Limit laws for area coverage in non Boolean models

Transcription

1 Limit laws for area coverage in non oolean models Tamal anerjee January 28, 2009 Abstract. Sensor nodes being deployed randomly, one typically models its location by a point process in an appropriate space. The sensing region of each sensor is described by a sequence of independent and identically distributed random sets. Hence sensor network coverage is generally analyzed by an equivalent coverage process. Properties of both area coverage and path coverage are well known in the literature for homogeneous sensor deployments. We study two models where the sensor nodes are deployed according to a stationary ox point process and a non homogeneous Poisson point process respectively. We derive asymptotic properties of vacancy in both the models. Key words. ox point process, non homogeneous Poisson point process, oolean model, area coverage, sensor networks. AMS subject classifications 2000). Primary 60D05; 60G70 Secondary 60K37; 60G55 Department of Mathematics, Indian Institute of Science, angalore , India, banerjee@math.iisc.ernet.in 1

2 1 Introduction 1.1 Motivation overage by a sensor network has always been a challenging problem from both theoretical and application viewpoint. A sensor is a device that measures a physical quantity over a region and converts it into a signal which can be read by an instrument or an observer. The union of all such sensing regions in the sensor field is the coverage provided by the sensor network. overage of a sensor network provides a measure of the quality of surveillance that the network can provide. Some of the common applications of sensor network includes environmental monitoring, emergency rescue, ambient control and surveillance networks. Adopting homogeneous scenarios in modeling the sensor locations is often too simplistic. For example, in applications like battlefield surveillance, an exact area of deployment is not known, at the same time a finite number of sensors are to be deployed over a large area. Even after deploying sensors, its location may change over time due to environmental factors like wind, river stream, rain, etc. Sometimes, a priori knowledge of the sensor field can be used to determine the concentration of sensors in the sensor field. This may result in higher concentration of sensors in some parts and lower concentration in others. Non uniform operational characteristics like interference, frequency of data collection and communication, etc., also results in non uniform degradation of the network [9]). An appropriate way to model such deployments is to assume a non homogeneous distribution for the location of sensors. For instance, the stochastic environmental heterogeneity for the distribution of the sensor nodes may be modeled by a ox point process. This motivates us to consider two such models for sensor deployment and study their coverage properties. In the first model we assume that the sensor locations are distributed according to a stationary ox point process and in the later model we consider a non homogeneous Poisson distribution for the sensor locations. 2

3 1.2 Model Description Let P {ξ i, i 1} be a stochastic point process in d, d 1 and {S 1, S 2,...} be i.i.d random sets in d, independent of P. Then {ξ i + S i, i 1} is called a coverage process. If in addition, P is a stationary Poisson point process then is a oolean model. The points of P may be interpreted as the location of sensors in a random sensor network and the shapes S i may be thought of as the sensing area about the ith sensor. Instead of working with random sets S i, we assume the sets S i s to be a fixed non random) non empty, orel measurable subset say S) of d with finite content c i.e, 0 < c = S < ). We consider the following two modes of deployment of sensors:- i) Model I P {ξ i, i 1} be a ox point process in d. We have the following definition of ox point process from [5] pp ). Let x), x d, be a non negative random field i.e, a non negative stochastic process indexed in d ) defined over some probability space. onditional on x) = λx), for x d, let Pλ) be a non homogeneous Poisson process with intensity function λ. Then P P ) is a ox point process. We assume that is stationary, i.e, E[ x)] = λ, a constant not depending on x. ii) Model II P {ξ i, i 1} be a non homogeneous Poisson process in d with intensity function λx). We assume the following bounds on the intensity function λx), for every fixed d, with finite content 0 λ l ± S) λx) λ u ± S) < x ± S, 1.1) the addition and substraction being understood in the Minkowski sense. 1.1) may be interpreted as the constraints on the number of sensors to be deployed in an operational 3

