The Marginal Utility of Cooperation in Sensor Networks

Transcription

1 e Maginal Utility of Coopeation in Senso Netwoks Yu-Ching Tong, Geg Pottie Depatment of Electical Engineeing Univesity of Califonia, Los Angeles Los Angeles, Califonia Abstact We pesent aguments that a small numbe of sensos within the netwok povide most of the utility. at is, coopeation of moe than a small numbe of nodes has little benefit. We pesent two scenaios. In the fist scenaio, all sensos povide identical utility, and thei utilities ae aggegated sequentially. e second scenaio is senso fusion with signal stength deceasing with distance. In that scenaio the souce is at the oigin and the sensos ae distibuted, eithe unifomly o accoding to a plana standad nomal distibution. We also vay the total numbe of sensos distibuted in both scenaios to obseve the utility/density tade off. Localization using the Fishe Infomation as the utility metic is used to demonstate that few sensos ae sufficient to deive most of the utility out of the senso netwok. ulation esults back up an ode statistics analysis of the behavio. e implication is that while co-opeation is useful fo some objectives such as combating fading and uncetainty of individual sensos, it is inefficient as a mean to incease the utility of a senso netwok if the best senso s utility is significantly shot of the desied utility. I. INTRODUCTION e usual paadigm of senso netwok eseach assumes thee will be a lage numbe of sensos pesent within the netwok. An impotant question fo minimizing esouce usage fo eithe scalability o extension of senso lifetime is how many of those sensos should be used at any given time. In many situations, afte the fist few sensos the utilities impovement apidly decease and each subsequent sensos added yield only maginal impovement [, [, [, [, [, [, [, [. At one exteme of the spectum, all the sensos have identical utility. us the oveall utility gows linealy. Although the oveall utility is unbounded, each additional senso contibutes less than sensos that wee aleady consideed. eefoe those sensos that ae selected ealie have a lage impact on the oveall utility. On the othe hand, typically sensos that ae close to the souce will be weighted moe heavily than those sensos that ae fathe away simply because of the highe signal stength that will be obseved by the sensos that ae close. Suppose sensos ae distibuted unifomly in a unit disk with a souce located at the oigin. We will show a small numbe of good sensos will povide most of the utility, and even in fading channels few ae equied. Localization is a common task fo senso netwoks. Senso netwoks can solve this type of poblem effectively by incopoating multiple views of the souce using diffeent sensos and multiple types of obsevations (e.g. ange, angle of aival, time diffeence of aival.) Senso fusion fo localization is staightfowad and can be solved efficiently. Using Fishe Infomation as the utility function, we will show that fo localization in a plana scenaio, a small numbe of sensos (on the ode of sensos) will povide most of the utility. In section II, we will assume each senso has identical utility. We evaluate how each added senso contibutes elatively less than those added peviously. Howeve, the oveall utilities can each an abitay value. We show how this plays out fo a localization poblem in section III. A non-unifom senso utility due to distance losses is consideed in section IV. We obseve that in a benign envionment, the best senso contibutes significantly moe than even the next best senso, and coopeation is not citical when the netwok is sufficiently dense. is is diven by the ode statistics of the sensos placement. e few good sensos utility is diven by the density of the deployment. e elative impovement of utility falls off quickly afte the fist few sensos. On the othe hand, even in a fading envionment, coopeation among the few good neighbos is sufficient to avoid outage. In section V, we summaize the two types of senso utility and thei implications fo the coopeation stategy in a senso netwok. II. IDENTICAL UTILITY Suppose each senso contibutes an identical amount of utility, and the oveall utility of the fused data will be the sum of the individual utilities, i.e. utility fo n sensos will be simply n. At the n-th iteation of data fusion, the existing senso set povides n units of utility, and the elative utility u of the existing set to the n-th iteation is e diffeence is u (n) = n n u (n) = n n(n + ) To incease the utility n by facto k, it will equie nk sensos, and the utility will appoach nk in O(/n). One simple model is coveage aea. Let each senso povide unit aea coveage and thee is no ovelap in coveage when we place each senso. eefoe n sensos povide utility n. We can cove an abitaily lage aea