NETWORK SECURITY: DESIGN, ANALYSIS AND TRADEOFF EVALUATION ALINA OLTEANU A DISSERTATION

Transcription

1 NETWORK SECURITY: DESIGN, ANALYSIS AND TRADEOFF EVALUATION by ALINA OLTEANU A DISSERTATION Submitte in partial fulfillment f the requirements fr the egree f Dctr f Philsphy in the Department f Cmputer Science in the Grauate Schl f The University f Alabama TUSCALOOSA, ALABAMA 009

2 Cpyright Alina Olteanu 009 ALL RIGHTS RESERVED

3 ABSTRACT Energy efficiency is an essential requirement fr all wireless evices. Recent evelpments in wireless sensr netwrs (WSNs), wireless lcal area netwrs (WLANs) an wireless persnal area netwrs (WPANs) have raise a eman fr energy-efficient algrithms an energy-efficient meium access cntrl (MAC) prtcls. When cnsiering security in this cntext, aitinal verhea is ae t the netwr an effrts must t be mae t minimize the extra la while at the same time achieving the esire level f security. Security attacs in the Internet are line t a ifferent set f vulnerabilities. The cmplex architecture f the Internet spanning ver ifferent aministrative mains an legal systems maes it easy fr attacers t cnceal the surce f the attac an preserve their annymity. This issertatin aresses several imprtant issues in netwr security an perfrmance incluing intrusin etectin, cipher esign, security verhea analysis an tracing, as fllws. We first prpse a mel fr intrusin etectin in WSNs, which ptimizes netwr cverage an etectin while minimizing the number f sensrs an energy cnsumptin. We then integrate a security mechanism int the sensr netwr in rer t achieve secure cmmunicatin. Specifically, we prpse a lightweight blc cipher base n a multiple recursive generatr (MRG) which is suitable fr WSN an RFID where pwer cnsumptin, banwith, memry an strage space are critical. ii

4 Next, we cnsier security in WLANs an WPANs an we apply the avance encryptin stanar (AES) cipher t ensure secure transmissin f frames. We integrate AES encryptin at the MAC layer f 80. WLANs an UWB WPANs, respectively, an stuy the verhea intruce by AES in this cntext. Finally, we analyze a type f security attac in the Internet where the intruer uses a chain f hst machines befre attacing the target. We iscuss tw mechanisms fr tracing intruers in the Internet, ne base n thumbprinting, an the ther n a timestamping technique f transmissin activities. iii

5 DEDICATION This issertatin is eicate t my parents. iv

6 ACKNOWLEDGMENTS I cannt give enugh thans t my issertatin avisr Dr. Yang Xia, wh supprte an avise me thrughut my research. His cntinuus guiance an encuragement ramatically imprve my research capabilities an prvie me with exciting pprtunities fr my future career. I am very grateful t all my issertatin cmmittee members, Dr. Marcus Brwn, Dr. Xiayan Hng, Dr. Zhijian Wu an Dr. Jingyuan Zhang fr their input an supprt f bth my issertatin an my acaemic prgress. I wul lie t express my appreciatin t the Cmputer Science Department, which ffere me a grauate assistant psitin an fstere my evelpment in the fiel f Cmputer Science. Finally, I wul lie t than my parents fr their unening lve an supprt. v

7 CONTENTS ABSTRACT... ii DEDICATION... iv ACKNOWLEDGMENTS...v LIST OF TABLES... xi LIST OF FIGURES... xii. INTRODUCTION.... WEAVING A PROPER NET TO CATCH LARGE OBJECTS IN WIRELESS SENSOR NETWORKS...9. Relate Wr...0. A Naive Apprach....3 Prbabilistic Meths fr Detecting Intrusin The Prperties f Detectin Prbability Cnstraine Optimal Detectin Prbability Detectin Prbability Relate T The Shape Of The Intrusin Object The Prperties f Intrusin Cverage Intensity Optimizing Netwr Lifetime Cnclusin...40 vi

8 3. A LIGHTWEIGHT BLOCK CIPHER BASED ON A MULTIPLE RECURSIVE GENERATOR FOR WIRELESS SENSOR NETWORKS AND RFID Bacgrun an Relate Wr The Blc Cipher Security Analysis Cnfientiality Authenticity Data Aggregatin Privacy Prtectin Analysis f The Numbers f Basic Operatins f ur Cipher Number f Basic Operatins fr Generating One Pseu-ranm Number Number f Basic Operatins fr the MRG Cipher Perfrmance Analysis Special Classes f Generatrs FMRG DX--s Generatrs Seeing the MRG RFID vs. WSN Cnclusin SECURITY OVERHEAD AND PERFORMANCE FOR AGGREGATION WITH FRAGMENT RETRANSMISSION (AFR) IN VERY HIGH-SPEED WIRELESS 80. LANS Bacgrun an Relate Wr...68 vii

9 4. Fragmentatin with Encryptin Per pacet MAC efficiency with AES encryptin Per frame MAC efficiency with AES encryptin Zer-waiting Scheme, Maximum Efficiency an Maximum Thrughput Analysis f the Mel Saturatin Thrughput Optimal Frame Size Optimal Fragment Size MAC Delay Analysis Mel Evaluatin Cnclusin OPTIMIZATION BETWEEN AES SECURITY AND PERFORMANCE FOR IEEE WPAN Relate Wr Cnstraine Thrughput Optimizatin Thrughput Optimizatin fr CFP CTA Immeiate ACK N ACK Delaye ACK Thrughput Optimizatin fr CAP Immeiate ACK N ACK Delaye ACK...0 viii

10 5.3 Unrestricte Optimum Thrughput Unrestricte Optimum fr CFP CTA Immeiate ACK N ACK Delaye ACK Unrestricte Optimum fr CAP Immeiate ACK N ACK Delaye ACK Perfrmance Evaluatin Cnstraine Optimum Unrestricte Optimum Cnclusin NON-INTRUSIVE TRACING IN THE INTERNET Relate Wr Thumprinting Algrithm Minimizing The Thumbprinting Interval Prperties f the Thumbprinting Functin Tracing the True Destinatin f a Surce Algrithm Analysis Generalizatin...8 ix

11 6.4 Evaluatin Cnclusin CONCLUSIONS...3 REFERENCES...34 x

12 LIST OF TABLES 3. Number f Basic Operatins in an MRG-base Cipher Basic Operatins fr Generating a 3 -bit MRG Pseu-ranm Number Basic Operatins in an LCG-base Cipher an the Operatins fr Generating an 8 -bit LCG Pseu-ranm Number Basic Operatins in an LCG vs. an MRG-base Cipher Basic Operatins in LCG an MRGs-base Ciphers vs. Peri Length Ntatins Use Parameters Use in Figures 4. an Parameters Use in Figures 4.4 an xi