4 area, which we assume to be known for each fixed. However in Section 3 we study the limit laws keeping fixed. Hence we drop the argument and denote λ l ± S) respectively λ u ± S)) as λ l respectively λ u ) for the rest of the paper. 1.3 Previous Work The coverage problem is one of the oldest in geometric probability. overage processes arising naturally in stochastic geometry have been studied in [12] along with their applications. However the oolean model seems to be the most famous random set model in stochastic geometry. The area coverage properties for the oolean model have been extensively studied in the literature, most notably in [5]. One may also see [8] for some recent results in the case of one coverage. The coverage of a line by a two dimensional oolean model was investigated in [13]. In [10] the authors considers a coverage process where the shapes S i have distributions that depend on the locations of their centers. They analyze the probability of the event that the whole of d will be covered by such a coverage process. The growth of tumor cells have been modeled using coverage process in [2]. The statistical properties of the coverage of a one dimensional path induced by a two dimensional non homogeneous random sensor network have been studied in [9]. In [7] coverage by a finite number of heterogeneous sensors is analyzed using integral geometry. The ox process have been actively applied in Finance and isk Theory [1,6] and in forestry [11]. To the best of our knowledge, the problem of area coverage in the two models, described above has not been addressed before. 1.4 Organization of the Paper and Summary of esults Our paper is in the same sprit as those that study area coverage as in [5]). In Section 2 we consider a coverage process arising out of a ox point process Model I). We derive the expectation and variance of vacancy. Finally we study the asymptotic vacancy under suitable scaling and obtain a central limit theorem for the vacancy. 4

5 We carry out similar type of analysis in Section 3 when P is a non homogeneous Poisson process Model II). We study the asymptotic vacancy under suitable conditions on the intensity function of the non homogeneous Poisson process and derive a central limit theorem for the vacancy. The techniques of the proofs in both cases are in general similar to those in hapter 3 of [5]. We provide a detailed proof for the ox process and indicate a sketch in the other case. 2 overage in a ox point process Model I) 2.1 Expectation and Variance of Vacancy onsider a coverage process {ξ i +S i, i 1} when P is as in i) of Section 1.2. Let be a orel subset of d, 0 < < and S i S i 1, S being a fixed non random) non empty, orel measurable subset of d with finite content c. Define the following indicator function for a point x d 1 if x / ξ i + S, i 1, χx) = 0 otherwise. We define the vacancy V within, to be the d-dimensional volume of the part not covered by, i.e., V = V ) = χx) dx. 2.1) 5

6 y Fubini s theorem and stationarity of the ox point process we obtain from [5] [ ] EV ) = E χx)dx = Pr[ x / ξ i + S, i 1] dx = Pr[ ξ i / x S, i 1] dx = Pr[ ξ i / S, i 1] dx = E [e S x) dx ]. Similarly, we have [e x1 x2 S) S) x) dx] E[χx 1 ) χx 2 )] = E )] = E [e x1 x2 S x) dx+ S x) dx x1 x2 S) S) x) dx = E [ e 2 S x) dx x.e 1 x 2 S) S) x) dx]. 2.2) The last line following from the stationarity of the ox point process. ov[χx 1 ) χx 2 )] = E[χx 1 ) χx 2 )] E[χx 1 )]E[χx 2 )] [ = E e 2 S x) dx x.e 1 x 2 S) S) x) dx] [ E e ]) 2 S x) dx. 2.3) Hence V AV ) = = ov[χx 1 ) χx 2 )] dx 1 dx 2 [ E e 2 S x) dx x.e 1 x 2 S) S) x) dx] E [e ]) ) 2 S x) dx dx 1 dx ) 6

7 2.2 Limit Laws for Model I onsider a coverage process δ) with fixed shapes S and P same as in i) of Section 1.2. We scale the shapes S by δ δ < 1) in the δ) model. Let V be the vacancy within the region 0 < < ) arising from the δ) model. An excellent discussion for studying limit laws in scaled models can be found in Section 3.4 of [5]. ecall that λ = E[ x)]. Theorem 2.1. onsider the scaled coverage process δ). Let δ 0 as a.s. ) ) such that δ d δz) dz δz) dz ρ a.s. 0 < ρ < ), V A e ) δd S δz) dz 0 a.s. d, 0 < <, x) > 0 a.s. and δd λ l, l being any positive constant. Then, δz) dz i) EV ) e ρ S. 2.5) ii) V AV ) ) iii) E V EV ) p 0, 1 p <. 2.7) 7

