Performance of Stochastically Intermittent Sensors in Detecting a Target Traveling between Two Areas

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1 American Journal of Operaions Research, 06, 6, 99- Published Online March 06 in SciRes. hp:// hp://dx.doi.org/0.436/ajor Performance of Sochasically Inermien Sensors in Deecing a Targe Traveling beween Two Areas Hongyun Wang *, Hong Zhou * Deparmen of Applied Mahemaics and Saisics, Baskin School of Engineering, Universiy of California, Sana Cruz, CA, USA Deparmen of Applied Mahemaics, Naval Posgraduae School, Monerey, CA, USA Received 6 November 05; acceped March 06; published 5 March 06 Copyrigh 06 by auhors and Scienific Research Publishing Inc. This work is licensed under he Creaive Commons Aribuion Inernaional License (CC BY. hp://creaivecommons.org/licenses/by/4.0/ Absrac We sudy he problem of deecing a arge ha moves beween a hiding area and an operaing area over muliple fixed roues. The research is carried ou wih one or more cookie-cuer sensors wih sochasic inermission, which urn on and off sochasically governed by an on-rae and an off-rae. A cookie-cuer sensor, when i is on, can deec he arge insanly once he arge comes wihin he deecion radius of he sensor. In he hiding area, he arge is shielded from being deeced. The residence imes of he arge, respecively, in he hiding area and in he operaing area, are exponenially disribued and are governed by raes of ransiions beween he wo areas. On each ravel beween he wo areas and in each ravel direcion, he arge selecs a roue randomly according o a probabiliy disribuion. Previously, we analyzed he simple case where he sensors have no inermission (i.e., hey say on all he ime. In he curren sudy, he sensors are sochasically inermien and are synchronized (i.e., hey urn on or off simulaneously. This happens when all sensors are affeced by he same environmenal facors. We derive asympoic expansions for he mean ime o deecion when he on-rae and off-rae of he sensors are large in comparison wih he raes of he arge raveling beween he wo areas. Based on he mean ime o deecion, we evaluae he performance of placing he sensor(s o monior various ravel roue(s or o scan he operaing area. Keywords Sochasically Inermien Sensors, Moving Targe wih Consrained Pahways, Mean Time o Deecion, Opimal Search Design * Corresponding auhors. How o cie his paper: Wang, H.Y. and Zhou, H. (06 Performance of Sochasically Inermien Sensors in Deecing a Targe Traveling beween Two Areas. American Journal of Operaions Research, 6, 99-. hp://dx.doi.org/0.436/ajor.06.60

2 . Inroducion Search and deecion heory has a hisory of principal imporance in operaions research. I has fundamenal miliary and civilian applicaions such as ani-submarine warfare, couner-mine warfare, and search and rescue operaions []-[4]. Nowadays when combaing piracy a sea, deecing and inerceping hrea objecs (boas such as erroriss and drug or weapon smugglers, or securing coaslines and rade roues, i is imporan o undersand he behavior of a arge and plan a search and deecion sraegy accordingly. In a recen work [5] we considered he problem of searching for a arge ha ravels beween a hiding area and an operaing area via muliple roues. By assuming cerain behaviors of he moving arge, we obained analyic expressions for he mean ime o deecion and hereby were able o deermine he opimal placemen of m cookie-cuer sensors (i.e. how many sensors should we place on which roues and he res is placed o search he operaing area. Ineresingly, we found ha he opimal placemen, someimes, is no he one suggesed by inuiion. In his paper we would like o exend our earlier sudy o include sochasically inermien sensors in deecing a arge moving beween a hiding area and an operaing area.. Mahemaical Formulaion for he Case of Sensors wihou Inermission We consider he search problem in which a arge moves beween a hiding area and an operaing area via consrained pahways, as depiced in Figure. The arge can say in he hiding area where he arge is shielded from being deeced by he sensors. There are N given roues connecing he hiding area and he operaing area. The arge can ravel along one of hese N given roues from he hiding area o he operaing area. The arge can spend ime in he operaing area o carry ou cerain aciviies/asks. Aferwards, he arge can reurn o he hiding area via possibly a differen roue. In he hiding area, he arge is no deecable. Ouside he hiding area, he arge is deecable along he roues and in he operaing area if i comes ino he deecion range of a sensor. In he search problem, variable number of synchronized inermien cookie-cuer sensors are used o deec he arge. We will firs inroduce he mahemaical model for he simple case of non-inermien sensors (i.e., hey say on all he ime and hen exend he model o accommodae he sochasic inermission of he sensors. We sar by specifying he arge behavior. The arge moves sochasically beween he hiding area and he operaing area according o he following rules. The dwell ime of he arge in he hiding area is exponenially disribued wih rae r f, he forward rae of he arge going from he hiding area o he operaing area. On is ravel from he hiding area o he operaing area, he arge akes roue k wih probabiliy N saisfies he consrain where N is he oal number of given roues. p k k p k, which Figure. A arge raveling beween wo areas. The arge may say in he hiding area where i is no deecable; i may ravel from he hiding area o he operaing area along one of he N given roues; i may spend ime in he operaing area before reurning o he hiding area via possibly a differen roue. 00