13 LIST OF FIGURES. Detectin f a rectangular intrusin bject.... Detectin f a circular intrusin bject Intersectin f sensrs sensing areas with the surface f an intrusin bject Sensing area efine by cnstraints Sensing area efine by cnstraints insie rectangle R....6 Sensing area f triangle T MNP efine by cnstraints... S, r, a r Strip = [ ) ( ).8 Rectangle 3 S...30 R insie strip = [, ) r, ( a r ) 3. Encryptin f a 4 byte pacet Hash chain base scheme fr privacy prtectin a) Per frame MAC efficiency an b) MAC an PHY parameters use Frame size vs. the level f the la α, R = 54 Mbps Frame size scales vs. PHY rate R, α = (a) AFR vs. AFR with AES with increasing frame sizes an (b) Clse-up f the saturatin thrughput in the AFR with AES scheme, when BER = Thrughput vs. fragment size The basic superframe xii

14 5. CTA uner ifferent ACK plicies Thrughput vs. payla size in Imm-ACK in the cntentin free CTA peri Thrughput vs. payla size in N-ACK in the CAP Thrughput vs. payla size in Imm-ACK in the cntentin free CTA Thrughput vs. payla size in N-ACK in the cntentin free CTA Thrughput versus payla size in Dly-ACK in the cntentin free CTA Vlume f the sli that lies belw functin f ( a, t) α ( a, t) n [, L ] [ 0,T ] 6. Vlume f the sli that lies belw functin f ( Ly Tτ ) α ( Ly, Tτ ) [,] [ 0,], n Apprximatin f the vlume efine by the uble integral...30 xiii

15 CHAPTER INTRODUCTION Energy efficiency is an essential requirement fr all wireless evices. Recent evelpments in wireless sensr netwrs (WSNs), wireless lcal area netwrs (WLANs) an wireless persnal area netwrs (WPANs) have raise a eman fr energy-efficient algrithms an energy-efficient meium access cntrl (MAC) prtcls. WSNs are a ppular netwr system cmpse f a large number f multifunctinal sensr nes, representing the integratin f sensr technlgy, wireless cmmunicatin, an a hc netwring. Majr issues in evelping efficient cverage-preserving algrithms fr sensr netwrs inclue the limite energy supply, netwr cverage an lifetime, banwith, limite cmputatinal pwer, memry an strage. With respect t WLANs an WPANs, effrts t achieve ptimal perfrmance at the MAC layer face challenges such as scaling with increasing physical rates an parameter ptimizatin. When cnsiering security in this cntext, extra verhea is ae t the netwr an effrts must t be mae t minimize the extra la while at the same time achieving the esire level f security. If we thin abut security attacs in the Internet this time, a ifferent set f issues must be cnsiere. The cmplex architecture f the Internet maes it easy fr attacers t cnceal the surce f the attac an preserve their annymity. One way t achieve this invlves lgging-in thrugh a series f machines befre attacing the target. Tracing bac the rigin f such an

16 attac must be ne efficiently, preferably withut invlving hst machines r impsing excessive la n netwr cmpnents. My issertatin first fcuses n wireless sensr netwrs. Sensr netwrs have fun varius applicatins in civil mnitring an military surveillance, incluing fr instance envirnment mnitring, intelligent transprtatin, smart hme, an intrusin etectin. In general, a large number f wireless sensr nes are eplye in the fiel an reprt events (e.g., appraching f enemy tans) t a prcessing center. Wireless sensr nes are nrmally pwere by battery an thus their energy cnsumptin becmes ne f the majr cncerns. In many cases, the ensity f wireless sensr nes is high, leaing t high reunancy in sensr nes' mnitring areas. In ther wrs, a sensr ne s mnitring area is usually verlappe with its nearby sensrs mnitring areas. Such reunancy is a blessing feature f wireless sensr netwrs fr energy saving because it is unnecessary t mae all sensr nes active at the same time. Sensr nes shul be scheule t wr alternatively within certain Quality f Service (QS) cnstraints in rer t save energy cnsumptin []. One f the imprtant QS cnstraints is the chance f etecting an interesting event. Uner this cnstraint, numerus reunancy-base sensr scheuling schemes [-4] have been prpse fr energy efficiency. The state-f-the-art mathematical mels [-3] are targete at the gal that each pint in the fiel is mnitre with a certain prbability at any time instance. By scheuling sensr nes t wr r sleep, we are actually weaving a "net" with its gri size ynamically ajuste with the number f wring sensrs. Analgusly, if we nly want t catch fish, we shul nt weave a net that can actually catch shrimp. Since in real applicatins, the size an the shape f the mnitre bjects shul nt be ignre, it is clear that scheuling schemes base n existing pint cverage mels will frm a netwr that is unnecessarily ense. Further imprvement n

17 this braly investigate prblem is pssible an is f great imprtance fr bth acaemic research an inustrial practice. In this wr, we buil a mathematical mel fr intrusin etectin which ptimizes netwr cverage an etectin while minimizing the number f sensrs an energy cnsumptin. We als stuy the prblem f maximizing netwr lifetime uner sme QS cnstraints. The mel taes int cnsieratin the impact f the size an the shape f mnitre bjects an is therefre mre accurate fr applicatins where the shape f mnitre bjects (e.g., tans) shul nt be mele as pints. Scheuling schemes base n ur new etectin mel can create a wring netwr that accurately matches the applicatin requirement. Such a netwr can be useful when it is emplye in a particular envirnment, as in etecting enemy tans, r in the brer, fr etecting human beings crssing the brer. In such cases the size f the intrusin bjects, whether they are tans r human bies, can be easily apprximate. In such an envirnment as a battlefiel r military surveillance, where an aversary can eavesrp n traffic, a security mechanism must be integrate int the sensr netwr in rer t achieve secure cmmunicatin. At the same time, we must tae int accunt the limite resurces in terms f cmputatinal pwer, banwith, an size f sensr netwrs. We prpse a lightweight blc cipher base n a ranm generatr, which is suitable fr WSNs, Rai Frequency Ientificatin (RFIDs), an their integratin, as well as applicatins such as wireless telemeicine [35-37, 4]. Fr sme evices, such as sensr nes in WSNs an RFID tags, with limite resurces, using a ranm generatr wul nly imply the transfer an strage f a shrt see. Hwever, since any algrithm uses a finite state machine, the pruce sequence is nt truly ranm; ue t the finite number f states which the generate sequence becmes periic. The peri length f such a sequence (i.e. the maximum length f the sequence befre it starts t 3