8 iv) ) δz) dz V AV ) a.s. σ 2 ρ e 2ρ S d [ e ρ y S) S) 1 ] dy. 2.8) Theorem 2.2. onsider the scaled coverage process δ). Let δ 0 as a.s. ) ) such that δ d δz) dz δz) dz ρ a.s. 0 < ρ < ), V A e ) δd S δz) dz 0 a.s. d, 0 < <, x) > 0 a.s. and δd λ l. Then, δz) dz ) 1/2 δz) dz {V EV )} d N0, σ 2 ) a.s. 2.9) where σ 2 is same as in 2.8). emark 2.1. From the scaling laws we have δz) dz a.s. λ. Hence the condition δ d λ l is needed for the above results to hold. emark 2.2. If x) γ, a constant then V A e ) δd S δz) dz = 0. Therefore the condition ) V A e ) δd S δz) dz 0 a.s. is not required in [5] to prove the central limit δz) dz theorem for vacancy in a oolean model. emark 2.3. The limit laws of vacancy for the oolean model can be derived from the above theorems by choosing x) γ, a constant. 8

9 3 overage in a Non Homogeneous Poisson Process Model II) 3.1 Expectation and Variance of Vacancy onsider a coverage process {ξ i + S i, i 1} when P is as in ii) of Section 1.2. Let be a non empty orel subset of d with finite content and S i S i 1, S being a fixed non random) non empty, orel measurable subset of d with finite content c. Define the following indicator function for a point x d 1 if x / ξ i + S, i 1, χx) = 0 otherwise. We define the vacancy V N within, to be the d-dimensional volume of the part not covered by, i.e., V N = V N ) = χx) dx. 3.1) y similar calculations as in Section 2.1 we obtain [ ] EV N ) = E χx)dx = Pr[ x / ξ i + S, i 1] dx = Pr[ ξ i / x S, i 1] dx = e ) x S λt) dt dx. 3.2) 9

10 Similarly, we have E[χx 1 )χx 2 )] = Pr[ x 1 / ξ i + S, i 1 and x 2 / ξ i + S, i 1 ] = Pr[ ξ i / x 1 S, i 1 and ξ i / x 2 S, i 1 ] = Pr[ ξ i / x 1 S) x 2 S), i 1 ] = e x 1 S) x 2 S) λt) dt = e x 1 S λt) dt+ x 2 S λt) dt x 1 S) x 2 S) λt) dt ). 3.3) ov[χx 1 ) χx 2 )] = E[χx 1 ) χx 2 )] E[χx 1 )]E[χx 2 )] ) = e x 1 S λt) dt+ x 2 S λt) dt x 1 S) x 2 S) λt) dt e x 1 S λt) dt) e x 2 S λt) dt). 3.4) Hence V AV N ) = ) e x 1 S λt) dt+ x 2 S [e λt) dt ] x 1 S) x 2 S) λt) dt 1 dx 1 dx ) We have the following bounds for V AV N ) from 1.1) e 2λu S [ e λ l x 1 x 2 +S) S 1 ] dx 1 dx 2 V AV N ) [ e 2λl S e λ u x 1 x 2 +S) S 1 ] dx 1 dx ) 3.2 Limit Laws for Model II onsider a coverage process δ) with fixed shapes S, P same as in ii) of Section 1.2 and the intensity function λx) of the non homogeneous Poisson point process satisfying the bound 10