3 The dwell ime of he arge in he operaing area is exponenially disribued wih rae r b, he backward rae of he arge going from he operaing area back o he hiding area. On is ravel from he operaing area back o he hiding area, he arge chooses roue k wih probabiliy q k, N which saisfies he condiion q k k. The ravel ime beween he operaing area and he hiding area is negligible in comparison wih he dwell imes in he hiding area and he operaing area. Mahemaically, we rea he ravel ime along a roue as zero. The arge s ravel beween he wo areas is mahemaically described by a Markov process of wo saes, wih forward rae r f and backward rae r b, as illusraed in Figure. Nex, we describe he ineracion beween he arge and sensors. A non-inermien cookie-cuer sensor is an ideal sensor which deecs he arge insanly once he arge comes wihin disance R o he cener of he sensor where he radius R is called he deecion radius of he sensor. When he arge is ouside he deecion radius, i is no deeced. In his sudy, we assume ha he deecion radius of sensors is large enough o cover he full widh of any one of he given roues. Consequenly, if a non-inermien sensor is assigned o monior a roue and he arge happens o move along ha roue, he arge will definiely be deeced by he sensor. Of course, he siuaion will be differen for an inermien sensor ha urns on and off sochasically. When a sensor is used o search he operaing area, when he sensor is on, and when he arge is in he operaing area, he ineracion beween he arge and he sensor is modeled using a deecion rae r d (probabiliy of deecion per ime. r rae of deecing he arge in he operaing area ( d This deecion rae is affeced by he size of he operaing area, and by he deecion radius and he speed of he sensor. When one or more non-inermien sensors are deployed o monior one or more roues or o search he operaing area, he ransiions and deecion of he arge are governed by a 3-sae Markov process of he same parameer form as he one shown in Figure 3. The values of p, q, and d depend on how many sensors are used and which roue(s and area are moniored/searched. Values of p, q, and d are given below for several cases. When only one sensor is deployed and i is placed o monior roue k, his gives p p, q q, d When only one sensor is deployed and i is used o search he operaing area, his corresponds o p, q, d rd When wo sensors are placed o monior, respecively, roues k and j ( k j, i follows ha p p + p, q q + q, d k k j k j When wo sensors are boh used o search he operaing area, we find k Figure. Markov ransiions of he arge beween he hiding area and he operaing area in he absence of any sensor. 0

