Adjustable Trajectory Design Based on Node Density for Mobile Sink in WSNs

Transcription

1 sensors Article Adjustable Trajecry Design Based on Node Density for Mobile Sk WSNs Guisong Yang 1,2, Shuai Liu 1, Xgyu He 3,4, *, Naixue Xiong 1, * and Chunxue Wu 1 1 Department Computer Science and Engeerg, School Optical-Electrical and Computer Engeerg, University Shanghai for Science and Technology, Shanghai 293, Cha; gsyang@usst.edu.cn (G.Y.); shuailiu@st.usst.edu.cn (S.L.); wcx@usst.edu.cn (C.W.) 2 Shanghai Key Lab Modern Optical Systems, Shanghai 293, Cha 3 Public Experiment Center, University Shanghai for Science and Technology, Shanghai 293, Cha 4 College Electronic and Information Engeerg, Tongji University, Shanghai 2184, Cha * Correspondence: xy_he@usst.edu.cn (X.H.); xiongnaixue@gmail.com (N.X.); Tel.: (X.H.); (N.X.) Academic Edir: Yu Hen Hu Received: 3 September 216; Accepted: 1 December 216; Published: 9 December 216 Abstract: The design movement trajecries for sk plays an important role garg for Wireless Sensor Networks (WSNs), as it affects network coverage, and packet delivery ratio, as well as network lifetime. In some scenarios, whole network can be divided subareas where s are randomly deployed. The densities se subareas are quite different, which may result a decreased packet delivery ratio and network lifetime if movement trajecry sk cannot adapt se differences. To address se problems, we propose an adjustable trajecry design method based on density for sk WSNs. The movement trajecry sk each subarea follows Hilbert space-fillg. Firstly, trajecry is constructed based on network size. Secondly, adjustable trajecry is established based on density specific subareas. Fally, trajecries each subarea are combed acquire whole network s movement trajecry for sk. In addition, an adaptable power control scheme is designed adjust s transmittg range dynamically accordg movement trajecry sk each subarea. The simulation results demonstrate that proposed trajecries can adapt network changes flexibly, thus outperform both packet delivery ratio and energy consumption trajecries designed only based on network size and whole network density. Keywords: movement trajecry; sk; density; Hilbert space-fillg ; power control 1. Introduction With fast spread communication technology and Internet Thgs applications, wireless sensor networks (WSNs) are beg widely used many applications such as medical care, military surveillance and environmental monirg [1,2], where y have become an important and effective means formation garg various fields. One ma goal WSNs is gar sensg and send m sk. In traditional WSNs, a static sk is used gar sensg. However, with limited energy and communication range a large scale sensor network, it is not practical for all s send sensg sk directly. Hence, static sk concept has some defects like poor network connectivity and low packet delivery ratio, as well as high energy consumption as a result unbalanced flow distribution. More seriously, it also curs hot-spot problem [3], which will cause network partition that furr degrades network lifetime. Sensors 216, 16, 291; doi:1.339/s

2 Sensors 216, 16, To address above problems, designg predefed trajecry that sk can move along collect sensg [4 1] has attracted extensive attention many researchers. By fully exploitg predefed mobility, network coverage is improved and number transmissions is decreased some extent. However, this case, sk may not visit all areas frequently, which will evitably cause packet loss and high end--end delay. Therefore how cover whole network and gar efficiently becomes a big challenge. To cope with this challenge, Viana focused on space-fillg [11]. Space-fillg s have property that y ensure fillg a d-dimension space by traversg every pre-defed pot a region once and only once, a specific order. Furrmore, Ghafoor proposed a sk trajecry design method based on Hilbert space-fillg [12], which trajecry was designed by analyzg network size. To furr deal with networks with high density, Ghafoor improved previous trajecry by takg density whole network consideration [13]. However, networks with non-uniform or uneven deployment, this method would result a significant decrease packet delivery ratio for high re-transmission rates. To address this problem, we propose a path-constraed sk trajecry design method, which trajecry is designed based on density each subarea. To ensure full coverage network and efficient garg, Hilbert space-fillg [14] is employed and sk moves along it. In addition, proposed movement trajecry sk can adapt networks with different densities. The trajecry is designed as follows: firstly, by analyzg maximum communication range sensor s and network size L, order k Hilbert s can be calculated ensure full coverage network, and n network is divided 4 k subareas. Secondly, by calculatg a proper Hilbert order each subarea, trajecry each subarea is determed based on density. Thirdly, trajecry is formed by combg above Hilbert s from all subareas. For example, by analyzg density a subarea, proper order d i Hilbert this subarea can be calculated and form correspondg Hilbert as movement trajecry it. After acquirg orders Hilbert all subareas and gettg ir correspondg Hilbert s, all se trajecries are combed a contuous fillg, which is fal movement trajecry for sk, so essence, sk can adjust its movement trajecry based on density each subarea so as adapt network status. In addition, we also propose an adaptable power control scheme. In each subarea, transmission range sensor s can be adjusted order crease energy efficiency, thus creasg network lifetime. The ma contributions work are as follows: 1. We design an adjustable trajecry (that follows Hilbert ) based on density each subarea for sk; 2. We propose an adaptable power control scheme adjust s transmittg range each subarea accordg order Hilbert it order reduce energy consumption. Comprehensive simulations are conducted with different densities or different kds deployments determe performance proposed method both terms packet delivery ratio and average energy consumption, which is compared with classic methods such as network size-based and whole network density-based strategy. The rest paper is organized as follows: Section 2, related works are presented. In Section 3, system model is troduced. Section 4 gives detailed design proposed method. The performance evaluation proposed method is presented Section 5, and fally we conclude paper Section Related Work In traditional WSNs, all sensor s as well as sk are static, which would result hot-spot problem [3]. To address this problem, many research works are drawn on leveragg

3 Sensors 216, 16, mobility sk make energy be consumed evenly. The first thought is make sk move accordg a predefed path. Many studies have been done usg sks with predefed paths [4 1]. In [4], mobility pattern a path-constraed sk was analyzed. The results showed that path-constraed approaches can reduce network overhead. In [5], authors explaed advantage path-constraed sk mobility, and analyzed performance five different sk deployment methods. In [6], authors proposed a garg scheme usg path-constraed sks, which network was divided two parts accordg communication range sks and distance sensor s crease flexibility. In [7], a one-hop garg scheme was proposed, which a three-layer network structure was used crease scalability and network lifetime. In [8], a low-complexity garg procol with one or more sks was proposed reduce routg overhead and prolong network lifetime. In [9], by resolvg Geometric Sk Trajecry problem, authors proposed a trajecry planng method for multiple sks, which packet delay and network lifetime were balanced. In [1], authors proposed a predefed trajecry for multiple sks a network divided hexagonal tiles, which trajecry is designed make sk act as bottleneck network and thus reduce risk energy holes. These approaches can all reduce energy consumption sensor s, thus prolongg network lifetime, but y are not flexible enough deal with issues complicated networks, for example, address problem that a movement trajecry has cover whole network so as improve packet delivery ratio network. Therefore space-fillg concept was troduced make garg more efficient. In [11], authors first troduced notion space-fillg s WSNs and proposed opportunistic delivery algorithm crease packet delivery ratio. In [12], a sk trajecry design which is based on Hilbert space-fillg was proposed by Ghafoor for full coverage network and less delivery delay, but a network with high density, this trajecry would lead poor delivery ratios and high average delivery delays. To solve this problem, Ghafoor also proposed a novel method [13] which improves upon previous one by takg density whole network consideration. By considerg both network size and density, efficiency garg is ensured, while network coverage is guaranteed. However, Ghafoor only considered density whole network or than that a specific subarea, that is, a large scale sensor network that can be divided many subareas where s are randomly deployed, density subareas may be non-uniform or uneven, and when density some subareas is relatively high, while rest subareas it is relatively low, above method can hardly adapt this situation, which it can lead a decrease packet delivery ratio and an creased energy consumption. To furr crease flexibility network, anor kd approaches usg random or controllable trajecries were proposed [15 19]. In [15], authors proposed a high-reliability garg procol based on sks, which adopts concept state mache achieve efficient routg and reduce overhead. In [16], a sk trackg method was proposed, which s track sk by receivg broadcast message sent by a sk periodically. In order furr reduce overhead, a sk trackg method called DAMST [17] was also proposed, which energy efficiency was high while overhead is cut down significantly. In [18], a contuous and optimal trajecry was proposed, which concept support vecr regression is used determe optimal trajecry a sk maximize network lifetime event-driven sensor networks. In [19], authors proposed a Greedy Scanng Data Collection Strategy (GSDCS), which network is divided many grids and sensor s same grid form a cluster crease efficiency garg. In each cluster, a cluster head is selected based on residual energy each as well as distance between each and center pot correspondg grid cell collect sensg. The trajecry sk is dynamically adjusted accordg network status by usg a scanng trajecry. Fally, by trackg

