

q gm gvac

pi s si s0 svac sP sTSP

then transmits data (via either a single hop or multiple hops) to the base station B. We assume that the exact positions of all sensor nodes and the base station are known. Table 1 lists the notation used in this paper. We consider a dynamic (time-varying) multi-hop routing scheme. Denote g ij ðtÞ and g iB ðtÞ as flow rates at time t from sensor node i to sensor node j and to the base station B, respectively. We have the following flow balance constraint. j–i k–i X X g ki ðtÞ þ Ri ¼ g ij ðtÞ þ g iB ðtÞ ði 2 N ; t P 0Þ

ð1Þ

j2N

k2N

Each sensor node’s battery has a capacity Emax and is fully charged initially. We consider energy consumption for data transmission and reception. Denote pi ðtÞ as the energy consumption rate of sensor node i at time t. In this paper, we use the following energy consumption model [1,3]: j–i k–i X X pi ðtÞ ¼ q g ki ðtÞ þ C ij g ij ðtÞ þ C iB g iB ðtÞ ði 2 N ; t P 0Þ; k2N

j2N

ð2Þ where q is the energy consumption for receiving a unit of data, C ij (or C iB ) is the energy consumption for transmitting a unit of data from sensor node i to sensor node j (or to the P base station B). In this model, q k–i k2N g ki ðtÞ is the energy

P consumption rate for reception and j–i j2N C ij g ij ðtÞ þ C iB g iB ðtÞ is the energy consumption rate for transmission. We assume that once a sensor node’s energy is less than a threshold Emin , this node cannot work any more. We employ WPT [12] to extend the sensor network lifetime (the time till the energy level of any sensor node in the network falls below Emin ).1 A WCV can recharge each sensor node’s battery periodically. It starts from a service station S, visits and charges each sensor, and returns the service station (see Fig. 2). We call such a trip a renewable cycle. We assume that there is no limitation on WCV’s moving path and only consider one WCV for charging in this paper. In reality, the WCV can be a car driven by someone following the planned route using a satellite navigation system, so that if she sees obstacles she avoids them. The Witricity Corp. has suitable small sized and light weighted products [38] that can be equipped on a WCV to perform WPT. For example, WiT3300 has a size of 20 cm  28 cm  7 cm and a weight 3.6 kg, A WCV can sustain such devices and batteries easily. We do not consider the energy required for moving the vehicle because it is not the bottleneck. We assume that the WCV has enough energy to charge all the sensor nodes in a renewable cycle. Denote V as the speed of the WCV and U as the full charging rate. When the WCV arrives at a sensor node i, it spends a time of si to

1 The network lifetime defined in [1,4,5] is a special case of our lifetime definition, i.e., the case of Emin ¼ 0.

37

L. Shi et al. / Computer Networks 74 (2014) 34–52

Service station

WCV

scheme, which is a special case of the dynamic routing scheme considered in this paper. As a result, we expect a better performance by our scheme but the problem is much harder. Numerical results and more discussions on dynamic routing are given in Section 6. 2.2. Problem formulation

Base station

Sensor node

Fig. 2. A sample traveling path for the WCV.

charge node i’s battery wirelessly via non-radiative power transfer [12]. After the WCV visits all the sensor nodes in the WSN, it will return to its service station to be serviced (e.g., replacing or recharging its battery). We call this period vacation time, denoted as svac . After this vacation, the WCV will go out for its next trip. Denote s as the overall time spent during such a renewable cycle. Ideally, we want to charge each sensor node’s battery periodically such that its energy never falls below Emin . Thus, the sensor network will never cease to be operational. Then the traditional lifetime performance bottleneck may be removed and we can move on to some other useful objectives. In this paper, we aim to maximize the percentage of the time that the WCV can take vacation in a cycle (i.e., svac s ) [21], or equivalently, to minimize the percentage of time that the WCV is out in the field. We consider how to optimize a dynamic multi-hop routing scheme for data transmission, a traveling path for the WCV to visit all the sensor nodes, and a charging schedule to charge all the sensor nodes in the WSN. Note that in [21], Shi et al. considered a static multi-hop routing

We focus on modeling the first renewable cycle, since subsequent renewable cycles can follow the same charging schedule and routing. We note that there is an initialization cycle before the first renewable cycle (the grey area in Fig. 3), which can also be constructed after we obtain a solution for the first renewable cycle (see Section 5). Therefore, we only need to solve one optimization problem to obtain a solution for the first cycle. Based on this solution, we can derive solutions for all other cycles. Fig. 3 shows the energy level of a sensor node i during the first two renewable cycles. We formally define a renewable cycle as follows. Definition 1. A renewable cycle with a length s should meet the following three requirements: (i) each sensor node’s energy starts and ends at the same level over a period of time s; (ii) each sensor node’s energy never falls below Emin ; and (iii) each sensor node’s energy never exceeds Emax . We now formulate constraints for the first renewable cycle during time ½s; 2s. Denote ei ðtÞ as the energy level at time t. The first requirement in Definition 1 is R 2s ei ð2sÞ ¼ ei ðsÞ. Since ei ð2sÞ ¼ ei ðsÞ  s pi ðtÞdt þ U si , we have the following energy balance constraint.

Z 2s s

pi ðtÞdt ¼ U si

ði 2 N Þ;

ð3Þ

i.e., the total amount of charged energy in a renewable cycle must be equal to the total amount of energy consumed in this cycle. Denote ai as the arrival time of the WCV at a sensor node i in the first renewable cycle. Then during ½ai ; ai þ si , the WCV is charging sensor node i with a charging rate U while during ½s; ai Þ or ðai þ si ; 2s, the WCV is not charging sensor node i. We have the following relationship between pi ðtÞ and ei ðtÞ.

@ þ ei ðtÞ ¼ @t



U  pi ðtÞ if t 2 ½ai ;ai þ si Þ ði 2 N Þ if t 2 ½s;ai Þ or ½ai þ si ; 2s pi ðtÞ

Fig. 3. Sensor node i’s energy ei ðtÞ during the first two renewable cycles.

ð4Þ

38

L. Shi et al. / Computer Networks 74 (2014) 34–52

During a renewable cycle (see Fig. 3), sensor node i’s energy level is decreasing when the WCV is not at node i (i.e., non-charging period). We have ei ðai Þ 6 ei ðtÞ 6 ei ðai þ si Þ for any t 2 ½s; ai  or ½ai þ si ; 2s. Thus, both the minimal and maximal energy levels occur at some time t 2 ½ai ; ai þ si .2 Therefore, for the second and third requirements in Definition 1, we only need to ensure that

ei ðtÞ P Emin

ði 2 N ; t 2 ½ai ; ai þ si Þ

ð5Þ

ei ðtÞ 6 Emax

ði 2 N ; t 2 ½ai ; ai þ si Þ:

ð6Þ

Denote P ¼ ðp0 ; p1 ; . . . ; pN ; p0 Þ as the traveling path by the WCV during a renewable cycle, where N ¼ jN j is the number of sensor nodes. The WCV starts from and ends at the service station (i.e., p0 ¼ S) and the i-th sensor node visited by the WCV along path P is pi , 1 6 i 6 N. Denote Dij as the distance between nodes i and j, DP as the total distance of path P, and sP as the time used for traveling over distance DP . We have

DP ¼

N1 X Dpk pkþ1 þ DpN p0

ð7Þ

k¼0

DP : V X s ¼ sP þ svac þ si :

sP ¼

ð8Þ ð9Þ

i2N

Note that power transfer efficiency decreases rapidly with distance. Even the state of the art wireless charging technique [12] can only charge a node within a short distance, e.g., two meters. Thus, in (7), we use Dpk pkþ1 to approximate the traveling distance between the charging locations for nodes pk and pkþ1 . For the i-th visited sensor node pi , we have

api ¼ s þ

i1 i1 X Dpk pkþ1 X þ spk V k¼0 k¼1

ð1 6 i 6 NÞ:

ð10Þ

Denote P as the set of all possible traveling paths for the WCV. Our problem can be formulated as follows.

OPT max

svac s

s:t: constraints ð1Þ—ð10Þ g ij ðtÞ; g iB ðtÞ P 0 ði; j 2 N ; j – i; s 6 t 6 2sÞ pi ðtÞ P 0; Emin 6 ei ðtÞ 6 Emax

ði 2 N ; s 6 t 6 2sÞ

P 2 P; DP ; sP ; s; svac ; si ; ai P 0 ði 2 N ; s 6 t 6 2sÞ; where flow rates g ij ðtÞ; g iB ðtÞ, energy consumption rate pi ðtÞ, energy level ei ðtÞ, traveling path P, distance DP , time sP , time intervals s; svac , and si , and arrival time ai are variables, and all the other parameters are constants. The following property shows the existence of ‘‘bottleneck’’ node in the WSN. 2

We consider the case that charging speed is fixed but energy consumption is dynamic, due to dynamic routing (or dynamic traffic load). As a result, it is possible that at the beginning of the charging period (right after time ai ), energy consumption rate is larger than charging speed. Then the minimal energy level is not at time ai , but at some time later than ai . For a similar reason, the maximal energy level may be at some time earlier than a i þ si .