4 Figure 3. Markov ransiions of he arge in he presence of non-inermien sensor(s. Noe ha he parameer form of he Markov process is he same for all cases while values of parameer p, q, and d vary from case o case. p, q, d r d When one sensor is placed o monior roue k and a second sensor is used o search he operaing area, his gives p p, q q, d r k k d To faciliae our analysis, we divide all ransiion raes by he sum ( rf rb + o non-dimensionalize he Markov process depiced in Figure 3. Addiionally, we inroduce wo dimensionless parameers: rf d α, r + r r + r f b f b The parameer form of he normalized Markov process is shown in Figure 4. In he absence of sensors, a equilibrium, he probabiliies of he arge being in he hiding area or he operaing area are, respecively, ( α and α. This fac will be used laer in he calculaion of mean ime o deecion. 3. Mean Time o Deecion in he Case of Sensors wihou Inermission Figure 4 describes he general model for he case of non-inermien sensors. I can accommodae arbirary number of sensors. For mahemaical convenience, we label he hiding area as sae and he operaing area as sae. We solve for he mean ime o deecion. Le T denoe he ime o deecion (random variable, and S( denoe he sae of he arge a ime. Time 0 is defined as he ime when he sensors are deployed. We examine he condiional mean ime o deecion given ha he arge is in sae j a ime 0. We derive wo equaions for and S is he probabiliy disribuion of ( ( ( ETS 0 j, j, (3 j based on he Markov process in Figure 4. Saring wih ( α o( ( ( p o(, wih probabiliy + ( S, wih probabiliy α p + o deeced, wih probabiliy α + Using he law of oal expecaion, we have ( α α( ( + + p + o ( S 0, (4 0

5 which, when divided by This is an equaion for wo equaions for and Figure 4. The normalized Markov process governing he ransiions of he arge. and in he limi of Solving linear sysem (5, we obain going o zero, yields α + α( p and. Similarly, anoher equaion can be derived saring wih ( are α + α( p (( α + + ( α( q p + + α( α α α + p + q pq α q + α α + p + q pq α Before he deploymen of sensors, he equilibrium disribuion of he arge is ( S( α ( S( Pr 0, Pr 0 α Thus, he overall mean ime o deecion has he expression ( p+ q + α α( α E( T ( α + α + p + q pq α 4. Mahemaical Formulaion for he Case of Synchronized Sochasically Inermien Sensors S 0. The We exend above discussions o consider synchronized inermien sensors ha sochasically urn on or off simulaneously. We model he sochasic evoluion of sensors as a -sae Markov process wih an on-rae r on and an off-rae r off, as illusraed in Figure 5. As in he previous secion, we divide all raes by ( rf + rb o normalize he problem. Afer he nondimensionalizaion scaling, we wrie he on-rae and he off-rae as ron on-rae r + r f b (5 (6 (7 03

6 Figure 5. The -sae Markov process governing he inermien sensors. r off-rae r + r f off b (8 where rf + rb ron, r + r r + r on off on off Noe ha he parameer defined above corresponds o he equilibrium probabiliy of sensors being on, which is also he fracion of ime ha sensors are on. The quaniy ( is he equilibrium probabiliy of sensors being off. The choice of noaion indicaes ha we will focus on he parameer regime of. r + r r + r r + r is he rae of sensors relaxing Tha is, we will focus on he case of ( on off ( f b o equilibrium beween he on- and off-saes; ( rf rb. Noe ha ( + is he rae of he arge relaxing o equilibrium beween he hiding area and he operaing area. Parameer in (9 measures he raio of hese wo relaxaion raes. We now combine he sochasic ravel of he arge and he sochasic inermission of he sensors ino a 5-sae Markov process for he arge-sensors sysem as illusraed in Figure 6. The normalized version of his 5-sae Markov process where all ransiion raes are divided by ( rf rb +, is shown in Figure 7. In he model, he sensors inermission and he arge s ravel beween he wo areas are independen of each oher. The argesensors sysem can reside in any of 4 saes before he deecion. For mahemaical convenience, we number he 4 saes as follows: (Figure 7. Sae : he arge is in he hiding area and he sensors are on. Sae : he arge is in he operaing area and he sensors are on. Sae 3: he arge is in he hiding area and he sensors are off. Sae 4: he arge is in he operaing area and he sensors are off. Figure 7 gives he general model for he arge-sensors sysem wih synchronized sochasically inermien sensors. I can accommodae arbirary number of sensors. We sudy he mean ime o deecion in his 5-sae Markov process. Again, le T represen he ime o deecion (random variable, and (. We consider he condiional mean ime o deecion We derive four equaions for (,, 3, 4 S, he probabiliy disribuion of S( ( ( on off (9 S denoe he sae of he arge a ime ETS 0 j, j,,3,4 (0 j based on he 5-sae Markov process in Figure 7. Saring wih is 04