4 Sensors 216, 16, location sks, sensg are relayed sks by cluster heads prolong network lifetime. However, for a network where sks can move randomly or uncontrollably, sensg has be sent it by trackg its real time location, and trackg sks brgs extra overhead computation and srage. In our work, movement trajecries Sensors are 216, predefed, 16, 291 and only additional overhead is troduced by power 4 24 control messages, which can be ignored compared with location updatg message overhead. move randomly or uncontrollably, sensg has be sent it by trackg its real time 3. System location, Model and trackg sks brgs extra overhead computation and srage. In our work, movement trajecries are predefed, and only additional overhead is Introduced many real by applications, power control sensor messages, swhich are deployed can be ignored unevenly compared or non-uniformly with location due an unfavorable updatg message geographical overhead. location. In order make movement trajecry sk adapt networks with uneven deployment or different densities, network needs be partitioned 3. System Model many subareas, so we can design an adjustable movement trajecry based on density In many eachreal specific applications, subarea. sensor Therefore, s are deployed networkunevenly needs or benon-uniformly partitioned first due accordg an network unfavorable size geographical ensure location. full coverage In order make network. movement trajecry sk adapt networks with uneven deployment or different densities, network needs be 3.1. Network partitioned Model many subareas, so we can design an adjustable movement trajecry based on density each specific subarea. Therefore, network needs be partitioned first accordg In this paper, network we size troduce ensure full followg coverage assumptions: network. sensor s with limited itial energy E are deployed a square size L, communication range sensor s is adjustable with a maximum 3.1. Network valuemodel r max. The network is divided some subareas accordg network size, and each subarea is ain square this paper, size we l, troduce tal number followg subareas assumptions: is N. sensor A s sk with used limited itial gar energy sensg periodically E are deployed starts a from square botm size L, leftcommunication networkrange and travels sensor at s a constant is adjustable speedwith v followg a a predefed maximum movement value rmax. trajecry. The network For is divided sk, some subareas when garg accordg network a subarea, size, and each sensor subarea is a square size l, tal number subareas is N. A sk used gar only sends one packet with a fixed size it directly. sensg periodically starts from botm left network and travels at a constant speed v followg a predefed movement trajecry. For sk, when garg a subarea, 3.2. Hilbert Space-Fillg Curve each sensor only sends one packet with a fixed size it directly. In this paper, address problem full coverage network, trajecry is designed based 3.2. Hilbert Space-Fillg Curve on a space-fillg. By usg a space-fillg, sk can traverse every defed pot In network this paper, once address and only problem once, thus full all coverage sensg network, can trajecry be gared is designed abased timely and efficient on way. a space-fillg. By usg a space-fillg, sk can traverse every defed pot network once and only once, thus all sensg can be gared a timely and The Hilbert space-fillg [9] is a space-fillg proposed by David Hilbert efficient way. Hilbert s The can Hilbert map space-fillg a one-dimensional [9] is le a space-fillg l on a two-dimensional proposed by square David Hilbert S. For stance, if we map a le Hilbert l on s square can map S, firstly a one-dimensional le l is split le l on four a two-dimensional identical sub-tervals square S. For and stance, square if we S four identical map a le sub-squares, l on square where S, firstly each le sub-terval l is split four can be identical contuously sub-tervals mapped and on square ones sub-square. If this contues, four identical l and sub-squares, S partitioned where each sub-terval 4 k identical can duplications be contuously formapped k = 1, 2, on 3, where one subsquare. k defes order If this Hilbert contues,. l and Fally S are partitioned s traverses 4 k identical over duplications 4 k sub-squares for k = 1, 2, two 3, where dimensions. k defes order Hilbert. Fally s traverses over 4 The correspondence between le tervals and sub-squares is shown k sub-squares two Figure 1, which dimensions. The correspondence between le tervals and sub-squares is shown first, second Figure and 1, third which order first, second Hilbert and third are order depicted. Hilbert are depicted. (a) (b) (c) Figure 1. The first three orders Hilbert space-fillg : (a) 1-order (k = 1, N = 4); (b) 2-order Figure 1. The first three orders Hilbert space-fillg : (a) 1-order (k = 1, N = 4); (b) 2-order (k = 2, N = 16); (c) 3-order (k = 3, N = 64). (k = 2, N = 16); (c) 3-order (k = 3, N = 64).

5 Sensors 216, 16, Sensors 216, 16, The pots on show correspondence between sub-squares and sub-tervals, so adjacent Thesub-tervals pots on always correspond show correspondence adjacent sub-squares. betweenthe sub-squares transformation and sub-tervals, a order so adjacent k 1 sub-tervals a always order correspond k can be viewed adjacent as a replacement sub-squares. The each transformation sub-square a order k order 1 with k a 1-order 1 a. order k can be viewed as a replacement each sub-square order k 1Besides with a 1-order Hilbert s,. re are many or space-fillg s such as Z, Grey-coded, Besides etc. [2]. Hilbert In [21], s, re authors are many compared or space-fillg above space-fillg s suchs as Zand, found Grey-coded that Hilbert, etc. [2]. outperforms In [21], authors ors compared locality above preservation space-fillg property s and that found can crease that Hilbert garg outperforms efficiency, which ors is ma locality reason preservation Hilbert property is adopted that can as crease sk movement garg trajecry efficiency, which this paper. is ma reason Hilbert is adopted as sk movement trajecry this paper Hilbert Curves with Heterogeneous Orders 3.3. Hilbert Curves with Heterogeneous Orders After partitiong accordg network size, network can be divided many subareas. Here, After number partitiong subareas accordg is defed network by size, order network which can be dividedsk traverses many subareas. m. Recallg Here, number that tal subareas number is defed subareas by is order N, which subarea that sk traverses sk traverses m. Recallg first is called that C1 tal while number subarea subareas is N, sk subarea traverses that last is called sk CN. traverses In this work, first is called C 1 while sk moves subarea an area from sk traverses botm left lastand is called exits Cfrom N. In thisbotm work, right. sk moves an area fromfigure botm 2a shows left and Hilbert exitss from with botm heterogeneous right. orders, where Hilbert with 1-order C1, Figure C3 and 2a C4 shows while Hilbert Hilbert s with with heterogeneous 2-order C2, orders, for where subarea C2 Hilbert has relatively with high 1-order density, C 1, C 3 C2 and so Cadopts 4 while 2-order Hilbert Hilbert with 2-order while ors C 2, foradopt subarea 1-order C 2 Hilbert has relatively. high Similarly, Figure density, 2b C 2 and so adopts Figure 2c 2-order show Hilbert s while with ors heterogeneous adopt 1-order orders Hilbert different. Similarly, subareas accordg Figure 2b,c show Hilbert density s with m. heterogeneous orders different subareas accordg density It should m. be noted that, when orders Hilbert s two adjacent subareas are different from Iteach should or, be noted re that, exists when a gap orders at border Hilbert s two two adjacent adjacent subareas subareas ( ared different le Figure from each 2), or, and so re movement exists a gap trajecry at border becomes discontuous. two adjacent subareas To get a ( contuous red le movement Figure 2), trajecry, and so movement gap on trajecry border becomes needs discontuous. be connected To get by ausg contuous red movement les as shown trajecry, Figure gap 2. on border needs be connected by usg red les as shown Figure 2. C 2 C 3 C 2 C 3 C 2 C 3 C 1 C 4 C 1 C 4 C 1 C 4 (a) (b) (c) Figure 2. Three different kds Hilbert s with heterogeneous orders: (a) 1-order C1, C 1, C3, C4 and 2-order C2; (b) 1-order C1, C2, C4 C and 3-order C3; (c) 2-order C1, 3, C 4 and 2-order C 2 ; (b) 1-order C 1, C 2, C 4 and 3-order C 3 ; (c) 2-order C2, C C3 1, C 2 and, C 3 3-order and 3-order C4. C Energy Energy Model Model Considerg Considerg communication communication range range is is relatively relatively short, short, this this paper paper adopted adopted free-space free-space channel model [22]. Let ε f s fsbebe energy energy require require by by amplifier amplifier free-space free-space channel, channel, α tx and txα rx and be energy rx be consumed energy consumed durg transmittg durg transmittg and receivg and receivg per bit respectively. per bit respectively. The energy consumed durg i transmits β bit j is described as follows: The energy consumed durg transmits bit j is described as follows: where where D (i,j) is (, i j) is distance distance between between i and i and j. j. E 2 E ( tx (i, j) = (α i, j ) ( tx + ε f s D (i,j) 2 )β, (1) D ), (1) tx tx fs ( i, j )

6 Sensors Sensors 216, 216, 16, 16, The energy consumed by j that receives bit is described as follows: The energy consumed by j that receives β bit is described as follows: Erx( j) rx, (2) E rx (j) = α rx β, (2) The tal energy consumed by i and j durg above two processes can be defed as follows: The tal energy consumed by i and j durg above two processes can be defed as follows: 2 Etal( i, j ) ( tx rx fsd ) E tal (i, j) = (α tx + α rx + ε f s D(i,j) 2, ( i, j ) (3) )β, (3) 4. Adjustable Trajecry Design Based on Node Density The design adjustable trajecry for sk consists three ma steps. In first step, ensure full coverage network, we should get proper order Hilbert (that is k) accordg network size. Then, prepare for for second step, step, whole whole network is divided is divided 4 k 4subareas. k Once Once subareas are aredetermed, y will rema unchanged. In second step, adapt networks with different densities, especially highly dense networks with non-uniform or uneven deployment, we design trajecry each subarea by analyzg density it. To distguish from order k Hilbert accordg network size, order Hilbert used subarea Ci C i based on density is denoted by di. d i. Fally third step, adjustable trajecry based on density for sk can be acquired by combg trajecries all subareas and trajecries on all borders. In addition, order relieve network congestion and prolong network lifetime, we propose an adaptable power control scheme accordg characteristics adjustable trajecry, which can adjust transmission power sensor s specific subareas The Order Hilbert Curve Accordg Network Size In first step our method, guarantee full coverage network, proper order order Hilbert should should be be selected accordg network network size. size. The The proper proper order order here here means means mimum mimum order order Hilbert Hilbert that that can can fill fill full full network. network. Recall that network is a square area size L, and suppose that network can be divided N subareas and subareas are smaller sub-squares size l, as shown Figure 3. The followg Equations (4) (8) calculatg order k Hilbert are cited from [11]. l l L Figure3. 3. The Thelength length networkand subarea. The l can be defed by and as follows: The l can be defed by L and N as follows: L l, (4) l = L N, (4) N

