Random v. Determinitic Deployment of Senor in the Preence of Failure and Placement Error Paul Baliter Univerity of Memphi pbalitr@memphi.edu Santoh Kumar Univerity of Memphi antoh.kumar@memphi.edu Abtract Although random deployment i widely ued in theoretical analyi of coverage and connectivity, and evaluation of variou algorithm (e.g., leep-wakeup), it ha often been conidered too expenive a compared to optimal determinitic deployment pattern when deploying enor in real-life. Roughly peaking, a factor of log n additional enor are needed in random deployment a compared to optimal determinitic deployment if n enor are needed in a random deployment. Thi may be an illuion however, ince all real-life large-cale deployment trategie reult in ome randomne, two prime ource being placement error and enor failure, either at the time of deployment or afterward. In thi paper, we conider the effect of placement error and random failure on the denity needed to achieve full coverage when enor are deployed randomly veru determinitically. We compare three popular trategie for deployment. In the firt trategy, enor are deployed in an optimal lattice but enough enor are colocated at each lattice point to compenate for failure and placement error. In the econd, only one enor i deployed at each lattice point but lattice pacing i ufficiently hrunk to achieve a deired quality of coverage in the preence of failure and placement error. In the third, a random deployment i ued with appropriate denity. We derive explicit expreion for the denity needed for each of the three trategie to achieve a given quality of coverage, which are of independent interet. In comparing the three deployment, we find that if error in placement are half the ening range and failure probability i 5%, random deployment need only around % higher denity to provide a imilar quality of coverage a the other two. We provide a comprehenive comparion to help a practitioner decide the lowet cot deployment trategy in real-life. I. INTRODUCTION Given the ever increaing number of new application that are becoming poible with wirele enor, the importance of educated deployment of wirele enor continue to increae. Knowing the appropriate number of enor to deploy i critical to enuring that a deired quality of monitoring i achieved at the lowet poible cot. Although optimal pattern of deployment to enure full coverage (i.e., covering every point in the region) when deploying enor determinitically, and critical denity needed to achieve full coverage when deploying randomly, ha been tudied extenively [2], [7], [9] [], joint effect of placement error and enor failure on the optimal/critical denitie ha not been addreed before. It i known that triangular lattice i an optimal pattern of deployment to achieve full coverage [2]. It i alo known that if enor are placed accurately and they are % reliable, optimal determinitic deployment will need O(log n) time fewer enor veru a random deployment, where n i the number of enor needed in a random deployment [7]. Thu it appear that determinitic deployment are far uperior. However, uch a concluion i illuory. For one, placing enor in uch a pattern i cumberome, at bet. Second, placement error are inherent, and they introduce coverage hole in the region. Third, failure of enor, which i alo quite likely (epecially in an outdoor unattended deployment [5]), will caue even larger coverage hole. One intuitive method to addre both of thee perturbation i to place multiple enor at each point of the triangular lattice (OPT). Another approach i to place enor in a lattice (or grid), but cale down the grid pacing ufficiently o a to achieve full coverage with thi fine lattice (FLAT). Thi model wa firt tudied in [8] for unreliable enor and ubequently refined in [7], []. Placement error are, however, not conidered in either work. The third approach, which i widely ued in theoretical tudie and imulation, i to deploy enor randomly according to a uniform Poion proce (RAND). In thi cae, placement error ha no effect, ince a Poion deployment with placement error i till a Poion deployment. Similarly, Poion deployment with random