Property 1. In an optimal solution, there exists at least one sensor node in the WSN with its maximal and minimal energy equal to Emax and Emin , respectively. Although this property is intuitive, its proof is far from straightforward. A proof based on contradiction is given in the Appendix. 3. From general time-varying solutions to ðN þ 1Þ-phase solutions Problem OPT formulated in the last section has many integration and differentiation terms in (3) and (4) and thus is a non-polynomial program, which is very challenging. Such integration and differentiation terms are caused by the dynamic (time-varying) routing scheme, where flow rates g ij ðtÞ and g iB ðtÞ are all functions of time. Then energy consumption rate pi ðtÞ and energy level ei ðtÞ are also functions of time, which in turn yield non-polynomial terms in OPT. To remove these non-polynomial terms, in Section 3.1, we introduce the concept of ðN þ 1Þ-phase solution, which has a special dynamic routing scheme. In Section 3.2, we prove that an optimal ðN þ 1Þ-phase solution has the same objective value as that by an optimal time-varying solution. 3.1. Problem formulation for ðN þ 1Þ-phase solutions Note that the WCV is designed to charge sensor nodes but does not participate in data routing. Therefore, instead of considering infinite number of WCV’s locations in a dynamic solution, it may be sufficient to consider ðN þ 1Þ phases, where one phase corresponds to not charging any node (i.e., traveling on the road or staying at the service station) and each of the other phases corresponds to charging a different sensor node. Such an ðN þ 1Þ-phase solution is defined as follows. Definition 2. In an ðN þ 1Þ-phase solution, phase i includes all time instances that the WCV is charging sensor node i 2 N , and phase 0 includes all time instances that the WCV is not charging any sensor node. Multi-hop data routing in an ðN þ 1Þ-phase solution is allowed to be changed only when one phase is changed to another (i.e., when the WCV begins or stops charging a sensor node). Denote T m as the set of all time instances in phase m, i.e.,

8 sm  < ½am ; am þ[ T m ¼ ½s; 2s  T i :

if

m2N

if

m ¼ 0:

ð11Þ

i2N

Recall that the time length for phase m 2 N is Denote s0 as the time length for phase 0. We have

s¼

N X

sm :

sm .

ð12Þ

m¼0

By (12), the objective function svac s can be re-written as PsNvac and constraint (9) can be re-written as m¼0

sm

s0 ¼ sP þ svac :

ð13Þ

39

L. Shi et al. / Computer Networks 74 (2014) 34–52

In an ðN þ 1Þ-phase solution, flow rates and energy consumption rates can be changed only when one phase is changed to another. Therefore, within each phase m; 0 6 m 6 N, flow rates and energy consumption rate are constants. For phase m, denote flow rates from sensor m node i to sensor node j and the base station B as f ij and m f iB , respectively, and energy consumption rate at sensor node i as pm i . Then constraints (1)–(3) become j–i k–i X X m m m f ij þ f iB  f ki ¼ Ri j2N

ði 2 N ; 0 6 m 6 NÞ

ð14Þ

k2N

j–i k–i X X m m m pm f ki þ C ij f ij þ C iB f iB i ¼ q k2N

ði 2 N ; 0 6 m 6 NÞ

sm pmi  U si ¼ 0 ði 2 N Þ:

ð16Þ

m¼0

For an ðN þ 1Þ-phase solution, it is easy to see that a sensor node i’s energy level is lowest at time ai and is highest at time ai þ si . Thus, constraints (5) and (6) become ei ðai Þ P Emin and ei ðai þ si Þ 6 Emax . Then we have ei ðai þ si Þ  ei ðai Þ 6 Emax  Emin . Since ei ðai þ si Þ  ei ðai Þ ¼ P m U si  pii si ¼ m–i 06m6N sm pi , where the second equality holds by (16), we have m–i X

sm pmi 6 Emax  Emin ði 2 N Þ:

ð17Þ

06m6N

Note that given a value for ei ðai þ si Þ, we can determine ei ðtÞ values for t 2 ½s; 2s based on pm i values and (4). Furthermore, when (17) holds, there exist at least one value for ei ðai þ si Þ, i.e., Emax , such that the corresponding ei ðtÞ values meet (5) and (6). Thus, (5) and (6) can be replaced by (17). Now the set of constraints are (14), (15), (16), (4), (17), (7), (8), (13), (10) and (12). Since s only appears in its constraint (12), we can remove s and (12) from the problem formulation. Similarly, we can remove ei ðtÞ and its constraint (4). We have the following optimization problem for ðN þ 1Þ-phase solutions.

OPT-Phase

svac m¼0 sm

max PN

s:t: ð14Þ—ð17Þ; ð7Þ; ð8Þ; ð13Þ; ð10Þ m

m

f ij ; f iB ; pm ði; j 2 N ; i – j; 0 6 m 6 NÞ i ; ai P 0 P 2 P; DP ; sP ; svac ; sm P 0 ð0 6 m 6 NÞ; m

m

where flow rates f ij and f iB , energy consumption rate pm i , traveling path P, distance DP , time sP , time intervals svac and sm , and arrival time ai are variables, and all the other parameters are constants. We now have a polynomial program, which does not have any integration or differentiation term. Once we have a solution to OPT-Phase, we can derive s and ei ðtÞ to obtain a solution to OPT as follows. We can determine s by (12). We note that Emax is always a feasible value for ei ðai þ si Þ such that ei ðtÞ values meet (5) and (6). Then we can determine ei ðtÞ by (4) and

ei ðai þ si Þ ¼ Emax :

We have a much simpler optimization problem OPTPhase than OPT by considering ðN þ 1Þ-phase solutions. Now the question is whether or not OPT-Phase can provide the same optimal objective value as that by OPT. We note that in an ðN þ 1Þ-phase solution, data routing cannot be changed within the same phase. Such a constraint does not exist in the original problem. In general, additional constraints yield a smaller optimization space and thus a sub-optimal solution. But for ðN þ 1Þ-phase solutions, we can prove the following theorem.

ð15Þ

j2N

N X

3.2. Optimality of ðN þ 1Þ-phase solutions

ð18Þ

Theorem 1. An optimal ðN þ 1Þ-phase solution has the same objective value as that by an optimal time-varying solution. i.e., OPT-Phase can achieve the same optimal objective value as that by OPT. Theorem 1 enables us to focus on ðN þ 1Þ-phase solutions without any performance loss. The proof of Theorem 1 is based on the following lemma. Lemma 1. Given any time-varying solution u ¼ ðg ij ðtÞ, g iB ðtÞ; pi ðtÞ; ei ðtÞ; P; DP ; sP ; s; svac ; si ; ai Þ, we can construct an ðN þ 1Þ-phase solution with the same objective value. Proof. For solution u, we can determine T m by (11) and s0 by (12). Then we construct an ðN þ 1Þ-phase solution m m  ¼ ðf m u ij ; f iB ; pi ; P; DP ; sP ; svac , sm ; ai Þ by letting

R

m

f ij ¼ R

m

f iB ¼

g ij ðtÞdt

t2T m

sm g iB ðtÞdt

t2T m

sm

R t2T m

pm i ¼

pi ðtÞdt

sm

ði; j 2 N ; j – i; 0 6 m 6 NÞ

ð19Þ

ði 2 N ; 0 6 m 6 NÞ

ð20Þ

ði 2 N ; 0 6 m 6 NÞ:

ð21Þ

m  , we define flow rates f m That is, to construct u ij ; f iB in a phase by the average of g ij ðtÞ; g iB ðtÞ in this phase, define energy consumption rate pm i in a phase by the average of pi ðtÞ in this phase, and keep all other variables unchanged.  and u have the same objective It is clear that solutions u  is feasible, the proof value. Once we verify that solution u is complete.  is feasible, i.e., conWe now verify that solution u  . Since u is straints (14)–(17), (7), (8), (13), (10) hold by u feasible, it meets (7)–(10). Moreover, (9) is equivalent to  , it also meets (7), (13). Then based on the construction of u (8), (13) and (10).  meets (14) as follows. We can verify that u

j–i k–i X X m m m f ij þ f iB  f ki j2N

¼

j–i X j2N

R ¼

k2N

R

t2T m

t2T m

g ij ðtÞdt

sm

R þ

t2T m

g iB ðtÞdt

sm

P j–i

j2N g ij ðtÞ þ g iB ðtÞ 

sm



k–i X

R t2T m

k2N  g k2N ki ðtÞ dt

g ki ðtÞdt

sm

Pk–i

R ¼

t2T m

Ri dt

sm

¼ Ri ; where the first equality holds by (19) and (20), and the third equality holds by (1) in solution u.

40

L. Shi et al. / Computer Networks 74 (2014) 34–52

 meets (15) as follows. We can verify that u

R t2T m

pm i ¼

pi ðtÞdt

sm  P  R Pj–i q k–i k2N g ki ðtÞ þ j2N C ij g ij ðtÞ þ C iB g iB ðtÞ dt t2T m ¼ sm ¼q

k–i X

R

g ki ðtÞdt

t2T m

sm

k2N

R

 ¼q

j–i X þ C ij

R

t2T m

g ij ðtÞdt

sm

j2N

þ C iB

g iB ðtÞdt

t2T m

4.1. Optimal traveling path

sm

j–i k–i X X m m m f ki þ C ij f ij þ C iB f iB ; j2N

k2N

where the first equality holds by (21), the second equality holds by (2) in solution u, and the last equality holds by (19) and (20).  meets (16) as follows. We can verify that u N X

sm pmi  U si ¼

m¼0

R

N X

sm

Z 2s s

pi ðtÞdt

t2T m

sm

m¼0

¼

pi ðtÞdt 



Z 2s s

Z 2s s

pi ðtÞdt

pi ðtÞdt ¼ 0;

where the first equality holds by (21) and (3) in solution u, and the second equality holds by (11) in solution u.  meets (17) as follows. We can verify that u m–i X 06m6N

s

m m pi

mal traveling path for the WCV, and an optimal charging schedule. In Section 4.1, we prove that the optimal traveling path must follow the shortest Hamiltonian cycle and thus can be determined separately by solving a traveling salesman problem (TSP). In Section 4.2, the remaining problem for data routing and charging schedule is formulated as OPT-LP, which is an LP and can be optimally solved in polynomial-time. Combining both results, we have an optimal ðN þ 1Þ-phase solution (see Fig. 4).