7 Sensors 216, 16, To ensure sensor each subarea can communicate with sk directly, equality between size a subarea l and maximum communication range sensor r max holds: l 1 2 r max, (5) Supposed that proper order Hilbert is k, network can be divided N = 4 k subareas, and n size a subarea l can be defed as: l = L 2 k, (6) By combg Equations (5) and (6), we have: m2 k After some manipulation, k can be obtaed as follows: k = log 2 { 2L r max 2L r max, (7) }, (8) Through above calculation, mimum order k Hilbert which can fill whole network can be obtaed. Through Equation (8), it can be known that value k depends on network size L and sensor s maximum communication range r max. When k = 2, trajecry sk is as shown Figure 3. It is clear that, if movement trajecry for sk is designed only accordg network size when usg Hilbert space-fillg s, and sce sk travels at a constant speed, time it stays each subarea must be a fixed value. In a highly dense network, sk can hardly collect sensg completely with a limited traversg time due communication congestion, and reby this results a decrease packet delivery ratio and network lifetime The Order Hilbert Curve Based on Node Density Each Subarea In Section 4.1, mimum order k Hilbert which can fill whole network is obtaed accordg network size. However, a highly dense network with uneven deployment, this kd trajecry accordg network size may hardly complete garg with a limited traversg time, and this will result a significant decrease packet delivery ratio. In second step our method, furr improve trajecry design accordg network size, difference density each subarea is taken consideration make trajecry adaptable networks with uneven deployment high density situations. For this we can design an adaptable trajecry based on densities specific subareas. After gettg proper order k Hilbert accordg network size, whole network can be divided N = 4 k subareas. In an adjustable trajecry design based on density, density each subarea is adopted calculate a proper Hilbert order this subarea. And hence, movement trajecries for sk whole network are acquired by combg all trajecries se subareas. After acquirg order k Hilbert accordg network size, it can be seen that each subarea (except first and last subarea), length this subarea is same as its side length. Therefore, traversg time sk subarea C i can be calculated as follows: t i = l i v, (9) where l i stands for side length subarea C i and v stands for travel speed sk.

8 Sensors 216, 16, Sce Sensors 216, each16, sensor 291 only sends one packet sk durg process 8 24 garg, thus subarea C i, we defe n i_max as maximum number sensor s that sk can where communicate li stands for with side durg length its traversg subarea Ci time, and v n stands n i_max for can travel be calculated speed from Equation sk. (1), where t avg Sce represents each sensor average only time sends forone transmittg packet sensg : sk durg process garg, thus subarea Ci, we defe ni_max as maximum number sensor s that sk can communicate with durg its traversg n i_max = t i time, n ni_max can be calculated from, (1) Equation (1), where tavg represents average time tfor avg transmittg sensg : ti Subarea C i has its own Hilbert order ni based on, _ max its density, and order (1) all tavg subareas are itialized with k, so a order that is not less than k will guarantee full coverage network. Subarea Here Ci has n i_max its own is used Hilbert as threshold order based on decide its wher density, and skorder can complete all garg subareas are itialized its traversg with k, time so a C i or order not. that is not less than k will guarantee full coverage In a network. with Here ni_max unevenly is used deployed as threshold s, if decide number wher s n i sk C i obeys can complete n i < n i_max, garg its traversg time Ci or not. order d i Hilbert equals itialized order k Hilbert accordg In a network with unevenly deployed s, if number s ni Ci obeys ni < ni_max, network size, which means that sk can complete garg durg its traversg time order di Hilbert equals itialized order k Hilbert accordg network this size, subarea. which Orwise, means that if number sk can s complete n i C i garg obeys n i durg n i_max its, we traversg would time try use this k + 1 as subarea. order Orwise, C i if first, number and consequently s C i is furr divided four smaller sub-squares, ni Ci obeys ni ni_max, we would try use k + 1 as and neworder traversg Ci first, time and t i_new consequently Ci is furr sk divided subarea Cfour i is extended smaller sub-squares, : and new traversg time ti_new sk subarea ( Ci ) is extended : 1/2li t i_new = 4 1 = 2t vl i, (11) i t 2 i _ new 4 ti v 2 (11) Sce average transmission time for a packet t, avg is a fixed value and new traversg time t i_new Sce average sktransmission takes becomes time twice for a t i, hence packet tavg is maximum a fixed value number and new sensor traversg s time sk ti_new can gar issk alsotakes doubled. becomes twice ti, hence maximum number sensor s Based on sk can analysis gar is above, also doubled. proper traversg time sk subarea C i can be acquired by Based usg on Equation analysis (11). above, Therefore, proper traversg order d i time Hilbert sk based subarea Ci on can density be acquired by usg Equation (11). Therefore, order di Hilbert based on density C i can be calculated from proper traversg time sk C i as follows: Ci can be calculated from proper traversg time sk Ci as follows: { di = d i k, nk i, < ni n i_max ni _ ( ) d i = k + log ni, (12) 2 n n, n i_max i n i_max i (12) d i k log2, ni ni _ max ni _ max, Fally, trajecries all subareas are combed form fal movement trajecry for sk. Fally, The adjustable trajecries trajecry all subareas basedare oncombed density form thisfal network movement is shown trajecry Figure for 4. sk. The adjustable trajecry based on density this network is shown In this case, suppose that network is divided four subareas (C 1 C 4 ) accordg its network Figure 4. In this case, suppose that network is divided four subareas (C1 C4) accordg size. The density C 1 is lowest one where sk moves followg a 1-order Hilbert its network size. The density C1 is lowest one where sk moves followg a ; density C 2 and C 3 is higher than that C 1 where sk moves followg 1-order Hilbert ; density C2 and C3 is higher than that C1 where sk a 2-order moves Hilbert followg ; a 2-order and Hilbert ; density and C 4 is density highest one, where sk moves C4 is highest one, where followg a sk 3-order moves Hilbert followg. a 3-order Hilbert. C 2 C 3 C 1 C 4 Figure 4. An example an adjustable trajecry based on density specific subareas. Figure 4. An example an adjustable trajecry based on density specific subareas.

9 Sensors 216, 16, 291 Sensors 216, 16, Algorithm 1 gives detailed calculation order Hilbert based on density a specific Algorithm subarea. 1 gives detailed calculation order Hilbert based on density a specific subarea. Algorithm 1. The Order Hilbert Curve Based on Node Density a Specific Subarea 1. Algorithm Input L, 1. rmax, The v, Order tavg; Hilbert Curve Based on Node Density a Specific Subarea Output Input L, di (i r max = 1,, v, 2, t3, avg, ; N.); Initialize Output d i (i variables = 1, 2, 3,.(e.g.,.., N.); k, N ); Calculate Initialize order variables k (e.g., Hilbert k, N ); accordg Equation (8); 4. Calculate order k Hilbert accordg Equation (8); 5. Calculate number subareas N; 5. Calculate number subareas N; for for each each subarea subarea Ci: C i i : = i 1, = 1, 2, 2,,.. N.., N Calculate density ni; n i ; //n i is //ni number is number s s C i Ci Calculate traversg time ti t i sk accordg Equation (9); Calculate threshold n i_max accordg Equation (1); ni_max accordg Equation (1); 1. Calculate order d i Hilbert accordg Equation (12); 1. Calculate order di Hilbert accordg Equation (12); 11. End for 11. End for It should be noted that usg a Hilbert with heterogeneous orders would result It should be noted that usg a Hilbert with heterogeneous orders would result a a discontuous trajecry. If two adjacent subareas adopt different orders, re exists a gap discontuous trajecry. If two adjacent subareas adopt different orders, re exists a gap on border two adjacent subareas, and so movement trajecry become discontuous on border two adjacent subareas, and so movement trajecry become discontuous sce exit location previous subarea is not entrance location next subarea. To keep sce exit location previous subarea is not entrance location next subarea. To keep trajecry contuous, we should design movement trajecry for sk on border trajecry contuous, we should design movement trajecry for sk on border two adjacent subareas. two adjacent subareas Trajecry Design on Borders 4.3. Trajecry Design on Borders The trajecry design on borders will be discussed third step our method this section. It canthe be trajecry known from design Section on borders 4.2, that will forbe andiscussed adjustable trajecry third step design our based method on this density section. It specific can be subareas, known from order Section 4.2, that Hilbert for an adjustable each trajecry subarea may design be based different. on Consequently, density specific as shown subareas, Figure 5, re order will be Hilbert a gap ( red each le subarea Figure may 5) on be different. trajecry Consequently, border as shown two adjacent Figure subareas 5, re with will different be a gap ( orders. red le The border Figure here 5) on means trajecry common edge border two two adjacent adjacent subareas. subareas with different orders. The border here means common edge two adjacent subareas. C 2 C 3 C 1 C 4 Figure 5. Two adjacent subareas with with different different orders orders will have will a have gap a on gap iron border ir ( border red le). ( red le). To combe se s and make whole trajecry a contuous, we n discuss how design To combe trajecry se on s border and make two adjacent whole trajecry subareas a contuous followg, paragraphs. we n When discuss how order k design Hilbert trajecry ison acquired border accordg two adjacent network subareas size first followg step our paragraphs. method, When number order subareas k Hilbert N is also determed. is acquired Thus accordg number network trajecries size on borders first that step need our method, be designed is number N 1. subareas N is also determed. Thus number trajecries on borders that need be designed is N 1.

