Multi-Round Sensor Deployment for Guaranteed Barrier Coverage

Transcription

1 Tis full text paper was peer reviewed at te direction of IEEE Communications Society subject matter experts for publication in te IEEE INFOCOM 21 proceedings Tis paper was presented as part of te main Tecnical Program at IEEE INFOCOM 21. Multi-Round Sensor Deployment for Guaranteed Barrier Coverage Guanqun Yang and Daji Qiao Iowa State University, Ames, IA 511 {gqyang, Abstract Deploying wireless sensor networks to provide guaranteed barrier coverage is critical for many sensor networks applications suc as intrusion detection and border surveillance. To reduce te number of sensors needed to provide guaranteed barrier coverage, we propose multi-round sensor deployment wic splits sensor deployment into multiple rounds and can better deal wit placement errors tat often accompany sensor deployment. We conduct a compreensive analytical study on multi-round sensor deployment and identify te tradeoff between te number of sensors deployed in eac round of multi-round sensor deployment and te barrier coverage performance. Bot numerical and simulation studies sow tat, by simply splitting sensor deployment into two rounds, guaranteed barrier coverage can be acieved wit significantly less sensors comparing to singleround sensor deployment. Moreover, we propose two practical solutions for multi-round sensor deployment wen te distribution of a sensor s residence point is not fully known. Te effectiveness of te proposed multi-round sensor deployment strategies is demonstrated by numerical and simulation results. I. INTRODUCTION Recently, barrier coverage [1] wit wireless sensor networks as received great attention. Te goal is to deploy a cain of wireless sensors in te barrier, wic usually is a long belt region, to prevent mobile objects from crossing te barrier undetected. Applications of barrier coverage include intrusion detection and border surveillance [2]. In tis paper, we study te problem of guaranteed barrier coverage and te goal is to guarantee tat a barrier is covered wit probability one using as few sensors as possible. Various scemes ave been proposed to reduce te number of sensors needed to cover a barrier, suc as information barrier coverage wic exploits te collaboration and information fusion between nearby sensors [3]. We approac tis problem from a different angle via investigating sensor deployment strategies, more specifically, line-based sensor deployment strategies. Wit line-based sensor deployment, sensors are deployed along a line, e.g., sensors are airdropped by an aircraft along a deployment line, and te final residence points of te sensors are distributed along te line wit random offsets. Recent work in [4] sows tat tis is a more realistic sensor placement model and sensor deployment strategies based on tis model can acieve more efficient barrier coverage wit Te researc reported in tis paper was supported in part by te Information Infrastructure Institute icube of Iowa State University and te National Science Foundation under Grants CNS and CNS less sensors tan tose based on te Poisson point process sensor placement model. In practice, due to wind and oter environmental factors, large placement errors often accompany sensor deployment. For example, wen sensors are airdropped into te target region from an aircraft, te final residence points of te sensors may deviate muc from te intended deployment points. To provide guaranteed barrier coverage wit te conventional single-round sensor deployment, sensors need to be deployed in a conservative manner i.e., wit small deployment interval and ence more sensors are needed to counter te randomness during te sensor deployment process. In comparison, by splitting sensor deployment into multiple rounds, it is safe to be more aggressive i.e., wit larger deployment intervals in te earlier rounds and ten deploy sensors more conservatively in te final round to fill te coverage gaps generated in te previous rounds. As a result, a significant number of sensors may be saved wit multi-round sensor deployment. Splitting sensor deployment into multiple rounds may incur iger deployment cost, e.g., an aircraft needs to fly along te deployment line multiple times to accomplis te task. From te analytical and simulation studies, we ave an interesting discovery tat te optimal two-round sensor deployment strategy yields te same barrier coverage performance as oter optimal strategies wit more tan two rounds. Tis result is particularly encouraging as it implies tat te best barrier coverage performance can be acieved wit low extra deployment cost by deploying sensors in two rounds. Furtermore, we propose two practical solutions, te tworound minimax solution and te pilot deployment solution, to deal wit realistic situations wen te knowledge about te deviation of sensors residence points wit respect to teir intended deployment points is not fully available. Te pilot deployment solution performs particularly well and te idea is to introduce an additional pilot round wic deploys a small number of sensors to estimate te distribution of sensors residence points and ten use tis information to aid te following rounds of sensor deployment. Te rest of te paper is organized as follows. We discuss te related work in Section II. Ten, we give te system models and te problem statement in Section III. In Section IV, we analyze multi-round sensor deployment in detail. Section V describes te two-round minimax solution and te pilot deployment solution. Numerical and simulation results are sown in Section VI. Section VII discusses future work and related issues, and Section VIII concludes te paper /1/$ IEEE