18 repeat) is thus an imprtant factr in achieving ranmness. A shrter than expecte peri might lea t the sequence failing statistical pattern etectin tests. We use a Multiple Recursive Generatr (MRG) t generate sequences f numbers with very lng peris, i.e., pseu-ranm sequences. We effectively use the MRG as the basis f the lightweight blc cipher with the purpse f satisfying imprtant quality requirements such as security, lng peri, ranmness, an efficiency. With respect t efficiency in wireless LANs, ne f the main challenges nwaays is t evelp a meium access cntrl (MAC) layer that will nt ecrease the efficiency f the MAC layer when PHY rates are increase. As bserve in [4-44], a theretical thrughput upper limit exists, inicating that by simply increasing the ata rate withut reucing verhea, the enhance perfrmance, in terms f thrughput an elay, is bune even when the ata rate ges infinitely high. The Aggregatin with Fragment Retransmissin (AFR) scheme, which was initially prpse in the IEEE 80.n tas grup [45], an then evelpe mre cmprehensively in [46], intruces a mel in which multiple frames are aggregate int a larger frame befre being transmitte t the physical layer (PHY). If the size f a frame is larger than a pre-establishe threshl, the frame is ivie int fragments befre being aggregate. Transmissin errrs are hanle by retransmitting nly the fragments f the frame that ha been crrupte. Hwever, the wr in [46] es nt cnsier security, i.e., the avance encryptin stanar (AES) algrithm, which is use in IEEE 80.i. In ther wrs, when IEEE 80.n an IEEE 80.i are bth apte, AES ver the high spee WLANs must be cnsiere. With this mtivatin in min, we analyze the verhea intruce by AES, when ae t the AFR aggregatin scheme. The stuy is imprtant ue t the significance f security as well as the fact 4

19 that amng the current huge number f papers n IEEE 80. perfrmance analysis, nne f them cnsiers AES verhea in their analysis. Wireless PANs, althugh nt as wiely evelpe as WLANs, cnstitute a very recent an expaning technlgy. Ultra wieban (UWB) WPANs enable wireless cnnectivity with cnsistent high ata rates acrss multiple evices an PCs within the igital hme an the ffice. This emerging technlgy prvies the high banwith that multiple igital vie an aui streams require thrughut the hme. The evices amng which UWB maes transmissins pssible range frm HDTV receivers, TV sets, cmputers, printers an igital cameras, t meical mnitring evices an vehicular raar systems. One f the critical challenges in UWB netwrs is crinating multiple accesses t the channel where, fr example, a receiver may nee a relative lng time t synchrnize with ther transmitte signals. As a result f strng effrts fr regulatin an stanarizatin, the IEEE MAC mechanism, prpse in the IEEE a tas grup, has emerge as the prtcl use t reslve the timing acquisitin prblem. IEEE supprts quality f service (QS) fr real time multimeia applicatins an insures reliability f elivery by apting errr cntrl techniques uner UWB errr channel cnitins. Mre precisely, the stanar apts three acnwlegement (ACK) schemes nwn as: N-ACK, Immeiate-ACK (Imm-ACK) an Delaye-ACK (Dly-ACK). The stanar is base n the ntin f a picnet, cntrlle by a picnet crinatr (PNC), an cnsisting f evices (DEVs) which cmmunicate with the PNC within given timeframes. Amng these timeframes, we istinguish the cntentin-free channel time access peri (CTAP) an the cntentin access peri (CAP). T ensure secure cmmunicatin in UWB netwrs, an encryptin mechanism must be emplye t prtect the transmitte frames frm ptential attacers. We again cnsier AES as 5

20 ur encryptin scheme, as it is the mst ppular encryptin cipher apte in recent years, an stuy the verhea intruce when AES is use t encrypt pacets transmitte at the MAC layer. Specifically, we analyze the traeff between thrughput, payla size an channel errr when AES is use t encrypt the frames. When cnsiering a wrlwie netwr f netwrs such as the Internet, the prblem f security hls ifferent challenges. Since the public expansin f the Internet in 990, many new challenges have surface; amng them, peratins between un-truste en-pints, mre emaning applicatins an less sphisticate users have pse severe stress n the Internet requirements. Furthermre, the number f attacs n netwre cmputer systems has been grwing expnentially frm year t year. When cnsiering the tas f tracing intruers in the Internet, we have t tae int accunt three main challenges. First, attacers hie their rigin by maing use f the Internet s architecture. By using ifferent hsts, belnging t ifferent cuntries an aministrative mains, t rute malicius acts, intruers actins becme extremely ifficult t trace bac. Secn, the ata cllecte frm an Internet trace is usually incmplete r has missing values. Fr example, ifferent mains f Internet service prviers (ISPs) may nt share ata ue t access issues, such as in ifferent cunties. Finally, rutes change frequently, pacets are lst an the netwr latency an time t cnvergence can be significantly increase ue t ruting instability. T eal with such prblems, tw tracing mechanisms exist [79]: ) meths f eeping trac f all iniviuals an accunting all activities, an ) reactive tracing, in which n glbal accunting is attempte until a prblem arises an then tracing bac its surce is attempte. In wn view, the first mechanism is much relate t netwr accuntability an the secn mechanism is much relate t netwr frensics. Netwr-base tracing an hst-base tracing are tw main appraches fr reactive 6

21 tracing in cnnectin chains. Hst-base tracing invlves ne tracing system per netwr hst [79], an ) a chain f cnnectin hsts can be nwn via hst cmmunicatins [78] r ) reversing the attac chain by breaing the hsts in the reverse rer [80]. Hst-base tracing schemes suffer when an extene cnnectin crsses a hst nt running the system [79]. Netwr-base tracing has the avantage that it es nt rely n hsts which can be untrusty an it es nt require hst participatin. Instea, netwr-base tracing uses the invariance f cnnectins at higher prtcl layers (such as the transprt layer) an base n this bservatin can establish whether tw cnnectins are part f the same cnnectin chain. In this wr, we stuy tw appraches fr netwr-base tracing. The first apprach is base n the iea f a thumbprint f a cnnectin. Thumbprinting has the avantage that it preserves the characteristics f a cnnectin while at the same time being cheap t cmpute an requiring little strage. Thumbprints f relate cnnectins are similar an can be use t cnstruct a cnnectin chain leaing bac t the surce f an attac. The secn apprach fr tracing intruers in the Internet, invlves passively mnitring flws between surce an estinatin pairs similar t [76]. This apprach is base n mnitring transmissin activities f nes an es nt interfere with netwr peratins. The rest f the issertatin is rganize as fllws. Chapter intruces the etectin mel fr ptimizing netwr cverage in a WSN. The lightweight blc cipher suitable fr secure cmmunicatin in WSNs an RFIDs is presente in Chapter 3. The integratin f the AES cipher at the MAC layer f 80. WLANs an UWB WPANs is cvere by Chapters 4 an 5, respectively. Chapter 6 iscusses tw mechanisms fr tracing intruers in the Internet while Chapter 7 cnclues the issertatin with a summary f results an future wr. 7