by deploying a sufficiently lage numbe of sensos. Howeve, the ate of each senso s actual contibution to the oveall utility deceases geometically as the total utility inceases, as seen in figue. e implication is that in a vey dense deployment, some ovelap o othewise a eduction in individual senso utility will not be noticeable. Figue shows that when individual senso utilities ae unifomly distibuted between. to., as the numbe of sensos incease, new senso contibution to the oveall utility still diminishes geometically. In addition, the vaiation in ealization of this pio utility ate due to individual senso utility vaiations diminishes as the numbe of sensos inceases due to law of lage numbes. Figue displays the esult of k tials, each with sensos. e ba limit is the maximum and the minimum utility fo a given numbe of sensos out of the k tials. A moe in-depth example can be seen in [. Even in thei complex deployments the satuation effect is eadily seen. ee ae situations that the utility fom sensos inceases at moe than a linea ate. ey aise when the numbe of sensos used is less than necessay to povide the desied quality of sevice and the undelying utility has ambiguity. A typical example is localization () ()

2 atio of existing contibution at next iteation Numbe of sensos.9 Fig.. Pio utility ate to the total utility can typically be achieved by thee techniques: tiangulation, scenes analysis and poximity sensing [. In this section, we will focus on a sub class of the tiangulation technique: ange/time of tavel (RNG), angle of aival (AOA) o time diffeence of aival (TDOA) measuement. A. Obsevation uncetainty model We model the obsevation as a Gaussian distibution centeed aound the tue eading. ρ i N( ρ i, σ i) fo i-th ange sensos, i =,..., k R, θ i N( θ i, σ i) fo i-th AOA sensos, i =,..., k A and τ i N( τ i, σ i) fo i-th TDOA sensos, i =,..., k T. B. Senso Utility Due to the obsevation uncetainty, localization will have limited accuacy. A common citeion fo accuacy of estimating paametes is the Came-Rao bound (CRB). e CRB matix tells us the best we can expect fom an unbiased estimato in tems of co-vaiance. A elated quantity, the Fishe Infomation Matix (FIM), which is the invese of the CRB, will be used as the basis of the localization poblem utility function. In paticula, the tace of the FIM will be the utility measue. e oveall FIM F of S S, whee S is the set of sensos selected and S is the set of all available sensos, is: F S = X k Sh kh k ()..... Fig.. Pio utility ate to the total utility with andom senso utility when the obsevations available ae less than equied to fom a unique solution. Such utility functions exhibit supe modula behavio [9. Unde such utility functions, the benefit of coopeation among sensos ae application specific. Suppose ou goal is to meet a cetain fixed amount of utility, and each additional senso povides additional utility, i.e., u n+ = ( + ǫ)u n, ǫ > whee u n is the utility of the n-th senso. e oveall utility U with N sensos is then simply X N U = u ( + ǫ) i () i= is geometic seies in eqn. is also unbounded and can also each any abitay utility fo sufficiently lage N. Even in this case, we can see that cetain sensos povide the majoity of the utility, but in this case it is the last few sensos as opposed to the fist few sensos. III. LOCALIZATION CASE STUDY In this case study, we will conside the utility that can be deived fom each senso in a localization poblem. Localization of souces Fom [,[,[, based on the obsevation uncetainty model used in III-A, in a plana localization poblem, h k is ( s k σ h k = k s k RNG () AOA y s y k,x s x k σ k s k whee s = [x s, y s is souce location and i = [x k, y k is k-th sensos location. σ k is the k-th sensos obsevation vaiance. With time diffeence of aival (TDOA) localization, in the case of unknown popagation velocity, the CRB matix fo location paamete and popagation velocity can be expessed as follows [: TDOA: h = v» s i σ i s i «s ef, τ s ef i whee ef = [x ef, y ef is the efeence senso location, v is the popagation velocity and τ ief is the time diffeence between the i-th senso and the efeence senso. Note that the above deivation did not explicitly assume any distance loss in signal quality, i.e. σ i in eality may also be a function of distance. is simplification has little impact when the souce is fa away fom the sensos goup and most sensos actually obseve simila signal stength. Howeve this simplification has consideable impact when the elative distances between the souce and the sensos vay significantly among sensos, which will be the case when the souce is nea the sensos goup. is distance dependent inteaction is affected by the distibution of sensos aound the souce and will be consideed below in section IV. C. Localization Utility ulation In the following simulation, the souce was placed within the field of sensos. We an tials of the expeiment. In each tial, thee wee RNG and AOA sensos, with. standad deviation on obsevation fo both types of sensos. eefoe the d be is. AOA sensos within adius will be selected fist, followed by the entie set of RNG sensos followed by the emaining AOA sensos. e ()