2 Tis full text paper was peer reviewed at te direction of IEEE Communications Society subject matter experts for publication in te IEEE INFOCOM 21 proceedings Tis paper was presented as part of te main Tecnical Program at IEEE INFOCOM 21. II. REATED WORK Te area coverage problem based on uniform and random distribution of sensor deployment as been well-studied in te past [5] [9]. In tese works, a sensor s landing position is assumed to be uniformly and randomly distributed over te monitored area. Controlled sensor deployment for target detection as been studied in [1] [16]. In [1] [14], a sensor s position can be arbitrarily controlled, i.e., te placement of eac deployed sensor is not subject to any placement error. Based on tis assumption, various scemes ave been proposed to determine te optimal sensor placement strategy to cover te target region wit as few sensors as possible. By comparison, in [15], [16], sensor deployment is partially controlled, i.e., a sensor s position is subject to random offsets wit respect to te deployment point. Sensor mobility as been incorporated into sensor deployment framework [17] [19], wic offers more flexibility for designing more efficient sensor deployment strategies for area coverage. Incremental sensor deployment as been studied in [2], [21], were algoritms are designed to deploy extra sensors after te initial deployment to improve te area coverage performance. Te problems of sensor deployment for oter purposes, e.g., connectivity, data aggregation, or energy efficiency, ave also been investigated [22] [26]. None of te above works addresses barrier coverage. Recently, barrier coverage [1], [5] as attracted great attention. [4] studies te line-based sensor deployment strategies and provides bot analytical results and interesting observations about ow line-based sensor deployment can improve te barrier coverage performance tan two-dimensional uniform sensor deployment. Teoretical foundations for weak and strong barrier coverage of a randomly deployed sensor network are studied in [1], [27]. Te autors of [28] and [29] study te coverage of a finite-size barrier and provide analytical metods to estimate te required density of deployed sensors for acieving barrier coverage or measuring te quality of barrier coverage. Centralized and distributed algoritms for providing barrier coverage are proposed and evaluated in [3] and [31] [33], respectively. III. MODES AND PROBEM STATEMENT A. Sensing and Coverage Models We consider a network of wireless sensors deployed to monitor a barrier wic is a long belt region of lengt l wit two parallel sides: entrance side and destination side as sown in Fig. 1. An object or intruder may cross te barrier via an intruding pat starting at te entrance side and ending at te destination side. We assume tat sensors are aware of teir locations. We also assume tat sensors communication range is reasonably large so tat network is connected and sensors can report teir location information to te sink. For simplicity, we assume an ideal /1 disc sensing model to demonstrate ow a well-designed multi-round sensor deployment sceme may elp reduce te number of sensors needed to guarantee barrier coverage. Specifically, we assume tat eac sensor as a sensing disc wit a radius of R s. An object witin outside a sensor s sensing disc is detected by te sensor wit probability one zero. In Section VII, we will discuss ow to extend our work under oter sensing models. destination side tleft entrance side Fig. 1. Illustration of barrier coverage and sensor deployment. Sensors are deployed along te deployment line wic is parallel to te destination side of te barrier. Deployment points are along and sown as small squares. Residence points of te sensors are sown as small dots and tey deviate from te deployment points. Tese sensors form a barrier coverage set because no intruding pat can cross te barrier witout being sensed by one of te sensors. Two special points, t left and t rigt, will be explained in Teorem 2. DEFINITION 1 Coverage of a Pat: A pat across a barrier is covered if tere exists a point along te pat tat is witin at least one sensor s sensing disc. DEFINITION 2 Barrier Coverage: A barrier is covered if all possible pats across te barrier are covered. Tis is also known as strong barrier coverage. DEFINITION 3 Barrier Coverage Set: A barrier coverage set is a set of sensors tat can togeter cover te barrier. DEFINITION 4 inkage: We say tat two sensors a and b ave linkage if tere exist a series of line segments connecting a and b, and every point on te line segments is witin at least one sensor s sensing disc. Particularly, we say tat a and b ave direct linkage if every point on te line segment connecting tem is witin te sensing disc of eiter a or b. B. Deployment Model We assume tat sensors are static once tey are deployed. We assume tat sensors are deployed along a deployment line wic is parallel to te destination side of te barrier. Te assumption of a straigt deployment line is due to practical considerations. For example, wen using an aircraft to airdrop sensors, it is more plausible tat te aircraft flies along a straigt line rater tan a zigzag route. We define deployment point as te point location were a sensor is to be deployed. In practice, due to environmental factors and terrain caracteristics, te deployment point likely is not te location were te sensor finally resides. Te sensor may reside at points around te deployment point according to a certain probability density function pdf. We define residence point as te point location were a sensor finally resides. Fig. 1 sows an example of deployment line, deployment points and residence points. et x i,y i denote te coordinates of te deployment point t i. Assume tat te residence point of a sensor wose deployment point is t i follows te pdf fx i,y i t i=f err x x i,y i y i. We assume tat f err as a finite closed support wic is a disc wit a radius of R err and circular-symmetric wit respect to x i,y i. An example of f err is a truncated two-dimensional Gaussian distribution. For te clarity of presentation, we put te deployment line along te X-axis of trigt