11 in 1.1). We scale the shapes S by δ δ < 1) in this δ) model. Let V N be the vacancy within the region 0 < < ) arising from the δ) model. To obtain non trivial coverage we assume λ l > 0 and λ u > 0. We obtain the following results by same techniques as in [5]. Lemma 3.1. If δ 0, as λ l and λ u, such that δ d λ l ρ l and δ d λ u ρ u, where 0 < ρ l ρ u <, then for the scaled process δ), lim sup EV N ) e ρ l S λ l,λ u 3.7) and lim inf EV N) e ρu S. 3.8) λ l,λ u If ρ l = ρ = ρ u, then EV N ) e ρ S. 3.9) Lemma 3.2. If δ 0, as λ l and λ u, such that δ d λ l ρ l and δ d λ u ρ u, where 0 < ρ l ρ u <, then for the scaled process, i) V AV N ) 0, even if ρ l ρu). 3.10) ii) E V N EV N ) p 0, 1 p <. 3.11) iii) lim sup λ l V AV N ) ρ l e 2ρ l S e ρ u y+s) S 1 ) dy 3.12) λ l,λ u d and lim sup λ u V AV N ) ρ u e 2ρ l S e ρ u y+s) S 1 ) dy. 3.13) λ l,λ u d 11

12 iv) lim inf λ lv AV N ) ρ l e 2ρu S e ρ l y+s) S 1 ) dy 3.14) λ l,λ u d and lim inf λ uv AV N ) ρ u e 2ρu S e ρ l y+s) S 1 ) dy. 3.15) λ l,λ u d Lemma 3.3. If ρ l = ρ = ρ u then under the same scaling law as in Lemma 3.1 we have for the scaled process, λ i V AV N ) σ 2 ρ e 2ρ S d e ρ y+s) S 1 ) dy for i {l, u}. 3.16) Lemma 3.4. onsider the scaled coverage process δ). If δ 0, as λ l, and λ u, such that δ d λ l ρ and δ d λ u ρ, 0 < ρ <, then for the scaled process, λ 1/2 i {V N EV N )} d N0, σ 2 ) for i {l, u} 3.17) where σ 2 is same as in 3.16). emark 3.1. As mentioned in Section 1.2 both ρ u and ρ l will depend on and S. However, since and S are kept fixed in the entire analysis, we do not indicate their dependence in the notation. emark 3.2. As λ l and λ u, the intensity function λx), pointwise for all x in the area of interest. Hence the scaling law implies δ d λx) ρ, pointwise for all x in that region. 12

13 4 Proof of Main results 4.1 Proof of esults in Model I Proof of Theorem ) follows trivially by dominated convergence theorem. y 2.4) and the inequality e x 1 xe x, x 0, we have, V AV ) [ = E e 2 δs x) dx.e [ E e 2 ) δs x) dx [ E x 1 x 2 δs) δs) x 1 x 2 δs) δs) x) dx] x 1 x 2 δs) δs) ] x) dx [ 2 δ d λ x 1 x 2 S) S) + V A δ E [e ]) ) 2 δs x) dx dx 1 dx 2 x x) dx) e 1 x 2 δs) δs) x) dx)] dx 1 dx V A e ) δs x) dx dx 1 dx V A e ) δs x) dx e δs x) dx ) ]. 4.1) The last line following from Fubini s theorem and the fact that E[ x)] = λ. oth the terms in 4.1) converges to zero under the given scaling law and the fact that S has a finite content. Hence 2.6) follows. y hebychev s inequality, P[ V EV ) > ɛ] V AV ) ɛ 2 0. Hence V EV ) 0 in probability. Since 0 V, the dominated convergence theorem gives us the L p convergence in 2.7). Applying the change of variables x 1 x 2 = y and x 2 = x, we obtain from 2.4) V AV ) = δ d [ dx E e 2δd S δz) dz.e δd y S) S) δz) dz] E [e ]) ) 2 δd S δz) dz dy. x ) δ 4.2) 13