10 Sensors 216, 16, Sensors 216, 16, To deal with trajecry design on borders, key is get length trajecry on border and traversg direction sk on border trajecry, respectively The Length Trajecry on Border To get length trajecry on a border, we should firstly determe order ddi i Hilbert subarea CCi i and order ddi+1 Hilbert subarea CCi+1,, and fill specific subarea usg Hilbert, n exit location previous subarea Ci C i and entrance location next subarea CCi+1 can be be obtaed. Once obtaed specific exit and entrance locations on border two adjacent subareas, length trajecry on border can be becalculated as asfollows: l border_trajecry = 2 d i+1 L L l _,, d i d 2 i (13) 2 d i+1+1 where ddi, i, di+1 d i+1 stand for for orders based on on density Ci, C i, Ci+1, C i+1 respectively., The Traversg Direction a Mobile Sk on Border Trajecry The current traversg direction sk is set as a reference direction. The traversg direction we need design for for sk sk on border on border trajecry trajecry is direction is direction relative relative reference reference direction. direction. Considerg Considerg that whenthat when sk leavessk one leaves subarea, one itssubarea, traversg its direction traversg is direction perpendicular is perpendicular border, traversg border, direction traversg direction sk on border sk trajecry on border could trajecry only be one could two only directions: be one left two ordirections: right. left or right. The traversg direction sk sk on on border border trajecry must must fulfill fulfill followg followg rules: rules: case case a a order order crease, crease, which which means means that dthat i+1 > di+1 d i >, if di, if current traversg direction sk sk is is vertical border, traversg direction for for sk sk on on border is is left, left, and and if if current current traversg traversg direction direction sk is horizontal sk is horizontal border, border, traversg traversg direction for direction for sk on border sk on is right; border is case right; case order reduction, order which reduction, means which that d i+1 means < d i, that traversg di+1 < di, direction traversg we need direction design we need is opposite design is opposite direction designed direction designed case order case crease. order crease. Figure 6 shows a network divided four four subareas accordg its network its network size, size, where where C 1 and C1 and C 4 adopt C4 adopt 1-order 1-order Hilbert Hilbert s, s, C 2 C2 adopts a a 2-order Hilbert and CC3 3 adopts a 3-order Hilbert accordg ir density. Recallg that if two adjacent subareas have different orders, re will be a gap on border m, this case, sce C1 1 adopts a 1-order Hilbert while C2 C 2 adopts a 2-order Hilbert, exit location trajecry CC1 1 does not not meet entrance location trajecry C 2, C2, thus thus trajecry becomes discontuous, and and accordgly border border trajecry (red (red le le between C 1 C1 and and C 2 C2 Figure Figure 6) is6) employed is employed connect connect discontuous discontuous trajecry. trajecry. Similarly, Similarly, gaps between gaps between C 2 and CC2 3, Cand 3 and C3, CC3 4 are and also C4 are filled also with filled a border with a trajecry. border trajecry. C 2 C 3 C 1 C 4 Figure The The traversg trajecry design on on border ( red le). It is worth noticg that traversg direction sk on border trajecry is associated with followg two facrs: current traversg direction sk and wher order is creased or reduced between two adjacent subareas.

11 Sensors 216, 16, It is worth noticg that traversg direction sk on border trajecry is associated with followg two facrs: current traversg direction sk and wher order is creased or reduced between two adjacent subareas. The detailed design border trajecry is shown Algorithm 2. Algorithm 2. The Detailed Design Trajecry on a Border 1. Input d i, d i+1, cur_dir; //cur_dir is current travelg direction sk 2. Calculate length trajecry on border l border_trajecry accordg Equation (12); 3. if (d i! = d i+1 ) //when order d i = d i+1 4. if (d i < d i+1 ) // order is creased 5. if (cur_dir = left or cur_dir = right) 6. //current traversg direction sk is horizontal 7. direction_on_border right; 8. else 9. direction_on_border left; 1. end if 11. Mobile sk traverses along direction direction_on_border with length l border_trajecry ; 12. else // order is decreased 13. if (cur_dir = up or cur_dir = down) 14. //current traversg direction sk is vertical 15. direction_on_border left; 16. else 17. direction_on_border right; 18. end if 19. Mobile sk traverses along direction direction_on_border with length l border_trajecry ; 2. end if 21. end if In Algorithm 2, determe wher employ border trajecry, two order d i, d i+1 C i, C i+1 are compared firstly. When d i = d i+1, trajecry C i and C i+1 can connect each or directly and re is no necessity employ border trajecry. Orwise border trajecry is employed accordg method presented above. Fally sk can move followg border trajecry on border when re is a gap between two adjacent subareas, thus entire traversg trajecry becomes a contuous Adaptable Power Control We recall that Section 4.1, a given network size L, value k is determed, order k and number subarea N holds N = 4 k, and so number subareas is also a determed value. In this case, when sk moves a subarea, s subarea can communicate with it usg a constant transmission range. In Section 4.2, order d is selected based on density each subarea, so some highly dense subareas, a relatively high order Hilbert s will be selected. Sce sk will traverse all subareas collect sensg m, refore, some subareas that s are densely deployed, sk moves followg high order Hilbert s. It takes more time traverse se subareas, which can decrease distance between a sensor and sk, and accordgly, we can use power control adjust transmission range sensor uses communicate with sk such a subarea, which can effectively reduce energy consumption, so it can be deduced that subareas where s are densely deployed, higher order d, shorter transmission range a needs communicate with sk, and accordgly, energy consumed can also be reduced.

12 Sensors 216, 16, needs communicate with sk, and accordgly, energy consumed can also be Sensors reduced. 216, 16, After first step our method, assumg that a subarea C1 will transmit sk, distance between a and sk is l1. Here we note that C1 is a subarea where After s are first densely step deployed, our method, so a assumg order that a subarea C 1 will transmit d1 that is larger than k is selected based on density sk, distance between a and sk is l 1. Here we note that C 1 is a subarea C1 design movement trajecry sk, and this case, distance where between s are a and densely deployed, sk becomes so a l2, order and d 1 that is larger than k is selected based on l2 holds l1 > l2. In density free-space C 1 channel, design energy movement saved trajecry for a transmittg sk, and bits this case, distance sk between a and sk becomes l 2, and l 2 holds l 1 > l 2. can be defed as: In free-space channel, energy saved for a transmittg β bits sk can be defed as: 2 2 Esave ( l1 l ) E save = l1 2 2 fs, (14) l2 2 ε f s β, (14) Therefore, it can be seen that energy consumption sensor is reduced by usg power Therefore, it can be seen that energy consumption sensor is reduced by usg power control adjust its transmission range. control adjust its transmission range. For example, Figure 7, sk follows a 2-order Hilbert subarea C2 where For example, Figure 7, sk follows a 2-order Hilbert subarea C 2 where s are densely deployed, and we can use power control adjust transmittg range sensor s are densely deployed, and we can use power control adjust transmittg range sensor used communicate with sk, so transmission range sensor C2 used communicate with sk, so transmission range sensor C can be reduced from L accordg Equation (5). Thus sensor s C2 can adjust ir can be reduced from 2L 2 2L 2L 4 accordg Equation (5). Thus sensor s C 2 can adjust ir transmission range order reduce energy consumption when transmittg ir ir sensg sk. sk. C 2 C 3 C 1 C 4 Figure An An example adjustable power control. To enable sensor s such subareas adjust ir transmission range communicate with To enable sensor s such subareas adjust ir transmission range communicate with sk timely fashion, sk periodically broadcasts power control message sk a timely fashion, sk periodically broadcasts a power control message sensor s as it moves subarea. The format power control message is as shown sensor s as it moves a subarea. The format power control message is as shown Table 1, which cludes message type, target subarea ID and new transmission range. Table 1, which cludes message type, target subarea ID and new transmission range. Table 1. The format power control message. Table 1. The format power control message. Message Type Target Subarea ID New TX Range Message Type Target Subarea ID New TX Range The Message type field dicates type message, which is maly used differentiate power The Message control messages type fieldfrom dicates or type messages used message, network. which is Target maly used subarea differentiate ID field is used power dicate control which messages subarea from this message or messages is supposed used be network. sent. The Target New subarea TX range ID field is used set dicate new which transmission subarea this range message for issensor supposed s be sent target. The subarea. New TX range field is usedthe set new transmission new transmission range range for sensor sensor rtx_i s is determed target by subarea. network size L and The order newdi, transmission and can be defed range as follows: sensor r TX_i is determed by network size L and order d i, and can be defed as follows: 2L r tx _ i d 2 i (15) 2L r TX_i = 2 d, (15) i Each sensor has its own subarea ID. On receivg power control message, a will decide wher accept it or not by comparg Target subarea ID message with its