7 Figure 6. The 5-sae Markov process governing he argesensors sysem. Figure 7. The normalized 5-sae Markov process governing he arge-sensors sysem. In he non-dimensionalizaion scaling, all r + r. ransiion raes are divided by ( f b S ( The law of oal expecaion gives us Dividing boh sides by, wih probabiliy α + + o, wih probabiliy α ( p + o( 3, wih probabiliy + o( deeced, wih probabiliy α p + o( ( + α + + α p o and aking he limi as ( ( going o zero, we arrive a ( p α + + α + 3. Three oher equaions are derived using he same approach, saring wih, are This is an equaion for (,, 3, 4 respecively, S, S 3, and S 4. Puing ogeher, hese four equaions for (,,, ( p α + + α + 3 ( ( ( q α α + 4 α α ( ( (3 (4 05

8 ( α ( α 3 + Analyical soluions o he above linear sysem is hard o obain. Insead, in he nex secion, we use his linear sysem o calculae asympoic expansions for (,, 3, 4 and for he overall mean ime o deecion when is small. 5. Asympoic Soluions for he Mean Time o Deecion in he Case of Synchronized Inermien Sensors We derive asympoic soluions for he mean ime o deecion for small. We sar by wriing he condiional mean ime o deecion in he asympoic form of ( j j j (5 + + (6 Subsiuing his asympoic form ino Equaions (-(5 and examining erms of he order are of he larges magniude, we obain 3 4 O, which From Equaions (-(5, we will derive wo equaions ha do no conain any coefficien of he order O. These wo equaions will hen be used o calculae he leading coefficiens in he asympoic form: and. We rewrie Equaions (-(5 o separae erms wih coefficiens The wo equaions are consruced by The resuling wo equaions are O from oher erms. (7 ( 3 α + α( p (8 ( 4 ( ( α + + ( α( q (9 ( 3 α3 + α (0 4 ( 4 ( α 4 + ( α ( 3 ( ( ( equaion 8 + equaion 0 ( ( ( equaion 9 + equaion ( ( ( α + α p α + α ( 3 4 ( ( α + + ( α( q ( ( α 4 + ( ( α 3 Subsiuing asympoic form (6 ino he wo equaions above, keeping only erms of order O ( resul (7, we arrive a α α( p (( ( 0 ( ( ( 0 α α q (3, and using (4 06

9 Sysem (4 is of he same parameer form as sysem (5 excep ha p, q, and in (5 are replaced by q, and in (4. The soluion of (4 immediaely gives us he leading coefficiens: We inroduce quaniy σ has he expression p + + α( α α α α q + α α α ( α ( p q pq p q pq p, α σ, which will be useful in he subsequen calculaion. Quaniy ( ( ( α α αp+ α q σ (6 + p + q pq α ( ( ( ( Nex, we calculae coefficiens (,,,. Subsiuing he asympoic form (6 ino equaion (0- (, and using he fac ha all leading coefficiens 3 4 ( ( are already known, we find expressions for ( and ( (. 3 ( 4 ( (5 ( ( σ 3 α + α (7 α ( ( ( ( ( ( ( 0 0 σ 4 α + α + (8 α ( ( ( ( ( 4 3 σ ( α( α Subsiuing he asympoic form (6 ino Equaions (-(3, collecing all erms of he order ( using resuls (7-(8, we obain wo equaions for ( and (. ( ( ( p σ α( α + ( ( ( q α α( α The soluion of (30 gives us expressions for coefficiens ( ( ( ( (,,, p ( α σ + p+ q pq α( α α ( q σ p q pq α( α + + α ( ( σ 3 + α ( ( σ α σ (9 O, and Before he sensors are assigned o monior/search roues, he equilibrium disribuion of he arge-sensors (30 (3 07

