Sensor Scheduling for Multiple Parameters Estimation Under Energy Constraint

Transcription

1 Sensor Scheduln for Mulple Parameers Esmaon Under Enery Consran Y Wan, Mnyan Lu and Demoshens Tenekezs Deparmen of Elecrcal Enneern and Compuer Scence Unversy of Mchan, Ann Arbor, MI {yws,mnyan,eneke}@eecs.umch.edu ABSTRACT We consder a sensor scheduln problem for esman Gaussan random varables under an enery consran. The sensors are descrbed by a lnear observaon model, and he observaon nose s Gaussan. We formulae hs problem as a sochasc sequenal decson problem. Due o he Gaussan assumpon and he lnear observaon model, he sochasc sequenal decson problem s equvalen o a deermnsc one. We presen a reedy alorhm for hs problem, and dscover condons suffcen o uaranee he opmaly of he reedy alorhm. Furhermore, we presen wo specal cases of he ornal scheduln problem where he reedy alorhm s opmal under weaker condons. We llusrae our resul hrouh numercal examples. I. INTRODUCTION Advances n neraed sensn and wreless echnoloes have enabled a wde rane of emern applcaons, from envronmenal monorn o nruson deecon, o roboc exploraon, ec. In parcular, unaended round sensors (UGS) have been ncreasnly used o enhance suaonal awareness for survellance and monorn ype of applcaons. In hs paper we focus on he use of sensors for he purpose of parameer esmaon. Specfcally, we consder he follown scheduln problem. Mulple sensors are sequenally acvaed by a cenral conroller o ake a measuremen of one of many parameers, and hen ransm he observaon daa back o he he conroller. The laer combnes successve measuremen daa o form an esmae for each parameer. A snle parameer may be measured mulple mes over me. Each acvaon ncurs a cos (e.., sensn and communcaon coss) whch may be boh sensor and parameer-dependen. Ths process connues unl a ceran creron s sasfed, e.., when he oal esmaon error s suffcenly small, when he me perod of neres has elapsed, ec. Assumn ha sensors may be of dfferen qualy (.e. hey may have dfferen snal o nose raos) and he acvaon of dfferen sensors may ncur dfferen coss, our sensorscheduln problem s o deermne he sequence of sensors o be acvaed and he correspondn sequence of parameers o be measured so as o mnmze he sum of he oal ermnal parameer esmaon errors and he oal sensor acvaon cos. In hs paper we resrc aenon o he case of N saonary scalar parameers, modelled by Gaussan random varables wh known mean and varance, measured by M sensors, each descrbed by a lnear Gaussan observaon model. Whou loss of eneraly, we assume ha each sensor can only be used once. Ths s because mulple uses of he same sensor can be effecvely replaced by mulple dencal sensors, each wh a snle use. We formulae he above sensor scheduln problem as a sochasc sequenal decson problem. Because of he Gaussan assumpon and he lneary of observaons, hs sochasc sequenal decson problem s equvalen o a deermnsc one. Sequenal allocaon problems have been exensvely suded n he leraure (see [1]). In eneral, s dffcul o explcly deermne opmal sraees or even qualave properes of opmal sraees for sequenal allocaon problems. The mul-armed band problem and s varans (see e.., [6] and [5]) are one class of sequenal allocaon problems where he opmal soluon has been explcly deermned. In [?](refer echncal repor) we compare our problem wh he mul-armed band problem and some of s varans. We show ha our problem does no concepually belon o he class of mul-armed bands. I appears dffcul o deermne he naure of an opmal soluon for he eneral problem under consderaon. Therefore, o oban some nsh no he naure of he problem, we consder a reedy alorhm, and derve condons suffcen o uaranee he opmaly of hs alorhm. We hen presen wo specal cases of he eneral problem under consderaon for whch he reedy alorhm resuls n an opmal sraey under condons weaker han he suffcen condons menoned above. Fnally we llusrae he naure of our resuls hrouh a number of numercal examples. Sensor scheduln problems assocaed wh saonary parameer esmaon were also nvesaed n [3] and 1 of 7