failure i again a Poion deployment but with lower denity. The denity needed to achieve full coverage for random deployment ha been extenively tudied [7], [9], []. However, thee give only aymptotic reult, and it i not clear how large n mut be for the etimate to become ufficiently accurate, limiting their uefulne in practice. In thi paper, we invetigate the three deployment trategie decribed in the preceding for their relative efficiency. To make our reult more ueful for practitioner, we ue the number of coverage hole a our metric. Traditionally, area of vacant region ha been ued to indicate the extent of coverage [9], []. Notice that even if the area of vacant region i mall, there could be many coverage hole. On the other hand, if the number of coverage hole <, it implie full coverage. We make the following contribution in thi paper: ) Comprehenive treatment of one dimenion: We give reliable, explicit etimate of the denity needed in each of the three deployment trategie for given failure probability and given degree of placement error in the -dimenional cae. (Section III)
2 2) Reliable denity etimate in two dimenion: We give reliable etimate of denity needed for RAND, but for the other two trategie only when the placement error i zero. For FLAT and OPT with non-zero placement error, the analyi become highly involved. Our etimate for RAND are much more accurate than known aymptotic critical condition, making them ueful for practical deployment. Previouly, only aymptotic critical condition for the FLAT deployment under failure were known [7], []. For OPT, we are not aware of any exiting work that conider enor failure. (Section IV) 3) Guidance on when to ue which: We compare each of the three deployment for variou value of failure probabilitie and placement error and point out the difference in denity therein. II. MODEL AND KEY OBSERVATIONS Conider a wirele enor network deployed in a large area. For implicity we hall aume all enor detect event within a dik, with common ening range r =. (Similar reult will in fact hold for mot plauible hape of ening region.) We model the error in enor placement by a ymmetric bivariate gauian ditribution with probability denity function 2πσ exp( x x 2 2 /2σ 2 ), where x R 2 i the actual enor poition, x R 2 i the target poition, and σ i the tandard deviation in each coordinate of the vector x x. We hall alo aume enor are active with probability p (i.e., fail with probability p). Finally, we aume the error in location and failure event of each enor are independent, and independent of all the other enor. Note that failure independence ha been oberved in real-life deployment [5]. If we wih to minimize the number of enor required, then triangular grid placement i known to be optimal with an aymptotic denity of 2/ 27.3849 [2]. Note the obviou lower bound of /π.383, ince each enor can only ene a region of area π. It i alo known that for random uniform deployment, the denity of enor required i aymptotically about π ( + o()) log A where A i the area of the deployment region A. Since thi tend to infinity a A, it eem that determinitic deployment i far uperior to random deployment. Thi would indeed hold if there were no failure or placement error. However, for any contant value of p<the denity of enor needed to enure a reaonable chance of coverage will be roughly proportional to log A. Indeed, if we divide the region A into about A /9 quare of ide length 3, and if each of thee quare ha at mot ε log A enor, then we would expect that about (/9) A ( p) ε log A =(/9) A +ε log( p) of thee quare not to contain any active enor. Hence if ε i mall enough, one would expect many empty quare, and hence many coverage hole. Similarly, for any contant σ>, the denity of enor needed will tend to infinity a A (poibly more lowly than log A ). Thu for full coverage of a large area we alway need a large denity in practice. In addition to the three trategie introduced in Section I, namely OPT, FLAT, and RAND, we hall alo conider trategie that are intermediate between OPT and FLAT, namely FLAT(k): Deploy k enor at each lattice point (for ome k ), but cale the lattice a in