14 It is easy to see under the given scaling law [ f δ x) = E e 2δd S δz) dz.e δd y S) S) δz) dz] E [e ]) ) 2 δd S δz) dz δ 1 x ) m e 2ρ S d e ρ y S) S) 1 ) dy. 4.3) dy y dominated convergence theorem, ) δz) dz V AV ) ρ m dx ρm, as δz) dz a.s. Hence we have 2.8). Proof of Theorem 2.2. Let r be a large positive constant. We divide all of d into a regular lattice of d dimensional cubes of side length τrδ),where τ = 2c = 2 S, so that each cube is separated from its adjacent cube by a spacing strip of width 2τδ). Let A 1 denote the union of all the cubes which are wholly contained within. A 2 be the union of all the spacing strips that are wholly contained within. Finally we denote A 3 as the intersection with of all the spacing strips and the cubes that are contained only partially within. The above configuration is illustrated in Figure 1 for dimension d=2. Let V i) the region A i. The vacancy within can be expressed as, be the vacancy within V = V 1) + V 2) + V 3). As δ 0, under the assumption of the theorem the cubes gets finer and we have A 3 0 a.s. 4.4) 14

15 Figure 1: The region shaded in dark represents A 1 whereas the lightly shaded region represents A 2. The unshaded part within the region represents A 3. We also have, A 2 l, where l is a constant independent of r. 4.5) r The vacancy V i) within the region A i satisfies V AV i) ) [ = E e 2 δs x) dx.e A i A i [ e 2 ) δs x) dx A i A i A i E d E [ x 1 x 2 δs) δs) A i δ 2d λ S ) 2 + A i 2 V A x 1 x 2 δs) δs) x) dx] x 1 x 2 δs) δs) ] x) dx E [e ]) ) 2 δs x) dx dx 1 dx 2 x x) dx) e 1 x 2 δs) δs) x) dx)] dx 1 dx 2 + A i 2 V A e ) δs x) dx dx 1 dx 2 + A i 2 V A e ) δs x) dx e δs x) dx ). 4.6) The last line following from Fubini s theorem and stationarity of the ox process. From 4.4), 4.5) and emark 2.1, we have, lim δz) dz ) δz) dz 15 V AV 3) ) = 0 a.s. 4.7)

16 and lim lim sup r δz) dz ) δz) dz V AV 2) ) = 0 a.s. 4.8) Hence we observe that the only significant term involves V 1) theorem it is enough to show that, and to prove the central limit {V 1) 1) 1) EV )}/V AV ))1/2 d N0, 1) a.s. 4.9) and lim lim sup r δz) dz δz) dz V AV 1 ) V AV ) ) = 0 a.s. 4.10) Let n denote the number of small cubes of length τrδ which make up the region A 1, and let D i denote the ith of these cubes, for 1 i n. We further denote by U i the contribution to V 1) from D i. Then we have V 1) = n i=1 U i. Since the shape δs is contained within a sphere of radius τδ, and the cubes D i are least 2τδ distance apart, the shape δs can not intersect more than one cube. Due to stationarity of the ox point process the variables U i are identically distributed. Hence U i s are i.i.d random variables and we have n V AV 1) ) = V AU i ) = nv AU i ). i Let D be a d dimensional cube of side length τr having the same orientation as D 1. For any two real sequences a n and b n a n b n implies a n /b n 1 as n. From 2.4) we obtain V AV 1) = n D i ) D i nδ 2d e 2ρ S [ E e 2 δs x) dx.e e ρ x 1 x 2 S) S) 1 ) dx 1 dx 2. D D x 1 x 2 δs) δs) x) dx] E [e ]) ) 2 δs x) dx dx 1 dx 2 16

17 Since n = O δz) dz) a.s. we have i E U i EU i ) 3 i V aru = E U i EU i ) 2 E U i EU i ) i)) 3/2 i V aru i)) 3/2 2 D i V aru i ) i V aru i)) 3/2 = 2τrδ) d i V aru i)) 1/2 = O ) ) 1/2 δz) dz ) as δz) dz a.s. Hence 4.9) follows by Lyapunov s central limit theorem. y auchy-schwarz inequality we have V AV ) V AV 1) ) [ )] 2 = E [V E V )] 2 E V E V 1) [ )) = E V 2) + V 3) E V 2) + V 3) [ ) 4 V A + V A V 2) V 3) 2V V 2) V 3) E )] 1/2 [ V A V ) + V A V 2) ))] 2V V 2) V 3) ) )] 1/2 + V A. V 3) 4.12) 4.10) follows from 2.8), 4.7), 4.8) and 4.12). Hence as δz) dz a.s. we have Theorem 2.2. emark 4.1. The condition showing 4.3), 4.7) and 4.8). ) δz) dz V A e ) δd S δz) dz 0 a.s. is used only in 4.2 Proof of esults in Model II y using asymptotic properties of vacancy for the oolean model as in [5], pp ), one can derive all the results for asymptotic vacancy in Section 3.2. While investigating the asymptotic properties of vacancy for the ox point process in Section 2.2, we have modified the proofs given in [5] and illustrated the detailed techniques. Hence to avoid repetitions we 17

