µ Surveillane overage o sensor networks under a random moility strategy eorge Kesidis & CS Depts, Penn State University University Park, PA, USA kesidis@engrpsuedu Takis Konstantopoulos C Dept, University o Texas Austin, TX, USA takis@eeutexasedu Shashi Phoha ARL, Penn State University University Park, PA, USA sxp26@psuedu Astrat "$ %&$ &( $$ " &$,$ 0&1$$" 23$ 4 $56872&$ 93$ %%:<5&= % (&?@ "< % 0 A($ = 5& $ 3 B C $D $3F" @ &=% &$ 1 @& %%: <% 3 H% 3 6I2 K 4&8LM=NO @& PQ$$ R5& S3$% T:<5&= $UV$W Q $W$XDK F<%"$ Q " %%Z [% \ =NO%A3=H% \K % [`& $ M@ :" a6lm :NMQ&$ A %3 <$ K& ḦH%$ 3 5 & $ K<@ X68 $& d e@ %V$("@&$% R %3 <$ K2& U$" O% 4:NM6 INTRODUCTION Sensor network appliations range rom passive surveillane (searh to simultaneous traking o multiple targets and distriuted identiiation o their oordinated movement The targets will have speii spaetime neighorhoods (rom stationary pointtargets to large, rapidly moving ones xample targets range rom individual humans roaming through a rowd to platoons o tanks in a desert war theatre to point explosions to ativation sites o hemial or iologial weapons A wide variety o sensors ould also e used inluding image (video, 3dimensional, inrared, omnidiretional, aousti mirophones, seismi or radiation meters, 72g2W7 Wg27 and h The sensor nodes themselves ould e ixed (to, eg, monitor an airport or power plant or ould orm a moile expeditionary sensor grid Sensor nodes may have limited amounts o availale energy stored in atteries whih may require energy eiient routing and ommuniation medium aess ontrol The environment in whih the sensors and their targets operate ould range rom land (desert, jungle, mountains, sea and air Furthermore, environmental onditions ould inlude high levels o noise, low levels o amient light, loud over and other olusions, and enemy ativity that an ompromise target searh and traking and network ommuniation Individual sensors may need to oordinate in a distriuted ashion to maintain ommuniation onnetivity [8, 13, 5, i5jxkmlonxpq:nrqs5prtukwvas5nxw%xmymzq<z {}2~rkmqqZwq:ymnrq z 5s5ymt:q:z ƒ q:nq<s5prtuk p 5ˆq:tZ~n [q:ymtz}(š@x ƒ Œas5ymzs5zm Oloylonr~rq<prq:zW{}2~kmq p `} ƒxq:nq<s5ptk Ž Ot:q Š ƒxž Œ xmymzmq:p a? ƒx vxs5pzw m X ` <š Vœ Z u Z Bœ[ [œ5žm y} [Ÿmloymlo [yn mymzmloym [n< s5ymz t: [ymt: oxmnrlo [ymn [p prq:t< [ O Oq:ymzms~lo [ymnxqz Ÿmpq:nnrq:zloy~rkmlon Ÿmxm{m olot<s~rlo [ys5prq ~km [nrq 5w~kmq s5x~km [prn s5ymzwzm ym 5~ ymq:t:q:nns5plo }pqz =q:tz~~rkmq loqzvn 5w~rkmq ƒ [px~rkmq ƒxž` j kq =pn~s5x~km [p< nxpq:nq<s5ptkwvxs5n nxmÿÿ$ [p~q:zloyÿ=s5p~ {}W A [ps5y~ Bj ƒa Vœ[ < [ [ [ª2s5ymz nrq<t: [ymzs5x~k [p nxpq:nq<s5prtukwvxs5n nrxmÿmÿ [pr~rq:zloyÿ=s5pr~ {} A [pus5y~ @ Vš[š[œ[ 5žš[ 1, target ontat, and surveillane overage to prespeiied degrees o onidene These are oten onliting goals requiring asi tradeos The ous o this paper is to study target detetion perormane or a sensor network under random moility This paper is organized as ollows We irst deine notation or our susequent prolem ormulation and give a rie overview o moility strategies in targetsearh mode Deterministi and random searh strategies are ompared Next, we desried the prolem o detetion o slowly moving (point targets in two and three dimensional environments, again under a random moility mehanism or the sensor grid For a twodimensional planar environment, we give a ound on the tail o the distriution o timeuntildetetion o a point target A more preise alulation is given or a threedimensional environment Our prolem ormulation allows us to onsider a large numer o sensors operating in a large region Finally, design issues pertaining to the single parameter o moility, the variane «a, are disussed TART SARCH STRATIS Consider a ontiguous region in the plane xtensions to threedimensions (undersea, air or spae environments are straightorward Let e the numer o moile sensors in that orm the nodes o our network The position o the ah node is assumed node at time is denoted y ² to have a maximum 5&$" <" %@& = At time, the total <"$ %@& =$ &$$ o the the intersetion o the set ²% with the region Here, ² [ V» ¾½ ÀK FÂ8»ÄÅ nodes is denotes a losed disk o radius and enter» The distane etween two points» \ is the ulidean distane»uâ X$ÈÇ ÉPÊ V» Ê Â8 Ê We assume in the ollowing that the maximum overage area (maximum assuming no overlapping disks is signiiantly less than the area o Thus, 3$%T: o the nodes is neessary to maintain surveillane overage o the whole region to some degree o onidene as desried elow