¼

N X

s

R m m pi

m¼0

¼ U si 

s

Z t2T i

i i pi

¼ U si  si

t2T i

pi ðtÞdt

si

pi ðtÞdt ¼ ei ðai þ si Þ  ei ðai Þ

6 Emax  Emin ; where the second equality holds by (16) (already proved) and (21), and the fourth equality holds by (4) in solution u, the last inequality holds by (5) and (6) in solution u. Since we have verified that all constraints (14)–(17),  , solution u  is feasible. h (7), (8), (13), (10) hold by u Now we can prove Theorem 1. Proof. Since an ðN þ 1Þ-phase solution is a special case of a general time-varying solution, it is clear that an optimal ðN þ 1Þ-phase solution cannot achieve a better objective value than that by an optimal time-varying solution. On the other hand, for an optimal time-varying solution u , there is an ðN þ 1Þ-phase solution with the same objective value by Lemma 1. Therefore, an optimal ðN þ 1Þ-phase solution can achieve at least the same objective value by an optimal time-varying solution. Combining both results, we prove it. h

The traveling path for the WCV is a part of OPT-Phase. It seems that we had to determine the optimal traveling path jointly with data routing and charging schedule. However, the following lemma shows that the optimal traveling path can be determined separately. Lemma 2. To maximize PsNvac

s

, the WCV must move along

m¼0 m

the shortest Hamiltonian cycle that crosses all the sensor nodes and the service station in an optimal solution. This result is intuitive since if the WCV moves along the shortest Hamiltonian cycle, then it consumes the minimal time for travel. Therefore, it can have a longer vacation time. A formal proof is given in the Appendix. The shortest Hamiltonian cycle can be obtained by solving TSP. Although TSP is NP-hard, fast algorithms (see, e.g., [26,27]) were developed and can solve TSP with thousand nodes quickly. We note that the shortest Hamiltonian cycle may not be unique. Moreover, to travel a cycle, there are two (opposite) directions for the WCV to start from its service station. Since any shortest Hamiltonian cycle and any direction yields the same traveling time sTSP , the selection of a particular shortest Hamiltonian cycle and direction does not affect the optimal objective value. 4.2. Optimal data routing and charging schedule After the optimal traveling path is obtained, we only need to optimize data routing for each phase and charging schedule (charging time for each sensor node). We now present the reduced optimization problem, where the traveling path follows the shortest Hamiltonian cycle. Denote DTSP as the total path distance in the shortest Hamiltonian cycle and sTSP ¼ DTSP =V. Then (13) becomes

s0  svac ¼ sTSP :

ð22Þ

Once we determine the traveling path P, variables DP ; sP are fixed as DTSP ; sTSP , respectively. These variables and corresponding constraints (7) and (8) can be removed from the optimization problem. Now the set of constraints are (14)–(17), (22) and (10). Since ai only appears in its constraint (10), we can remove

4. Obtain an optimal ðN þ 1Þ-phase solution We showed that problem OPT-Phase provides the same optimal objective value as OPT (see Fig. 4). Now we develop an optimal solution for OPT-Phase, which includes an optimal multi-hop data routing in each phase, an opti-

OPT

=

OPT−Phase TSP

OPT−LP

Fig. 4. A flowchart to obtain an optimal solution.

41

L. Shi et al. / Computer Networks 74 (2014) 34–52

ai and (10) from the optimization problem. Then we have a reduced optimization problem as follows.

svac m¼0 sm

max PN

m

ði; j 2 N ; i – j; 0 6 m 6 NÞ; f ij ; f iB ; pm i ; svac ; sm P 0 m

m

where flow rates f ij and f iB , energy consumption rate pm i , time intervals svac and sm are variables, and all the other parameters are constants. Although this problem is much simpler than OPT-Phase, it has both nonlinear objective PsNvac and nonlinear terms sm pm i in constraint (17). We m¼0

sm

now reformulate this problem to remove all the nonlinear terms. To remove the nonlinear objective PsNvac , we define m¼0

s gvac ¼ PN vac m¼0 sm sm gm ¼ PN ð0 6 m 6 NÞ k¼0 sk h ¼ PN

1

m¼0 sm

j–i N X N X X C ij qm C iB qm ij þ iB  U gi ¼ 0;

m¼0k2N

m¼0 j2N

qqmki þ

gvac 6 g0 

s:t: constraints ð14Þ—ð17Þ; ð22Þ m

N X k–i X

sm

ð23Þ

sTSP

ð24Þ ð25Þ

;

N X

gm ¼ 1:

ð26Þ

m¼0

Dividing both sides of (16), (17) and (22) by have

PN

s we

m¼0 m ,

Emax  Emin

k–i X

j–i X C ij qiij  C iB qiiB ;

k2N

j2N

qqiki 

gm pmi  U gi ¼ 0 ði 2 N Þ

ð27Þ

j2N

Now the set of constraints are (37), (15), (35), (36), (29) and (26). Since h only appears in its constraint (29), we can remove h and (29) from the problem formulation. Similarly, we can remove pm i and (15). Therefore, we have the following problem formulation. OPT-LP max

gvac

m ði; j 2 N ; i – j; 0 6 m 6 NÞ; qm ij ; qiB P 0; 0 6 gvac ; gm 6 1

m where data volumes qm ij and qiB , and normalized lengths

gvac and gm are variables, and all the other parameters are constants. OPT-LP is a linear programming, which can be optimally solved in polynomial-time [28, Chapter 8].   m to Once we have an optimal solution qm ij ; qiB ; gvac ; gm OPT-LP, we can derive an optimal solution   m m m f ij ; f iB ; pi ; svac ; sm for OPT-Phase by the following algorithm.

g

6 ðEmax  Emin Þh ði 2 N Þ

ð28Þ

charging time

sm ¼

06m6N

g0  gvac : sTSP



sTSP Emax  Emin

U gi  pii gi

j–i k–i X X m m m f ki þ C ij f ij þ C iB f iB

q

j2N

k2N

gvac 6 g0 

h

svac ¼ gvac by (23) and (25), and h by (24) and (25).

5. Initialization cycle



ði 2 N Þ:

ð30Þ

By (15), constraints (27) and (30) can be rewritten as

m¼0

gm

ð29Þ

Now we reformulate (28). We have U gi  pii gi ¼ Pm–i g0 gvac m 06m6N pi gm 6 ðEmax  Emin Þh ¼ ðEmax  Emin Þ sTSP , where the first equality holds by (27), the second equality is (28), and the last equality holds by (29). Thus, (28) can be rewritten as

gvac 6 g0 

sTSP Emax  Emin

" U gi 

!

gm  U gi ¼ 0 j–i k–i X X i i i f ki þ C ij f ij þ C iB f iB

q

k2N

ð31Þ ! #

gi ð32Þ

j2N

In this section, we show why a different solution is needed for the initialization cycle and how to construct such a solution. Each sensor node i has a starting energy level Emax in the initialization cycle. We now show that in the first renewable cycle, each sensor node i has a starting energy level less than Emax . Suppose that the optimal traveling path for the WCV is P and the optimal solution to OPT-LP is u ¼ ðqmij ; qmiB ; gvac ; gm Þ. By Algorithm 1, we can derive an m m optimal solution ðf ij ; f iB ; pm i ; svac ; sm Þ to OPT-Reduced. We can further determine s by (12) and ai by (10). Then the starting energy level in the first renewable cycle can be determined as

To remove nonlinear terms in (31) and (32), we define

qm ij ¼ qm iB ¼

m f ij m f iB

qm

Algorithm 1. We can determine flow rates f ij ¼ gij by (33), m qm m f iB ¼ giB by (34), energy consumption rate pm i by (15), h by (29), vacation time

m m pi

ð37Þ

k2N

m

m¼0 m–i X

ð36Þ

Multiplying both sides of (14) by gm , we have j–i k–i X X m qm qm ij þ qiB  ki  Ri gm ¼ 0 ði 2 N ;0 6 m 6 NÞ:

m

N X

N X

U gi 

!

s:t: constraints ð37Þ; ð35Þ; ð36Þ; ð26Þ

where gvac and gm are normalized length of vacation time and phase time, respectively. By (24), we have

h¼

ð35Þ

m¼0

 gm

ði; j 2 N ; i – j; 0 6 m 6 NÞ

ð33Þ

 gm

ði 2 N ; 0 6 m 6 NÞ;

ð34Þ

which are the total data volume in phase m from sensor node i to sensor node j and the base station B, respectively. Then constraints (31) and (32) can be rewritten as

ei ðsÞ ¼ ei ð2sÞ ¼ ei ðai þ si Þ  ¼ Emax 

Z 2s ai þsi

Z 2s ai þ si

pi ðtÞdt;

pi ðtÞdt ð38Þ

where pi ðtÞ ¼ pm i if t 2 T m and the last equality holds by (18). Then we have

42

L. Shi et al. / Computer Networks 74 (2014) 34–52

Fig. 5. Sensor node i’s energy during the initialization cycle and how it connects the first renewable cycle.

ei ðsÞ < Emax ;

ð39Þ

R 2s

since ai þsi pi ðtÞdt > 0. We showed that each sensor node i has a higher starting energy Emax in the initialization cycle than its starting energy ei ðsÞ in the first renewable cycle (see Fig. 5). If we use the solution for the first renewable cycle in the initialization cycle, then node i’s energy will exceeds Emax before the WCV stops charging this node. To avoid overcharge, it is necessary to determine a suitable charging rate ui (less than the full charging rate U) for each node i in the initialization cycle. We now construct a solution for the initialization cycle, which should meet the following criterion. Criterion 1. An initialization cycle should meet the following requirements: (i) each sensor node i’s energy starts at Emax and ends at ei ðsÞ (determined by (38) for the obtain renewable cycle solution); (ii) each sensor node’s energy never falls below Emin ; and (iii) each sensor node’s energy never exceeds Emax . To construct the initialization cycle, we use the same traveling path P (and thus the same pi ; DP ; sP ) and conm m struct a solution w ¼ ðf ij ; f iB ; pm i ; svac ; sm ; ui Þ for the initialization cycle by letting ewi ð0Þ ¼ Emax , where ewi ðtÞ for t 2 ½0; s denotes sensor node i’s energy at time t under solution w, and letting

R ai þ si pi ðtÞdt ui ¼ s ;

ð40Þ

si

i.e., the same data routing and charging scheme are used in

w. We have the following theorem.