22 The wr in Chapter has been the subject f tw cnference papers [84], [85]. The shrt versin f Chapter 3 has been presente at GLOBECOM 008 [83], while the shrt versin f Chapter 4 has been accepte in the ICC 009 cnference [8]. In aitin, lng versins f chapters, 3, 4 an 5 have been submitte t Cmputer Science jurnals. Out f these, the wr in Chapter 5, submitte at the IEEE Transactins n Wireless Cmmunicatins [8], has been revise an is being cnsiere fr publicatin prvie minr revisins are mae. 8

23 CHAPTER WEAVING A PROPER NET TO CATCH LARGE OBJECTS IN WIRELESS SENSOR NETWORKS In this chapter, we buil a etectin mel f an intrusin event that taes int cnsieratin the impact f the size an the shape f mnitre bjects. Mre specifically, we investigate the relatinship between the etectin prbability, the intrusin cverage intensity, the number f wring sensr nes, an the size an the shape f intrusin bjects. We prve many mathematical results relate t the etectin prbability an intrusin cverage intensity an stuy the asympttic prperties f these etectin metrics. Our mel als prvies analytical results n tw ptimizatin prblems: ne is t fin the minimum number f wring sensr nes that can etect a given bject f nn-negligible size with a given prbability; the ther is t fin the maximum etectin prbability given the rati f the number f sensr nes ver the size f the mnitre area. In aitin, we stuy the prblem f maximizing netwr lifetime uner sme QS cnstraints. We prve the existence f the slutin an erive the explicit frm f the slutin uner certain cnitins. Many results in this chapter remain vali fr any bject size. In particular, the tw ptimizatin prblems in Sectin.4 are inepenent f the bject s shape an size an are vali fr any bject f a given reasnable area. The prblem f maximizing netwr lifetime an the results relate t the intrusin cverage intensity prvie sme insights n hw the netwr can be aapte t the size an the shape f the intruer, specifically, hw many wring sensrs 9

24 shul be use an hw they can be scheule epening n the imensins f the bject we want t etect. The rest f the chapter is rganize as fllws. In Sectin., we intruce relate wr. In Sectin., we present a naive apprach that cvers a given area an etects an intrusin bject f circular r rectangular shape with the minimum number f sensr nes. In Sectin.3, we stuy a stchastic versin f the prblem, i.e., the prbabilistic apprach t etecting intrusin bjects. We further investigate tw ptimizatins relate t the etectin prbability an the shape an the size f intrusin bjects in Sectins.4. Sectin.5 fcuses n the intrusin cverage intensity (efine later in the chapter), an Sectin.6 fcuses n ptimizing netwr lifetime. We cnclue the chapter in Sectin.7.. Relate Wr Much wr has fcuse n sensr scheuling algrithms with the purpse f achieving energy efficiency withut reucing sensing cverage. One way t minimize energy cnsumptin an exten netwr lifetime is t put sme sensr nes int sleep but allw thers t remain active as lng as the whle given area is cvere. In the meantime, it is require that bth cverage an netwr cnnectivity shul be satisfie. T slve the abve prblem, prbabilistic cverage with ranmize scheuling algrithms is stuie. In many cases, existing research [-3] assumes that each pint in the fiel shul be mnitre. In [-3], a special case f a ranmize scheuling algrithm is investigate, where subsets f sensrs wr alternatively, with nly ne subset being active at a certain mment. Each subset cntains the same number f sensrs. The prblem f maximizing netwr lifetime uner Quality f Service cnstraints is analyze in [-3]. The results are 0

25 state an prve in terms f the netwr etectin prbability, the etectin elay, an the intrusin cverage intensity. The etectin prbability is the prbability that an intrusin event is etecte, e.g., an enemy bject enters the mnitre area. The etectin elay is efine as the average elay with respect t the scheuling runs t etect such an event. The intrusin cverage intensity is efine as the prbability that a given area is etecte at any given time by at least ne active sensr. Fr mre results cncerning energy efficient jint estimatin in sensr netwrs, please refer t [-3]. Xia et al. [4] stuy anther case f the ranmize scheuling algrithm t etect intrusin bjects with a large size, in which case treating the bjects as pints may be t cnservative. Using the same mel, in this chapter we perfrm a cmprehensive stuy n the etectin f an intrusin bject that has a nn-negligible size (such as a tan). Alng the line, the relevant ptimizatin prblems can be slve as well. This is the purpse f the wr presente in this chapter.. A Naive Apprach In this sectin, we escribe a simple apprach t answering the fllwing questin: hw t use the minimum number f sensr nes t cver a given rectangular area such that an intrusin bject with a nn-negligible size can be etecte. T simplify analysis, we assume that a sensr ne s sensing cverage can be apprximate as a circular area. Assume that the whle rectangular area is ivie by a virtual gri. Als assume that the sensr nes are place at the intersectins f the virtual gri. We first etermine the maximum size f the square cells in the virtual gri, with which we can then easily estimate the minimum number f nes neee t cver the area.

26 Let ente the iameter f the square cells in the virtual gri. Let ente the size f the intrusin bject. We use tw typical examples t illustrate ur calculatin. Case : Assume that the intrusin bject can be apprximate as a rectangular shape, with sie length f b an b, respectively. Let r be the size f the circular area mnitre by each sensr. It is easy t see that the raius f the circular area is ( r π ). We can then chse t ivie the whle area int squares such that the iameter f each square satisfies min{ b, b} + ( r π ) <, as shwn in figure.. Sensrs are place at the intersectins f a virtual gri. The circles represent the areas cvere by sensrs an the intrusin bject is apprximate as a rectangle. Fig... Detectin f a rectangular intrusin bject. Case : Assume that the intrusin bject can be apprximate as a is. We can chse ( π ) + ( r π ) <, as shwn in figure.. The circles represent the areas cvere by sensrs an the intrusin bject is apprximate as a is.