FLAT. Note that FLAT() i the ame a FLAT. For implicity, and to avoid dependency on the area A of the deployment region, we hall conider deployment in the infinite plane. Reliability will be aeed by conidering the aymptotic denity of coverage hole, i.e., the limit N A I = lim A A, where N A the expected number of uncovered region in a dik A, ay,ofarea A. It can be proved (uing the Stein- Chen method, ee []) that with any of the above trategie the number of coverage hole i approximately Poion ditributed with mean I A, and thi approximation improve for fixed I A a I and A. Thu the probability of full coverage i well approximated by e I A for mall I. For illutration, we hall compare the hole denitie I of the different trategie for two dimenion. Figure how diagram giving the hole denitie uing trategie OPT, FLAT(), FLAT(2), and FLAT(3) when the denity of enor ued i ufficient to give trategy RAND a failure rate of I = 2, i.e., one coverage hole on average per 2 unit of area. Contour line indicate by what factor the denity of enor need to be increaed when uing trategy RAND o a to match the given trategy. Thi lightly trange repreentation wa choen ince it allow imple comparion between the trategie, while avoiding the problem that the coverage hole denitie of trategie OPT and FLAT(k) can be rather erratic function of the parameter when σ i mall. (The coverage hole denity function I for trategy RAND i well behaved ee below.) In each cae, we vary the deployment poition error σ and enor reliability rate p. Figure alo give a diagram for trategy OPT when I = 4, and a diagram indicating which trategy i bet (for I = 2 ). We now make ome key obervation from thee reult. Obervation. The denity of enor required to achieve a given level of reliability doe not vary much between the different trategie unle p and σ. The difference between the trategie decreae a p (unreliable enor) or σ (large error in deployment) a one might expect. However, the enor denity for uniform deployment i only a few of percent more throughout much of thee diagram. The difference between trategie OPT and FLAT(k) are even maller in general. Obervation 2. The ratio between the enor denitie required for each trategy i largely independent of I. For example, there i very little difference in the diagram for I = 2 and I = 4. Obervation 3. The bet trategie eem to be OPT (for low p and low σ, orflat for large σ or large p. There are a few location with mall σ where one of the other FLAT(k) trategie win, but motly the choice i between OPT and FLAT.
3 p p p Bet OPT OPT (e-4).. OPT FLAT() (3) FLAT() (2)..2.3.4.5 FLAT(2)..2.3.4.5.2.3.4.5 FLAT(3).2.3.4.5 p p p..2.3.4.5 Fig.. Two-dimenional deployment. Ratio of denitie needed a a function of p and σ for a fixed choice of I RAND (= 4 for bottom right and 2 for the ret). Contour how by what factor denity mut be increaed in RAND to match reliability of trategy OPT or FLAT(k). Alo hown i a diagram (lower left) giving the bet choice of thee algorithm for different value of p and σ. FLAT()..2.3.4 p p p Bet OPT OPT (e-4)..2.3..2.3 FLAT().4.4 (2) (3).5 FLAT(2)..2.3.4 p p p Fig. 2. One-dimenional deployment. The etup and legend are ame a in Figure..5 FLAT(3)..2.3.4 In the following ection we derive etimate for the coverage hole denity which explain Obervation and 2, but alo how that Obervation 3 doe not tell the whole tory. In fact, a I, trategyflat alway win, while OPT i eventually wore than even the RAND trategy for mall value of p. One dimenional deployment. We alo conider the cae of one dimenional deployment where enor are deployed along a line. Simulation reult for thi cae are hown in Figure 2, which ue a imilar framework a for the two dimenional cae. There are a number of imilaritie between the reult for one and two dimenional cae, however, there are alo ome very ignificant difference. Obervation 4. Strategy OPT perform badly in the onedimenional cae when σ>, and there i a dicontinuity in.5 performance at σ =, while in the two dimenional cae OPT can often be quite good, and there i no uch dicontinuity. A we tated above, trategy OPT i only really good in the two dimenional cae when I i