3 sensos ae distibuted unifomly ove a [ -, [ -, squae and the souce is placed unifomly ove a [-.,. [-.,. box. λ in Fig. is the sum of the eigenvalues of the FIM and is the utility metic fo this localization simulation. Note the apid satuation of the utility afte a few sensos, egadless of the selection algoithm used in selecting sensos. In this case the minimum numbe of sensos is thee fo the RNG sensos. Seveal algoithms wee used to select sensos in a sequential fashion among the entie set of sensos. e diffeent selection algoithms show that the undelying localization poblem ende the senso selection poblem tivial when sensos ae sufficiently dense in deployment. e algoithms we used ae as follows: ) Random: We simply pick sensos andomly. is is the simplest method; it equies no pio infomation. e density of the deployment detemines the success of this method. In paticula, this method will be successful in a dense deployment, and will fail easily in a spase deployment. ) Entopy Diffeence: Fom the obsevation model, each sensos obsevation has a cetain pobability distibution. Fom [, we may use a heuistic based on infomation theoy to sot sensos accoding to thei potential benefit in impoving ou accuacy in the localization poblem. is method consides the poblem in its entiety; both the senso s obsevation vaiance and the geometic factos ae consideed. is method equies two pats. Fist the entie obsevable space has to be discetized once to compute H v i ((9) in [) fo each senso to compute the a pioi entopy of obsevation: Z Hi v = p(z) log p(z)dz whee z is the field of view of the sensos. is H v i only depends on sensos location and the geomety of the obsevable space. At each iteation, based on the peviously picked senso, acoss the entie obsevable space, we need to compute H s i (() in [ ), whee H s i epesents the entopy of the senso obsevation given that the souce location is estimated based on knowledge available up to pesent. Z Hi s = p(z ˆx) log p(z ˆx)dz whee ˆx is the latest maximum likelihood estimate of souce location. [ detailed how H v i H s i appoximates the mutual infomation compaison at each senso and the diffeence fom the actual mutual infomation. All of this is vey computationally intensive. In fact, accoding to [, it s O `w, assuming the obsevable space is gidded into a n w matix. If sensos ae to compute the entopy diffeence in a distibuted fashion, each sensos equies the pobability distibution of the souce location. At the end of each iteation the pobability distibution of the souce location will be updated with the selected senso s obsevation. ) Neaest Senso Fist: Anothe heuistic method is to sot the sensos accoding to thei distance fom the estimated souce location, then pick the closest senso fist. Pesenting the selection algoithm as an optimization poblem, we want to select the i-th sensos that min s i i All that is equied is some kind of estimation of souce location and all sensos locations. A typical ealization of utility pogess, with a vaiety of senso selection algoithms is shown in Fig.. Fig. shows the numbe of λ Tue initial Location Neaest andom Entopy ARA Numbe of sensos Fig.. Tue Neaest, el. metic Tue Entopy Diffeence, el. metic Fig.. One ealization of of localization utility Tue ARA, el. metic Random pick, el. metic Numbe of sensos to achieve 9% of total utility sensos needed fo a vaiety of senso selection algoithms to achieve 9% of the utility achived by using all sensos. ) Angle-Range-Angle: Yet anothe heuistic method is to select some AOA sensos fist, then select all the RNG sensos and then the emaining AOA sensos. Suppose fo all RNG sensos the obsevation vaiance is σ R and fo all AOA sensos the obsevation vaiance is σ A. All RNG sensos ae equivalent in tems of thei utility towad the localization application, since h RNG ae unit vecto scaled by σ R. On the othe hand, AOA sensos that ae close to the souce povide moe utility than the fathe away countepats. D. Localization Conclusion Clealy the fist few sensos contibuted the majoity of the utility of the data fusion, unde this simplified utility function. If some sensos actually have a highe utility function than othe sensos, even fewe sensos will contibute most of the utility. Consequently, while thee ae some small diffeences among the algoithms pefomance, the utility function hee penalizes the simple selection algoithm only slightly compaed to moe complex algoithms.