27 Fig... Detectin f a circular intrusin bject. By chsing the iameters as abve, we ensure that n matter hw we place an intrusin bject n the gri, its bunary will verlap with at least ne sensr s mnitring area, i.e., the bject cannt escape the etectin. Having a cnstraint n the iameter f the square cells, we can easily erive an upper bun n the ege length f the squares, square ( ) l, which is l square < min{ b, b} + ( r π ) ( ) when the intrusin bject has a rectangular shape, an l ( ) square π + ( r π ) < when the intrusin bject has a is shape. It is easy t see that the minimum number f sensr nes require t cver the whle rectangular area in the abve way is [ lsquare][ a lsquare] a, where a an a are the sie length f the rectangular area. While this meth f placing sensr nes can prvie eterministic intrusin etectin, it requires that all sensr nes t be active at any given time, which may eplete sensrs energy quicly. In aitin, placing sensr nes accurately n the intersectins f a gri may nt be an easy jb an may require high human labr. As such, we investigate the prbabilistic meths in the fllwing sectins. 3

28 .3 Prbabilistic Meths fr Detecting Intrusin We stuy a prbabilistic scheuling meth that uses alternatively wring subsets f sensrs [-3]. Assume that n sensrs are ranmly eplye in a fiel. We put n sensrs evenly int isjint subsets an then let the subsets wr accring t the run-rbin scheuling. We are intereste in the intrusin cverage intensity an the etectin prbability, which are efine later. We first summarize the analysis in [-3]. Let r be the size f the sensing area f each sensr an a the size f the whle sensing fiel. The prbability that each sensr cvers a given pint in the fiel is r a. Since every sensr is scheule in ne f the subsets, the prbability that the sensr is active an cvers a given pint in any run is [ ] n r a. Then r ( a) represents the prbability that a given pint is nt cvere by any f the active sensrs. Fllwing this line f reasning, the prbability that a pint in the fiel is etecte by at least ne sensr at any given time is [ r /( a) ] n. In the fllwing, we erive the prbability that a sensr s sensing area intersects the surface f an intrusin bject, ente as p. As shwn in figure.3, any sensr within the bunary f the grey lines can etect the intrusin bjects. 4

29 Fig..3. Intersectin f sensrs sensing areas with the surface f an intrusin bject. Case : Assume that the intrusin bject can be apprximate as a is f size. It is easy t see that the size f the area bune by the grey lines as shwn in figure.3 (right) cul be calculate by π ( π ) + ( r π ) [ ] = ( r + ) (.) p ( r ) a + =.. In this case: Case : Assume that the intrusin bject can be apprximate as a rectangular shape with sie length f b an b, respectively. The size f the area bune by the grey lines as shwn in figure.3 (left) cul be calculate by ( b) r π + 4r( 4) = + ( b + b) r r + b r π + π +, which is btaine by aing the areas f 4 smaller rectangles: tw with sie length f b an r π, an tw with sie length b an r π, plus 4 quarters f a is f area r (at the crners), t the initial bject area. In this case: ( r) a (.) = + ( b + b)( r ) p π +. With p calculate, we can btain the intrusin cverage intensity, which is efine as 5

30 the prbability that any intrusin bject is etecte by at least ne sensr at any given time. Fllwing the same reasning in [-3], p represents the prbability that a sensr is active an etects an intrusin bject in any run. Since there are n sensrs in ttal, the prbability that the intrusin bject is nt etecte by any active sensr, at any time, is [ p ] n. Then the Vn p intrusin cverage intensity is given by [ ] n =. The intrusin cverage intensity isclses hw well the whle area is cvere with respect t the etectin f intrusin bjects with nn-negligible size. Assume that L is the uratin f an intrusin event an T is the length f a run-rbin scheuling run. We nee t nw the prbability that the intrusin event is etecte by at least ne sensr. This prbability is calle etectin prbability an is ente as P. By replacing btain: r a with p in [-3], we (.3) P = n ( p ), L ( ) L T L T + ( ) s p s p, L < ( ) n T n. T This expressin epens n L T, n,, r a, an s, where s = ( L T + L T ) assumptin is that L may nt be a natural multiple f T. /. The.4 The Prperties f Detectin Prbability.4. Cnstraine Optimal Detectin Prbability Having erive the analytical expressin f the etectin prbability P, we nw investigate the cnstraine ptimal intrusin etectin by slving tw main ptimizatin prblems an their variatins. The results prvie by these tw ptimizatin prblems are 6

31 inepenent f the bject s shape an size an are vali fr any bject f a given area, reasnably assuming that is smaller than the ttal area t mnitr. We als etermine a lwer bun an an upper bun n the etectin prbability. The first prblem is cncerne with maximizing/minimizing the etectin prbability P uner ifferent cnstraints. Fr instance, we may put the cnstraint n the rati f n a (i.e., the rati f the number f sensrs ver the size f the whle area), since this rati represents the ensity f the sensr nes an thus the system cst. Fr cmparisn, we use r as a measure f benchmar, because this measure means the ieal situatin: ne sensr per area f size r. Optimizatin Prblem Determine the sufficient cnitins fr fining the maximum value an the minimum value fp uner the cnstraint that n a u0, where u 0 is a given psitive cnstant. By changing the value f u 0, particularly by checing whether r nt u 0 is larger than r, we btain the prperties f etectin prbability in ifferent scenaris. T ease erivatin, we will use x, a real number, instea f n, t mae the ifferential calculatin pssible. In the abve ptimizatin prblem, since the etectin prbability P is cnsiere as a functin f tw variables, the number f sensrs x an the size f the whle area a, we rewrite P as ( x a) P, an buil a Cartesian crinate system with the x-crinate representing the number f sensrs an the y-crinate representing the size f the whle area. T btain results meaningful in practice, we use a rectangle R [ 4, u a ] [ 4r a ] = n the Cartesian crinate 0 0, system t cnfine the variables, i.e., 4 x u0a0 an 4r a a0. This limitatin is t exclue certain uninteresting scenaris, e.g., the number f sensrs is t small (smaller than 4 ), r the 7 0

32 size f the whle area is t small (smaller than 4 r ). We put an upper bun a 0 n the size f the whle area, because it is impractical t cnsier an area having infinite size. The fllwing results hl fr intrusin bjects with either circular r rectangular shape. Therem prvies the maximum an minimum values f ( x a) P, n rectangle R uner cnstraint x a u0 where u0 r. It als prvies the maximum an minimum values f ( x a) P, with cnstraint x = au0 n the same rectangle. Cnstraints x a u0 an u r = (see figure.4). etermine trapezi MNPQ insie rectangle R [ 4, u a ] [ 4r a ] 0 0, 0 0 Area Fig..4. Sensing area efine by cnstraints. Sensrs Therem Cnsier the prblem f fining max ( x, a) an min ( x, a) R P R P with the variables = 0 0, 0 an with the cnstraint a u0 cnfine in R [ 4, u a ] [ 4r a ] cnstant greater than r. We have max P ( x, a) P ( 4ru0,4r) we change the cnstraint x a < u0 t a u0 min P R ( x, a) P ( u a a ) =. 0 0, 0 R 8 x < where u 0 is a given psitive = an min P ( x, a) P ( 4, a ) R =. If x =, we have P ( x, a) P ( 4ru,4r) max 0 R = an 0