13 Sensors 216, 16, own subarea ID. Sce each subarea may adopt different Hilbert orders and thus mimum required transmittg range sensor s differs, power control message has dicate target subarea case message was received by s a wrong subarea. If Target subarea ID equals s own subarea ID, a n updates its transmission range usg value New TX range, orwise, it just ignores this message. Once transmission range is updated, sensor can communicate with sk by usg it this subarea. 5. Performance Evaluation Our simulations are performed under OMNeT with INET framework [23]. The parameters used this simulation are listed Table 2. Parameters Table 2. Simulation parameters. Values Network Size 8 8 m 2 Number Nodes Network 5, 1, 2, 4, 8, 12 Number Mobile Sk 1 Movement Speed Mobile Sk 5 m/s Mobility Model Hilbert space-fillg Sensg Range 1 m MAC Initial Energy (E) 2 J Transmission Power (P tx ) mw Reception Power (P rx ) 1 mw By followg Hilbert design movement trajecry sk, this simulation, we compare our proposed adjustable trajecry based on density specific subareas with trajecry based on network size [12], and trajecry based on density whole network [13], evaluate its efficiency. In this section se three kds trajecries are denoted as a, k-based and trajecries, respectively. In this work, we maly concentrate on comparg parameters such as packet delivery ratio (PDR) and energy consumption. In this simulation, sensor s are deployed an 8 8 m 2 network. Two kds communication models which are sgle-hop model and cluster model are adopted test performance proposed method. The reason we choose se two kds communication models is that y can fit requirements different applications. In Table 2, transmission power sensor ranges from.5 mw 2.6 mw. In fact, transmission power also limits communication range sensor s. In cluster model, mimum transmission power is used tra-cluster communication and adaptive power control method can adjust transmission power cluster head (CH). In this scheme, CH is selected randomly and each can and only can belong one CH at a time. In process garg, sensor s send ir sensg CHs first and n CHs relay se sensg sk as sk passes by. We set six scenarios with different deployments. In Scenarios 1 and 3, sensor s are deployed uniformly, while Scenarios 2, 4, 5 and 6, sensor s are deployed unevenly. Then sk can move followg trajecry collect sensg Evaluation Packet Delivery Ratio a Sgle-Hop Model Scenario 1. Sensor Nodes Are Uniformly Deployed Network In Scenario 1, we uniformly deploy 5, 1, 2, 4, 8, and 12 s network. The sk gars sensg along trajecry that follows adjustable trajecry based on density specific subareas, trajecry based on network size, and trajecry based

14 Sensors 216, 16, density whole network. Then packet delivery ratio is evaluated accordg simulation density results. The whole orders network. Hilbert Thens packet selected delivery by each ratio trajecry is evaluated accordg accordg different number simulation s results. are shown orders Table Hilbert 3. selected by each trajecry accordg different number s are shown Table 3. Table 3. The order k-based, and adjustable d based for different number s. Table 3. The order k-based, and adjustable d based for different number s. No Number Nodes k-based Curve Order d-based Curve Order Adjustable d-based Curve Order No 1 Number5 Nodes k-based Curve 1 Order d-based Curve 2 Order Adjustable 1 ( d-based all subareas) Curve Order ( all all subareas) subareas) ( (all allsubareas) ( (all allsubareas) ( all subareas) ( (all allsubareas) ( It can be seen that when number s is set 4 or larger, our a achieved a 23.43% 29.7% It can be seenimprovement that when number PDR over s k-based is set 4 orand larger, a 1.3% 4.21% our aimprovement achieved PDR a 23.43% 29.7% over improvement, respectively, PDR over uniform k-based deployment. and a 1.3% 4.21% improvement PDRAs over shown Figure, 8, when respectively, number uniform s is less deployment. than 4, PDR under Asoutperforms shown Figure 8, ors, whensce number it begs s with isa less 2-order than 4, Hilbert PDR, under while or two trajecries outperforms start from ors, a 1-order sce ithilbert begs. with awhen 2-order Hilbert number, s while rises or 4, two PDR trajecries under start adjustable from a 1-order Hilbert trajecry. outperforms When number ors and s remas rises best 4, as number PDR under s creases adjustable 12. trajecry outperforms ors and remas best as number s creases There is 12. no doubt that a higher order Hilbert can achieve a better PDR densely deployed networks. There The is no doubt that and a higher adjustable order Hilbert s can achieve use higher a better PDRorders densely than deployed that a k-based networks. The accordg and adjustable density, which s is use ma higher reason that orders PDR thanunder that se a k-based two s accordg can outperform k-based density, which. is The reason ma reason that that adjustable PDR under se twocan s surpass can outperform k-based is that. adjustable The reason that adjustable uses adaptive power cancontrol surpass adjust transmission that range adjustable sensor s, and thus uses it adaptive relieves power communication control adjust congestion transmission and achieves range sensor desired s, PDR. and thus it relieves communication congestion and achieves desired PDR. 1 9 Packet Delivery Ratio (%) k-based a ,2 Number s Figure 8. The PDR k-based, and adjustable trajecries. Scenario 2. Sensor Nodes Are Unevenly Deployed Network Scenario 2. Sensor Nodes Are Unevenly Deployed Network In Scenario 2, verify our proposed adjustable trajecry based on density specific In Scenario 2, verify our proposed adjustable trajecry based on density specific subareas, sensor s are unevenly deployed, whereby some subareas are densely deployed, and subareas, sensor s are unevenly deployed, whereby some subareas are densely deployed, and or subareas are sparsely deployed. The number s network is set 8 and re or subareas are sparsely deployed. The number s network is set 8 and re are are three kds deployments adopted this scenario, which are shown Figure 9a c, three kds deployments adopted this scenario, which are shown Figure 9a c, respectively. respectively. Figure 1 shows correspondg k-based, and adjustable trajecries Figure 1 shows correspondg k-based, and adjustable trajecries each kd each kd deployment this scenario, respectively. deployment this scenario, respectively. In Figure 9a,b, only one or two subareas are densely deployed while or subareas are sparsely deployed. In Figure 9c, furr verify flexibility se three kds trajecries, densities C1, C2, C3 are different from each or and C3, C4 are densely deployed.

15 Sensors 216, 16, In Figure 9a,b, only one or two subareas are densely deployed while or subareas are sparsely deployed. In Figure 9c, furr verify flexibility se three kds trajecries, densities C 1, C 2, C 3 are different from each or and C 3, C 4 are densely deployed. Sensors 216, 16, Sensors 216, 16, C 2 C 3 C 2 C 3 C 2 C 3 C 2 C C 2 C C 2 C C 1 C (a) C 4 C C 4 C C 4 C 4 C 1 (b) C 4 C 1 (c) C 4 Figure 9. Three (a) kds deployments Scenario (b) 2: (a) C1, C2, C4 are deployed (c) with 5 s Figure 9. Three kds deployments Scenario 2: (a) C and C3 is deployed with 65 s; (b) C1, C3 are deployed with 1, C 5 s 2, C 4 are deployed with 5 s and C2, C4 Figure 9. Three kds deployments Scenario 2: (a) C1, C2, are deployed C4 are deployed with 5 s and Cwith 3 is deployed 35 s; with (c) 65 C1 is deployed s; (b) with C 1 5, Cs, 3 are deployed C2 is deployed with with 5 s 15 s, and C C3 2 and, C 4 are C4 are deployed and all C3 is deployed with 65 s; (b) C1, C3 are deployed with 5 s and C2, C4 are deployed with 35 deployed s; with (c) 3 C 1 s. is deployed with 5 s, C 2 is deployed with 15 s, C 3 and C 4 are all with 35 s; (c) C1 is deployed with 5 s, C2 is deployed with 15 s, C3 and C4 are all deployed with 3 s. deployed with 3 s. k-based a k-based (a) a (a) k-based a k-based (b) a (b) k-based a k-based (c) a Figure 1. Illustration k-based, and adjustable (c) trajecries Scenario 2: (a) three Figure kds 1. Illustration trajecries k-based, Figure 9a; (b) and three adjustable kds trajecries trajecries Figure 9b; Scenario and (c) 2: (a) three Figurekds 1. Illustration three kds trajecries trajecries k-based, Figure Figure 9c. and adjustable trajecries Scenario 2: (a) 9a; (b) three kds trajecries Figure 9b; and (c) three three trajecries Figure 9a; (b) three kds trajecries Figure 9b; and (c) three kds trajecries Figure 9c. kds trajecries Figure 9c.