Theorem 2. The constructed solution w is feasible for the initialization cycle. It is easy to see that the arrival time of the WCV at senPi1 Dpk pkþ1 sor node i in the initialization cycle is þ k¼0 V Pi1 s ¼ a  s . We can further find that during p i k¼1 k ½0; ai  sÞ, node i’s energy curve is above its energy curve during ½s; ai Þ but less than its starting energy Emax ; during ½ai  s; ai  s þ si , node i’s energy is increasing to Emax (due to (40)); and during ðai  s þ si ; s, node i’s energy curve is the same as its energy curve during ðai þ si ; 2s (see Fig. 5). Thus we can prove Theorem 2. A formal proof is given in the Appendix. 6. Numerical results In this section, we present numerical results to show the obtained optimal solutions for renewable WSNs. We will also demonstrate the advantage of dynamic routing over static routing by analyzing these optimal solutions. 6.1. Simulation settings We consider three randomly generated WSNs consisting of 20; 50, and 100 nodes, respectively. Each sensor node is randomly deployed over a square area of 1000  1000 (in meters) and has a data rate randomly generated within ½1; 50 kb/s. The base station is located at ð500; 500Þ and the service station for the WCV is at ð0; 0Þ (all in meters). The traveling speed of the WCV is V ¼ 5 m/s. The WPT rate is U ¼ 5 W, which is well within feasible range [12].

Table 2 Location and data rate Ri for each node in a 20-node WSN. Node index

Location (m)

Ri (kb/s)

Node index

Location (m)

Ri (kb/s)

1 2 3 4 5 6 7 8 9 10

(648, 652) (679, 541) (636, 870) (945, 265) (209, 318) (709, 119) (236, 940) (119, 646) (607, 479) (450, 639)

15 15 12 24 21 24 30 30 6 6

11 12 13 14 15 16 17 18 19 20

(459, 545) (662, 647) (770, 544) (350, 721) (662, 522) (416, 994) (842, 219) (833, 106) (256, 110) (613, 64)

21 3 18 18 27 15 12 21 24 18

43

L. Shi et al. / Computer Networks 74 (2014) 34–52

Y (m)

Y (m)

1000

1000 WCV

800

800

600

600

Base station

400

400

200

200

Service station

200

400

600

800

1000 X (m)

200

400

600

800

1000 X (m)

200

400

600

800

1000 X (m)

Fig. 6. An optimal WCV traveling path for the 20-node WSN.

Y (m) We use the following model for energy consumption coefficients [29]. The energy consumption coefficient for transmitting data is C ij ¼ b1 þ b2 Daij and C iB ¼ b1 þ b2 DaiB , where b1 ¼ 5  108 J/b, b2 ¼ 1:3  1015 J=ðb m4 Þ and a ¼ 4. The energy consumption coefficient for receiving data is q ¼ 5  108 J/b. We consider a regular NiMH battery at each sensor node. Since its nominal cell voltage is 1.2 V and its quantity of electricity 2.5Ah, we have Emax ¼ 1:2  2:5  3600 ¼ 10; 800 J ¼ 10:8 kJ [30, Chapter 1]. We let Emin ¼ 0:05  Emax ¼ 0:54 kJ.

1000

800

600

400

6.2. Optimal solutions and discussions 200

A 20-node WSN. We first present results for a 20-node WSN. Table 2 gives each node’s location and data rate for this 20-node WSN. Fig. 6 shows the obtained optimal traveling path for the WCV (i.e., the shortest Hamiltonian cycle) by the Concorde solver [26]. For this optimal path, we have DTSP ¼ 3960 m and sTSP ¼ 792 s. By solving OPTLP, we obtain the optimal objective value gvac ¼ 34:10%. The corresponding cycle time and vacation time can be derived by Algorithm 1, which are s = 4.40 h and svac = 1.50 h. We analyze the optimal solution to gain more insights on the benefits of dynamic routing. Table 3 shows the

Fig. 8. Multi-hop data routing for the 20-node WSN in different phases.

minimal energy at each sensor node, which is ei ðai Þ since node i’s energy is lowest at time ai (see Fig. 5). Note that in our solution, each sensor node has its maximal energy

...

Fig. 7. Sensor node 4s energy during the first two cycles.

44

L. Shi et al. / Computer Networks 74 (2014) 34–52

Table 3 Each node’s minimal energy ei ðai Þ for the 20-node WSN. Node index

ei ðai Þ (kJ)

Node index

ei ðai Þ (kJ)

Node index

ei ðai Þ (kJ)

1 2 3 4 5 6 7

10.76 10.38 10.17 0.54 0.61 10.54 7.46

8 9 10 11 12 13 14

8.69 9.94 10.49 10.51 10.77 10.47 10.14

15 16 17 18 19 20

10.70 10.41 0.89 10.50 9.77 10.73

Table 4 Energy consumption rate pm 4 of node 4 in each phase for the 20-node WSN. Phase

pm 4 (W)

Phase

pm 4 (W)

Phase

pm 4 (W)

0 1 2 3 4 5 6

0.47 1.52 1.52 1.52 1.52 0.37 0.37

7 8 9 10 11 12 13

1.52 1.52 1.52 1.52 1.52 1.52 1.52

14 15 16 17 18 19 20

1.52 1.52 1.52 1.52 1.52 1.52 0.37

ei ðai þ si Þ ¼ Emax . We find that sensor node 4 has its minimal energy e4 ða4 Þ ¼ 0:54 kJ ¼ Emin , i.e., sensor node 4 is the bottleneck node for the 20-node WSN (see Property 1). The energy level of sensor node 4 during the first two cycles is shown in Fig. 7 (the solid curve). We can see that a larger svac (and thus a larger objective value) is not available because sensor node 4s non-charging period cannot be further increased. We now analyze how sensor node 4s energy is consumed. Fig. 8 shows the multi-hop data routing schemes in phase 4 and phase 0. We observe that sensor node 4 forwards more data for other nodes in phase 4 (when node 4 is being charged). The flow rates associated with sensor 4 4 0 node 4 are f 17;4 ¼ 75 kb/s, f 4;13 ¼ 99 kb/s, f 17;4 ¼ 6:75 kb/s, 0 and f 4;13 ¼ 30:75 kb/s. Thus, sensor node 4s energy consumption rate in phase 4 is larger, i.e., p44 ¼ 1:52 W > p04 ¼ 0:47 W. Table 4 shows the energy consumption rate of sensor node 4 in each phase. We find that the energy

consumption rate of sensor node 4 is largest in phase 4. That is, during its charging period (phase 4), sensor node 4 forwards more data for other sensor nodes such that other sensor nodes’ energy consumption rates can be less; during its non-charging period (other phases), sensor node 4 forwards less data such that its energy consumption rate can be less and thus can have a longer non-charging period. Both are beneficial for maximizing the objective. On the other hand, the benefits of dynamical routing can be demonstrated by analyzing what will happen if we use a single multi-hop routing scheme in one phase for the entire renewable cycle. We have the following two cases.  Suppose that we want to use the multi-hop routing scheme in phase 4 (see Fig. 8(a)) for the entire renewable cycle. Since the energy consumption rate of sensor node 4 is largest in phase 4 among all phases (see Table 4), by using the multi-hop routing scheme in phase 4 for the entire renewable cycle, sensor node 4 will consume more energy in other phases. We show its new energy level in Fig. 7 (the dashed line). We can see that its energy will be less than Emin before being charged in the next cycle, which yields an infeasible solution.  Suppose that we want to use the multi-hop routing scheme in a phase other than phase 4 for the entire renewable cycle. We analyze the multi-hop routing scheme in phase 0 (see Fig. 8(b)) as an example. We

Table 5 Location and data rate Ri for each node in a 50-node WSN. Node index

Location (m)

Ri (kb/s)

Node index

Location (m)

Ri (kb/s)

Node index

Location (m)

Ri (kb/s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

(985, 326) (559, 630) (934, 230) (720, 580) (484, 603) (639, 600) (888, 448) (199, 35) (395, 514) (992, 408) (402, 108) (659, 460) (901, 451) (995, 551) (653, 805) (108, 701) (36, 872)

5 45 30 35 15 30 40 50 25 5 5 35 40 35 20 50 30

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

(618, 52) (756, 294) (962, 460) (746, 959) (663, 790) (282, 94) (260, 333) (962, 59) (540, 741) (30, 507) (696, 200) (520, 427) (59, 169) (890, 752) (330, 368) (230, 942) (114, 17)

50 50 15 40 50 25 15 10 45 40 15 15 20 45 45 30 30

35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

(311, 829) (228, 627) (652, 539) (66, 651) (275, 727) (523, 452) (880, 878) (444, 14) (567, 220) (603, 180) (783, 926) (114, 68) (979, 581) (849, 637) (51, 651) (466, 865)

15 35 40 25 25 40 15 35 5 50 40 15 35 10 10 10

L. Shi et al. / Computer Networks 74 (2014) 34–52

45

s  s17 ¼ 15; 850  1992 ¼ 13; 858 s. Thus, to ensure

Y (m)

that sensor node 17’s energy is always no less than Emin , its energy consumption rate cannot be larger than

1000 WCV

Emax Emin ss17

800

600

¼ 0:74 W. But we find that it consumes 1.10 W

in phase 0. Therefore, if we use the multi-hop routing scheme in phase 0 for the entire renewable cycle, then sensor node 17s energy will be less than Emin before being charged in the next cycle, which yields an infeasible solution.

