Reliable Identification of RFID Tags Using Multiple Independent Reader Sessions

Reliable Identificati on of RFID T ags Using Multiple Independent Reader Sessions Rasmus Jacobs en, Karsten Fyhn Nielsen, Petar Popovsk i and T orben Larsen Departmen t of Electronic Systems, Aalbo rg University , Denmark { raller, kfyhn , petarp, tl } @es.aau .dk Abstract —Radio Frequency Iden tification (RFID) systems are gaining momentum in v arious applications of logistics, in v entory , etc. A generic problem in such systems is to ensure that the RFID readers can reliably read a se t of RFID tags, such that the probability of missing tags stays below an acceptable value. A tag may be missing (left u nread) du e to erro rs in t he communication link towards the reader e.g. due to obstacles i n the radio path . The present p aper proposes techniqu es that use multi ple reader sessions, during which the system of readers obtains a runn ing estimate of the probability to hav e at least one tag missing. Based on such an estimate, it is decided wh ether an addi tional reader session is required. T wo methods are proposed, th ey rely on the statistical in dependence of the tag reading errors acros s different reader sessions, which is a plausible assump tion when e.g. each reader session is executed on different readers. The first method uses statistical relationships that are valid when the reader sessions are i ndependent. The second method is obtained by modifyin g an existing capture–re capture estimator . The results show that, when the reader sessions are independent, the proposed mechanisms provide a good approximation t o the probability of missing tags, such th at the number of reader sessions made, meets th e target specifi cation. If the assumption of independ ence is violated, th e estimators are still useful, but they should be corrected by a margin of additional reader sessions to ensure th at the target probability of missing tags is met. Index T erms —Mi ssing tag problem, set cardinality estimation, error probability estimation, RFID networks I . I N T RO D U C T I O N RFID tech nology featur es a growing set of app lications for identificatio n of various objects. The ap plications sp an from simply identifying objects, serv ing as more inf orma- ti ve b arcodes, gath ering of sensory data and ho lding pri- vate/confidential informa tion [1][2][3]. The advantages of RFID technolog y include the low co st per ta g a nd the low energy consumption , which lets them have a very long lifetime [4]. The p assi ve RFID tags repr esent a category of ta gs that does not have p ower supply , they rely solely on th e signal sent from a reader to power their circ uitry , and to respond by backscattering the sign al [4 ]. The commu nication parad igm in the passiv e RFID sy stems is based o n requ est/response: In the first step, the rea der send s an interro gation signal to the tags within its rang e. In the second step the tag s send their response to the reader by backscattering the signal. If mu ltiple tags simultaneo usly reply to the read er , then th e r eader exper iences tag collision. Hence, the reader sh ould run a certain anti-collision pro tocol (also called co llision-resolution or arb itration proto col) in o rder to successfully reso lve each tag in its pro ximity . Th ere are various anti-collision protoco ls, which are in gen eral divided into two gro ups, ALOHA–based [5][6] and tr ee-based [ 7][8]. Regardless of the actual arb itration protoco l used, after a single run of the pro tocol is terminated, the reader has the identities o f the tags in its pro ximity . In the ideal case, when there are no tr ansmission erro rs and the o nly erro r experienced at th e r eader is du e to the tag collisions, then on e can be certain that all the tag iden tities have b een collected durin g the a rbitration pro cess. Howe ver , er rors do occur if either the query fro m a read er is no t received correctly at a ta g or th e tag reply is n ot received at th e reader . In principle, if a tag is a t a b lind sp ot [9], then the communication between the tag an d the reader is always in erro r . The pro bability that a tag is at a blind spot can be su bstantial and is p rimarily determin ed by the physical disposition of the tag, but also by th e ma terial to which the tags are af fixed. I n [10] it is sho wn, that if a tag is attached to solar cream, the probability of not resolving a tag is 30% and with mineral water