16 Sensors 216, 16, Sensors 216, 16, Sensors 216, 16, In Figure 9a, it it can can be be seen seen that that s s subarea subarea C 3 are C3 densely are densely deployed deployed while while or or three subareas three In subareas Figure y are 9a, y sparsely it are can sparsely be deployed. seen that deployed. Sce s Sce network subarea network isc3 divided are densely is divided four deployed subareas four while subareas and C 3 cludes and or C3 65/8 cludes three subareas = 65/ % y = s 81.25% are sparsely s whole deployed. network, whole Sce network, PDR network PDR C 3 contributes is divided C3 contributes most four most subareas overall overall and PDR. C3 Similarly, PDR. cludes Similarly, 65/8 Figure = Figure 81.25% 9b,c, 9b,c, s PDR PDR subareas whole subareas with network, with largest PDR largest density C3 contributes density can affect can most affect overall overall PDR PDR. most. Similarly, most. Figure 9b,c, PDR subareas with largest density can affect overall PDR It can most. be seen from Figure 11 that PDR usg outperforms ors subareas It can CC1, 1 be, C2 seen 2 and from C4. 4. However, Figure 11 that subarea PDR C3, C 3 usg, where most s are outperforms deployed, adjustable ors subareas C1, C2 gets and much C4. However, better PDR PDR than than subarea or or C3, two where two s. s. most s Similarly, are Figures deployed, Figures 12 and 1213, and adjustable 13, PDR under PDR under a gets a much better outperforms outperforms PDR than that under that or under two or s. or two s Similarly, two s subareas Figures subareas with 12 dense and with 13, dense deployment. PDR under deployment. Therefore, a Therefore, as shown as outperforms shown Table 4, Table that adjustable 4, under adjustable or two s achieves subareas achieves highest with highest overall dense PDR overall among deployment. PDR se among three Therefore, se kds three as trajecries shown kds Table trajecries all4, three kds adjustable all three kds deployments. achieves deployments. The reason highest is that The reason overall adjustable is PDR that among adjustable se three adopts kds proper trajecries adopts order proper all Hilbert three order kds t adapt Hilbert deployments. highly adapt dense The deployment reason highly is that dense subarea, deployment adjustable while subarea, or two while s adopts or not. two proper The s adaptive order do not. power The Hilbert control adaptive power an adjustable adapt control trajecry an highly adjustable based dense trajecry on deployment density based subarea, on helps while relieve density or helps network relieve s traffic do network congestion, not. The adaptive traffic thus congestion, power adjustable control thus an adjustable can trajecry achieve based can desired achieve on PDR. density From desired lps PDR. simulation relieve From results network simulation and detailed traffic results congestion, analysis and detailed above, thus analysis a adjustable above, outperforms a can achieve k-based outperforms desired andk-based PDR. From and bysimulation 21.2% and results 12.76% by and 21.2% on detailed average and 12.76% analysis packet on above, delivery average ratio a packet Scenario delivery 2, outperforms respectively. ratio Scenario k-based 2, respectively. and by 21.2% and 12.76% on average packet delivery ratio Scenario 2, respectively. Packet Packet delivery delivery ratio ratio each each subarea(%) k-based k-based a a Subarea ID Subarea ID Figure Figure The The PDR PDR k-based, k-based, and adjustable and adjustable deployment deployment (a) Scenario (a) 2. Figure Scenario 11. The 2. PDR k-based, and adjustable deployment (a) Scenario 2. Packet Packet delivery delivery ratio ratio each each subarea(%) Subarea ID k-based k-based a a Subarea ID Figure 12. The PDR k-based, and adjustable deployment (b) Scenario 2. Figure Figure The The PDR PDR k-based, k-based, and adjustable and adjustable deployment deployment (b) Scenario (b) 2. Scenario 2.

17 Sensors 216, 16, Sensors 216, 16, Packet delivery Packet ratio delivery each ratio subarea(%) each subarea(%) 1 9 Sensors 216, 16, k-based a k-based Subarea ID 2 Figure 13. k-based, 1 and adjustable a Figure 13. The PDR k-based, and adjustable deployment deployment (c) Scenario (c) 2. Scenario Table 4. The overall PDR k-based, and adjustable s Scenario 2. Subarea ID Table Node 4. The Deployment overall PDR (a) k-based, Node Deployment and adjustable (b) Node s Deployment Scenario (c) 2. Figure 13. The PDR k-based, and adjustable deployment (c) Scenario 2. In Scenario 2 In Scenario 2 In Scenario 2 Node Trajecry Deployment Overall PDR Trajecry Overall PDR Trajecry Overall PDR Table 4. The (a) overall PDR k-based, Node Deployment and adjustable (b) Node s Deployment Scenario 2. (c) k-based In Scenario % k-based In Scenario75.38% 2 k-based In Scenario 77.13% 2 Node Deployment 82.38% (a) Node Deployment 81.75% (b) Node Deployment 86.75% (c) Trajecry Overall PDR Trajecry Overall PDR Trajecry Overall PDR a In Scenario % a In Scenario 293.5% a In Scenario % k-basedtrajecry 8.88% Overall PDR k-based Trajecry Overall 75.38% PDR Trajecry k-based Overall PDR 77.13% 5.2. Evaluation k-based on Network 82.38% 8.88% Lifetime a k-based Sgle-Hop Model 75.38% 81.75% k-based 77.13% 86.75% a 96.25% 82.38% a 81.75% 93.5% a 86.75% 93.13% Scenario 3. Sensor Nodes are Uniformly Deployed Network a 96.25% a 93.5% a 93.13% 5.2. Evaluation In Scenario on Network 3, Lifetime number as Sgle-Hop is set Model 8 and s are deployed uniformly 5.2. network. Evaluation The k-based, on Network Lifetime and adjustable a Sgle-Hop Model s are evaluated energy consumption. Scenario Figure Sensor shows Nodes residual are Uniformly energy Deployed network under Network Scenario 3. Sensor Nodes are Uniformly Deployed Network se three kds trajecries by time. In Scenario In Scenario 3, 3, number number 16 s s is setis set 8 and 8 and s s are deployed are deployed uniformly uniformly network. The k-based, network. The k-based, and adjustable and adjustable s ares evaluated k-based are evaluated energy energy consumption. consumption. Figure shows residual energy network under se three kds Figure 14 shows residual energy network under se three kds trajecries trajecries by time. by time. a Residual energy(j) Residual energy(j) Time(s) k-based a 2 Figure 14. The residual energy network Scenario From simulation results above, it can be seen that a achieves much better Time(s) results on residual energy than k-based and consumes 23.9% less energy than uniform Figure deployment. 14. The residual energy network Scenario 3. Figure 14. The residual energy network Scenario 3. Sce all 8 s are deployed uniformly network, both and adjustable From simulation use results a 3-order above, Hilbert it can be, seen while that a k-based uses achieves a 1-order much Hilbert better From results. It on can residual simulation be seen energy from results than Figure above, 14 k-based that it can be energy seen and consumption consumes that a 23.9% usg less adjustable energy achieves than much s better results on residual uniform energy deployment. than k-based and consumes 23.9% less energy than uniform Sce all 8 deployment. s are deployed uniformly network, both and Sce adjustable all 8 suse area 3-order deployed Hilbert uniformly, while network, k-based bothuses a 1-order Hilbert and adjustable. It can be seen from use Figure a 3-order 14 that Hilbert energy, consumption while k-based usg adjustable uses a 1-order s Hilbert

18 Sensors 216, 16, Sensors Sensors 216, 216, 16, 16, It can be seen from Figure 14 that energy consumption usg adjustable s is much less than that with k-based and, which makes network save more energy. The k-based dras battery much faster than or two s due high packet transmission rate, while usg or two two s can can reduce packet packet transmission rate rate by usg by usg a higher a higher order Hilbert order Hilbert, thus, reducg thus reducg energy consumption. energy consumption. The energy consumption The energy with consumption an a with an a is even less than is even that with less a than that with. a In an adjustable. In an adjustable, transmission, ranges transmission sensor ranges s sensor communicate s with communicate with sk are reduced sk by are adaptable reduced by power adaptable control, which power is control, ma which reason is that ma adjustable reason that adjustable achieves better performance achieves better than performance s than energy s consumption. energy consumption. Scenario 4. Sensor Nodes Are Unevenly Deployed Network We want evaluate energy consumption k-based, and adjustable s a scenario a scenario with with different different densities densities specific subareas. specific Therefore subareas. Therefore deployments used deployments Scenario used 4 are Scenario same 4 as are those same Scenario as those 2. The Scenario experimental 2. The results experimental three results kds three deployment kds are deployment shown Figures are shown 15 17, respectively. Figures 15 17, respectively. Residual energy(j) Time(s) Time(s) k-based k-based a a Figure 15. The energy (a): C1, C2, C4 Figure 15. The residual energy network deployment (a): CC1, with 1, CC2, 2, CC4 4 are deployed with 5 C3 5 s and C C3 3 is deployed with 65 s. Residual energy(j) Time(s) Time(s) k-based k-based a a Figure 16. The energy (b): C1, C3 Figure 16. The residual energy network deployment (b): C1, are with 5 C3 Figure 16. The residual energy network deployment (b): C are deployed with 5 s and C4 are with 35 s. 1, C 3 are deployed with s and C2, C4 5 s and C are deployed with 35 s. 2, C 4 are deployed with 35 s. It can be seen from results above that a consumed 89.2%, 89.2% and It can be seen from results above that a consumed 89.2%, 89.2% and 89.12% 89.12% less energy, respectively, than three kds uneven less energy, respectively, than three kds uneven deployment. It is deployment. It is observed that energy consumption under k-based scenario is much observed that energy consumption under faster than that under and a k-based s. By scenario usg is much adaptable faster than power that control under scheme and and selectg a proper s. By usg order each adaptable subarea, power control average scheme energy and selectg consumption with adjustable is less than that with. In simulation, when sk traverses a subarea with high density, k-based contues use a 1-order Hilbert, thus resultg a high transmission rate and high energy consumption. Similarly,