10 sysem is ( S( ( α ( S( Pr 0, Pr 0 α ( S( ( α( ( S( α( Pr 0 3, Pr 0 4 I follows ha he overall mean ime o deecion has he expression ( ( α + α + ( α ( 3 + α ( 4 (( ( ( ( 0 ( ( 0 α + α + α + α 3 4 E T ( ( ( ( ( ( ( ( ( α + α + α + α Therefore, he overall mean ime o deecion has he asympoic expansion where he coefficiens and ( ( are given by ( E T + + (3 p q+ α α ( α + p + q pq α ( ( p ( p q ( q ( p q ( + α + + α p + q pq α( α α The normalized ransiion raes and parameers (shown in Figure 7 are relaed o he physical ransiion raes (shown in Figure 6 as follows d r + r f f b rf α r + r r on on ron + r b off rf + rb r + r 6. Behaviors of he Mean Time o Capure ( We examine behaviors of he mean ime o deecion, E( T +, as a funcion of parameers, α,, p, q, and. Observaion : (0 is a decreasing funcion of. We differeniae as defined in (33 wih respec o and find ha p q+ d d d + p + q pq α off d α α ( α ( ( α p ( ( α q < 0 α ( α + p + q pq α (33 (34 (35 08

11 In he derivaive above, he righ-hand side is negaive because facors ( α H. Y. Wang, H. Zhou, p, and q are all posiive and bounded by. This propery of is no surprising. Recall ha is he normalized deecion rae in he operaing area. For a larger, i is reasonable o expec ha he arge will be deeced sooner. Observaion : (0 is a decreasing funcion of p and a decreasing funcion of q. We differeniae, respecively, wih respec o p and q and obain ha + p + q pq + q p q + α α α ( α < 0 + p + q pq α ( d dp + p + q pq + p p q + α α α ( α < 0 + p + q pq α ( d dq Boh d dp and d dq are negaive, which follows direcly from he fac ha all erms in he numeraors are posiive. In paricular, i is rue ha p q pq p q pq 0 α + + > + > p q+ > ( p+ q > 4 ( p+ q > 0 α α α α α ( ( This propery of is also expeced. Remember ha p and q are, respecively, he probabiliy of he arge being deeced on is way o he operaing area and he probabiliy of being deeced on is way back o he hiding area. Increasing eiher p or q while keeping he oher unchanged will speed up he deecion of he arge. Observaion 3: While (p + q keeps fixed, becomes an increasing funcion of (pq. In he expression of, while he sum ( p+ q is fixed, he combinaion ( pq appears only in he denominaor. p q+ α α ( α + p + q pq α When he produc ( pq is increased, he denominaor is decreased, and as a resul, is increased. This behavior ells us ha when evaluaing several opions wih he same value of ( p+ q, he opimal opion is he one wih he smalles value of ( pq. Observaion 4: (0 is a decreasing funcion of. We differeniae wih respec o o ge ( ( α pq + ( p + q pq d α d α( α + p + q pq α ( p + q ( + α( α pq < 0 α( α + p + q pq α This propery of is expeced. is he fracion of ime ha he sensors are on. Increasing he value of leads o an increase in he probabiliy ha he sensors are on a a random ime. and hus reduces he ime o deecion. 09