3 Tis full text paper was peer reviewed at te direction of IEEE Communications Society subject matter experts for publication in te IEEE INFOCOM 21 proceedings Tis paper was presented as part of te main Tecnical Program at IEEE INFOCOM 21. a Cartesian coordinate system wit its left end sitting at te origin. Tus, for eac deployment point t i,weavey i =. THEOREM 1: If R err is smaller tan te sensing radius R s, two sensors are guaranteed wit probability one to ave direct linkage between tem wen teir deployment points are less tan 2R s 2R err apart. On te oter and, if R err R s, direct linkage between two sensors cannot be guaranteed regardless of te distance between teir deployment points. Te proof of te teorem is straigtforward and omitted due to space limitation. It is interesting to see tat, if f err does not ave a finite closed support i.e., R err =, direct linkage between sensors cannot be guaranteed and, consequently, guaranteed barrier coverage cannot be acieved. Similarly, we ave te following teorem. THEOREM 2: If R err <R s, a sensor is guaranteed to cover te left rigt boundary of te barrier if its deployment point is t left t rigt, wic is R s R err away from te left rigt end of te deployment line. t left and t rigt are sown in Fig. 1. If R err R s, suc guarantee cannot be acieved. C. Problem Statement Given tat sensors can be deployed in multiple 2 rounds, our goal is to design a proper multi-round sensor deployment sceme to provide guaranteed barrier coverage wit as few sensors as possible. More specifically, we aim to strategically decide te deployment points of sensors at eac round so tat te total number of sensors needed to form a barrier coverage set is minimized. In Section VII, we will discuss ow to extend our work to acieve guaranteed k-barrier coverage. IV. MUTI-ROUND SENSOR DEPOYMENT In tis section, we first study te two-round sensor deployment problem, were sensors are deployed in two rounds to cover a barrier. Ten, we extend our analysis and solution to te general m-round m >2 sensor deployment problem. We assume tat te pdf of a sensor s residence point wit respect to its deployment point f err is known apriori. In Section V, we will loose tis assumption and study te problem under practical considerations were f err is only partially known. A. Overview of Two-Round Sensor Deployment Tere are infinite number of ways to deploy sensors in two rounds to cover a barrier. For example, we may coose to deploy different numbers of sensors in eac round, and tere are infinite number of possible deployment points to coose for eac sensor. In general, it is difficult to enumerate all te possibilities. In tis paper, we study a specific set of two-round sensor deployment strategies described below. DEFINITION 5 Gap Distance: Gap distance denoted as between two adjacent sensors is defined as te distance between te leftmost and rigtmost deployment points of te sensors deployed to fill te gap between tem. DEFINITION 6 Two-Round Sensor Deployment Strategy: We consider a special set of two-round sensor deployment strategies tat distribute sensors evenly in eac gap. Specifically, it operates as follows: In te first round, te entire barrier is a single big gap. According to Teorem 2, te leftmost and rigtmost deployment points of te sensors deployed to fill te gap must be t left and t rigt, respectively. We use a special symbol l to denote te corresponding gap distance wic is l = l 2R s R err.etn l, 2, 1 denote te number of sensors deployed in te first round. If N l, 2, 1 = 1, eiter t left or t rigt sall be selected as te deployment point. If N l, 2, 1 2, te deployment points sall be evenly distributed between t left and t rigt l N l,2,1 1 apart. wit a distance of I l, 2, 1 = Tere are tree possible outcomes after te first round of deployment: If N l, 2, 1 =, since no sensors are deployed in te first round, te entire barrier needs to be filled in te second round. Tis in fact is equivalent to singleround sensor deployment. If N l, 2, 1 = 1, only a single gap is generated to te rigt left of te deployed sensor if te deployment point of te sensor is at t left t rigt. Tis is because te sensor wit deployment point at t left t rigt is guaranteed to cover te left rigt boundary of te barrier. If N l, 2, 1 2, multiple gaps may be generated. In te second round, for eac gap generated in te first round, deploy a minimum number of evenly distributed sensors to guarantee tat it is covered. Note tat te key assumption in tis type of strategies is tat sensors are evenly distributed in eac gap wile different deployment intervals may be used to deploy sensors in different gaps. We coose tis specific strategy because it is quite common in realistic scenarios. et Ntotal l, 2,N l, 2, 1 denote te expected total number of sensors needed to cover te barrier wen a two-round sensor deployment strategy described above is used. Clearly, wen N l, 2, 1 is small, fewer sensors are deployed in te first round but larger coverage gaps may be generated. On te oter and, wen N l, 2, 1 is large, more sensors are deployed in te first round but smaller gaps are to be covered in te second round. So tere is a tradeoff. Our goal is to find te optimal N l, 2, 1 so tat Ntotal l, 2,N l, 2, 1 is minimized: l, 2,N l, 2, 1 N l, 2, 1 = arg min N l,2,1 = l, 2 = N total l, 2,N l, 2, 1. B. Teoretical Analysis of Two-Round Sensor Deployment We now investigate te relation between N l, 2, 1 and l, 2,N l, 2, 1. We first study te case wen N l, 2, 1 2. Consider two adjacent sensors a and b deployed in te first round, as sown in Fig. 2. et s a and s b denote teir residence points wit coordinates of x a,y a and x b,y b, respectively, and tere may be a coverage gap between tem. To fill te gap, additional sensors need to be deployed in te second round. According to Teorem 1, te leftmost and 1

4 Tis full text paper was peer reviewed at te direction of IEEE Communications Society subject matter experts for publication in te IEEE INFOCOM 21 proceedings Tis paper was presented as part of te main Tecnical Program at IEEE INFOCOM 21. Sa:xa,ya 2Rs-Rerr t tr 2Rs-Rerr Sb:xb,yb Fig. 2. Two adjacent sensors a and b are deployed in te first round and tere may be a coverage gap between tem. t and t R are te leftmost and rigtmost deployment points of te sensors tat are deployed in te second round to fill te gap. Sa: xa,ya ta: xa,ya u v 2Rs-Rerr Rerr t tr z 2Rs-Rerr zr Sb: xb,yb tb: xb,yb Fig. 4. Wen te number of sensors deployed in te first round is small, some big coverage gaps wit > may be generated and we need to deploy more tan one sensors in te second round to cover eac of tem. rigtmost deployment points denoted as t and t R of tese additional sensors must be at a distance of 2R s R err tos a and s b, respectively. Te distance between t and t R is te corresponding gap distance denoted as. Note tat, wen te left or rigt boundary of a gap is te left or rigt boundary of te barrier, t t left or t R t rigt. We analyze te following terms: i P zero I l, 2, 1 te probability tat tere is no coverage gap between sensors a and b, i.e., tere exists a direct linkage between tem; ii f HIl,2,1 te pdf of te gap distance between a and b; iii P one I l, 2, 1 te probability tat te gap between a and b can be filled by deploying at most one additional sensor in te second round. 