Base station

400

200

Service station 200

400

600

800

1000 X (m)

Fig. 9. An optimal WCV traveling path for the 50-node WSN.

Table 6 Each node’s minimal energy ei ðai Þ for the 50-node WSN. Node index

ei ðai Þ (kJ)

Node index

ei ðai Þ (kJ)

Node index

ei ðai Þ (kJ)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

10.54 6.53 10.43 8.43 3.54 7.89 10.66 9.86 8.16 10.59 5.22 6.71 10.70 10.18 10.48 7.36 9.01

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

9.27 0.54 9.66 10.71 9.95 0.54 10.39 10.34 7.87 9.85 9.48 10.11 10.59 8.85 6.49 10.19 106.59

35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

6.08 0.54 10.70 10.73 9.89 9.87 9.29 8.54 0.54 9.91 10.04 10.58 5.94 2.71 10.55 7.18

note that the same conclusion holds for other phases. Sensor node 17 consumes more energy in phase 0 than phase 4 (see Fig. 8(b)). Sensor node 17 is fully charged at a17 þ s17 and the WCV starts charging it in the next cycle at a17 þ s. The time of the non-charging period (from a17 þ s17 to a17 þ s) is

In summary, using a single multi-hop routing scheme in any phase will yield an infeasible solution. Then it is necessary to have a dynamic multi-hop routing scheme such that the optimal objective value can be achieved. A 50-node WSN. For the 50-node WSN, each node’s location and data rate are shown in Table 5. We obtain the optimal traveling path for the WCV by the Concorde solver, which is shown in Fig. 9. For this optimal path, we have DTSP ¼ 6097 m and sTSP ¼ 1219 s. By solving OPT-LP, we obtain the optimal objective value gvac ¼ 26:51%. The corresponding cycle time and vacation time (derived by Algorithm 1) are s = 10.51 h and svac = 2.79 h. We again analyze the optimal solution to gain more insights on the benefits of dynamic routing. Table 6 shows the minimal energy ei ðai Þ at each sensor node. From which we can see that node 19, 23, 36, and 43 are the bottleneck nodes for the 50-node WSN. They have their minimal energy 0:54 kJ ¼ Emin . We will analyze one of them, say node 19. The energy level of sensor node 19 during the first two cycles is shown in Fig. 10 (the solid curve). We can see that a larger svac (and thus a larger objective value) is not available because sensor node 19s non-charging period cannot be further increased. We now analyze how sensor node 19s energy is consumed. Fig. 11 shows the multi-hop data routing schemes in phase 19 (when sensor node 19 is being charged) and in phase 0. We can see that sensor node 19 forwards more data for other sensor nodes in phase 19 than that in phase 0. The flow rates associated with sensor node 19 are 19 19 0 f 28;19 ¼ 30 kb/s, f 19;12 ¼ 35 kb/s, f 28;19 ¼ 15 kb/s, and 0 f 19;12 ¼ 20 kb/s. Thus, the energy consumption rate of sensor node 19 in phase 19 is larger (p19 19 ¼ 0:65 W > p019 ¼ 0:37 W). We analyze the energy consumption rate of sensor node 19 in all phases and show the results in Table 7. We find that the energy consumption rate of sensor node 19 is largest in phase 19. That is, during its

Fig. 10. Sensor node 19s energy during the first two cycles.

46

L. Shi et al. / Computer Networks 74 (2014) 34–52

charging period (phase 19), sensor node 19 forwards more data for other sensor nodes such that other sensor nodes’ energy consumption rates can be less; during its noncharging period (other phases), sensor node 19 forwards less data such that its energy consumption rate can be less and thus can have a longer non-charging period. Both are beneficial for maximizing the objective. On the other hand, the benefits of dynamical routing can be demonstrated by analyzing what will happen if we use a single multi-hop routing scheme in one phase for the entire renewable cycle. We have the following two cases.  Suppose that we want to use the multi-hop routing scheme in phase 19 (see Fig. 11(a)) for the entire renewable cycle. Since the energy consumption rate of sensor node 19 is largest in phase 19 (see Table 7), by using the multi-hop routing scheme in phase 19 for the entire renewable cycle, sensor node 19 will consume more energy in other phases. We show its new energy level

in Fig. 10 (the dashed line). We can see that its energy will be less than Emin before being charged in the next cycle, which yields an infeasible solution.  Suppose that we want to use the multi-hop routing scheme in a phase other than phase 19 for the entire renewable cycle. We analyze the multi-hop routing scheme in phase 0 (see Fig. 11(b)) as an example. From Fig. 11(b), we can see that sensor nodes 2, 5, 12, 19, 23, 24, 26, 28, 32, 35, 39, 40, 44, and 50 consume more energy in phase 0 than phase 19. Among these nodes, we find that sensor nodes 23, 40, and 43s energy will be less than Emin before being charged in the next cycle, which yields an infeasible solution. In particular, for sensor node 23, its non-charging period (from a23 þ s23 to a23 þ s) is s  s23 ¼ 37; 837  2341 ¼ 35; 496 s. Thus, to ensure that its energy is always no less than Emin , its energy consumption rate cannot be larger than

Emax Emin ss23

¼ 0:29 W. But we find that it con-

sumes 0.62 W in phase 0. Therefore, if we use the multi-hop routing scheme in phase 0 for the entire renewable cycle, then sensor node 23s energy will be less than Emin before being charged in the next cycle, which yields an infeasible solution.

Y (m) 1000

In summary, using a single multi-hop routing scheme in any phase will yield an infeasible solution. Then it is necessary to use a dynamic multi-hop routing scheme such that the optimal objective value can be achieved. A 100-node WSN. Table 8 gives each node’s location and data rate for a 100-node WSN. The optimal traveling path for the WCV (by the Concorde solver) is shown in Fig. 12 with DTSP ¼ 8026 m and sTSP ¼ 1605 s. In our solution, we have the optimal objective value gvac ¼ 43:60%, with the corresponding cycle time s ¼ 19:24 h and vacation time svac ¼ 8:39 h. The analysis on the optimal solution and the discussion on the benefits of dynamic multihop routing is similar to that for the 20- and 50-node WSNs and thus is omitted here to conserve space.

50

35

800 36

26

39

2 5

600

23

Base station 12

400 32 24

43

200 44

19 28

40

Service station 200

400

600

800

1000 X (m)

(a) Phase 19. Y (m)

Table 7 Energy consumption rate pm 19 of node 19 in each phase for the 50-node WSN.

1000 50 35

800

26 36 39

2 5

600

23

Base station 12

400 32 24

19

43

200 44

28

40

Service station 200

400

600

800

1000 X (m)

(b) Phase 0. Fig. 11. Multi-hop data routing for the 50-node WSN in different phases.

Phase

pm 19 (W)

Phase

pm 19 (W)

Phase

pm 19 (W)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0.37 0.37 0.37 0.37 0.37 0.37 0.37 0.37 0.37 0.37 0.37 0.37 0.12 0.12 0.12 0.12 0.12

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

0.37 0.12 0.65 0.37 0.12 0.12 0.25 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.37 0.12 0.37

34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

0.37 0.12 0.37 0.37 0.12 0.12 0.37 0.12 0.12 0.09 0.12 0.12 0.12 0.12 0.37 0.37 0.37

47

L. Shi et al. / Computer Networks 74 (2014) 34–52

7. Related work In the early 1900s, Nikola Tesla invented his WPT technique [31] and thus showed the feasibility of transferring electric power from one storage to another without any wires and plugs. However, Tesla’s technique was never put into practice due to its large electric fields and low power transfer efficiency. There was a few progress after Tesla’s work (e.g., [32]) but practical applications were very limited, mainly due to inefficient power transfer. For example, [33] found that using the technique in [32], a receiver at 10 cm away can only obtain about 45 mW power when the transmission power is 3 W, which corresponds to a power transfer efficiency of 1.5%. Similar power transfer efficiency were also reported by [34,35]. A recent breakthrough was made by Kurs et al. [12], which is based on magnetic resonance. Their experiment showed that a 60 W light bulb at two meters away can be fully powered by WPT, with about 40% power transfer efficiency. Kurs et al.’s work showed that non-radiative power transfer is not only feasible but also practical, which paves the way for applying WPT in practice, e.g., [14, Chapter 4], [15–18]. Wireless Power Consortium [36] has been established to set the international standard for interoperable WPT. Some companies have shown their interests to WPT [37–39].