»» Â ½ Let e the mean density o nodes in the given region, ie, or i is random The ojetive o surveillane is to gather inormation aout ertain target ojets that move through the region These ojets exist in a soalled timespae neighorhood haraterized y a timevarying untion or times in some inite interval o (ontinuous time where takes values in the susets o For onveniene o notation, we an ` deine or In the ollowing, we simply assume that i [ Ê ² [ then sensor node ojet at time Certain targets may e :5&=% & in the sense that, or a ixed and» <, or all times where is the enter (ground zero o, or example, an explosion and is its duration An expanding ojet s detetion region grows as a untion o time, eg, a iologial weapon» < < whose agents diuse through the air ie, or some inreasing untion o Finally, a single 3=H ojet ould e haraterized y a (one dimensional K&= in,» or as well as a radius 2, ie, V» < We oviously assume that no sensor node is &$ aware o any suh attriutes o a target ojet In [7, a data usion ramework or sensor data is given and, in partiular, a mehanism is desried or estimation o ojet veloity involving a triangulation etween two nodes sensing the ojet Prolems in surveillane overage Moility strategies have two modes o operation depending on the presene (data aquisition, target traking or asene (searh o a [ target ojet(s In searh mode, moility strategies an e roughly lassiied into two groups: random and deterministi In deterministi strategies, region ould e partitioned a priori into piees and $3 a sensor node assigned to eah piee that eomes the area o the node The node sweeps its home area in, eg, a deterministi zigzag pattern o a lawnmower, searhing or targets Under random searh strategies, eah node moves at random or diuses" through Random moility an e made somewhat more eiient y adopting strategies wherein nodes loally repel" eah other and are then less likely to visit areas very reently visited Alternatively, proximal nodes an dynamially elet a luster head" whih remains ative while the other proximal nodes sleep" to onserve energy and redue ommuniation ongestion in the luster neighorhood [15, 4, 12 Clearly, hyrid and hierarhial moility patterns ould also e devised, eg, a oarser partition o in whih eah element o the partition has more than one node and eah node employs a random sweep pattern within its (now larger and shared home area In dataaquisition mode, the irst node to sense an ojet ould e deemed the oordinator" or that ojet and the alarm to the rest o the network Certain nodes would then deterministially drit toward the target In the ollowing, we ous on sensor networks only in searh mode Roustness o moility strategies A sensor node may ail eause o eletromehanial prolems, unoreseen ostrutions or hazards (inluding traps plaed y the enemy, exhaustion o its power supply, or y overt enemy ativity leading to destrution or apturinghijaking o the node When a node ails in the deterministi moility setting, the remaining nodes need to quikly eome aware o the ailure (note that this assumption would require a nontrivial intrusion detetion system An advantage o random moility strategies is that they require minimal oordination (ommuniation overhead espeially when a node ails and experiene graeul degradation in surveillane overage in that event On the other hand, random moility strategies will, in general, have poorer target detetion perormane Sensor networks with unontrolled random moility Reently, networks o extremely small sensors have een proposed, in partiular the smart dust" projet [9, 10 In suh networks, ontrol o sensor moility may highly limited y