2 [2]. Our resuls are dfferen from hose of [3] and [2] prmarly because he observaon model and performance crera n [3] and [2] are dfferen from ours. The res of he paper s oranzed as follows. In Secon II we sae he opmzaon problem and nroduce several prelmnares. In Secon III we provde he suffcen condons for he reedy alorhm o be opmal. Specal cases are presened n Secon IV, and numercal examples are analyzed n Secon V. Secon VI concludes hs paper. Due o he space lmaon, all proofs are omed; hey can be founded n [?]. II. PROBLEM FORMULATION In hs secon, we formulae he problem of esman mulple saonary parameers wh mulple sensors descrbed n he prevous secon, and presen a number of prelmnares. A. The Measuremen Model and Problem Formulaon Consder a se Ω p of saonary scalar parameers, ndexed by {1, 2,, N} ha have o be esmaed. Parameer s modeled by a Gaussan random varable, denoed by X, wh mean µ (0) and varance σ (0). There s a se Ω s of sensors, ndexed by {1, 2,, M}, whch we use o measure he parameers. The measuremen of parameer aken by sensor j s ven by Z,j = H,j X + V,j, (1) where Z,j s he observaon of parameer by sensor j, H,j s a known an, and V,j s a Gaussan random varable wh E(V,j ) = 0, V ar(v,j ) = R,j. A nonneave measuremen cos c,j s ncurred by acvan and usn sensor j o measure parameer. The avalable sensors are acvaed one a a me o ake a measuremen of a specfc parameer, upon reques from a conrol cener. The measuremen daa s hen ransmed back o he conrol cener, where he esmae of ha parameer and he oal accumulaed cos are updaed. The conrol cener hen decdes wheher o acvae anoher sensor from he se of remann avalable sensors, or o ermnae he process. Ths connues unl eher all M sensors are used, or unl he me perod of neres T has elapsed, or unl he conrol cener decdes o ermnae he process. For smplcy and whou loss of eneraly, we redefne T = mn{m, T }, mplyn ha a mos T sensors/parameers can be scheduled. Consequenly, he decson/conrol acon a each me nsan s a random vecor U := (U 1,, U 2, ), akn values n Ω p S {, }, where S s he se of sensors avalable a, and U = (, ) means ha no measuremen s aken a. A measuremen polcy s defned by where γ s such ha U γ = (U γ 1,, U γ 2, ) where γ := (γ 1, γ 2,, γ T ), = γ (σ 1 (0), σ 2 (0),, σ N (0), Ω p, Z γ, 1, U γ, 1 ), Z γ, 1 := (Z γ 1, Zγ 2,, Zγ 1 ), U γ, 1 := (U γ 1, U γ 2,, U γ 1 ), and he varable Z γ denoes he measuremen aken a me. Snce parameers are saonary, no akn a measuremen a some me nsan wll leave he parameers and her esmaes unchaned. Thus, whou loss of opmaly, we can resrc aenon o measuremen sraees wh he follown propery. Propery 1: For, = 1,, T 1, f U γ = (, ), hen for >, U γ = (, ). Le Γ be he se of all admssble measuremen polces ha sasfy hs propery. The opmzaon problem s Problem 1 (P1): mn J(γ) = N { [ E X ˆX ] } { T } 2 γ (T ) + E c U γ γ Γ =1 =1 { ˆXγ s.. (T ) = E[X Z U γ 1({U γ 1, = }), = 1,, T ] U γ 2, U γ 2, f,, = 1,, τ γ, where J(γ) s he cos of polcy γ Γ, ˆXγ (T ) s he ermnal esmae of parameer under sraey γ, and 1(A) s he ndcaon funcon such ha 1(A) = 1 f A s rue and 0 oherwse. Denoe by Z γ, he observaon daa se for parameer unl me under sraey γ, = 1, 2, T. Snce X s a Gaussan random varable, E{[X ˆX { { }} γ (T )]2 } = E E [X E(X Z γ,t ) 2 Z γ,t { = E [X E(X Z γ,t )] 2},.e. he error varance s ndependen of he observaon daa. Denoe he varance of parameer under sraey γ a me as σ γ () := E { [ X E(X Z γ, ) ] } 2, = 1,, N. Then a any me nsan, he varance of parameer evolves as follows. If a + 1, parameer and sensor j are seleced by γ, hen σ γ ( + 1) = σγ () (σγ ())2 H,j 2 σ γ ()H2,j + R ; (2),j f a me + 1, parameer s no seleced by γ, hen σ γ ( + 1) = σγ (). (3) (see [4]) 2 of 7