The WPT technique based on magnetic resonance has several advantages. First, it does not interfere with the wireless data transfer [40]. Second, it is safe to human body and can be used on wireless Body Sensor Network (wBSN) or medical implants [41]. Third, comparing with other energy-harvesting devices such as the solar cell, the size and the weight of the WPT devices are more suitable for small sensor nodes of WSNs [37]. Because of these advantages, researchers have conducted important steps on WSNs by using WPT. The feasibility of using the WPT to prolong the lifetime of WSNs was studied in [33], and a greedy algorithm was proposed. The authors later improved their algorithm and got a new scheme named J-RoC [20], which handled some weak points in [33]. However, the J-RoC scheme is still a heuristic algorithm. Optimal or near-optimal solutions on static routing and charging was developed in [21] for the case that the WCV can only charge one node each time and in [22,23] for the case that WCV can charge multiple nodes simultaneously. These works assume static routing while we consider a general dynamic routing case in this paper. 8. Conclusion and future work We exploited WPT for WSNs in this paper. We showed that if properly designed, the traditional lifetime

Table 8 Location and data rate Ri for each node in a 100-node WSN. Node index

Location (m)

Ri (kb/s)

Node index

Location (m)

Ri (kb/s)

Node index

Location (m)

Ri (kb/s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

(983, 294) (896, 27) (866, 93) (801, 798) (555, 711) (419, 783) (127, 624) (655, 825) (864, 35) (275, 405) (840, 250) (71, 481) (379, 881) (268, 281) (153, 599) (631, 26) (316, 155) (959, 834) (499, 195) (739, 830) (13, 338) (605, 671) (576, 52) (807, 734) (655, 499) (878, 943) (902, 290) (152, 377) (193, 114) (791, 965) (61, 433) (390, 85) (300, 717) (734, 507)

4 40 24 40 16 36 12 40 36 12 40 36 16 28 24 36 12 12 28 16 36 36 40 16 36 8 28 28 4 24 36 12 12 36

35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67

(104, 328) (793, 754) (783, 836) (532, 254) (253, 534) (71, 435) (626, 158) (25, 600) (62, 937) (130, 108) (451, 900) (672, 550) (856, 427) (498, 152) (49, 248) (314, 447) (642, 533) (786, 355) (289, 773) (498, 882) (818, 734) (595, 406) (536, 604) (331, 641) (412, 127) (794, 496) (343, 310) (463, 579) (368, 944) (680, 427) (598, 33) (652, 929) (491, 925)

20 8 12 28 24 12 28 32 12 20 16 32 8 40 12 20 32 28 8 28 8 40 36 24 28 40 8 32 36 32 8 20 4

68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

(398, 358) (477, 260) (67, 787) (411, 512) (969, 563) (781, 685) (729, 92) (766, 873) (757, 943) (843, 97) (770, 846) (979, 909) (111, 11) (396, 524) (492, 650) (258, 385) (37, 649) (974, 763) (726, 576) (148, 632) (148, 278) (705, 840) (381, 427) (76, 632) (411, 833) (143, 270) (799, 401) (930, 554) (5, 444) (650, 90) (679, 744) (254, 33) (843, 430)

16 8 20 24 32 20 24 40 36 24 40 4 36 16 12 36 16 8 4 40 28 36 40 28 40 12 36 40 16 4 32 4 32

48

L. Shi et al. / Computer Networks 74 (2014) 34–52

Appendix A

Y (m) 1000

Proof of Property 1. The proof is based on contradiction. That is, suppose that there exists an optimal solution

800





u ¼ g ij ðtÞ; g iB ðtÞ; pi ðtÞ; ei ðtÞ; P ; DP ; sP ; s ; svac ; si ; ai ; 600

where for each node i, either its maximal energy is less than Emax or its minimal energy is more than Emin . Then we can construct a solution with a larger objective value, which is a contradiction. Denote Eimin and Eimax as the minimal and maximal energy at node i in u , respectively. Denote

400

200

c¼ 200

400

600

800

1000 X (m)

Fig. 12. An optimal WCV traveling path for the 100-node WSN.

performance bottleneck for WSNs may be removed, which enabled us to consider other useful objectives. In this paper, we considered how to maximize the ratio of the WCV’s vacation time over the cycle time by optimizing a dynamic multi-hop data routing, a traveling path for the WCV, and a charging schedule. This problem was very challenging as dynamic data routing yields a non-polynomial programming formulation (with many integration and differentiation terms). We overcame this challenge by considering ðN þ 1Þ-phase solutions. We showed that an optimal ðN þ 1Þ-phase solution can achieve the same objective value as that by an optimal time-varying solution. Subsequently, we identified the optimal traveling path and formulated the problem for data routing and charging schedule as an LP. Comparing with previous result on the same problem but for static routing, we showed that our solution procedure can achieve a much larger objective value with a much lower complexity. There are a number of interesting open problems for future research. In general, with WPT, we may re-consider the design for many energy-constrained wireless networks. For the problem considered in this paper, we note that the number of phases in our solution increases linearly with the number of nodes in a WSN. Thus, one important question is scalability, i.e., how to decrease the number of phases for a large-sized WSN. There are also other open problems, e.g., considering multiple WCVs and/or multiple base stations in the WSN, extending the problem into 3-D area, considering that the WCV can also gather data. We will explore these problems in our future research.

Acknowledgments This paper is supported by National Natural Science Foundation of China (No. 61370088), International Science & Technology Cooperation Program of China (No. 2014DFB10060), and International S&T Cooperation Program of Anhui Province of China (1303063009).

Emax  Emin n o; max Eimax  Eimin

ð41Þ

i2N

Based on our assumption, we have either Eimax < Emax or Eimin > Emin for each node i. Thus, we have Eimax  Eimin > Emax  Emin for each node i, i.e.,

c > 1:

ð42Þ

Denote

T m

 8    < am ; am þ sm [  ¼ ½s ; 2s   T : i

s0 ¼ s 

X

if m 2 N ð43Þ

if m ¼ 0 ;

i2N

sm :

ð44Þ

m2N

Then we can construct a new solution

b D ;s ;s ^ ^ ^ ^ ^ ¼ ðg^ij ðtÞ; g^iB ðtÞ; p ^i ðtÞ; ^ei ðtÞ; P; u bP bP ; svac ; si ; ai Þ; b ¼ P ; D ¼ DP ; s ¼ sP , by letting P b b P

P

s^ ¼ c  s s^i ¼ c  si ði 2 N Þ X s^0 ¼ s^  s^m ;

ð45Þ ð46Þ ð47Þ

m2N

^i ¼ s ^þ determining a

Pi1

k¼0

Dp^ p^ k kþ1 V

þ

Pi1

8 ^m ; a ^m þ s ^m  if m 2 N < ½a [ Tb m ¼ ½s ; b ^ ^ : ; 2s  T i if m ¼ 0 i2N R g  ðxÞdx x2T m ij ^ g ij ðtÞ ¼ ðt 2 Tb m Þ; sm R g  ðxÞdx x2T m iB ðt 2 Tb m Þ; g^iB ðtÞ ¼ 

k¼1

sp^ k by (10), letting ð48Þ

ð49Þ ð50Þ

sm

P Pj–i ^i ðtÞ ¼ q k–i ^ ^ ^ determining p k2N g ki ðtÞ þ j2N C ij g ij ðtÞ þ C iB g iB ðtÞ by (2), ^ei ðtÞ by (4) and

^i þ s ^i Þ ¼ Emax ; ^ei ða

ð51Þ

P ^vac ¼ s ^  s  i2N s ^i by (9). and determining s bP ^vac P 0; s ^0 > 0. For First of all, we show the defined s s^vac , we have

s^vac ¼ s^  sbP 

X

X

i2N

i2N

s^i ¼ c  s  sP 

> c  ðs  sP 

X

c  si

si Þ ¼ c  svac P 0;

i2N

49

L. Shi et al. / Computer Networks 74 (2014) 34–52

where the second equality holds by (45), the construction of sb, and (46), the third inequality holds by (42), and the P fourth equality holds by (9) in solution u . Furthermore, ^0 and s ^vac , we have by comparing the construction of s s^0 ¼ s^vac þ sbP > 0. ^ is feasible, we need to verify that u ^ To show that u ^ , it meets constraints (1)–(10). By the construction of u meets (2), (4), (9) and (10). Since u is a feasible solution, it ^ , it also satisfies (7) and (8). Then by the construction of u meets (7) and (8). ^ meets (1), (3), (5) and (6). We now verify that u b m , we can verify that u ^ meets (1) as follows. Assuming t 2 T k–i k–i X X g^ki ðtÞ þ Ri ¼ k2N

R

x2T m

¼ R ¼ ¼

sm

P

j–i  j2N g ij ðxÞ

x2T m

g ij ðxÞdx

sm

R þ

t2b Tm

^i ðtÞdt ¼ c p

t2T m

s^0 ¼ s^ 

s^m ¼ cs 

m2N

X

ði 2 N ; 0 6 m 6 NÞ

X

csm ¼ c s 

m2N

ð53Þ

!

sm ¼ c  s0 ;

¼ Emax  c  U si þ c

m2N

t2b Tm

^i ðtÞdt ¼ p

t2b Tm

^m q ¼s

R

k–i X

x2T m

k–i Z X k2N

¼c

Z t2T m

j–i g ki ðxÞdx X C ij 

sm

k2N

¼c q

j2N

k2N

x2T m

g ki ðxÞdx

x2T m

þ

j–i X

C ij

j2N

k–i X g ki ðtÞ þ k2N

j2N

g ij ðxÞdx

sm

j2N

j–i X

q

R

Z x2T m

R þ C iB

g ij ðxÞdx

þ C iB

!

Z

t2b Ti

Z

t2T i

^i ðtÞdt p pi ðtÞdt

Emax  Emin n o  U si  ¼ Emax  max Eimax  Eimin i2N

! Z j–i k–i X X q g^ki ðtÞ þ C ij g^ij ðtÞ þ C iB g^iB ðtÞ dt ¼

Z

pi ðtÞdt

^i ; pi ðtÞdt ¼ c  U si ¼ U s

^i Þ ¼ ^ei ða ^i þ s ^ei ða ^i Þ  U s ^i þ

where the first equality is (47), the second equality holds by (45) and (46), and the last equality is (44). We can verify (53) as follows.