4 Fom the above execise, we can see that a few sensos in addition to the minimum equied to esolve the souce location uniquely will be sufficient to localize the souce, echoing a esult obseved in [. IV. NON-UNIFORM SENSOR UTILITY Now we will conside non-unifom senso utility that aises due to the sensos geometic distibution and the esulting distance loss effect on the expected utility. Ode statistics will play a key ole to tansfom the sensos distibution to the expected utility. In addition, we will obseve that a small numbe of sensos coopeating help in ovecoming outage due to fading. A. Ode Statistic e distibution of the k-th statistic out of n IID dawn andom vaiables can be witten as follows [: n f X(k) (x) = n k «F(x) k ( F(x)) n k f(x) () Fom eqn., we can obtain the distibution fo the neaest k-th senso distance to the souce given the numbe of sensos that ae dawn, along with the senso-souce distance distibution, assuming all the sensos ae dawn in an IID fashion. In the following we will conside two distibutions: sensos ae distibuted in the unit disk unifomly, and sensos ae distibuted accoding to a nomal distibution in a plane. B. Unifom Disk ) Layout and Assumptions: We will assume a souce is at the oigin and all sensos ae distibuted unifomly within the unit disk. e utility function fo a given senso is the distance of the senso and the souce, i.e. α i fo the i-th senso which is i away fom the souce and α. ) eoetical distibution of distance: All the sensos ae placed in an IID manne, and the distances ae distibuted accoding to f R() =, [,, a tiangula distibution. e odeed statistic fo the k-th closest senso distance fo n sensos total is as follows afte putting the appopiate tems into eqn.. f R(k) () = n n k «(k ) ( ) n k () As fig. shows, the fist few distibutions ae simila and eqn. matched well with the simulation. is similaity between the fist few distibutions is the key to undestanding how to ovecome fading as will be discussed in section IV-E. C. D Gaussian ) Layout and Assumptions: We continue to assume a souce is at the oigin but now all the sensos ae distibuted in a plana standad nomal distibution. e utility function fo a given senso is the distance of the senso and the souce, i.e. α i fo the i-th senso which is i away fom the souce and α. ) eoetical distibution of distance: All the sensos ae placed in an IID manne, and the distance is distibuted accoding to Rayleigh distibution, with f R() = e /. e exteme odeed statistics ae as follows. n f R(k) () = n k «( e / ) k (e / ) n k e / (9) As fig. shows, the fist few distibutions ae also simila to each othe. is distibution also shaes a simila shape with the unifom (E[U N E[U N )/E[U N E[U N E[U N Fig.. (E[U N E[U N )/E[U N E[U N E[U N 9 Numbe of sensos.. Numbe of sensos Neaest senso expected utility evolution fom n to n+ in a disk Numbe of sensos.. Numbe of sensos Fig.. Neaest senso expected utility evolution fom n to n+ in a D nomal distibution disk distibution, with the exception that we no longe have a had bounday limitation as in the disk model. Howeve, ou inteest is in those that ae close to oigin. us the tail of the distibution has little impact. D. Expected Utility by vaying k, n e expected utility of the k-th senso is Z E[u k = α f R(k) ()d () Fom eqn., and the espective ode statistic distibution fom eqn. and 9, we obtain the following figues, illustating evolution of utility deived fom the neaest senso as the total numbe of sensos incease in the espective envionments. As shown in figues and, both distibutions behave similaly. Both expeience a shap incease in utility initially, and then the elative utility gowth diminishes. As seen fom the figues 9 and, the utility is dominated by the neaest senso. e elative utility is plotted, whee the neaest senso