3 Therefore problem P 1 can be reformulaed as a deermnsc problem n he follown way. Denoe a scheduln sraey by := (P, S ), wh P = {a 1,, a T }, and S = {b 1,, b T }, whch ndcaes ha under sraey, parameer a s measured by sensor b a me, where a Ω p, b Ω s { }. If b =, no measuremen akes place a me and c a,b = 0. Smlarly o propery 1, we can resrc aenon o measuremen sraees wh he follown propery. Propery 2: For, = 1,, T 1, f b =, hen b =, for >. Le G be he se of all admssble measuremen polces wh he propery 2. The opmzaon problem s Problem 2 (P2): N J() = σ (τ τ ) + mn G s.. =1 =1 c a,b { a Ω p, and b Ω s { }, b b f, where τ denoes he number of measuremens aken under polcy. B. Prelmnares The follown defnons characerze he oodness of a sensor n erms of s qualy of measuremen. Defnon 1: The ndex of sensor j for parameer s ven by I,j = H2,j R,j. An ndex can be vewed as he snal-o-nose rao (SNR) of sensor j when measurn parameer. Ths quany reflecs he accuracy of he measuremen; he hher he ndex/snr, he more accurae he measuremen. Lemma 1: Assume sensor se A s used o measure parameer wh nal varance σ (0) and parameer s pos-measuremen varance s σ (A). Then we have σ (0) σ (A) = σ (0)Î,A + 1 where Î,A = j A I,j. Furhermore, σ (A) s an ncreasn funcon of σ (0) and a decreasn funcon of A,.e. f A 1 A 2, hen σ (A 1 ) > σ (A 2 ). The varance reducon of parameer wh nal varance σ (0) by usn sensor se A, denoed by R (σ (0), A), and ven by. R (σ (0), A) := σ (0) σ (A) = (4) σ2 (0)Î,A σ (0)Î,A + 1. (5) Under any polcy, we can re-wre he objecve funcon J() as follows. τ { [ ]} N J() = c a,b σ a ( 1) σ a () + σ (0) =1 =1 =1 τ = P a (σ a ( 1), b ) + N σ (0), (6) where P (σ, j) s ven by: =1 P (σ, j) = c,j R (σ, j) = c,j σ2 I,j σi,j + 1. (7) The quany P (σ, j) s referred o as he sep cos of usn sensor j o measure parameer, when he varance of parameer before he measuremen s σ. Thus he oal cos o be mnmzed s he sum of nal varance of each parameer and he sep coss. Defnon 2: The hreshold of a sensor j for parameer s ven by T H,j = 1 2 (c,j + c 2,j + 4 c,j/i,j ). Wh hs defnon, we have ha when σ = T H,j, P (σ, j) = c,j σ2 I,j σi,j + 1 = 0 ; (8) when σ > T H,j, P (σ, j) = c,j σ2 I,j σi,j + 1 < 0. (9) Therefore sensor j s hreshold for parameer can be vewed as he break-even pon n parameer s varance. Tha s, usn sensor j o ake a measuremen of parameer, whose varance σ s equal o T H,j, resuls n zero sep cos. If he curren varance of parameer exceeds he hreshold T H,j, hen usn sensor j wll resul n a neave sep cos (.e. benef), and vce versa. For sensors wh he same ndex, lower hreshold s equvalen o smaller measuremen cos; for sensors wh he same measuremen cos, lower hreshold s equvalen o hher ndex. Therefore, he hreshold combnes measuremen qualy and measuremen cos and reflecs he overall oodness of a sensor: he lower he hreshold of a sensor, he beer he sensor s qualy. III. SUFFICIENT CONDITIONS FOR THE OPTIMALITY OF A GREEDY POLICY We decompose he sensor-selecon parameeresmaon sequenal decson problem no wo subproblems. The frs one s o deermne he order n whch sensors should be used reardless of whch parameer s measured. The second problem s o deermne whch parameer should be measured a each me nsan ven he order n whch sensors are used. Such a decomposon s no always opmal. We presen condons ha uaranee he opmaly of he aforemenoned decomposon. Specfcally, we deermne wo condons under whch s opmal 3 of 7

4 o use he sensors n non-ncreasn order of her ndces (reardless of whch parameer s measured). Havn uaraneed he opmaly of he proposed decomposon, we propose a reedy alorhm for he selecon of parameers. We deermne a condon suffcen o uaranee he opmaly of he reedy alorhm. A. The Opmal Sensor Sequence Condon 1: The sensors can be ordered no a sequence s 1, s 2,, s M such ha I j,s1 I j,s2 I j,sm, j = 1, 2, N. (10) Ths condon says ha f we order he sensors n non-ncreasn order of her qualy for a parcular parameer, hen hs order remans he same for all oher parameers. For he res of our dscusson we wll