33 Prf: T prve that the minimum etectin prbability is reache at pint (, ) the ifference between the value f P ( 4, a ) 0 P in a ranm pint ( a) 4 a, we cmpute 0 x,, insie trapezi T, an, an shw that this ifference is always psitive. Fr this we first cmpute the partial erivatives f P with respect t bth x an a an establish their sign. P a P x x Q Q ( x, a) = ( s) ( + r ) ln ( + r ) Q + s a Q a a x Q + ( + r ) ln ( + r ) > 0. x a ( x, a) = x ( s) ( r + ) ( r + ) + s ( + r ) ( r + ) < 0. a Q a Q + a x Q + a P Next, using Lagrange s mean value therem [5], let us cnsier the ifference: P x P a ( x a) P ( 4, a ) = ( ρ, ρ )( x 4) + ( ρ, ρ )( a a ) 0 P ( x, a) P ( 4, a ) fr any( x, a)., T Hence, minimum P is reache at (, ) 4 a. 0 A mre cmplicate slutin is neee fr the maximizatin prblem. Since we want t shw that at ( ru, 4r) 4 0 P reaches the maximum value, the previus technique cannt be use as we nt nw the sign f x 4ru0. Therefre, t fin the maximum f P we first use the fact that, accring t Fermat s therem fr statinary pints [6], the maximum is reache n the trapezi s bunary, hence, n ne f T s sies. P ecreases with a alng the vertical sie QP, s that the maximum n QP is reache in Q. P increases with x s that P increases alng sies QM an PN. Therefre, the pssible maximum pints are n the line segment S MN. S all we nee t stuy is the maximum f P n MN. On the segment 9

34 MN, x = au0, s we can express P as a functin f nly ne real variable, a. Let this functin be ϕ. P 0 0 Q Q +, a a. ( au a) = ( s) ( r + ) s ( r + ) = ϕ( a) 0 au We tae the erivative f ϕ an stuy its sign in rer t establish the functin s mntny. ϕ au 0 ' Q Q + ( ) ( ) ( ) ( ) ( )( + ) 0 ln + Q a r a = s u r + + r + a a a ( Q a)( r + ) au 0 Q ( ) ( ) ( + ) )( + ) 0 Q + Q a r su r + ln r + + a 0. a a (( Q + ) a)( r + ) The abve is true because the sum f tw negative numbers is als negative. Here we have use the inequality: ( ) ln( ρ) + ρ 0 fr all ρ < ρ. au Therefre, P is nn-increasing n MN, which means that the maximum is reache at M : max P = P ( 4ru0,4r). T The mntny f P alng segment MN als slves the relate prblem, when x a = u0, state in the therem. We have: P = P ( M ) = P ( 4ru, r) an min P P ( N ) P ( u a a ) max 0 4 S MN =. = S MN 0 0, 0 As shwn in figure.4, the trapezi T = has vertices M ( 4ru 0, 4r), ( u a ) TMNPQ N, 0 0, a0 P ( 4, a ) an Q(, 4r) 0 minimum value f ( x a) 4. Therem inicates that when x a < u0, the maximum value an the P, is reache at vertex M an vertex P, respectively. When x a = u0, this cnstraint efines a line segment with en pints M an N, an ( x a) maximum value an the minimum value at pints M an N, respectively. P, reaches the 0

35 The fllwing crllary cnsiers the special case u0 = r, that is, the trapezi becmes triangle MNP (, ) 0 T, an the vertices M, N anp are given by ( 4,4r), ( r ) a 0, a 0 an 4 a, respectively, as shwn in figure.5. Cnstraint x a u r etermines triangle MNP insie rectangle [ 4, r] [ 4r a ] a. 0, 0 0 = Area Sensrs Fig..5. Sensing area efine by cnstraints insie rectangle R. Crllary Let the ptimizatin prblem be as in Therem. If x a < u0 an u0 = r, then max P R ( x, a) = P ( 4,4r) an min P ( x, a) P ( 4, a ) R = 0. If 0 x a = u an u0 = r, then max P R = P ( 4,4r) an min P P ( a r a ) R =. 0, 0 Therem an Crllary present the results when u0 > r an u0 = r, respectively. Therem cnsier the case where u0 < r.

36 Area Sensrs Fig..6. Sensing area f triangle T MNP efine by cnstraints. Therem Cnsier the prblem f fining max ( x, a) an min ( x, a) R P R P with the variables cnfine in R [ 4, u a ] [ 4r a ] = 0 0, 0 an with the cnstraint a u0 x < where u 0 is a given psitive cnstant smaller than r min P R max P R ( x, a) P ( 4, a ). We have max P ( x, a) P ( 4,4 / u ) = 0. If we change the cnstraint 0 ( x, a) = P ( 4,4 / u ) an min P ( x, a) P ( u a a ) 0 R =. 0 0, 0 R = an x a < u t x a = u0, we have 0 Cnstraints x a u0 an u0 < r etermine a triangle T MNP insie rectangle [ 4, u a ] [ 4r a ] R =, as shwn in figure.6. The vertices M, N an P f the triangle have 0 0, 0 crinates: ( 4, 4 u ), ( a u ) an (, ) u , a0 4 a, respectively. Therem inicates that when 0 x a <, the maximum value an the minimum value f P ( x, a) is reache at vertex M an vertex P, respectively. When x a = u0, the cnstraints efine a line segment with en pints M

37 an N, an ( x a) respectively. P, reaches the maximum value an the minimum value at pints M an N, It is interesting t see the practical meaning that Therems an eliver. In bth cases, the minimum etectin prbability is the case when the minimum number f sensrs is use t cver the entire area, as we wul have expecte intuitively. If we l at the maximum etectin prbability, in the case f Therem, maximum etectin is achieve when the area is minimal an the number f sensrs is an intermeiary value between 4 an a 0u0, while in the case f Therem, maximum etectin is reache when the number f sensrs equals 4 an the size f the area equals 4 u 0. The secn prblem we cnsier cnsists in the reverse ptimizatin prblem: minimize the rati f the number f sensrs ver the size f the whle area, given a certain etectin prbability r an upper bun n the etectin prbability. Optimizatin Prblem Fin ( x a) min, with cnstraint P = P0 where P 0 is a cnstant greater than zer. If pssible, fin tighter lwer an upper buns n ( < P ) P < The restrictin P = P0 efines implicitly x as a functin f a, if we nt assume that a is given a priri. We thus use the Implicit Functin Therem [7] t slve the prblem. As befre, we cnfine the variables, the number f sensrs x an the area t be cvere a, within a rectangle R : [ 4, n ] [ 4r a ] = 0, 0, where 0 n is a natural number an the upper bun n the number f sensrs. Let 0 := n0 m, where m is the number f sensrs in each f the subsets. T btain results, we nee t assure that the curve P = P0 intersects the interir f rectangle R. The cnitin > T ( 0 ) L allws us t express x explicitly as a functin f a. Here, 3