not too mall. In the one dimenional cae it i alway bad for σ>. Obervation 5. Strategy FLAT i uually beaten by ome FLAT(k), k>, in the one-dimenional cae, wherea thi i eldom true in the two dimenional cae. In the following ection, we provide analytic explanation for each of thee obervation. III. ONE-DIMENSIONAL DEPLOYMENTS We firt preent our reult for the one dimenional cae. We aim to tudy the number of enor gap in a long line where, for each i Z, we place k enor with a normal ditribution N(ih, σ 2 ), and each enor ha a probability p of being active. Here, h i the lattice cale, whoe value i decided by the particular deployment trategy. A dicued in Section II, we compare only the coverage hole denitie ince it uffice for comparing the probabilitie of full coverage. To count the expected number of coverage hole, conider the probability p(x) that an active enor (aumed to be at poition x) i immediately followed by a gap of length 2 which doe not contain another active enor. When σ> one obtain (ee [3] for more detail) ( p(x) = pφ ( ) ( 2+x ih σ + pφ x ih ) ) k δ i, σ () i= where φ(x) = 2π e x2 /2 i the probability denity function, and Φ(x) = x φ(t) dt the cumulative function, of a N(, ) random variable, and δ i =when i =and when i. Multiplying by the denity of enor at x and integrating over x give the denity of coverage hole per unit length a I(k, h, p, σ) = kp h p(x)φ( x σ ) σ dx. (2) Although no cloed form expreion for I(k, h, p, σ) eem likely, it can be evaluated numerically quite eaily. For each of the trategie we fix the denity of enor to be ρ enor per unit length. Thu we require that ρ = k/h. For trategy OPT, wefixh =2(o k =2ρ) ince it i clear that the optimal deployment involve placing enor at interval 2 apart. For trategy FLAT(k), wefixk (o h = k/ρ). For trategy RAND p(x) =e 2pρ. (The number of active enor in [x, x +2] i Poion with mean 2pρ for the uniform deployment.) Thu the intenity of coverage gap i given explicitly by the formula I RAND = pρe 2pρ. (3) A σ, all the above trategie converge to trategy RAND ince the ditribution of enor become more uniform. To explain the difference between the trategie, it therefore help to conider the cae when σ i mall, in particular we hall conider the cae σ =, when there are no error in poitioning the enor. Fix k, h and p a above and σ =. To obtain a coverage hole in the above model, we need at leat one of k enor at
4 one point to be active, but all of the next 2/h group of k enor to fail. Thi occur with probability ( ( p) k )( p) k 2/h, and can happen at each enor location. Hence the denity of hole i I(k, h, p, ) = h ( ( p)k )( p) k 2/h ( ( p)k ) pk pρ( p) 2ρ, (4) where the approximation i exact if 2/h i an integer. Fixing ρ and letting k vary, we ee that the approximation to I(k, h, p, ) above decreae with increaing k. In particular, comparing trategy FLAT with FLAT(k) we have I FLAT(k) /I FLAT = ( ( p) k )/(pk) which i alway le than. Of coure thi relie on 2/h being an integer, o in practice I FLAT(k) will fluctuate omewhat with k. Weeein Figure 2 that the optimal value of k varie erratically with the parameter. Thi i due to the rapid variation in the fractional part of the number 2/h. Increaing k however uually improve the trategy, and for the larget poible k (h =2) we obtain trategy OPT. The reaon i eentially that in all cae we need k 2/h 2ρ enor to fail for a coverage hole to form, but for large k, there are fewer choice for thee enor that actually reult in a hole. Now (3) and (4) can be inverted to find the approximate denity pρ of active enor needed when A i large: OPT pρ p log( p) log(2 A ) (σ =) FLAT pρ p log( p) (log(2 A ) + log log(2 A )) RAND pρ. (log(2 A ) + log log(2 A )) Note that p/ log( p) = p 2 +O(p2 ) i cloe to when p i mall, and there i little difference between FLAT and OPT for any p. For σ>but mall, trategie FLAT(k) are not uually ignificantly affected. However, trategy OPT i. The reaon for thi i that it i poible for each group of k enor to contain an active enor, yet a mall coverage hole form due to the fact that the gap between the two group i lightly more than 2. Suppoe σ i poitive, but very mall, o that