19 Sensors 216, 16, proper Sensors 216, 16, order 291 each subarea, average energy consumption with adjustable is Sensors less 216, than16, that 291 with. In simulation, when sk traverses a subarea with high uses density, a d-order k-based Hilbert contues that is based use on a 1-order Hilbert density, thus whole resultg network, a so high compared transmission with uses a rate d-order adjustable and high Hilbert energy consumption., that is it based is still Similarly, on not flexible enough density reflect whole uses areal network, d-order Hilbert so deployment compared with that a is specific based adjustable subarea, on while density, an adjustable it is whole still not network, flexible, so enough compared proper reflect with adjustable real order is deployment used, crease it a is specific still PDR not subarea, accordg flexiblewhile enough an reflect adjustable density real specific deployment, subareas proper and a specific transmission subarea, order while used ranges crease an adjustable s are PDR reduced accordg, accordg proper density order is specific used this subarea, subareas creaserefore and PDR transmission accordg a ranges achieves density s best specific are energy reduced subareas efficiency accordg and among transmission se three order ranges kds this trajecries, subarea, s are refore reduced and we can accordg a conclude that it achieves is better order this use best subarea, an energy a efficiency refore trajecry among a than se a k-based three kds achieves or trajecries, best trajecry energy and efficiency we such can a conclude network among se that with it uneven three is better kds use deployment. trajecries, an a andtrajecry we can conclude than a k-based that it isor better use trajecry an a such trajecry a network than awith k-based uneven or deployment. trajecry such a network with uneven deployment. Residual Residual energy(j) energy(j) k-based k-based a a Figure 17. The residual energy network Time(s) deployment (c): C1 is deployed with 5 s, Figure C2 is deployed 17. The residual with 15 energy s, C3 and network C4 are deployed deployment with 3 s. (c): C1 is deployed with 5 s, Figure 17. The residual energy network deployment (c): C 1 is deployed with 5 s, C2 is deployed with 15 s, C3 and C4 are deployed with 3 s. C 2 is deployed with 15 s, C 3 and C 4 are deployed with 3 s Evaluation on Packet Delivery Ratio a Cluster Model 5.3. Evaluation on Packet Delivery Ratio Cluster Model 5.3. Scenario Evaluation 5. Sensor on Packet Nodes Delivery Are Unevenly Ratio a Deployed Cluster Model Network Scenario In this 5. Sensor scenario, Nodes number Are Unevenly s Deployed is set 8 and Network sensor s are deployed usg three In kds this scenario, deployments number s Scenario is set 2. Then 8 and PDR sensor for all s three are deployed kds trajecries usg three which kds are proposed: deployments a Scenario a cluster 2. Thenmodel, PDR k-based for all three and kds trajecries are which evaluated. are are proposed: The results a se three a cluster kds a cluster model, model, k-based deployments k-based and are shown and Figures are evaluated. 18 2, are The evaluated. respectively. results The se results three kds se three deployments kds aredeployments shown Figures are shown 18 2, respectively. Figures 18 2, respectively. Packet Packet delivery delivery ratio ratio each each subarea(%) subarea(%) Time(s) k-based k-based a a Subarea ID Figure 18. The PDR k-based, and adjustable Subarea ID deployment (a): C1, C2, Figure 18. The PDR k-based, and adjustable deployment (a): C 1, C 2, Figure C4 are deployed 18. The PDR with 5 k-based, s and C3 is and deployed adjustable with 65 s. deployment (a): C1, C2, C 4 are deployed with 5 s and C 3 is deployed with 65 s. C4 are deployed with 5 s and C3 is deployed with 65 s.

20 Sensors 216, 16, Sensors 216, 16, 291 Sensors 216, 16, Packet delivery ratio each subarea(%) Subarea ID Subarea ID k-based k-based a a (b): C1, C3 Figure 19. The PDR k-based, and adjustable deployment (b): (b): CC1, 1, CC3 3 CC2, C4 2, C 4 are deployed with 5 s and C2, C4 are deployed with 35 s. Packet delivery ratio each subarea(%) Subarea ID Subarea ID k-based k-based a a Figure 2. The PDR k-based, and adjustable deployment (c): C1 is Figure 2. The PDR k-based, and adjustable deployment (c): CC1 1 is deployed with 5 s, C2 is deployed with 15 s, C3 and C4 are deployed with 3 s. deployed with 5 s, CC2 2 is deployed with 15 s, CC3 3 and C4 C 4 are deployed with 3 s. Table Table gives gives overall overall PDR PDR se se three three trajecries trajecries this this scenario. scenario. Table 5 gives overall PDR se three trajecries this scenario. Table 5. The Overall PDR k-based, and adjustable s Scenario 5. Table 5. The Overall PDR k-based, and adjustable s Scenario 5. Table 5. The Overall PDR k-based, and adjustable s Scenario 5. Node Deployment (a) Node Deployment (b) Node Deployment (c) Node Deployment (a) Node Deployment (b) Node Deployment (c) NodeIn Scenario In Scenario In Scenario In Deployment Scenario 5 (a) Node Deployment In Scenario 5 (b) NodeIn Deployment Scenario 5 (c) Trajecry Overall PDR Trajecry Overall PDR Trajecry Overall PDR Trajecry In Scenario Overall 5 PDR Trajecry In Scenario Overall 5 PDR Trajecry In Scenario Overall 5 PDR k-based 8.88% k-based 75.38% k-based 77.13% k-based Trajecry Overall 8.88% PDR Trajecry k-based Overall 75.38% PDR k-based Trajecry Overall 77.13% PDR 82.38% 81.75% 86.75% k-based 8.88% 82.38% k-based 75.38% 81.75% k-based 77.13% 86.75% a 86.% a 83.25% a 88.63% a 82.38% 86.% a 81.75% 83.25% a 86.75% 88.63% a 86.% a 83.25% a 88.63% It It can can be be seen seen from from Table Table 5 that that a a cluster cluster model model achieves achieves desired desired overall overall PDR and on average it outperforms k-based and s PDR by 11.4% and 3.64%, It respectively, can PDR be and seen on from average this Table it scenario. 5 outperforms that a k-based and cluster s model achieves PDR by 11.4% desired and overall 3.64%, respectively, In PDR subarea and on with average this scenario. sparse it outperforms deployment, k-based CHs and have collect s all sensg PDR by 11.4% ir and clusters 3.64%, In respectively, a subarea with first, and n this sparse send scenario. deployment, CHs have collect all sensg ir clusters first, and n send is is sk, sk, which which also also means means that that each each CH CH has has send send more more sensg In a sensg subarea with sparse sk sk than deployment, than a sensor sensor CHs sgle-hop have collect sgle-hop model, all model, where sensg where sensor sensor s ir s can clusters can send first, send sensg and n sensg send is sk sk, sk directly. which directly. However, also means However, sce that sce each time CH time has send sk more sk stays stays sensg one one subarea subarea is is same same as sk as that than that a sensor sgle-hop sgle-hop model, sgle-hop model, such such a time model, time period where period may may not not be sensor be enough s enough for for can send sk sensg sk collect collect all all sensg sk sensg directly. from from all However, all CHs, CHs, which sce which return time return results results sk decrease stays decrease PDR one subarea PDR for for this is this subarea. same subarea. In as In a subarea that subarea with sgle-hop with dense dense model, deployment, such a time deployment, period number may number not sensor be enough sensor s for s one one cluster sk cluster creases collect creases due all due sensg higher higher from density, all CHs, density, thus which thus CHs return CHs have have results send send even even more decrease more PDR for this sk. subarea. sk. However, In a subarea However, a with dense a can deployment, can dynamically dynamically select number select a higher sensor higher order order Hilbert s Hilbert, one cluster, which which results results an an crease crease stayg stayg time time sk sk this this subarea, subarea, and and thus thus re re may may be be enough enough

21 Sensors 216, 16, creases Sensors 216, due16, 291 higher density, thus CHs have send even more Sensors 216, 16, sk. However, a can dynamically select a higher order Hilbert, which results an time crease for stayg sk time fully collect all sksensg this subarea, from and all thus CHs, re refore, may beit enough can result time for sk fully collect all sensg from all CHs, refore, it can result time for crease sk PDR fully for collect this subarea. crease PDR for this subarea. all sensg from all CHs, refore, it can result creaseby comparg PDR for this subarea. overall PDR a with different communication models, By overall PDR a with different communication models, which By comparg are sgle-hop overall model PDR Scenario a and cluster with model different this scenario, communication it can be models, seen which are sgle-hop model Scenario 2 and cluster model this scenario, it can be seen which that are a sgle-hop model sgle-hop Scenario model 2 and outperforms cluster model that this cluster scenario, model it can PDR be seen by 9.69% that a a sgle-hop model outperforms that a cluster model PDR by 9.69% that on average. a on average. a sgle-hop model outperforms that a cluster model PDR by 9.69% on average. For cluster model, first place, this kd two-step communication has effects on For For a a cluster model, first place, this kd two-step communication has has effects on on performance features such as latency, each CH (relay ) curs additional delays, e.g., for queug performance features such as as latency, each CH (relay ) curs additional delays, e.g., e.g., for for queug. On or hand, communications over two-step model may ten be less reliable,.. On On or hand, communications over a two-step model may mayten be be less less reliable, enhancg communication time and overhead, sce same must be transmitted several enhancg communication time and overhead, sce same must be betransmitted several times (and limited channel capacity must be shared among relays for both receivg and times times transmittg), (and (and limited so maximum channel capacity achievable must throughput be shared among will also suffer. relays for Furr, forboth bothreceivg reliability and can be and transmittg), affected by havg so so multiple maximum s achievable (relays) volved throughput will also communication suffer. Furr, Furr, reliability which creases reliability can can be risk be affected affected disruptions by by havg havg due multiple multiple lk s s or (relays) (relays) failures. volved volved Therefore, communication PDR for a which which trajecry creases cluster risk disruptions disruptions model less duethan due lk that or lk or sgle-hop failures. failures. model. Therefore, Therefore, PDR PDR for a for a trajecry trajecry a cluster a cluster model model is less than that a sgle-hop model. is less than that a sgle-hop model Evaluation on Network Lifetime Cluster Model Evaluation Evaluation on on Network Network Lifetime Lifetime a a Cluster Cluster Model Model Scenario 6. Sensor Nodes Are Unevenly Deployed Network Scenario 6. Sensor Nodes Are Unevenly Deployed Network Scenario 6. Sensor Nodes Are Unevenly Deployed Network In this scenario, number s is set 8, and s are also deployed usg three In this scenario, number s is set 8, and s are also deployed usg three kds In this scenario, deployments number used s Scenario is set 2. 8, The and simulation s aresults also deployed are shown usg Figures three 21 23, kds kds deployments used Scenario 2. The simulation results are shown Figures 21 23, respectively. deployments used Scenario 2. The simulation results are shown Figures 21 23, respectively. respectively. Residual energy(j) k-based k-based a a Time(s) Time(s) Figure 21. The residual energy network deployment (a): C1, C2, C4 Figure are deployed with Figure The The residual energy network deployment (a): C1, C 1, C2, C 2 C4, Care 4 are deployed with with 5 s and C3 is deployed with 65 s. 5 5 s s and and C 3 C3 is is deployed with 65 s. Residual energy(j) k-based k-based a a Time(s) Time(s) Figure 22. The residual energy network deployment (b): C1, C3 are deployed with 5 Figure 22. The residual energy network deployment (b): C1, C3 Figure are deployed with 5 s 22. and The C2, residual C4 are deployed energy with 35 network s. deployment (b): C 1, C 3 are deployed with s and C2, C4 are deployed with 35 s. 5 s and C 2, C 4 are deployed with 35 s.