38 again, we have an ptimizatin prblem f tw variables: the number f sensrs x an the area t be cvere, a. Hwever, by expressing x as a functin f a, we are able t cnvert the prblem t an ptimizatin prblem in nly ne variable a. Therem 3 Assume that P ( 4, a ) < P P ( n a ) an als that P (, a ) P P ( 4, r) Assume that n 0 is such that < 0, satisfies the cnitin > T ( 0 ) 4r < a < a 0 such that in the interval[ a, a 0 ], we have: (.4) ( a) 4 ln( t ( a )) ln( t ( a) ) x 0 0 = an (.4 ) { ( a) a P ( x( a), a) = P, ( x( a), a) R } = 4 a where t ( a) ( r ) a 0 : = +. min x 0 < < 4 L. Then there exists a, 0, Prf : Since P x > 0 an a < 0 implicitly a functin x( a) P, at each pint ( a) x,, the relatin P = P 0 = ct efines x =, which is a C functin f a. Frm the sign f the partial erivatives we als have x a > 0. Therefre a is an increasing functin f x. We stuy the mntny f functin x ( a) a. The erivative f this functin is given by the frmula: x a a x a ( a), therefre ( g a) = sign[ x( a) ( t + ( t ) ln( t ))] 0 ( a) : = ( r + ) 0 < t a 0 < sign, where. It fllws that ( a) a x increases with a. Thus, t fin the minimum value f x ( a) a, we must l fr the minimum f a in R. Since a is an increasing functin f x, the minimum f a is reache when x is the minimum, an the minimum x is 4. Next we etermine the explicit frmula f x ( a), via Implicit Functin Therem [7] an elementary integratin. Using the hypthesis an Darbux prperty [5], there exists a pint 4

39 ( x ( ) ), an anther pint ( 4, a ), such that P ( x( a ) a ) = P = P ( 4, a ) a 0, a 0 ( a0 ) 0 4 < x < n. P x, a t0 =, fr On the ther han, the fact that ( ) ( ) x (.5) L > T ( ) x t0 ( a) leas t ( a) = x( a), in the interval [, a ] a ( t ( a) ) ln( t ( a) ) 0 After multiplying by x( a) 0 0, 0 0 a. 0, where an integrating, we have: ( a) = x( a ) ln( t ( a )) ln( t ( a) ) = 4 ln( t ( a )) ln( t ( a) ) where t ( a) x an 4r a < a0 t 0 <. Thus, fr small psitive real x = x( a) such that ( x) = x m have (.4). If we chse n 0 such that 0 = is as abve : satisfies (.5), we 0 = n0 m satisfies (.5), then (.5) leas t (.4), with the abve reasns. Frm (.4), an frm the facts that ( a) a min x is reache at a = a an that x ( a ) = 4, we fin (.4 ): ( ( a) a) = 4[ ln( t ( a )) ln( t ( a ))] a = 4 a min x 0 0. This cnclues the prf. The minimum rati f the number f sensrs ver the size f the whle area, given a esire etectin prbability, can be reache when the minimum number f sensrs is use fr a P 4 a = P certain area size a. This value can be cmpute easily, by slving the equatin (, ) 0 fr a, where P is given by (.3). The next result represents the upper an lwer buns fr P. 5

40 L L T T + Therem 4 The fllwing inequalities stan: p P < < p. n n n n + = s p + s p < s + s p Prf: ( ) L T L T + ( ) L T P Liewise, ( ) L T > + P s s p n. n. Therem 4 presents sme very simple, tight buns fr P. We can see that s has been eliminate frm the rather cmplicate expressin f P in (.3). Instea f using frmula (.3) we can nw use either ne f the tw buns in Therem 4 as an apprximatin f P..4. Detectin Prbability Relate T The Shape Of The Intrusin Object Intuitively, the etectin prbability increases as the size f the intrusin bject increases. But it is unclear fr tw rectangular intrusin bjects, which ne is easier t be etecte: a square bject r a lng thin strip? We answer this questin in this subsectin. Assume that the size f the rectangular intrusin bject is, an its ne sie length is b. In the fllwing we stuy P as a functin f b, shifting the prblem frm the size t the shape f the intrusin bject. Lemma etermines the signature f the erivative f p as a functin f b, an Lemma etermines the signature f the erivative f P as a functin f b. We shw that the minimum etectin prbability is reache when a square. We then give an asympttic result fr P min. 6 b =, i.e., when the intrusin bject is

41 Lemma The signature f b p p is: ( ) b b < 0, 0, b < b,. Therefre p, ecreases n[ 0, ], an increases n the right f. a r Prf: ( ) = r ' p b b. Hence: p ( b) Functin ( b) minimum. b π r < < 0, b =. a π b 0, b p first ecreases an then increases, s b 0 = is the pint fr p t becme the P p sign b = sign b. b b Lemma ( ) ( ) Prf: By using cmpse functin erivatin, we have: P b P p =. The first fractin n p b P the right han sie is strictly psitive, s the sign f b p is given by the sign f, fr b allb > 0. Lemma shws that P behaves similarly t p, s is als a minimum pint fr P. Therefre a square shape intrusin bject is mst liely t g unetecte. By replacing b with in the expressin f p fr a rectangle (see Sectin.3), we btain the fllwing expressin f P min : 7

42 P min n L L T T + ( s) p s p min, = min n where r p min = r. If m is the number f sensrs per subset (i.e., = n m ), a π we have the fllwing asympttic result: lim P n min ( s) L L m p min m p min T T + = e se..5 The Prperties f Intrusin Cverage Intensity We ente the intrusin cverage intensity as ( x ) 8 V, if we treat the intrusin cverage intensity as a functin f tw variables, the number f sensrs x an the size f the intrusin bject. Frm Sectin.3, by replacing p as a functin f, we have the fllwing expressins: (.6) V ( x, ) = ( r + ) fr a circular intrusin bject, an (.7) V ( x, ) = + r + r π ( b + b) a fr a rectangular intrusin bject, respectively. T stuy the prperties f ( x ) a ( ) x x V,, we buil a Cartesian crinate system with the x- crinate representing the number f sensrs an the y-crinate representing the size f a sensr s cverage area. Similar as befre, we cnfine the variables in an area, ente as S, ver