to etimate the probability of a coverage hole between, ay, x =and x =2, we can ignore the enor which were targeted at point other than x =and x =2. Then one can how (ee [3] and below) that for σ> OPT pρ p/2 log( p/2) (log(2 A )+ 2 log log(2 A )) which lie between trategie RAND and FLAT. Note that thi i almot independent of σ>, unle σ i large enough that the enor location error ha a reaonable chance of being more than 2 unit. We can ee thi on Figure 2, where the contour line jump from σ =to σ>, but then tay almot vertical until σ i large. A dicued in Section II, Figure 2 how how many additional enor are needed in RAND to match the reliability of trategy FLAT(k) and OPT. For mall σ the bet trategy i likely to be a light modification of trategy OPT, where we cale the deployment pattern o that adjacent group of enor are jut lightly le than 2 unit apart. Effectively thi i trategy FLAT(k) with a large, but carefully choen k. For larger σ, difference between the different trategie become negligible. IV. TWO DIMENSIONAL DEPLOYMENTS For the -dimenional cae, (2) (or (3) or (4)) allow adequate determination of the coverage hole intenity. For the 2-dimenional cae, no uch imple expreion eem likely, o in general we will need to reort to numerical imulation. However, we have obtained the following reult for the number of coverage hole in the cae of uniform deployment. Theorem 4.: The coverage hole denity for the uniform random deployment cae in 2 dimenion i given by I RAND = pρ(πpρ )e πpρ + pρ( + o())e 4πpρ. Due to pace limitation, we defer the proof of Theorem 4. to [3]. Although there are etimate in the literature for the probability of full coverage, and for the number of coverage hole, the above etimate i far more accurate and the error term can be explicitly bounded (ee [3]), o that they can be applied in ituation far from the n limit in which previou etimate have been derived. For ituation in which one ha a reaonable chance of full coverage, the econd term above i in practice extremely mall. A dicued in Section II, once hole intenity i known, probability of full coverage i given approximately by e I A (once again, the error can be explicitly bounded), and hence a reliable etimate for denity can be derived for any et of parameter. Looking at Figure we once again ee that the trategie converge when σ i large. To undertand the difference in trategie for mall σ we again look firt at the σ =cae. Firt, let u conider trategy OPT with enor denity ρ. Theorem 4.2: The coverage hole denity for trategy OPT when σ =i given by I OPT =(2/ 27)( p) k + O(( p) 2k ) where k = ρ 27/2 i the number of enor at each lattice point. Proof: For a coverage hole to exit, an entire group of k enor mut be inactive. For a fixed group thi occur with probability ( p) k. The number of coverage hole i at mot the number inactive group. On the other hand, if an inactive group i urrounded by ix active group then a hole i definitely formed. Thu the number of hole per lattice point i at leat ( p) k ( ( p) k ) 6 =( p) k + O(( p) 2k ). The reult follow a the area per lattice point i 27/2. For mall σ > there doe not eem to be a jump in reliability a there wa in the one dimenional cae. The derivation of I OPT in the σ> cae i much more involved than in the one dimenional cae, but a imple heuritic offer ome explanation. In the one dimenional cae, a mall coverage hole appear between lattice point with probability about ( p/2) 2k πk (ee [3]) which i more than the probability ( p) k that an entire group of enor are inactive. Ignoring the πk factor, ( p/2) 2k i jut the probability that the mid-point between two enor group i uncovered. Indeed, there are 2k enor that could cover it, and each