denoe s j as he j-h sensor n hs ordered se. Condon 2: For each parameer, T H,s1 T H,s2 T H,sM, where s, = 1,, N s defned n Condon 1. If Condons 1 and 2 boh hold, hen hey mply ha he ordern of sensors wh respec o her sensn qualy s he same as her ordern when cos s also aken no accoun. Furhermore, boh orderns are parameer nvaran. The nex heorem shows ha he opmal sequencn of sensors s accordn o non-ncreasn order of her ndces. Theorem 1: Under Condons 1 and 2, assume ha an opmal selecon sraey s = (P, S ), where P = {p 1, p 2,, p τ },S = {b 1, b 2,, b τ }. Then for each parameer, b k S, a Ω s S, we have I,bk I,a. The nuon behnd hs heorem s ha alhouh usn dfferen sensors may ncur dfferen coss, so lon as he coss are such ha hey do no chane he relave qualy of he sensors (represened by her ndces), he bes way o use he sensors s by non-ncreasn order of her ndces. The performance of an allocaon sraey s compleely deermned by he se of sensors allocaed o each parameer; he order n whch he sensors are used for a parameer s rrelevan. Thus, sraees ha resul n he same assocaon beween sensors and parameers may be vewed as equvalen sraees. From Theorem 1, we conclude ha for any opmal sraey, here exss one equvalen sraey, under whch sensors are used n nonncreasn order of her ndces. Therefore, whou loss of opmaly, we wll only consder sraees ha use sensors n non-ncreasn order of her ndces. Parameer Selecon Alorhm L: 1: := 0 2: whle < T do 3: k := ar mn =1,,N P (σ (), s +1 ) 4: f P k (σ k (), s +1 ) < 0 hen 5: p +1 := k 6: σ k ( + 1) := σ k() σ k ()I k,s : for := 1 o M do 8: f k hen 9: σ ( + 1) := σ () 10: end f 11: end for 12: else 13: BREAK 14: end f 15: := : end whle 17: reurn τ := and P := {p 1,, p τ } F. 1. A reedy alorhm o deermne he parameer sequence. Consequenly, problem P 2 s reduced o deermnn he soppn me τ and he parameer sequence correspondn o he sensor sequence S = {s 1, s 2,, s τ }. B. A Greedy Alorhm We consder he parameer selecon alorhm, ven n Fure 1. Under Condons 1 and 2, hs alorhm compues a sequence of parameers, P, by sequenally selecn a parameer ha provdes he mnmum sep cos (.e. he maxmum benef) amon all parameers. The alorhm ermnaes when he mnmum sep cos becomes nonneave, or he me horzon T s reached. The ermnaon me s he soppn me τ. The parameer selecon sraey resuln from hs alorhm, combned wh he ven sensor sequence, s denoed by := (P, S), where P = {p 1,, p τ }, and S = {s 1,, s τ }. Ths alorhm s reedy n naure n ha always uses he bes avalable sensor (n erms of s ndex), and for ha sensor always selecs he parameer whose measuremen provdes he maxmum an (mnmum sep cos). In he nex subsecon, we nvesae condons under whch hs reedy scheduln alorhm s opmal for problem P 2. C. Opmaly of Alorhm L In hs secon, our objecve s o deermne condons suffcen o uaranee he opmaly of he reedy alorhm ven n Fure 1. Below s a ls of noaons used. 4 of 7

5 σ (): The varance of parameer a me. Ths varance depends on he nal varance σ (0) and he se of sensors used for parameer up unl me. σ (, A): The varance of parameer afer sensor se A s used o measure parameer afer me. R (σ (), A): The varance reducon of parameer resuln from he use of sensor se A wh nal varance σ (), s.. R (σ (), A) = σ () σ (, A). R (σ (, A), B): The varance reducon of parameer resuln from he use of sensor se B wh nal varance σ (, A), s.. R (σ (, A), B) = R (σ (), A B) R (σ (), A) = σ (, A) σ (, A B). (11) Assume a some me nsan, he avalable sensor se s {s +1,, s M }. Then for any sensor subse E {s +1,, s M }, we defne he advanae of usn sensor s o measure p a me followed by E, denoed by B (p, E), as follows: B (p, E) :=R p (σ p ( 1), {s } E) R p (σ p ( 1), E) c p,s. (12) The advanae s essenally he addonal varance reducon resuln from sensor s measurn parameer p, afer has been measured by sensor se E, mnus he observaon cos. Because of (11), B (p, E) can be rewren as B (p, E) = R p (σ p ( 1), {s }) c p,s + p (E), (13) where p (E) := R p (σ p ( 1, {s }), E) R (σ p ( 1), E). We have he follown propery for p (E). Lemma 2: Consder he avalable sensor se A = {s +1,, s M } afer sae of he sequenal allocaon process. Le E 1 = {s +1, s +2,, s k }, E 2 = {s +1, s +2,, s j }, where j < k M. Then p (A) p (E 1 ) < p (E 2 ) 0. (14) (?)Based on Lemma 2 and (13), we can defne upper bound B u, (p ) and lower bound B l, (p ) on he aforemenoned advanae as follows: B u, (p ) := R p (σ p ( 1), {s }) c p,s (15) = R p (σ p ( 1), {s }) c p,s + max E A p (E) R p (σ p ( 1), {s }) c p,s + p (E) = B (p, E), where he equaly holds when E =, B u, (p ) s he upper bound of B (p, E), and B l, (p ) := R p (σ p ( 1), {s }) c p,s + p (A) (16) = R p (σ p ( 1), {s }) c p,s + mn E A p (E) R p (σ p ( 1), {s }) c p,s + p (E) = B (p, E) where he equaly holds when E = A and B u, (p ) s he lower bound of B (p, E). Therefore, B l, (p ) B (p, E) B u, (p ). Noe ha B u, (p ) s he same as he sep cos P p (σ p ( 1), s ). The use of he above upper and lower bounds allows us o oban he follown resul. Lemma 3: Consder any wo sraees 1 = (S 1, P 1 ) and 2 = (S 2, P 2 ), wh S 1 = S 2 = {s 1, s 2,, s }, P 1 = {p 1,, p 1, p, p +1,, p }, P 2 = {p 1,, p 1, p, p +1,, p }. If B l, (p ) > B u, (p ), hen J( 1) < J( 2 ). The nuon behnd hs lemma s ha reardless of whch allocaon sraey wll follow afer me, usn sensor s o measure parameer p a me wll resul n beer performance han usn sensor s o measure parameer p. The resul of Lemma 3 allows us o oban he follown condon (?) whch, oeher wh Condons 1 and 2, are suffcen o uaranee he opmaly of he reedy alorhm descrbed n Fure 1. Condon 3: A some me nsan, here exss some parameer p, such ha for any parameer p no equal o p, B l, (p ) B u, (p ), where B l, (p ) and B u, (p ) are defned n a manner smlar o (16) and (15) respecvely. Noe ha f Condon 3 holds a me nsan, p s unque. Furhemore, snce B u, (p ) B l, (p ) B u, (p ), and B u, (p ) s equal o he sep cos, p s he parameer, whch wll resul n he smalles sep cos, measured by sensor s. Theorem 2: If Condons 1 and 2 hold and Condon 3 s sasfed a each me nsan 1 τ, hen Alorhm L resuls n an opmal sraey for Problem (P2). IV. SPECIAL CASES We presen wo specal cases of he eneral formulaon ven n Secon II. In he frs case, here s only one parameer o be esmaed, whch means he second subproblem n he decomposon of P 2 does no exs. In he second case, M sensors are dencal, whch means he frs subproblem n he decomposon of P 2 does no exs. For boh cases, we show ha he reedy sraey s opmal under condons weaker han hose n Secon III. 5 of 7

6 A. 1 Parameer and M Dfferen Sensors Consder problem P 2, when only one sac parameer has o be esmaed. Then he observaon model for sensor j s Z j = H j X + V j (17) The cos of usn sensor j s c j. In hs case we only need o deermne whch sensors should be used o measure he parameer. Thus, he second subproblem of he decomposon n Secon III does no exs. Furhermore, Condon 1 s sasfed auomacally. Then under Condons 1 and 2, s opmal o use he sensors accordn o non-ncreasn order of her ndces by Theorem 1. Noe ha f he observaon cos for every sensor s equal,.e. c j = c, j = 1,, M, Condon 2 s equvalen o Condon 1. In hs suaon, s opmal o use he sensors accordn o non-ncreasn order of her ndces whou any consran. B. N Parameers and M Idencal Sensors wh Sensor- Independen Observaon Model Consder problem P 2 n he case where he M sensors are dencal. Then he observaon model for parameer s sensor-ndependen, ha s, Z = HX + V, for all sensors. (18) The cos of measurn parameer by any sensor j s c. Snce he sensors are dencal, Condons 1 and 2 are sasfed auomacally. Therefore, n hs case we only concern he second subproblem of he decomposon n Secon III. Thus, we can vew he M dencal sensors as one processor whch can be used a mos M mes, and he N dfferen parameers as N ndependen machnes. The sae of every machne/parameer s s varance. A