12 Observaion 5: When p q, (0 is a decreasing funcion of α. When sensors are deployed only o search in he operaing area and no sensor is used o monior any of he roues, we have p q. In his case, has he simple expression + α which is a decreasing funcion of α. When only he operaing area is searched, he arge can only be deeced in he operaing area; he arge canno be deeced on is ravels beween he wo areas. Consequenly, he deecion probabiliy increases wih he fracion of ime ha he arge spends in he operaing area, which is given by α. Observaion 6: In general, (0 is no necessarily a decreasing funcion of α. When he condiion p q is no saisfied, may no be monoonic wih respec o variable α. Figure 8 shows a plo of vs α while oher parameers are fixed a.55, p.45, q.3, and.4. iniially decreases as α is increased from 0. I aains a minimum around α.6. For α > 0.6, appears o be an increasing funcion of α. I is clear ha in general, is no monoonic wih respec o α. Nex we sudy how he mean ime o deecion changes wih, he normalized ime scale of sensors relaxing ( o equilibrium beween he on- and off-saes. For ha purpose, we examine coefficien in he asympoic ( expansion E( T + +. Observaion 7: ( is always posiive. ( We wrie as ( h ( α( α + p + q pq α α,, pq,,, given by where h is a funcion of ( h ( + p α ( p+ q ( + q α( p+ q + α( α + p+ q pq α ( To prove ha is always posiive, we only need o show h > 0 is always rue. As a firs sep, we esablish ha h is an increasing funcion of, which is reached by differeniaing h wih respec o. Figure 8. Plo of vs α for.55, p.45, q.3, and.4. Noe ha in general, is no monoonic wih respec o α. 0

13 dh + ( p + q α pq > ( p + q pq > 0 d α Thus, o prove h > 0, we only need o show h > 0. This can be verified direcly: 0 In he above, we have used he facs ( ( ( ( ( ( pq( α( α ( p q α( α pq 0 h p α p+ q q α p+ q + α α p+ q pq > > α( α and ( p+ q < 4 Therefore, we conclude ha ( is always posiive. I follows ha as a funcion of while oher parameers are fixed, he mean ime o deecion E( T decreases as is reduced. In oher words, when he fracion of ime ha sensors are on is fixed (i.e., is kep a a consan, and when he normalized ime scale of sensors relaxing o equilibrium beween he on- and off-saes is reduced (i.e., is made smaller, he deecion performance of sensors is improved (i.e., he mean ime o deecion is shorer. Finally, we demonsrae he accuracy of he asympoic soluion for he mean ime o deecion. Figure 9 compares an accurae numerical soluion and he asympoic expansion + ( for E( T, in he range of [ 0, 0.3]. I is clear ha he dependence of E( T on is well capured in he asympoic soluion ( + even a Conclusion We have addressed he performance of sochasically inermien sensors when used o deec a arge ha moves beween a hiding area and an operaing area via muliple roues. We have derived asympoic expansions for he mean ime o deecion when he on-rae and off-rae of he sensors are large in comparison wih he raes of he arge moving beween he hiding area and he operaing area. Using he mean ime o deecion, we have evaluaed he performance of placing sensor(s o monior various ravel roue(s or o scan he operaing area. Figure 9. Comparison of an accurae numerical soluion of ET ( and he asympoic soluion ( + for α.4,.5, p.37, q., and.4.

14 Acknowledgemens and Disclaimer Hong Zhou would like o hank Naval Posgraduae School Cener for Muli-INT Sudies for supporing his work. Special hanks go o Professor Jim Scrofani and Deborah Shiffle. The auhors also hank Mr. Ed Walz and Dr. Will Williamson for heir inspiraional suggesions. The views expressed in his documen are hose of he auhors and do no reflec he official policy or posiion of he Deparmen of Defense or he U.S. Governmen. References [] Koopman, B.O. (999 Search and Screening: General Principles wih Hisorical Applicaions. The Miliary Operaions Research Sociey, Inc., Alexandria, VA. [] Sone, L.D. (989 Theory of Opimal Search. nd Ediion, Operaions Research Sociey of America, Arlingon, VA. [3] Wagner, D.H., Mylander, W.C. and Sanders, T.J. (999 Naval Operaions Analysis. 3rd Ediion, Naval Insiue Press, Annapolis, MD. [4] Washburn, A. (04 Search and Deecion. 5h Ediion, CreaeSpace Independen Publishing Plaforms, Locaion. [5] Wang, H. and Zhou, H. (05 Searching for a Targe Traveling beween a Hiding Area and an Operaion Area over Muliple Roues. American Journal of Operaions Research, 5, hp://dx.doi.org/0.436/ajor