18 indicate only a sketch of the proofs in the non homogeneous model. In all the proof that follows the operational area is kept fixed. Proof of Lemma 3.1. We have from 3.2) e λu S EV N ) e λ l S. 4.13) Thus the expected vacancy is bounded from above respectively from below) by the expectation of vacancy in a oolean model driven by a Poisson point process with intensity λ l respectively λ u ) and generated by the same shapes S. Now scaling the shapes S by δs we have, e λuδd S EV N ) e λ lδ d S. Hence 3.7) and 3.8) follows under the scaling law in Lemma 3.1. In the case when ρ u = ρ = ρ l, 3.9) follows trivially. Proof of Lemma 3.2. From equation 3.6) for the scaled process we have V AV N ) e 2λ lδ d S [ e λuδd x 1 x 2 δ +S) S ) 1 ] dx 1 dx ) 3.10) now follows by the same argument as in 2.6) of Theorem 2.1. y hebychev s inequality, P[ V N EV N ) > ɛ] V AV N) ɛ 2 0. Hence V N EV N ) 0 in probability. Since 0 V N, the dominated convergence theorem gives us the L p convergence in 3.11). y making the change of variable, x 1 x 2 = y and x 2 = x, as in the proof of 2.8) in Theorem 2.1 one can prove 3.12), 3.13), 3.14) and 18

19 3.15). Lemma 3.3 follows trivially from Lemma 3.2 when ρ l = ρ = ρ u. Proof of Lemma 3.4. We proceed exactly as in the proof of Theorem 2.2. Unless otherwise stated all the notations that are used in the proof of Theorem 2.2 will have an analogous meaning for Model II. We obtain the following equations for the scaled process from 3.6) and [5]. V AV i) N ) [ e 2λ l δs e λ u x 1 x 2 +δs) δs ) 1 ] dx 1 dx 2 A i A i λ u δ 2d e λuδd S A i S ) Hence and lim lim λ uv A λ l,λ u lim sup r λ l,λ u V 3) N λ u V A ) = ) V 2) N ) = ) Hence to prove the central limit theorem it is sufficient to prove that {V 1) N 1) 1) EV N )}/V AV N )1/2 d N0, 1), 4.18) and lim lim sup λ u V A r λ l,λ u V 1) N ) V AV N )) = ) 19

20 n i=1 E U i EU i ) 3 τrδ)d n i=1 V AU 3/2 i)) n i=1 V aru i)) 1/2 τrδ) d ne 2λu δs D i D i [e λ l x 1 x 2 +δs) δs 1] dx 1 dx 2 ) 1/2 = Oλ 1/2 u ) 0 under the given scaling law. 4.20) Therefore 4.18) follows by Lyapunov s central limit theorem. The rest of the proof follows as in [5] pp ). The other statement in Lemma 3.4 can be proved in a similar way. 5 elated Problems The results of [5] for asymptotic vacancy in a oolean model have been generalized in [4], using the notion of associated random measures. It could be interesting to obtain similar generalization results for the coverage processes studied in our paper. It is not clear if one can obtain a central limit theorem for the vacancy in non homogeneous deployment of sensors without imposing condition 1.1) on the intensity function. One can carry out a similar type of analysis of path coverage as in [9,13] for the ox point process. Acknowledgements The present author would like to thank Debleena Thacker for drawing the figure and providing important suggestions in preparing this report. 20

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