Z

t2T m

where the first equality holds by (3) (already proved), the ^i ðtÞ is a constant for any third equality holds because p bi. t2T Since a node i’s energy level is decreasing when the WCV is not charging this node and is increasing when the WCV is charging this node, we only need to prove ^i Þ P Emin and ^ ^i þ s ^ei ða ^i Þ 6 Emax for (5) and (6). Note ei ða ^i þ s ^i Þ 6 Emax holds by (51). that ^ei ða ^i Þ P Emin . We have Now we verify ^ei ða

We can verify (52) as follows.

X

c

m¼0

U¼

ð52Þ pi ðtÞdt

Z N X

Z Z 1 2s^ 1 ^i ðtÞdt > ^ ðtÞdt p p s^i s^ s^i t2bT i i 1 ^ ðtÞs ^ ¼p ^i ðtÞ ðt 2 Tb i Þ; ¼ p s^i i i

sm

where the first equality holds by (49), the third equality holds by (1) in solution u , and the last equality by (49) and (50). ^ meets (3), we prove the Before we verify that u following results.

Z

Z 2 s

^i ðtÞdt ¼ p

where the first equality holds by (48), the second equality holds by (53), the third equality holds by (43), the fourth equality holds by (3) in solution u , and the last equality holds by (46). ^ meets (5) and (6), we prove that Before we verify that u b i , i.e., node i’s energy is increasing ^i ðtÞ for any t 2 T U>p ^i ; a ^i þ s ^i . Note that g^ij ðtÞ and g ^iB ðtÞ are conduring t 2 ½a b i by (49) and (50). Then based on the stants for any t 2 T bi. ^i ðtÞ, it is also a constant for any t 2 T construction of p Thus, we have

Ri dx

j2N

Z

t2b Tm

m¼0

s

j–i X g^ij ðtÞ þ g^iB ðtÞ;

s^0 ¼ c  s0

N Z X

¼c

g iB ðxÞdx

x2T m

^i ðtÞdt ¼ p

s^

 þ g iB ðxÞ dx

sm

j2N

¼

Z 2s^

 k–i  k2N g ki ðxÞ þ Ri dx

x2T m

j–i X

t2 T m

of time t, the fourth equality holds by (46), the last equality holds by (2) in solution u . ^ meets (3) as follows. Now we can verify that u

sm

P

x2T m

R

x2T m

þ

sm

k2N

R

R

g ki ðxÞdx

^i ðtÞ, where the first equality holds by the construction of p the second equality holds by (49) and (50), the third equalR ity holds because terms within b ðÞdt are independent

q

t2b Tm

k–i X

R

k2N

g iB ðxÞdx

x2T m

x2T m

!

sm Z x2T m

C ij g ij ðtÞ þ C iB g iB ðtÞ dt ¼ c

!

g iB ðxÞdx

Z t2T m

pi ðtÞdt;

g ki ðxÞdx

sm

j–i X þ C ij j2N

R

x2T m

g ij ðxÞdx

sm

R þ C iB

!

Z t2T i

x2T m

pi ðtÞdt

g iB ðxÞdx

sm

! dt

50

L. Shi et al. / Computer Networks 74 (2014) 34–52

E  Emin max  

max ei ðai þ si Þ  ei ai i2N ! Z

^ and u , we have For the objective values by u

P Emax   U si  ¼ Emax 

 U si 

t2T i

D svac þ sP  sbP svac þ DVP  VbP s^vac s ¼ ¼ > PN vac ; PN PN P N    ^ m¼0 sm m¼0 sm m¼0 sm m¼0 sm

pi ðtÞdt

^m ¼ sm in the where the first equality holds by (54) and s ^ , the second inequality holds by (8), and construction of u b follows the shortthe third inequality holds because that P

E E n max R min o max U si  t2T  pi ðtÞdt i i2N ! Z t2T i

est Hamiltonian cycle while P does not (i.e., the traveling

pi ðtÞdt P Emax  ðEmax  Emin Þ ¼ Emin ;



s distance DP > Db). But Ps^Nvac > PNvac  contradicts the P s^m sm m¼0

ei ðtÞ, where the first equality holds by the construction of ^ the second equality holds by (51), (46) and (53), the third equality holds by (41), the fourth inequality holds by (5) and (6) in solution u , and the fifth equality holds by (4) in solution u . ^ meets (1)–(10). Thus, u ^ is a feasible We verified that u ^ can offer a better (larger) solution. We now show that u objective value. We have

P

P

s^vac s^  sbP  i2N s^i cs  sP  i2N csi ¼ ¼ cs s^ s^ P P cs  csP  i2N csi s  sP  i2N si svac > ¼ ¼  ; cs s s ^vac , where the first equality holds by the construction of s the second equality holds by (45) and (46), and the construction of sb, the third equality holds by (42), and the last

m¼0

assumption that u is optimal. h Proof of Theorem 2. Since U is the full charging rate of WCV, we need to ensure that

ui 6 U;

ð55Þ

for each sensor node i. Moreover, to prove that w is feasible for the initialization cycle, we need to verify that its meets (14), (15), (17), (22), and Criterion 1. Since u is a feasible renewable cycle and thus meets (14), (15), (17) and (22), we know that w also meets (14), (15), (17) and (22) by its construction. For (55), we have

R ai þ si R 2s pi ðtÞdt p ðtÞdt ui ¼ s 6 s i ¼ U;

si

si

equality holds by (9) in solution u . This contradicts the assumption that u is an optimal solution and completes the proof. h

where the first equality is (40), the second equality holds Dp p since 2s  ðai þ si Þ P VN 0 P 0, and the last equality holds by (3). Now we verify that w meets Criterion 1. For (i), we have ewi ð0Þ ¼ Emax by the construction of w. We also have

Proof of Lemma 2. This proof is based on contradiction. m  m  That is, if there is an optimal solution u ¼ ððf ij Þ ; ðf iB Þ ,

ewi ðsÞ ¼ ewi ð0Þ 

P

   ðpm i Þ ; P ; DP ;

sP ; svac ; sm ; ai Þ, where the WCV does not move along the shortest Hamiltonian cycle, then we can ^m ^m b ^ ¼ ð^f m construct a feasible solution u ij ; f iB ; pi ; P; Db; sb; P

P



s^vac ; s^m ; a^i Þ with an improved objective Ps^Nvac ^ > PsNvac  , m¼0

sm

0

Z 2s s

pi ðtÞdt þ ui si pi ðtÞdt þ

¼ ei ðai þ si Þ 

Z 2s ai þsi

s

m¼0 m

which is a contradiction. m  ^m m  ^ by letting ^f m We construct u ij ¼ ðf ij Þ ; f iB ¼ ðf iB Þ , m  b ^m P being the shortest Hamiltonian cycle (by p ¼ ðp Þ ; i i following either direction), determining Db by (7), sb by P P (8), letting

s^vac ¼ svac þ sP  sbP ;

¼ Emax 

Z s

ð54Þ

s^m ¼ sm , and determining a^i by (10). ^ is feasible, we need to verify that u ^ To show solution u meets constraints (14), (15), (16), (17), (7), (8), (13), and ^ , it meets (7), (8), and (10). Based on the construction of u (10). Since u is a feasible solution, it meets (14)–(17). ^ , it also meets (14)–(17). We Then by the construction of u ^ meets (13) as follows. can verify that u

s^0 ¼ s0 ¼ sP þ svac ¼ sbP þ s^vac ; ^ , the where the first equality holds by the construction of u second equality is (13) in solution u , and the last equality ^ meets all constraints and thus is holds by (54). Thus, u feasible.

Z

ai þ si

s

pi ðtÞdt

pi ðtÞdt ¼ ei ð2sÞ ¼ ei ðsÞ;

ð56Þ

where the second equality holds by the construction of w and (40), the third equality holds by (18), and the last equality holds because solution u meets (i) in Definition 1. Thus, we verified that w meets (i) in Criterion 1. Note that ewi ðtÞ achieves local minima at t ¼ ai  s and t ¼ s. We need to show ewi ðai  sÞ P Emin and ewi ðsÞ P Emin for (ii) in Criterion 1. For ewi ðai  sÞ, we have

ewi ðai  sÞ ¼ ewi ð0Þ  > e i ð sÞ 

Z

Z

ai  s

0 ai

s

pi ðtÞdt ¼ Emax 

Z

ai

s

pi ðtÞdt

pi ðtÞdt ¼ ei ðai Þ P Emin ;

where the second equality holds by the construction of w, the third inequality holds by (39), and the last equality holds because solution u meets (ii) in Definition 1. For ewi ðsÞ, we have

ewi ðsÞ ¼ ei ðsÞ P Emin ; where the first equality is (56) (already proved) and the second inequality holds because solution u meets (ii) in