every me nsan, we mus selec one machne/parameer a o process/ esmae. The varance of machne/parameer a s updaed and all he oher machnes /parameers sae/varance s frozen. The reward a each me nsan s he varance reducon of parameer a mnus he observaon cos c a. Vewed hs way, problem P 2 s a fne horzon mul-armed band problem wh dscoun facor equal o one. For fne horzon mul-armed band problems, he Gns Index rule(see [1]) s no enerally opmal. However, n he problem under consderaon, he reward sequence for each machne/parameer s deermnsc and non-decreasn wh me. Thus, for each machne/parameer, he Gns Index s always acheved a τ = 1, whch concdes wh he one-sep look-ahead polcy resuln from Alorhm L descrbed n Secon machn percenae(%) σ (1,10), I (1,5), loop= observaon cos F. 2. performance percenae(%) σ (1,10), I (1,5), loop= averae 4.5 upperbound observaon cos Performance of Greedy Alorhm. III. Consequenly, he Gns Index rule s opmal for hs specal case. V. NUMERICAL EXAMPLES We llusrae he performance of Alorhm L wh numercal expermens(examples). We denoe by L he reedy sraey correspondn o Alorhm L, and by o he opmal sraey. We defne he performance devaon (PD) of sraey as P D() := J() J( o). (19) J( o ) The seup of he numercal expermen s as follows. There are 7 sensors and 3 parameers. The observaon cos s a consan for all he sensors and parameers; 51 observaon coss are ncremened from 0 o 0.5 wh ncremenal sze For each cos selecon, we run he expermen 1000 mes wh 7 ndces, each chosen accordn o a unform dsrbuon on (1, 5) and 3 nal varances, each chosen accordn o a unform dsrbuon on (1, 10). We adop he follown hree performance crera. he number ha L=o 1) Machn Rae: 1000 ; P D(L ) 1000 ; 2) Averae Devaon: 3) Maxmum Devaon: max P D( L ). The resuls are shown n Fure 2. When he observaon cos s suffcen lare, sraey L s always opmal. Ths s conssen wh he nuon. When he observaon cos s lare, each parameer can be measured a mos once. In hs case one can show ha usn sensor wh he lares ndex o measure he parameer wh he lares varance a presen s an opmal sraey. When sraey L s no opmal, he averae devaon and he maxmum devaon are always below 5%. We observe very smlar resuls when he expermen s repeaed wh dfferen values of sensors ndces and parameers nal varance. From he numercal expermens, we can conclude ha Alorhm L s a very ood approxman alorhm, 6 of 7

7 whch can be opmal when he observaon cos s suffcen lare. VI. CONCLUSION We consdered a sensor scheduln problem for mulple parameer esmaon under an enery consran. We decompose he sequenal decson problem no wo subproblems. The frs one s o deermne he sequence of he sensors o be used, whch s ndependen of he parameer selecon, and he second one s o deermne he sequence of parameers o be measured for a ven sensor sequence. We denfed condons suffcen o uaranee ha a reedy polcy, defned by Alorhm L, s opmal for he problem under consderaon. The numercal examples we consdered ndcae ha for lare values of he measuremen cos, he reedy polcy performs well. We presened an explanaon as o why such behavor of he reedy polcy should be expeced for lare measuremen cos. REFERENCES [1] J. C. Gns. Band process and dynamc allcoaon ndces. Journal of he Royal Sascal Socey. Seres B (Mehodolocal), 41:pp , [2] G. P. H. Wan, K. Yao and D. Esrn. Enropy-based sensor selecon heursc for are localzaon. In Proceedns of The Thrd Inernaonal Symposum on IPSN, paes pp [3] V. Isler and R. Bajcsy. The sensor selecon problem for bounded uncerany sensn models. In Proceedns of The Fourh Inernaonal Symposum on IPSN, paes pp [4] P. R. Kumar and P. Varaya. Sochasc Sysems: Esmaon, Idenfcaon and Adapve Conrol. Prence Hall, [5] J. C. W. Pravn P. Varaya and C. Buyukkoc. Exenons of he mularmed band problem: The dscouned case. IEEE Transacons on Auomac Conrol, 30:pp , [6] P. Whle. Mul-armed bands and he ns ndex. Journal of he Royal Sascal Socey. Seres B (Mehodolocal), 42:pp , of 7