5 enor cover it with probability p/2 (p to be active, and /2 due to poitioning error). In the cae of two dimenion, the mid-point of a lattice triangle i at ditance r from three lattice point, but i till covered by active enor with probability /2. Thu we have a probability of ( p/2) 3k that thi mid-point i uncovered. It turn out that the probability of a mall coverage hole near the center of a lattice triangle i about ( p/2) 3k time ome mall order term (power of k). However, for p<.7639, ( p/2) 3k i le than ( p) k, which i the probability that a hole appear due to an entire group of enor being inactive. Thu there i generally not o much difference between the mall σ> and σ =cae in two dimenion. Finally, conider trategy FLAT (or FLAT(k)) in the cae when σ i mall (or zero). Auming the enor denity ρ i large, the enor are deployed on a very fine grid with pacing much le than r. One can follow the argument ued to prove Theorem 4. to derive an etimate for the coverage hole denity. Theorem 4.3: The coverage hole denity for trategy FLAT (or FLAT(k) for fixed k) igivenforσ, ρ, by I FLAT = pρ(πpρ )( p) πρ+o(ρ/2). The main difference i that the probability of a given point being uncovered (i.e., no active enor within ditance r) i e πpρ in the cae of RAND, but( p) πρ+o(ρ/2) in the cae of FLAT. (The error term due both to the dicretene of the lattice, and alo due to the placement error.) Alo, the error in the exponent dominate the econd term that one would expect by comparion with Theorem 4.. Alo, the only difference between FLAT and FLAT(k) i that the contant implicit in the error term increae with increaing k due to larger boundary effect when the grid i coarer. Similarly the error term increae with σ. A in the one dimenional cae we can invert the formulae above to find the denity of active enor needed to obtain A I when A i large: OPT pρ = p 2 log( p) 27 FLAT pρ = p log( p) π RAND pρ = π Unlike the one dimenional cae, the dominant term in the OPT and FLAT cae i different. The reaon i that trategy FLAT need about πρ enor to fail in a mall region becaue the minimum number of enor failure for a hole occur when all the enor in a roughly circular region of radiu. However trategy OPT only need about ( 27/2)ρ enor to fail. The ratio (2π/ 27).29 i jut the denity factor we loe in OPT a a reult of the fact that dik do not tile the plane. Strategy FLAT alo ha a better dominant term than trategy RAND, o at firt ight it eem trategy FLAT hould alway win. In the limit a I ( A ) thi i indeed the cae, but the imulation reult in Figure 2 appear to how trategy OPT a better for a nontrivial range of value of p and σ. To explain thi dicrepancy, we conider more carefully the expreion for I OPT and I FLAT above. Roughly peaking I OPT (2/ 27)( p) ( 27/2)ρ, while I FLAT (pρ) 2 ( p) πρ. The ratio i I FLAT /I OPT 2.6(pρ) 2 ( p).5435ρ 2.6(pρ) 2 e.5435pρ. Thu we ee a ignificant polynomial factor and a mall negative exponential factor. For modet target hole denitie, trategy OPT can indeed beat trategy FLAT for mall value of p and σ. However for larger value (e.g., pρ i enough), trategy FLAT i better. V. FUTURE WORK Although we provide a comprehenive comparion of variou deployment here, everal problem till remain open. Firt, derivation of denity etimate for determinitic deployment in the preence of both failure and placement error remain open. Our etimate for determinitic deployment i reliable only for zero placement error cae. Second, we only conider -full coverage here. Extenion to the cae of k-full coverage remain open. Third, we conidered only the full coverage model here. A comparion of imilar deployment trategie for other coverage model uch a barrier coverage [6] and Trap Coverage [4] remain open. ACKNOWLEDGMENT Thi work wa ponored partly by NSF Grant CNS- 72983, CCF-728928, NIH Grant UDA2382 from National Intitute for Drug Abue (NIDA), and FIT at the Univerity of Memphi. The content i olely the reponibility of the author and doe not necearily repreent the official view of the ponor. REFERENCES [] R. Arratia, L. Goldtein, and L. Gordon, Two Moment Suffice for Poion Approximation: The Chen-Stein Method, Annal of Probability, vol. 7, pp. 9 25, 989. [2] X. Bai, S. Kumar, D. Xuan, Z. Yun, and T. H. Lai, Deploying Wirele Senor to Achieve Both Coverage and Connectivity, in ACM MobiHoc, 26. [3] P. Baliter and S. Kumar, Random v. Determinitic Deployment of Senor in the Preence of Failure and Placement Error, Univerity of Memphi, CS-9-2, Tech. Rep., 28. [4] P. Baliter, Z. Zheng, S. Kumar, and P. 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