L. Shi et al. / Computer Networks 74 (2014) 34–52

Definition 1. Thus, we verified that w meets (ii) in Criterion 1. Note that ewi ðtÞ achieves local maxima at t ¼ 0 and t ¼ ai  s þ si . Since ewi ð0Þ ¼ Emax , we only need to show ewi ðai  s þ si Þ 6 Emax for (iii) in Criterion 1. We have

ewi ðai  s þ si Þ ¼ ewi ð0Þ  ¼ Emax 

Z

Z s

ai sþsi

pi ðtÞdt þ ui si

0 ai þ si

pi ðtÞdt þ

Z s

ai þ si

pi ðtÞdt

¼ Emax ; where the second equality holds by the construction of w and (40). Thus, we verified that w meets (iii) in Criterion 1. Therefore, the constructed solution w is feasible for the initialization cycle. h References [1] J. Chang, L. Tassiulas, Maximum lifetime routing in wireless sensor networks, IEEE/ACM Trans. Networking 12 (4) (2004) 609–619. [2] A. Giridhar, P.R. Kumar, Maximizing the functional lifetime of sensor networks, in Proc. ACM/IEEE International Symposium on Information Processing in Sensor Networks, Los Angeles, CA, 2005, April 25–27, pp. 5–12. [3] Y.T. Hou, Y. Shi, H.D. Sherali, Rate allocation and network lifetime problems for wireless sensor networks, IEEE/ACM Trans. Networking 16 (2) (2008) 321–334. [4] Y. Shi, Y.T. Hou, Some fundamental results on base station movement problem for wireless sensor networks, IEEE/ACM Trans. Networking 20 (4) (2012) 1054–1067. [5] W. Wang, V. Srinivasan, K.C. Chua, Using mobile relays to prolong the lifetime of wireless sensor networks, in: Proc. ACM MobiCom, Cologne, Germany, August 28–September 2, 2005, pp. 270–283. [6] Y. Ammar, A. Buhrig, M. Marzencki, B. Charlot, S. Basrour, K. Matou, M. Renaudin, Wireless sensor network node with asynchronous architecture and vibration harvesting micro power generator, in: Proc. of Joint Conference on Smart Objects and Ambient Intelligence, Grenoble, France, October 12–14, 2005, pp. 287–292. [7] N. Bulusu, S. Jha (Eds.), Wireless Sensor Networks: A Systems Perspective, Artech House, Norwood, MA, 2005. [8] X. Jiang, J. Polastre, D. Culler, Perpetual environmentally powered sensor networks, in: Proc. ACM/IEEE International Symposium on Information Processing in Sensor Networks, Los Angeles, CA, April 25–27, 2005, pp. 463–468. [9] A. Kansal, J. Hsu, S. Zahedi, M.B. Srivastava, Power management in energy harvesting sensor networks, ACM Trans. Embeddded Comput. Syst. 6 (4) (2007) (article 32). [10] S. Meninger, J.O. Mur-Miranda, R. Amirtharajah, A.P. Chandrakasan, J.H. Lang, Vibration-to-electric energy conversion, IEEE Trans. Very Large Scale Integration (VLSI) Syst. 9 (1) (2001) 64–76. [11] G. Park, T. Rosing, M.D. Todd, C.R. Farrar, W. Hodgkiss, Energy harvesting for structural health monitoring sensor networks, J. Infrastructure Syst. 14 (1) (2008) 64–79. [12] A. Kurs, A. Karalis, R. Moffatt, J.D. Joannopoulos, P. Fisher, M. Soljacic, Wireless power transfer via strongly coupled magnetic resonances, Science 317 (5834) (2007) 83–86. [13] A. Kurs, R. Moffatt, M. Soljacic, Simultaneous mid-range power transfer to multiple devices, Appl. Phys. Lett. 96 (4) (2010) (article 4102). [14] K. Finkenzeller, RFID Handbook: Fundamentals and Applications in Contactless Smart Cards and Identification, second ed., Wiley, New York, 2003. [15] B. Jiang, J.R. Smith, M. Philipose, S. Roy, K. Sundara-Rajan, A.V. Mamishev, Energy scavenging for inductively coupled passive RFID systems, IEEE Trans. Instrum. Meas. 56 (1) (2007) 118–125. [16] A.K. RamRakhyani, S. Mirabbasi, M. Chiao, Design and optimization of resonance-based efficient wireless power delivery systems for biomedical implants, IEEE Trans. Biomed. Circ. Syst. 5 (1) (2011) 48–63.

51

[17] G. Wang, W. Liu, M. Sivaprakasam, M. Humayun, J. Weiland, Power supply topologies for biphasic stimulation in inductively powered implants, in: Proc. IEEE International Symposium on Circuits Systems, Kobe, Japan, May 23–26, 2005, pp. 2743–2746. [18] F. Zhang, X. Liu, S.A. Hackworth, R.J. Sclabassi, M. Sun, In vitro and in vivo studies on wireless powering of medical sensors and implantable devices, in: Proc. IEEE/NIH Life Science Systems and Applications Workshop (LiSSA), Bethesda, MD, April 7–8, 2009, pp. 84–87. [19] E. Giler, Eric Giler Demos Wireless Electricity. . [20] Z. Li, Y. Peng, W. Zhang, D. Qiao, J-roc: a joint routing and charging scheme to prolong sensor network lifetime, in: Proc. IEEE International Conference on Network Protocols (ICNP), Vancouver, BC, Canada, October 17–20, 2011, pp. 373–382. [21] Y. Shi, L. Xie, Y.T. Hou, H.D. Sherali, On renewable sensor networks with wireless energy transfer, in: Proc. IEEE INFOCOM, Shanghai, China, April 10–15, 2011, pp. 1350–1358. [22] L. Xie, Y. Shi, Y.T. Hou, W. Lou, H.D. Sherali, S.F. Midkiff, On renewable sensor networks with wireless energy transfer: the multi-node case, in: Proc. IEEE Communications Society Conference on Sensor, Mesh and Ad Hoc Communications and Networks (SECON), Seoul, Korea, June 18–21, 2012, pp. 10–18. [23] L. Xie, Y. Shi, Y.T. Hou, W. Lou, H.D. Sherali, On traveling path and related problems for a mobile station in a rechargeable sensor network, in: Proc. the Fourteenth ACM International Symposium on Mobile Ad Hoc Networking and Computing, ACM, New York, NY, USA, 2013, pp. 109–118. [24] R.C. Shah, S. Roy, S. Jain, W. Brunette, Data MULEs: modeling a threetier architecture for sparse sensor networks, in: Proc. IEEE International Workshop on Sensor Network Protocols and Applications (SNPA), Anchorage, AL, May 11, 2003, pp. 30–41. [25] W. Zhao, M. Ammar, E. Zegura, Controlling the mobility of multiple data transport ferries in a delay-tolerant network, in: Proc. IEEE INFOCOM, Miami, FL, March 13–17, 2005, pp. 1407– 1418. [26] Concorde tsp solver. . [27] M. Padberg, G. Rinaldi, A branch-and-cut algorithm for the resolution of large-scale symmetric traveling salesman problems, SIAM Rev. 33 (1) (1991) 60–100. [28] M.S. Bazaraa, J.J. Jarvis, H.D. Sherali, Linear Programming and Network Flows, fourth ed., John Wiley & Sons Inc., New York, 2010. [29] W. Heinzklman, Application-Specific Protocol Architectures for Wireless Networks, Ph.D. dessertation, Dept. of EECS, MIT, Cambridge, MA, June 2000. [30] D. Linden, T.B. Reddy (Eds.), Handbook of Batteries, third ed., McGraw-Hill, 2002. [31] N. Tesla, Apparatus for Transmitting Electrical Energy, US patent number 1,119,732, issued in December 1914. [32] Powercast. . [33] Y. Peng, Z. Li, G. Wang, W. Zhang, D. Qiao, Prolonging sensor network lifetime through wireless charging, in: Proc. IEEE Real-Time Systems Symposium, San Diego, CA, November 30–December 3, 2010, pp. 129–139. [34] S. He, J. Chen, F. Jiang, D.K.Y. Yau, G. Xing, Y. Sun, Energy provisioning in wireless rechargeable sensor networks, in: Proc. IEEE INFOCOM, Shanghai, China, April 10–15, 2011, pp. 2006–2014. [35] B. Tong, Z. Li, G. Wang, W. Zhang, How wireless power charging technology affects sensor network deployment and routing, in: Proc. IEEE International Conference on Distributed Computing Systems, Genoa, Italy, June 21–25, 2010, pp. 438–447. [36] Wireless Power Consortium. . [37] eCoupled. . [38] WiPower. . [39] JustinCumming. . [40] J. Johnson, E. Basha, C. Detweiler, Charge selection algorithms for maximizing sensor network life with UAV-based limited wireless recharging, in: Proc. IEEE 8th International Conference on Intelligent Sensors, Sensor Networks and Information Processing (ISSNIP), Melbourne, VIC, Australia, April 2–5, 2013, pp. 159–164. [41] Z. Fei, L. Jianbo, M. Zhihong, S. Mingui, Mid-range wireless power transfer and its application to body sensor networks, Open J. Appl. Sci. 2 (1) (2012) 35–46.

52

L. Shi et al. / Computer Networks 74 (2014) 34–52 Lei Shi was born in 1980. He received the B.S. degree in 2002, M.S. degree in 2005, and Ph.D. degree in 2012, all from Hefei University of Technology, Hefei, Anhui, China. He is currently an assistant professor in School of Computer and Information, Hefei University of Technology. His main research area lies in wireless network optimization.

Xu Ding was born in 1984. He received the B.S. degree from School of Computer and Information, Hefei University of Technology in 2002, and he is now pursuing his Ph.D. degree in Hefei University of Technology. His research fields mainly lies in wireless communications and wireless sensor networks.

Jianghong Han received his B.S. degree in 1982 and M.S. degree in 1987 from Hefei University of Technology, China. He is now a professor and a research lab leader in School of Computer and Information, Hefei University of Technology. His research interests includes computer networks, wireless sensor networks, discrete-time control systems, embeded systems and smart homes.

Zhenchun Wei was born in 1978. He received the B.S. degree in 2000, and Ph.D. degree in 2007, all from Hefei University of Technology, Hefei, Anhui, China. He is currently an associate professor in School of Computer and Information, Hefei University of Technology. His main research area lies in wireless sensor networks and internet of things.

Dong Han received his B.Sc. Eng. degree in Computer Science and Technology from Zhejiang University of Technology, China, and his M.S. degree in Telecommunication Engineering from the University of Sydney, Australia. He is a Ph.D. candidate at University of Houston, USA. His current areas of research focuses on revealing network protocols, application workloads, and topology information by fine-grained energy instrumentation on the wireless networks.