1 P zero I l, 2, 1: et t a and t b denote te deployment points of sensors a and b, and te coordinates of t a and t b are x a, and x a + I l, 2, 1,, respectively. Te residence points s a and s b are independently and identically distributed according to te pdf of f err x a x a,y a and f err x b x a + I l, 2, 1,y b, respectively. To guarantee a direct linkage between two sensors, s b must be witin te circle centered at s a wit a radius of 2R s, as sown in Fig. 3. et A 1 x a,y a,i l, 2, 1 denote te intersection region between te circle centered at s a wit a radius of 2R s and te circle centered at t b wit a radius of R err, wic is sown as te saded area in te figure. Ten we ave: P zeroi l, 2, 1 = f err A 1x a,y a,i l,2,1 a a, a f err x a xa,y a a a a x b xa + I l, 2, 1,y b dx s b b b err b b, b b dy b dx a dy a. Fig. 3. To guarantee a direct linkage between two adjacent sensors a and b deployed in te first round, s b must be inside te saded area. t a and t b are te deployment points, and s a and s b are te residence points. 2 f HIl,2,1: As sown in Fig. 4, t R is te intersection point of te deployment line and te circle centered at s b wit a radius of 2R s R err.etz and z R denote te distance between t a and t, and between t b and t R, 2 respectively. Ten we ave = I l, 2, 1 z z R.Drawa circle wit t at te center and a radius of 2R s R err.etu and v denote te intersection points of tis circle and te circle centered at t a wit a radius of R err. Note tat t remains te same as long as s a falls on te arc ûv. Terefore, te pdf of z is a line integral: f Zz = x a,y a ûvz ferr x a xa,y a dx a dy a. 3 Since te relation between t b and t R is te same as tat between t a and t, z R and z ave te same pdf. Terefore, te pdf of is f HIl,2,1 =f HIl,2,1I l, 2, 1 z z R = f Zz f ZI l, 2, 1 z dz. 3 P one I l, 2, 1: Basedonteresultoff HIl,2,1, we can obtain P one I l, 2, 1 witout muc difficulty. We know tat, wen, a linkage between tem can be guaranteed by deploying at most one additional sensor at eiter t or t R. On te oter and, suc guarantee cannot be acieved if >. Terefore, we ave: PoneI l, 2, 1 = f HIl,2,1 d. 5 Note tat P one I l, 2, 1 P zero I l, 2, 1 is te probability tat tere exists a coverage gap between two adjacent sensors deployed in first round and te gap can be filled by deploying exactly one additional sensor in te second round. 4 Calculation of Ntotal l, 2,N l, 2, 1: Now we are ready to analyze te relation between l, 2,N l, 2, 1 and N l, 2, 1.et N 2 I l, 2, 1 denote te average number of additional sensors needed to guarantee a linkage between two adjacent sensors, say a and b, deployed in te first round. We ave te following cases: Wen, we need at most one additional sensor to guarantee a linkage between a and b. Tere are two subcases: wit probability P zero I l, 2, 1, tere already is a direct linkage between a and b and no additional sensor is needed; wit probability P one I l, 2, 1 P zero I l, 2, 1, one additional sensor is needed. Wen >, we need at least 1+ 2R s R err additional sensors to guarantee a linkage between a and b. 4

5 Tis full text paper was peer reviewed at te direction of IEEE Communications Society subject matter experts for publication in te IEEE INFOCOM 21 proceedings Tis paper was presented as part of te main Tecnical Program at IEEE INFOCOM 21. Terefore, we ave: N 2I l, 2, 1 = P zeroi l, 2, 1 +P onei l, 2, 1 P zeroi l, 2, f HIl,2,1 d 2R s R err = P onei l, 2, 1 P zeroi l, 2, P onei l, 2, 1 + f HIl,2,1 d 2R s R err = 1 P zeroi l, 2, 1 + f HIl,2,1 d. 2R s R err 6 Finally, Ntotal l, 2,N l, 2, 1 can be calculated by: l, 2,N l, 2, 1 = N l, 2, 1+N l, 2, 1 1 N 2 I l, 2, 1. 7 Now let s consider te case wen N l, 2, 1 = 1. Witout loss of generality, assume te sensor is deployed at t left and a single coverage gap is generated to te rigt of te deployed sensor, as sown in Fig. 5. According to Teorem 2, to guarantee tis gap is filled, te rigtmost deployment point for te additional sensors deployed in te second round must be at t rigt. So te gap distance = l z. Since z as te same pdf as 3 and all oter deployment points are evenly distributed between t and t rigt, we can calculate l, 2,N l, 2, 1 as: l, 2,N l, 2, 1 =1+ 2R s R err l z =1+ 2R s R err +1 f Zz dz +1 f Zz dz. For te case wen N l, 2, 1 =, te entire barrier needs to be covered in te second round. Hence, we ave l, 2,N l, 2, 1 a a a s err left err = l 2R s Rerr Fig. 5. Wen N l, 2, 1 = 1, a single coverage gap is generated after te first round. Witout loss of generality, assume te sensor is deployed at t left and a single coverage gap is generated to te rigt of te deployed sensor. In te second round, te leftmost and rigtmost deployment points to fill te gap are at t and t rigt, respectively. To summarize, we ave te following result: l, 2,N l, 2, 1 = l +1, if N 2Rs Rerr l, 2, 1 =, 1+ l z +1 f 2Rs Rerr Zz dz, if N l, 2, 1 = 1, N l, 2, 1 + N l, 2, 1 1 N 2I l, 2, 1, if N l, 2, By plugging 1 back into 1, we ave completed te definition of te optimization problem for two-round sensor rigt deployment, wic can be solved witout muc difficulty using numeric metods. Tis is because te searc space is limited: N l, 2, 1 is an integer and te number of possible coices for N l, 2, 1 is finite. C. Multi-Round Sensor Deployment Now we study te general multi-round sensor deployment problem were sensors are deployed in m>2 rounds to cover a barrier. Similar to te two-round sensor deployment strategy described in Section IV, we assume even distribution of sensors in eac gap. Te goal is to, at eac of te m rounds say, round j were 1 j m and for eac coverage gap wit a gap distance of, determine te optimal number of sensors to be deployed inside te gap, denoted as N, m, j, so tat te barrier can be covered wit fewest sensors. We acieve tis goal by using