energy supply and environmental onditions, an example o the latter eing turulent air For suh networks, moility may e impliitly random, ie, moility may e modeled with diusion dynamis together with a drit imparted at the time o deployment POINT TART DTCTION UNDR BROWNIAN MOVMNT Consider an aritrary point in or, representing a stationary point target, taken to e the origin and taken to ome into existene at time zero (oth without loss o generality sine our sensor moility models are spatial and temporal translation invariant In this setion, we study the distriution o time required y a sensor grid, under a Brownian moility strategy, to detet the ojet Our prolem ormulation allows us to onsider a large numer o sensors operating in a large region d We assume the initial positions o the sensors 5Å" are distriuted as a Poisson point proess with intensity nodesm or nodesm (depending on whether the network is two or threedimensional Reall, at this point, L`H % & $ the notion o a [14 It is onstruted y drawing iid opies o some random set (eg, a disk or a all that are known as $5& and y translating eah opy at the point o a Poisson proess these points are known as $ A and are taken to e independent o the grains LM=NO &$ 2L`H % & % Below we model the network as a (BBM wherey the nodes exeute independent Brownian (, motions,, & The assumption o iid node displaements means that the sensor positions orm a Poisson proess or all [6
k The threedimensional environment In this setion, eah stohasti proess is a 3dimensional Brownian motion with independent Cartesian oordinates and variane oeiient «(measured in m s Without loss o generality, suppose that a target is loated at the origin Let = e the all o radius (the surveillane radius entered at the origin At eah point o time, the area that an e monitored y the sensor network is represented y the set ( where : < denotes Minkowski addition o two sets ½ ÀÄ Thus, at eah, the random set ( W Å = is a Boolean model We are interested in the time required or the target detetion: Deine We have 3ä½ " FÀ 3( m [Å or all Now note that, or eah, is itsel a Boolean model Indeed, we may take as enters (or germs the initial Poisson points, and as sets around them (or grains iid opies o the set $ %m& < [ ie, a all arried around y the trajetory o a % % Brownian motion etween and that is a soalled Wiener sausage Hene, is atually the volume ration o the Boolean model with Wiener sausage grains [14: ( d e : $ r where is the target hitting time o a single grain Clearly, y Fuini s theorem, $ <,021 ½» [Å d» Note that» 5476 89 476 89 476 89 476 89 30 V» :8H U ÀW»Â À3»Â À3T» =««&  << d» < À»U %mm %mp < P =«< m &0 where is a standard Brownian motion (variane 1 with Letting <"= = denote the irst hitting time o the all y a standard Brownian motion started at», we thereore have Hene, $»?4@6 < < = < = Now onsider again the Bessel proess C or a standard Brownian motion started at» It is known ` that C is a onedimensional diusion with C»F d3% ½ QÀ Ä Å Deine <IH C and note that sine there is isotropy (rotational invariane, the hitting time depends on the initial position only through its magnitude, ie, <I= or all» suh that QÈ» Thus, $ r < F K The random variale < H MA < H We see that ON P MA < H ALB Hene we may take «< H ALB 2 d»»d < H d $ has a wellknown distriution [2: Q H$R S XWX e[z$â Q H$R HUT LV MA < H \ S K " d % «, to alleviate notation, and in the end replae y «X With this in mind, and using the expression aove, we have $ r < d `a H((Â8 Â8 ehixâ Ç Ude d d V A The irst integral is learly, the volume o the all = For the seond È integral we use Fuini and a hange o variales (setting Â3» to get that the seond integral equals g ((Â8= R ADh ((Â8 H e  g i d d R A h»v»d H e:âo» m g»9i d d R A$j d Let» e :ÂO» m