43 the Cartesian crinate system. Therem 5 fins a pint ver S that minimizes ( x ) the intrusin bject has a is shape. V, when Therem 5 If we efine an area ver the Cartesian crinate system, [ ) ( ) S =, r, a r, ( a r ) > S (, r) = ( 4r a) minv ( x, ) = V. an > r, then V x V r V x V Prf: ( ) (, ) ( θ, θ )( ) + ( θ, θ )( ) 0, = x r the left sie is the sum f tw nnnegative terms fr any ( x ) S. The inequality hls since,. Obviusly, x an r are nnnegative, s all that remains t be shwn is that the first rer partial erivatives f V are psitive. Recall that fr an intrusin bject shape as a is, the analytic expressin f V is: V ( ) ( r + ) x = x V,. V is an increasing functin f bth x an, s > 0 a x an V > 0. It fllws that V ( x, ) V (, r) fr any ( x ) S,. The reasn that we cnfine the variables within S as shwn in figure.7 is as fllws. When the intrusin bject has a is shape, we have ( r + ) a = p ( a r ). This leas t. In aitin, it is nly interesting t assume r because we nly want t etect bjects with a large size. We als assume that the number f sensrs is large enugh, e.g., larger than. 9

44 Area Sensrs S =. Fig..7. Strip [, ) r, ( a r ) Area Fig..8. Rectangle 3 Sensrs S =. R insie strip [, ) r, ( a r ) The fllwing crllary taes the supremum ver all pints in the set S an shws that it is upper-bune by a finite number. V x Crllary sup V ( x, ) (, r)( x ) (, r)( r) V (, r) V 4r a 8r ln( 4r a ) 4r a ( 4r a) = e e = e a 4r =. 30

45 3 Prf: By Taylr s frmula, we have: ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( )( ) ( )( ).,,,,,,, = r t t V r x t t x V r t t x V r r V x r x V r V x V The last term is negative, since all secn rer partial erivatives f V are negative. The cnclusin fllws. If we put an upper bun n the number f sensrs, i.e., 0 n x, we reuce the area S t a rectangle [ ] ( ) = 0 3,, r a r n R ver the Cartesian crinate system, as shwn in figure.8. The fllw therem shws that with variables cnfine in 3 R, the maximum an minimum intrusin cverage intensity are reache. Therem 6 ( ) ( ) ( ) 0 3,, max 0, n R x r a n V x V = = an ( ) ( ) ( ) R x a r r V x V, 4,, min 3 = =. Prf: ( ) ( ) ( )( ) ( ) ( ) + = 0 0,,,, r a V n x x V r a n V x V θ θ θ θ, where ( ),θ θ is a pint situate n the segment with en pints ( ) x, an ( ) 0, r a n. The first rer partial erivatives f ( ) x V, are psitive, an are multiplie by negative quantities. Hence, the ifference ( ) ( ) 0,, r a n V x V is negative. Liewise, it can be

46 shwn that the ifference V ( x ) V (, r), is psitive. The cnclusin fllws. Base n the previus result, we btain the fllwing buns. 8r mn0 Crllary 3 e a V ( x, ) e nm, fr all (, ) R3 x. Prf: n 0 0 n n ln m n m n m n mn = m n = e e = e. 0 0 ( ) (.8) ( ) ( ) mn maxv 0 Accring t Therem 6 we can write: ( ) nm R 3 = n n (.7) e mn Fr the left-han sie inequality, we use the well-nwn inequality: ( + x) x ln 4r a 8r ( ) an we btain: ( 4r a) = e e a ln, fr all x >, Crllary 4 Let < x < n0 an let n0 =. The fllwing asympttic result stans: m 8r a m ( x, ) e e V. Prf: Accring t Crllary 3, we have: e 8r a 4r = lim a V m n n0 m ( x, ) lim = e. n0 0 3

47 33 In the fllwing, we cnsier V as a functin f x an b an we establish the upper an lwer buns fr ( ) b x V,. Therem 7 Let be a fixe cnstant, b b =, where b is variable. Assume that [ ] 0, 4 n x an [ ] b,b. We have ( ) ( ) 0, 4 4 n b x V r r a + + π. Prf: Cnsier the expressin f ( ) b x V, given by (.7). ( ) 0, b x x V fr all ( ) 4, R b x an ( ) < = > = =.,, 0,,,, b b b b sign b x b V sign. Therefre, ( ) ( ) 4 4 4,, = r r a r r a x V b x V x π π. On the ther han, ( ) ( ) ( ) 0, n x b x V. The last result f this sectin uses sme well-nwn inequalities t btain the upper an lwer buns fr V nly as a functin f the number f sensrs, x. Therem 8 ( ) m p x mx p x m p m p m p e V e e e 4 4 3, max. Prf: Frm Sectin.3 we nw that n n x p n m p V = =. We use the

48 n α fllwing well-nwn inequality: α e, α 0. V x is therefre bune by: n mp mp (.9) V e V e. On the ther han, it is well nwn that (.0) ( V ) x V x x. 4 x It fllws that: V x 4V x (.9) mp ( e ) 4. Therefre: (.) V x pm 3 4e. 4 = mp pm ( e ) 4( e ) Frm (.9) an (.) we btain the lwer bun f the cnclusin. Fllwing, we prve the upper bun. V x m p n = x. Using inequality: ln( + ) x x, x + fr all x >, we btain: x ln p m p m x x x m p x mp = m p x mxp = x mp. It fllws that: x mxp m xmp p e. This leas t the cnclusin xmp V x e x mxp. We summarize the analytical results f this sectin an their practical meaning as fllws. When the intrusin bject has a is shape, we have fun the minimum intrusin cverage intensity (Therem 5) an have shwn that this value can be upper bune by a finite 34

49 number (Crllary ). By putting a limit n the number f sensrs, we further fun the minimum an maximum cverage intensities (Therem 6). When the intrusin bject has a rectangular shape, the intrusin cverage intensity becmes a functin f the number f nes an the sie length f the rectangle. We have fun upper an lwer buns fr the cverage intensity (Therem 7). By fixing the size f the bject, the intrusin cverage intensity is nly a functin f the number f sensr nes an its upper an lwer buns can be calculate as well (Therems 8). Crllary an Therem 7 imply that if the intrusin bject is large, then the rati x (the ttal number f sensrs ver the number f subsets) shul be ept small. Liewise, if the intrusin bject is small, the rati x cul be large. These results prvie sme insights n the relatinship between the size f the bject an the active sensrs require in each run f scheuling..6 Optimizing Netwr Lifetime Maximizing the netwr lifetime leas in fact t maximizing the number subsets f sensrs ( ) that wr alternatively [-3]. Fllwing the iea in [-3], we l t maximize as a functin f the size f the intrusin bjects,, with the fllwing cnstraints: ) P = P0, ) fixe number f sensrs n, 3) cnstant between 0 an an QS V n, an 4) buns n the bject size, where P 0 is a given QS V n is a preefine Quality f Service (QS) cnstraint. We are able t btain the frm f the slutin, nt nly its existence as in [-3]. In aitin, by chsing equality cnstraints fr the etectin prbability, we erive the explicit expressin f 35