a recursive algoritm wic is described below. Firstly, consider te general case wen 1 i<m. Suppose tat, at round i, N, m, i sensors are deployed inside te gap wic may satter te gap into multiple smaller gaps. Assume tat eac newly generated smaller gap will be filled wit te optimal strategy in te next round, i.e., round i +1. Ten, te expected total number of sensors needed to fill te gap in m i +1 rounds, i.e., from round i to round m, can be calculated as:, m i +1,N, m, i = N total, m i, if N, m, i =, 1+ N total z,m i fzz dz, if N, m, i =1, N, m, i +[N, [ m, i 1] [P one I, m, i P zero I, m, i] ] +[N, m, i 1] N total,m i f HI,m,i d, if N, m, i were I, m, i = N, m, i f HI,m,i and f Z are given in 4 and 3, respectively. f HI,m,i is te pdf of te gap distances between adjacent sensors deployed at round i wen N, m, i 2. Moreover, P zero I, m, i and P one I, m, i are te probability tat tere is no coverage gap between adjacent sensors deployed at round i and te probability tat te coverage gap between adjacent sensors deployed at round i can be filled by deploying at most one additional sensor in te next round, respectively, and tey are given in 2 and 5. Ten we ave: N, m, i =arg min N,m,i, m i +1,N, m, i, 13 and, m i +1 =, m i +1,N, m, i. 14 Now consider te special case wen i = m, i.e., during te final round of sensor deployment. To guarantee tat all gaps are filled, according to Teorem 1, we ave: N, m, m = R s Rerr So, by using tis special case as te boundary condition, we ave fully specified ow to obtain te optimal multi-round

6 Tis full text paper was peer reviewed at te direction of IEEE Communications Society subject matter experts for publication in te IEEE INFOCOM 21 proceedings Tis paper was presented as part of te main Tecnical Program at IEEE INFOCOM 21. sensor deployment strategy tat uses fewest sensors to cover a barrier by 13, 14, 11, 12 and 15. Remarks: As will be sown in Section VI-B, tere is an interesting discovery tat optimal m-round m 2 sensor deployment strategies all yield te same barrier coverage performance regardless of m. In oter words, te best barrier coverage performance can be acieved wit te optimal tworound sensor deployment. For tis reason, we only study tworound sensor deployment in te practical considerations next. V. PRACTICA CONSIDERATIONS In practice, te distribution of a sensor s residence point, i.e., te f err function, may not be fully known. For example, if we assume tat f err is a truncated two-dimensional Gaussian function wit respect to a deployment point at x, y =,as described in 16 below, te parameter σ real may not be known. ferrx x, y = 1 A I x x 2 +y 2 9σ 2 real were and A = x x 2 +y 2 9σ 2 real e e x x 2 2σ real 2 + y 2 2σ real 2, 16 x x 2 2σ real 2 + y 2 2σ real 2 dx dy, 17 1, if x x 2 + y 2 9σ 2 real, I x x 2 +y 2 9σ 2 = real, oterwise. Instead, we may only know te range of σ real based on te istorical information, i.e., σ real [σ min,σ max ]. We assume tat R err =3σ max is smaller tan R s ; oterwise, barrier coverage cannot be guaranteed according to Teorem 1. In tis section, we present two practical solutions to deal wit tis situation. Note tat te proposed solutions can be extended to oter f err functions as long as te requirements on f err specified in Section III-B, i.e., circular symmetry and finite closed support, are satisfied. A. Te Two-Round Minimax Solution Te two-round minimax solution works as follows: σ et l, 2,N l, 2, 1 denote te expected total number of sensors needed to cover te barrier wen i N l, 2, 1 sensors are deployed in te first round; and ii sensors are deployed in te second round by setting R err =3σ max to guarantee filling of all coverage gaps. Recall tat l, 2 = l, 2,N l, 2, 1 is te minimum expected total number of sensors needed to cover te barrier wen σ real is known. Te difference between σ l, 2,N l, 2, 1 and l, 2 is ten te number of extra sensors deployed due to te lack of knowledge about σ real, and it varies wit N l, 2, 1. Wit te two-round minimax solution, te following number of sensors are deployed in te first round so tat te maximum number of extra sensors can be minimized: N minimax l, 2, 1 = arg min max N l,2,1 σ real [ σ l, 2,N l, 2, 1 N ] total l, 2, wic can be obtained using numerical metods because te number of possible coices for N l, 2, 1 is finite. B. Te Pilot Deployment Solution Now we present a more efficient solution by introducing an additional pilot round prior to te two rounds of sensor deployment. Te basic idea is to use te residence points of te sensors deployed in te pilot round to estimate σ real, wic is ten used to guide te next two rounds of sensor deployment. et N pilot and I pilot denote te number of sensors deployed in te pilot round and te deployment interval. Te deployment could start from eiter t left or t rigt. Recall tat, in order to guarantee coverage of te left rigt boundary of te barrier, t left t rigt sould be at a distance of R s 3σ max from te left rigt end of te deployment line. After te pilot round, te residence points of deployed sensors are collected and used to estimate σ real as follows. Te sample variance of te deviation of te residence points along X-axis and Y-axis wit respect to te corresponding deployment points is calculated as follows: S = N pilot i=1 x i xi 2 + N pilot i=1 2N pilot 1 y 2 i, 2 based on wic we propose te following estimator for σ real : σ min, if S σ 2 min, ˆσ real = S, if σ 2 min <S<σ2 max, σ max, if S σ 2 max. Tis estimator makes sense because te truncated twodimensional Gaussian distribution symmetrically at 3σisvery similar to te non-truncated version. Terefore, te deviations along X-axis and Y-axis, i.e., x i x i and y i, can be treated as two sets of independent samples to collectively contribute to te estimation of σ real. Te conditional pdf of ˆσ real is approximately: were fˆσreal ˆσ σ real =B 1 δˆσ σ min +B 2 δˆσ σ max + Iˆσ f Γ ˆσ 2 ; 2N pilot 1 2σ 2 real, 2ˆσ, 2 2N pilot 1 σ min B 1 = f Γ B 2 = f Γ ˆσ 2 ; 2N pilot 1, 2 ˆσ 2 ; 2Npilot 1, 2 σmax { 1, if σmin ˆσ σ max, Iˆσ =, oterwise, 2σ 2 real 2N pilot 1 2σ 2 real 2N pilot 1 2ˆσdˆσ, 2ˆσdˆσ, and f Γ is te Gamma distribution. Te derivation of Eq. 22 is omitted due to space limitation and details can be found in [34]. Ten, te next two rounds of sensor deployment are planned as follows: For eac coverage gap wit a gap distance of generated after te pilot round, deploy N ˆσ real, 3, 2 sensors in te first round, were N ˆσ real, 3, 2 is obtained based on te assumption tat σ real =ˆσ real ;