j R  k» g e the density o, where is a standard normal distriuted random variale Thus, m»»@ H e:âo» m d» A»V»D 1 ½» U 0 1 ½,0 1 ½ Åm&0 Colleting the alulations together, we get $ < g Å R A» Å d» 1 ½ a A Åm m Now reall that the atual variane is «(hene replae y «X in the last expression and inally sustitute in the expression m to get: e  «This is a distriution with a Weiulltype tail The twodimensional environment In a twodimensional (planar environment, the expression or the distriution o the hitting time < given in (3 aove is not availale We an, however, use the Cherno upper ound: (O < H < H OÂ8 e  " where the expression or ex ½ < H ex «< H is given as a ratio o modiied Bessel untions in 201, p 297, o [2 This lower ound ould then e sustituted into (2 to otain a lower ound on $$ and thus otain an upper ound on the tail o the target detetion time distriution (1 Random moility design A design ojetive here ould e as ollows: Suppose that a point target is required to e deteted within a prespeiied time with an error proaility less than R, where is a positive numer Our design variale is the variane «o the node moility When is very small, the target detetion goal an e : ahieved i we hoose «X roughly larger than «[Å d This is ound y solving a quadrati inequality and taking asymptotis when eomes small We omit the details o the omputation We next ompute the expetation o the target hitting time as ollows First rewrite ( R RT P with the ovious hoie o the onstants The integration d an e perormed to give O % A V» where is the tail o the standard normal distriution untion Using the inequality V» R = ounds an e otained Finally, y sustituting the values o the onstants, we otain an exat expression: R  R «This is a rapidly dereasing untion o A, (exponential deay SUMMAR We onsidered the prolem o surveillane o a potentially large region undertaken y a potentially large group o moile sensors Under a random moility strategy or the sensor grid, the distriution o the ontat time etween two nodes and the distriution o the timeuntildetetion o slowly moving (point targets were studied Both two and three dimensional environments were onsidered Finally, design issues pertaining to the single parameter o moility, the variane «a, were disussed RFRNCS [1 Bettstetter, C On the minimum node degree and H $6 onnetivity o a wireless multihop network " $$&%, Lausanne, Switzerland, une 2002 W& KH$,? [2 Borodin, AN and Salminen, P LM=NO &$ $$ ( & [ &$ $("&$ Birkhäuser, Boston, 1996 [3 Canny Complexity o Root Motion Planning MIT Press, Camridge, MA, une 1988 [4 C Chevally, R Van Dyk and TA Hall, Selorganization or Wireless Sensor Networks In 6,, Prineton, Marh 2002 [5 Dowell, L and Bruno, ML Connetivity o random graphs in moile sensor networks: validation o Monte 6" 72L " Carlo simulation results, Berlin, 2000 [6 Frey, A and Shmidt, V Marked point proesses in the plane I 6 <V A& K 2 &$r6, 65110, 1998 [7 Friedlander, D and Phoha, S Semanti inormation usion or oordinated signal proessing in moile
h,z Ẍ66 2$ V$3&$ sensor networks `$3"% 26 Speial issue on sensor networks,, August 2002 [8 upta, P and Kumar, PR Critial power onnetivity 5& : & <@D in wireless networks In: `$ $D%W3 &=% ẌD& KU &=$ KÀ $"3 % $ M$ 6 6 3, WM Mneany, in, and Q Zhang (eds Birkhäuser, Boston, 547556, 1998 [9 Kahn, M, Katz, RH and Pister, KS Moile H $6", $ ` 3" &$ :NM$,H% networking or smart dust In `$ :6$, Seattle, 1999 [10 Kahn, M, Katz, RH and Pister, KS merging Challenges: moile networking or smart dust $ " & `$3F" @ &=% & :NM$,H,Z 6, Vol 2, No 3, 2000 [11 Kesidis, R Rao and S Phoha, "Moility management o adho sensor networks using distriuted annealing", CS Dept, Penn State University, Tehnial Report, CS 03017, Sept 29, 2003 [12 Krishnamahari, B, Wiker, SB, and Bejar, R Phase transition phenomena in wireless ad ho networks 6, 3 %" $ H d 2 :Ǹ,D :" $ \NOT, % L %, San Antonio, TX, Novemer 2001 [13 Q Li, R Peterson, M DeRosa, and D Rus Reative Behavior in Selreoniguring Sensor Networks in 6" % L, %, Atlanta, 2002 [14 Santi, P, Blough, DM, and Vainstein, F A proailisti analysis or the range assignment 6" $ % prolem in ad ho networks, Long Beah, CA, 2001 [15 Stoyan, D, Kendall, WS, and Meke, Q?& K,u522%@ & $ 6 5& : Wiley, New ork, 1996 [16 L Suramanian and RH Katz An arhiteture or 6, uilding selonigurale systems In $$H, 2000