5 k=, n=. k=, n= k=, n= (a) k=, n= (b) k=, n= (c) k=, n= Fig.. ulation and theoetical ode statistic of distance to oigin, unifom disk. k=, n=.. k=, n=.9. k=, n= (a) k=, n= (b) k=, n= (c) k=, n= Fig.. ulation and theoetical ode statistic of distance to oigin, plana nomal k th sensos 9 eoetical elative utility by each senso Total numbe of senso k th sensos 9 eoetical elative utility by each senso Total numbe of senso Fig. 9. Relative utility in disk Fig.. Relative utility in D nomal distibution is the efeence, and the level cuve is the utility below, the efeence, in db. e implication of the above esult is that the one o two sensos that ae closest to the souce will geneate most of the utility. is futhe implies that coopeation will not be an effective means to incease the utility of sensos. us coopeating beyond necessay, e.g. the minimum numbe of sensos equied to uniquely localize a taget, will not povide much incease in utility. Howeve, some level of coopeation should be consideed in a senso netwok to defend against uncetain envionments as discussed below. E. Coopeation One such infelicitous envionmental facto is fading. Fading can occasionally cause significant degadation to signal stength. Hee we conside the utility function ( α ) is multiplied by a fading facto g distibuted accoding to the Rayleigh distibution, f G(g) = g/σ f exp( g /(σ f)). Fo a given geomety, we will daw a set of

6 Expeimental Outage Pobability Expemential elative utility by each senso... k th neaest sesno selected n, total numbe of sensos k th sensos.... Total numbe of senso..... Fig.. Fading outage in a disk Fig.. Fading outage in a disk, fixed goal Expeimental Outage Pobability Expemential elative utility by each senso k th neaest sesno selected n, total numbe of sensos k th sensos.... Total numbe of senso.. Fig.. Fading outage in D nomal distibution Fig.. Fading outage in D nomal distibution, fixed goal Rayleigh distibuted andom vaiables to simulate the fading effect and collect the statistics ove multiple instances of the fading. We declae an outage if the sum of the faded utility is less than the neaest utility when thee is no fading. In figues and, outage pobability, as defined above, is shown with σ f =.. Both types of distibutions expeience impovement with a small numbe of collaboatos and diffe only in the tail egion when outage is below %. As the numbe of sensos needed to mitigate fading inceases, the diffeence between the two ode statistics become appaent. Not supisingly, the outage is independent of the numbe of sensos in the entie deployment, as seen in figues and that the given pobability of outage depends only on numbe of sensos used (k), and not on the total numbe of sensos (n). at is due to the outage definition above, whee outage is elated to the neaest senso utility in a non-fading envionment. Fo a given fading envionment, a few sensos coopeating is necessay to povide acceptable pefomance. is is in contast to the pevious scenaio. In the scenaio whee thee is no fading, the neaest senso alone is sufficient. Nonetheless, even in this case a small numbe of sensos suffice. Suppose we define outage as a cetain quality of sevice (QoS), in this case as the expected utility fom two sensos unde non-fading envionment. e outage is shown in figues and. With the fixed QoS, it is not supising that as the total numbe of sensos o the numbe of sensos used in coopeation inceases the outage deceases. Note also that as the total numbe of sensos inceases, the numbe of sensos fo actual coopeation can be educed in ode to each the taget QoS. at is achieved by sensos being close to the souce such that those sensos can povide the tageted QoS even in a fading envionment. V. CONCLUSION Coopeation among sensos is not an effective means to incease senso netwok utility fom individual senso utility o scaling of coveage. Howeve, coopeation is an effective means to defend against fading and is necessay to povide coveage. In the case of using coopeation against fading, a small numbe of sensos coopeating is sufficient. e numbe is mostly a function