22 Sensors 216, 16, Sensors 216, 16, Residual energy(j) k-based a Time(s) C1 Figure 23. The residual energy network deployment (c): C is 5 1 is deployed with 5 s, C2 is C3 C4 C are with 3 2 is deployed with 15 s, C 3 and C 4 are deployed with 3 s. It can be seen from above results that a cluster model consumed It can be seen from above results that a cluster model consumed 94.82%, 94.6% and 94.5% less energy than three kds uneven 94.82%, 94.6% and 94.5% less energy than three kds uneven deployment, respectively. deployment, respectively. The simulation results show that Scenario 6, k-based is also first one dra its The simulation results show that Scenario 6, k-based is also first one dra its battery. There is no doubt that high retransmission rate k-based case results high battery. There is no doubt that high retransmission rate k-based case results high energy consumption. By usg cluster model and adaptive power control, transmission energy consumption. By usg cluster model and adaptive power control, transmission power sensor s is mimized while transmission power CHs is adjusted accordg power sensor s is mimized while transmission power CHs is adjusted accordg order trajecry a cluster model. Therefore, a order trajecry a cluster model. Therefore, a cluster model achieves best energy efficiency among se three trajecries. cluster model achieves best energy efficiency among se three trajecries. By comparg energy consumption for a sgle-hop model By comparg energy consumption for a sgle-hop model Scenario and a cluster model this scenario, it can be calculated that Scenario 4 and a cluster model this scenario, it can be calculated that a cluster model consumed 52.85%, 41.8% and 45.35% less energy than that a cluster model consumed 52.85%, 41.8% and 45.35% less energy than that sgle-hop model three kds deployment, respectively. sgle-hop model three kds deployment, respectively. By usg cluster model, garg process a consists two By usg cluster model, garg process a consists two steps. steps. In first step, most sensor s only need transmit ir sensg CHs In first step, most sensor s only need transmit ir sensg CHs which which are close m or rar than sk that is far from m. Therefore, are close m or rar than sk that is far from m. Therefore, reduction reduction transmission distance can ultimately result an energy consumption decrease based on transmission distance can ultimately result an energy consumption decrease based on our energy our energy model. In second step, CHs can send gared sensg sk model. In second step, CHs can send gared sensg sk usg usg transmission power adjusted by adaptive power control method, however, due transmission power adjusted by adaptive power control method, however, due crease crease size and communication distance, energy consumption for CHs is higher than size and communication distance, energy consumption for CHs is higher than that that sgle-hop model, but it can be known that, cluster model, most sensor s sgle-hop model, but it can be known that, cluster model, most sensor s only need only need communicate with CHs over a small distance while only a few s (which are communicate with CHs over a small distance while only a few s (which are CHs) need CHs) need communicate with sk over a long distance, and refore, overall communicate with sk over a long distance, and refore, overall energy consumption energy consumption for whole network can be reduced dramatically. for whole network can be reduced dramatically. From above work, we can draw conclusion that, a sgle-hop From above work, we can draw conclusion that, a sgle-hop communication model has highest PDR and a cluster model has communication model has highest PDR and a cluster model has highest energy efficiency. Therefore, communication model can be properly selected maximize highest energy efficiency. Therefore, communication model can be properly selected maximize PDR or network lifetime, accordg application requirement. PDR or network lifetime, accordg application requirement. 6. Conclusions 6. Conclusions In this paper, an adjustable trajecry design method is developed by usg Hilbert spacefillg, which full coverage network and desired packet delivery ratio are In this paper, an adjustable trajecry design method is developed by usg Hilbert space-fillg, which full coverage network and desired packet delivery ratio are achieved. achieved. By selectg proper order Hilbert based on density specific By selectg proper order Hilbert based on density specific subareas, subareas, more sensg can be gared and network performance is improved densely more sensg can be gared and network performance is improved densely deployed deployed networks with uneven deployment. In addition, proposed trajecry also saves networks with uneven deployment. In addition, proposed trajecry also saves energy and energy and prolongs network lifetime. Simulation results demonstrate that proposed trajecry is flexible and can adapt networks with different density or deployment. It is

23 Sensors 216, 16, prolongs network lifetime. Simulation results demonstrate that proposed trajecry is flexible and can adapt networks with different density or deployment. It is also verified that proposed trajecry outperforms existg classic trajecries packet delivery ratio and network energy efficiency with designed adaptable power control scheme. One future project is how design a more flexible method network partition, so sensor s a subarea can work cooperatively collect sensg usg a smaller transmission range, and we can consider usg more sophisticated routg mechanisms, such as opportunistic routg, deliver sensg sk, order decrease average packet delay each round garg, which is very important event-driven WSNs. Anor future direction this work is how use multiple sks crease garg efficiency, and hoe apply an energy rechargeable algorithm and energy consumption model considerg energy consumptions both transmission and computg [24 26]. Acknowledgments: The authors would like appreciate all anonymous reviewers for ir sightful comments and constructive suggestions polish this paper high quality. The research was supported by National Natural Science Foundation Cha (No ), Shanghai Engeerg Research Center Project (No. 1414) and program for Pressor Special Appotment (Eastern Scholar) at Shanghai Institutions Higher Learng jotly. All funds declare cover costs publish open access. Author Contributions: Guisong Yang and Naixue Xiong origated this work and proposed scheme; Shuai Liu conceived experiments and designed simulation; Xgyu He contributed ory studies and refement scheme; Chunxue Wu and Shuai Liu helped Guisong Yang improve quality this work and prepared manuscript. Conflicts Interest: The authors declare no conflict terest. References 1. Gu, Y.; Ren, F.; Ji, Y.; Li, J. The Evolution Sk Mobility Management Wireless Sensor Networks: A Survey. IEEE Commun. Surv. Tur. 216, 18, [CrossRef] 2. Wang, T.; Peng, Z.; Liang, J.; Wen, S.; Bhuiyan, M.Z.A.; Cai, Y.; Cao, J. Followg Targets for Mobile Trackg Wireless Sensor Networks. ACM Trans. Sens. Netw. 216, 12, 24. [CrossRef] 3. Perillo, M.; Cheng, Z.; Hezelman, W. An analysis strategies for mitigatg sensor network hot spot problem. In Proceedgs International Conference on Mobile and Ubiquius Systems: Networkg and Services, San Diego, CA, USA, July Wang, B.; Xie, D.; Chen, C.; Ma, J.; Cheng, S. Employg sk event driven wireless sensor networks. In Proceedgs Vehicular Technology Conference, Mara Bay, Sgapore, May Vlajic, N.; Stevanovic, D.; Spanogiannopoulos, G. Strategies for improvg performance IEEE /ZigBee WSNs with path-constraed sk(s). Comput. Commun. 211, 34, [CrossRef] 6. Gao, S.; Zhang, H.; Das, S.K. Efficient Data Collection Wireless Sensor Networks with Path-Constraed Mobile Sks. IEEE Trans. Mob. Comput. 211, 1, [CrossRef] 7. Ma, M.; Yang, Y. Data Garg Wireless Sensor Networks with Mobile Collecrs. In Proceedgs International Parallel & Distributed Processg Symposium, Miami, FL, USA, April Wichmann, A.; Korkmaz, T. Smooth path construction and adjustment for multiple sks wireless sensor networks. Comput. Commun. 215, 72, [CrossRef] 9. Poe, W.Y.; Beck, M.; Schmitt, J.B. Planng trajecries multiple sks large-scale, time-sensitive WSNs. In Proceedgs 211 International Conference on Distributed Computg Sensor Systems and Workshops (DCOSS), Barcelona, Spa, June Pradeepa, K.; Anne, W.R.; Duraisamy, S. Improved sensor network lifetime usg multiple sks: A new predetermed trajecry. In Proceedgs International Conference on Computg Communication and Networkg Technologies (ICCCNT), Karur, India, July Viana, A.C.; de Amorim, M.D. Sensg and actg with predefed trajecries. In Proceedgs HeterSanet, Hong Kong, Cha, 3 May Ghafoor, S.; Cho, S.; Park, S.H. Dynamic Trajecry Design Mobile Sk Wireless Sensor Network. In Proceedgs ITC-CSCC 29, Jeju, Korea, 5 8 July 29.

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