7 Tis full text paper was peer reviewed at te direction of IEEE Communications Society subject matter experts for publication in te IEEE INFOCOM 21 proceedings Tis paper was presented as part of te main Tecnical Program at IEEE INFOCOM 21. For eac coverage gap of size generated after te first round, deploy 2R s 3σ max +1 sensors in te second round to guarantee coverage of te gap. Finally, we can find te optimal <Npilot,I pilot > tat minimizes te maximum number of extra sensors deployed wen te pilot deployment solution is used: <N pilot,i pilot >=arg min max <N pilot,i pilot > σ real N p total l,n pilot,i pilot,σ real, ˆσ fˆσreal ˆσ σ real dˆσ N total l, 2, were N p total l,n pilot,i pilot,σ real, ˆσ is te expected total number of sensors needed to cover te barrier by following te above pilot deployment solution, wose analysis is similar to tat of in Section IV and omitted due to space limitation. VI. PERFORMANCE EVAUATION We develop a custom simulator based on MATAB and use it to evaluate te performance of our proposed multi-round sensor deployment scemes for guaranteed barrier coverage. In te numerical and simulation studies, we consider a barrier wit a lengt of l = 1 units. Sensing radius R s is1unit. A sensor s residence point follows a truncated two-dimensional Gaussian distribution wit respect to its deployment point, as described in 16, and te parameter σ real is witin te range of [1/3, 3/1] units, wic corresponds to R err [.1,.9] units. Results wit oter σ real values or oter distributions of sensors residence points yield similar trends and are not included ere. A. Two-Round Sensor Deployment wen R err is Known We first study te proposed two-round sensor deployment strategy wen R err is known. Fig. 6a sows te results wen R err =.3 units; it plots te expected total number of sensors needed to cover te barrier l, 2,N l, 2, 1 wen te number of sensors deployed in te first round N l, 2, 1 varies from to 1. We also plot te simulation results as marks, were eac point is averaged over 1 simulation runs. As sown in te figure, simulation results matc analytical results closely. We ave te following observations. In general, as N l, 2, 1 increases starting from zero, Ntotal firstly decreases wit fluctuations till reacing te minimum, and ten increases almost linearly as N l, 2, 1 increases furter. As sown in Fig. 6a, Ntotal reaces te minimum of 593 wen N l, 2, 1 is 572, meaning tat te numbers of sensors deployed in te two rounds is 572 and 21, respectively. On te oter and, wen N l, 2, 1 is zero, te corresponding is 7, meaning tat if sensors are deployed in one round, 7 sensors are needed to cover te barrier. Tis sows tat, wit two-round sensor deployment, 15.3% of te sensors are saved. Now let s take a look at te optimal two-round deployment strategy in more detail. One salient feature of te strategy is tat a large number 572 of sensors are deployed in te first round, wile a very small number 21 of sensors are deployed in te second round to fill te gaps. To provide guaranteed barrier coverage wit single-round sensor deployment, sensors 24 N total Analytical results Simulated results N l,2,1 a R err =.3 units N total Analytical results Simulated results N l,2,1 b R err =.5 units Fig. 6. Expected total number of sensors needed to cover te barrier l, 2,N l, 2, 1 vs. te number of sensors deployed in te first round N l, 2, 1 wit te proposed two-round sensor deployment strategy. Note tat Ntotal fluctuates in bot figures. Tis is because, wen N l, 2, 1 increases from small, te number of sensors to fill eac gap generated in te first round decreases approximately according to a staircase function. So fluctuates wit respect to N l, 2, 1 until N l, 2, 1 gets sufficiently large. need to be deployed in a conservative manner i.e., wit small deployment interval and ence more sensors are needed to counter te deviation of te sensors residence points wit respect to teir deployment points. In comparison, wit tworound sensor deployment, it is safe to be more aggressive i.e., wit larger deployment interval in te first round and ten deploy sensors more conservatively in te second round to fill te limited number of gaps generated in te first round. As a result, a significant number of sensors are saved by simply splitting sensor deployment into two rounds. In te optimal two-round sensor deployment strategy, te deployment interval in te first round is I l, 2, 1 = l N l,2,1 1 = units. Besides, we find tat i te number of gaps generated in te first round is small; and ii all te gaps are small gaps wic can be filled by deploying one extra sensor in te second round. Te latter one is an important observation as it sows tat, as long as te sensors communication range is reasonably large e.g., 4R s, wen te optimal two-round sensor deployment strategy is used, connectivity among sensors deployed in te first round can be guaranteed, wic is a critical assumption discussed in Section III for te strategy to function properly. In Fig. 6b, R err is increased from.3 to.5 units and all te above observations still old. Particularly, wit te proposed two-round sensor deployment strategy, Ntotal reaces te minimum of 647 wen N l, 2, 1 is 6. In comparison, wen N l, 2, 1 is zero, te corresponding becomes 1, meaning tat 1 sensors are needed to cover te barrier if deployed in a single round. Tis sows tat, a larger portion of sensors 35.3% can be saved wen R err is increased to.5units.infig.7,wevaryr err from.1 to.9 units and sow te benefit of two-round sensor deployment in terms of te percentage of sensors saved. It can be seen clearly from te figure tat altoug more sensors are needed to cover te barrier as R err goes up, it saves more sensors percentage wise by increasing te number of sensor deployment rounds from one to two. Tis is because a larger R err requires a more conservative sensor deployment round i.e., te only round in single-round sensor deployment vs. te second round in tworound sensor deployment to deal wit.