7 of the fading paamete, assuming the netwok is sufficiently dense such that the coopeating neighbos also have simila utility. In the case of inceased coveage, only the neaest few sensos ae needed. is equies the netwok be deployed at sufficient density so that the few closest sensos will povide the desied quality of sevice. An example is the localization poblem, whee a cetain minimum numbe of sensos ae needed to poduce a unique estimate. e equied numbe based on geomety and a few additional sensos to mitigate poo geomety placement and/o fading will be sufficient to povide most of the utility. If the few closest sensos ae not sufficient, a lage numbe of sensos will not help, especially afte consideing the distance loss and the maginal utility povided by the late sensos. ese conclusions follow easily fom consideation of the ode statistics. REFERENCES [ A. Kause and C. Guestin, Nonmyopic active leaning of gaussian pocesses: an exploation-exploitation appoach, in ICML : Poceedings of the th intenational confeence on Machine leaning. New Yok, NY, USA: ACM,, pp. 9. [ S. Meguedichian, F. Koushanfa, M. Potkonjak, and M. B. Sivastava, Coveage poblems in wieless ad-hoc senso netwoks, in Poc. IEEE INFOCOM, vol., Anchoage, AK, Ap., pp.. [ K. Yedavalli, B. Kishnamachai, S. Ravula, and B. Sinivasan, Ecolocation: a sequence based technique fo f localization in wieless senso netwoks, in IPSN : Poceedings of the th intenational symposium on Infomation pocessing in senso netwoks. Piscataway, NJ, USA: IEEE Pess,, p.. [ L. Dohety, K. S. J. Piste, and L. E. Ghaoui, Convex position estimationin wieless senso netwoks, in Poc. IEEE INFOCOM, vol., Anchoage, AK, Ap., pp.. [ D. Niculescu and B. Nath, Ad hoc positioning system (aps),. [Online. Available: citesee.ist.psu.edu/niculescuad.html [ N. Bulusu, J. Heidemann, and D. Estin, Adaptive beacon placement, in Distibuted Computing Systems,. st Intenational Confeence on., Ap., pp [Online. Available: citesee.ist.psu.edu/ aticle/bulusuadaptive.html [ Y. Sung, L. Tong, and H. V. Poo, Senso configuation and activation fo field detection in lage senso aays,. [Online. Available: [ Y. Mostofi, T. H. Chung, R. M. Muay, and J. W. Budick, Communication and sensing tade-offs in decentalized mobile senso netwoks: A coss-laye design appoach, in Infomation Pocessing in Senso Netwoks,. IPSN. Fouth Intenational Symposium on, Ap., pp.. [9 F. Bian, D. Kempe, and R. Govindan, Utility-based senso selection, in Infomation Pocessing in Senso Netwoks (IPSN ), Nashville, TN, Ap.. [ J. Hightowe and G. Boiella, Location systems fo ubiquitous computing, IEEE Compute, vol., no., pp.,. [Online. Available: citesee.ifi.unizh.ch/eylocation.html [ A. Savvides, Design implementation and analysis of ad-hoc localization methods, Ph.D. dissetation, Univesity of Califonia, Los Angeles,. [ N. Patwai, Location estimation in senso netwoks, Ph.D. dissetation, Univesity of Michigan,. [ C. Chang and A. Sahai, Estimation bounds fo localization, in IEEE Confeence on Senso and Ad Hoc Communications and Netwoks, Oct.. [ J. C. Chen, K. Yao, T. L. Tung, C. W. Reed, and D. Chen, Souce localization and tacking of a wideband souce using a andomly distibuted beamfoming senso aay, e Intenational Jounal of High Pefomance Computing Applications, vol., pp. 9, Fall. [ H. Wang, K. Yao, G. Pottie, and D. Estin, Entopy-based senso selection heuistic fo taget localization, in Infomation Pocessing in Senso Netwoks (IPSN ), Bekeley, CA, Ap.. [ V. Isle and R. Bajcsy, e senso selection poblem fo bounded uncetainty sensing models, in Infomation Pocessing in Senso Netwoks (IPSN ), Los Angeles, CA, Ap.. [ N.Balakishnan and A. C. Cohen, Ode Statistics and Infeence. Academic Pess, 99.