8 Tis full text paper was peer reviewed at te direction of IEEE Communications Society subject matter experts for publication in te IEEE INFOCOM 21 proceedings Tis paper was presented as part of te main Tecnical Program at IEEE INFOCOM 21. Fig. 7. sensors saved % Sensors saved N * total R err units Percentage of sensors saved and N total l, 2 vs. R err. B. Two-Round vs. M-Round M 3 Sensor Deployment In Section VI-A, we observe tat significant performance improvement may be acieved by increasing te number of sensor deployment rounds from one to two. In tis section, we study ow or weter te performance may be improved furter by increasing te number of deployment rounds more. Fig. 8 compares te performances of two-round and treeround sensor deployment strategies wen R err is.3 units. Results reveal a rater surprising discovery tat te optimal tree-round sensor deployment strategy produces te same = 593 as te optimal two-round strategy. As sown in te figure, wit te tree-round sensor deployment strategy, = 593 is reaced wen N l, 3, 1 = or 572. Wen N l, 3, 1 =, te number of sensors deployed in te tree rounds is, 572 and 21, respectively, wile wen N l, 3, 1 = 572, te numbers are 572, 21 and. We can see tat, te optimal tree-round strategy essentially only deploys sensors in two out of tree rounds. We ave also done simulations wit m-round m 4 sensor deployment and remains te same. Tis means tat te minimum number of sensors needed to cover te barrier is te same regardless of te number of sensor deployment rounds, as long as te sensors are deployed in multiple rounds. Tis discovery as ig practical significance because it implies tat te best barrier coverage performance can be acieved wit low extra deployment cost by deploying sensors in two rounds. N total N * total 4 Tree round deployment 3 Two round deployment te number of sensors deployed in te first round Fig. 8. Comparison of te expected total number of deployed sensors wit two-round and tree-round sensor deployment strategies. C. Two-Round Sensor Deployment wit R err Partially Known Finally, we study te scenario wen R err is only partially know, i.e., te range of R err is known.1 R err.9 but te actual value of R err is unknown. Using te proposed tworound minimax solution, Nminimax is 587, wile wit te pilot deployment solution, Npilot is 51 and I pilot is 1.69 units. Fig. 9 compares te performances of tese two solutions against te ideal solution wen R err is known, wit te actual value of R err varying from.1 to.9 units. As sown in te figure, te two-round minimax solution works well in most scenarios except wen te actual value of R err is close to.1 or.9 units. In tese scenarios, as many as 5 extra sensors may be needed to guarantee barrier coverage. In comparison, wit te pilot deployment solution, te number of extra sensors is reduced significantly to no more tan 9. Tis is due mainly to te extra pilot round of sensor deployment wic collects some preliminary statistics about R err and ten uses tem to aid te following two rounds of sensor deployment. We ave also simulated te pilot deployment solution. Te results are averaged over 1 simulation runs and a close matc between analytical and simulation results can be observed. N * total Two round deployment, ideal case Two round minimax deployment Pilot deployment Pilot deployment simulation R err units Fig. 9. Expected total number of deployed sensors vs. te actual value of R err unknown wit te proposed practical solutions. D. Summary We summarize te key findings from our numerical and simulation studies as follows: By splitting sensor deployment into multiple 2 rounds, te number of sensors needed to provide guaranteed barrier coverage can be reduced significantly. Te performance gain of multi-round sensor deployment over single-round sensor deployment becomes more significant as te deviation of a sensor s residence point wit respect to its deployment point gets larger, i.e., wen more randomness is present during sensor deployment. Wit te optimal two-round sensor deployment strategy, all te coverage gaps generated in te first round are small gaps. Tis means tat, as long as te communication range of sensors is reasonably large, te sensors deployed in te first round are connected, wic supports te practical feasibility of te strategy. Optimal m-round m >2 sensor deployment strategies yield te same performance as te optimal two-round sensor deployment strategy. Tus, te best barrier coverage performance can be acieved wit low extra deployment cost by deploying sensors in two rounds. Wen te information about te deviation of a sensor s residence point wit respect to its deployment point is not fully available, te pilot deployment solution performs particularly well and is comparable to te ideal solution.

9 Tis full text paper was peer reviewed at te direction of IEEE Communications Society subject matter experts for publication in te IEEE INFOCOM 21 proceedings Tis paper was presented as part of te main Tecnical Program at IEEE INFOCOM 21. VII. DISCUSSIONS AND FUTURE WORK In tis section, we discuss ow our proposed multi-round sensor deployment may be extended and some related issues. 1 Different Sensing Models. We assume te disc sensing model in tis paper. In practice, te sape of a sensing area could be irregular. In general, it is difficult to ave a rigorous analysis of te coverage performance in tis situation. One simple way is to approximate te sensor area wit te largest disc centered at a sensor and contained in its sensing area, wic would result in conservative sensor deployment strategies. We also assume te /1 sensing model in tis paper. A potential extension to te work is to incorporate te more realistic probabilistic sensing model and te concept of information coverage to exploit te collaboration between sensors into multi-round sensor deployment, wic could save even more sensors wen covering a barrier. 2 k-barrier Coverage. Te barrier coverage studied in tis paper is te simple 1-barrier coverage. Te work can be extended to k-barrier coverage by making te following modifications in te analysis. Wit k-barrier coverage, a coverage gap between two adjacent sensors deployed in te previous rounds exists if tere are less tan k disjoint linkages between tem. Terefore, coverage gaps can be classified into multiple categories according to te number of additional disjoint linkages tat are needed to fill te gap. Different types of coverage gaps need to be analyzed differently. 3 Oter Issues. Recall tat according to our numerical and simulation studies, wit te optimal two-round sensor deployment strategy, as long as te sensors communication range is at least four times te sensing range, te sensors deployed in te first round are connected. In practice, if under certain circumstances wen te communication range is small, an additional constraint on te deployment interval sould be included so te connectivity requirement could be satisfied. Note tat we study te problem of guaranteed barrier coverage in tis paper. A related but distinct problem is ow to provide barrier coverage in a probabilistic sense wit multi-round sensor deployment. Tis problem may require different tecniques to deal wit. Tradeoff between te desired probability of barrier coverage and te deployment interval needs to be investigated. Tis is part of our future work. VIII. CONCUSION In tis paper, we conduct extensive analytical and simulation studies on reducing te number of sensors needed to provide guaranteed barrier coverage wit multi-round sensor deployment strategies. We study te performance of multiround sensor deployment and derive te optimal strategies tat use fewest sensors to cover a barrier. We find tat te efficient barrier coverage can be acieved wit te simple two-round sensor deployment. In addition, two practical solutions are presented to deal wit realistic situations wen te distribution of a sensor s residence point is not fully known. Te effectiveness of te proposed solutions is supported by numerical and simulation results. REFERENCES [1] S. Kumar, T. H. ai, and A. Arora, Barrier Coverage Wit Wireless Sensors, in Proc. of ACM MobiCom, Aug. 25. [2] Article id: ttp:// [3] G. Yang and D. Qiao, Barrier information coverage wit wireless sensors, in Proc. of IEEE Infocom, Apr. 29. [4] A. Saipulla, C. Westpal, B. iu, and J. Wang, Barrier coverage of linebased deployed wireless sensor networks, in Proc. of IEEE Infocom, Apr. 29. [5] B. iu and D. Towsley, A Study of te Coverage of arge-scale Sensor Networks, in Proc. of IEEE MASS, Oct. 24. [6] S. Meguerdician, F. Kousanfar, G. Qu, and M. Potkonjak, Exposure In Wireless Ad-Hoc Sensor Networks, in Proc. ACM MobiCom, 21. [7] S. Kumar, T. H. ai, and J. Balog, On k-coverage in a Mostly Sleeping Sensor Network, in Proc. of ACM MobiCom, Sept. 24. [8] H. Zang and J. C. Hou, On deriving te upper bound of α-lifetime for large sensor networks, in Proc. of ACM MobiHoc, June 24. [9], Is Deterministic Deployment Worse tan Random Deployment for Wireless Sensor Networks? in Proc. of IEEE Infocom, Mar. 26. [1] S. Dillon and K. Cakrabarty, Sensor placement for effective coverage and surveillance in distributed sensor networks, in Proc. of IEEE WCNC, Mar. 23. [11] Z. Yuan, R. Tan, G. Xing, C. u, Y. Cen, and J. Wang, Fast sensor deployment for fusion-based target detection, in Proc. of IEEE Real- Time Systems Symposium RTSS, Nov. 28. [12] A. Krause, C. Guestrin, A. Gupta, and J. Kleinberg, Near-optimal sensor placements: Maximizing information wile minimizing communication cost, in Proc. of IPSN, Apr. 26. [13] H. Gonzalez-Banos and J. atombe, A randomized art-gallery algoritm for sensor placement, in Proc. of ACM Symp. on Computational Geometry, June 21. [14] J. Zang, T. Yan, and S. Son, Deployment Strategies for Differentiated Detection in Wireless Sensor Networks, in Proc. of IEEE SECON, 26. [15] M. eoncini, G. Resta, and P. Santi, Analysis of a wireless sensor dropping problem in wide-area environmental monitoring, in Proc. of ACM/IEEE IPSN, Apr. 25. [16] P. Balister and S. Kumar, Random vs. deterministic deployment of sensors in te presence of failures and placement errors, in Proc. of IEEE Infocom Miniconference, Apr. 29. [17] Y. Zou and K. Cakrabarty, Sensor deployment and target localization based on virtual force, in Proc. of IEEE Infocom, Apr. 23. [18] G. Wang, G. Cao, P. Bermn, and T. Porta, Bidding protocols for sensor deployment, IEEE Transactions on Mobile Computing, pp , May 27. [19] G. Wang, G. Cao, and T.. Porta, Movement-assisted sensor deployment, in Proc. of IEEE Infocom, Mar. 24. [2] T. Clouqueur, V. Pipatanasuporn, P. Ramanatan, and K. K. Saluja, Sensor Deployment Strategy for Target Detection, in Proc. of ACM WSNA, Sept. 22. [21] R. Gau and Y. Peng, A dual approac for te worst-case-coverage deployment problem in ad-oc wireless sensor networks, in Proc. of IEEE MASS, Oct. 26. [22] X. Bai, S. Kuma, D. Xua, Z. Yun, and T. H. ai, Deploying Wireless Sensors to Acieve Bot Coverage and Connectivity, in Proc. of ACM MobiHoc, May 26. [23] X. Han, X. Cao, E. loyd, and C.-C. Sen, Deploying directional sensor networks wit guaranteed connectivity and coverage, in Proc. of IEEE SECON, June 28. [24] S. Sakkottai, R. Srikant, and N. Sroff, Unreliable sensor grids: coverage, connectivity and diameter, in Proc. of IEEE Infocom, 23. [25] B. Seng, Q. i, and W. Mao, Data storage placement in sensor networks, in Proc. of ACM MobiHoc, May 26. [26] W. i and C. Cassandras, A minimum-power wireless sensor network self-deployment sceme, in Proc. of IEEE WCNC, Mar. 25. [27] B. iu, O. Dousse, J. Wang, and A. Saipulla, Strong barrier coverage of wireless sensor networks, in Proc. of ACM MobiHoc, May 28. [28] P. Balister, B. Bollobas, A. Sarkar, and S. Kumar, Reliable density estimates for acieving coverage and connectivity in tin strips of finite lengt, in Proc. ACM MobiCom, Sept. 27. [29] A. Cen, T.-H. ai, and D. Xuan, Measuring and guaranteeing quality of barrier-coverage in wireless sensor networks, in Proc. of ACM MobiHoc, May 28. [3] S. Kumar, T. H. ai, M. E. Posner, and P. Sina, Optimal sleepwakeup algoritms for barriers of wireless sensors, in Proc. of IEEE BROADNETS, Sept. 27. [31] A. Cen, S. Kumar, and T. H. ai, Designing localized algoritms for barrier coverage, in Proc. of ACM MobiCom, Sept. 27. [32] X.-Y. i, P.-J. Wan, and O. Frieder, Coverage in wireless ad-oc sensor networks, IEEE Transactions on Computers, vol. 52, no. 6, June 23. [33] G. Veltri, Q. Huang, G. Qu, and M. Potkonjak, Minimal and maximal exposure pat algoritms for wireless embedded sensor networks, in Proc. of ACM SenSys, Nov. 23. [34] Tec. Rep., ttp:// gqyang.