it is 67%. The er ror proba bility can vary a lot, increasing the p robability of missing one or more tags complete ly . In summ ary , if during th e arbitra tion protoco l the link between a read er and a tag is in err or , th en this tag is n ot ide ntified at the en d o f the pro tocol run . This is defined a s th e missing tag pr o blem . There ar e mu ltiple appro aches to min imize the prob ability of m issing a tag. In [11], a meth od for determinin g group completen ess in an RFID n etwork is described , based on each tag storing o ne o r more re ferences to surrou nding tags. The resolved tags and the refer ences ar e compar ed, and if no t all referenc es are resolved, the reading /comparison is re peated. Thereby the reader kn ows with high prob ability if tag s are missing. This method is targeting r ather static co nstellations of tags, e.g. good s o n pallets. Another ap proach is presented in [9], where a method fo r resolving a set o f RFID tag s is presented. This is do ne by u sing two in dependen t samp les, in this case a database and RFID readin gs. These two samples are used as in a classical capture– recapture mo del [12] to d eriv e estimators for th e tag set cardinality . The p aper is organize d as fo llows. An overvie w of th e problem and an intuitive explan ation is presented in the next section in th e c ase of two reader sessions, followed by the system mo del in Section III. Deriv ation of estimator s for two read er sessions is in Sectio n I V. T he estimators are generalized to mu ltiple reader sessions in Section V, and the estimators in th e case o f two rea der sessions are ev a luated analytically in Section VII. I n Section VIII, simu lations sh ow the perfor mance of the prop osed estimator s in scenarios with both depe ndent and inde pendent reader sessions. The work is conclud ed in Section IX. I I . P RO B L E M D E FI N I T I O N The main idea in this paper is to use se veral in dependen t readings o f the tag set that consists o f N tag s. One reading of the tag set co nsists of on e ru n of the arb itration pr otocol, and is denoted a reader session. In each r eader session the probab ility that a given tag is no t read is p . Reader sessions are indepen dent, when the prob ability that each tag is read in on e reader session is in dependen t of it being read in another . Th e value of th e error pr obability p and tag card inality N are not known a prio ri. At this p oint it is natura l to ask: How ca n we assure, or at least attemp t, to make the read ings indep endent? Here are two plausib le examples: 1) Before the n ext read er session with the same reader, the tagg ed items are phy sically displaced/sh uffled and it can be assumed that such a n action “resets” the physical links and ge nerates e rror with pr obability p . 2) If one r eader with multiple anten nas or multiple reader s are located at different positions, but re main in commu- nication range with the same tags, the r eader sessions may be assume d ind ependen t. A scen ario that enc ompasses b oth cases is the one with a con- veyor belt, alon g which several readers are dep loyed. It should be no ted that these are ways to aim for indepen dence and simulations show how the methods introd uced unde rperfor m when the ind ependen ce assump tion do es not ho ld. The b asic id ea o f ou r app roach leverages on the recent ideas about co operative readers [ 13] that can jo intly infer statistical inform ation abou t the set of tags S in ran ge. In order to illustrate the idea, co nsider the case with two r eaders each having a read er session, r 1 and r 2 respectively . The probab ility that a tag is n ot re ad in reader session r i is p . Af ter the two re ader session s ar e terminated, the readers exchang e informa tion abou t the tag identities they have g athered. Let k 1 denote the sub set of tag s th at have b een r ead in bo th reader session r 1 and r 2 . Let k 2 a ( k 2 b ) be the subset of tags that are read only in r 1 ( r 2 ) . This is schematically represented in Fig. 1. There is a lso a set of k 3 tags th at are no t read in either of the reader sessions. Let ˆ p and ˆ N denote the estimates of p and N , respectively . Based on the expected values for k 1 , k 2 a , and k 2 b , o ne can wr ite: k 1 = ˆ N (1 − ˆ p ) 2 k 2 a + k 2 b = 2 ˆ N (1 − ˆ p ) ˆ p (1) Using these two equations, we can obtain v alues fo r ˆ p and ˆ N . Based on that, we can estimate th e expected value of the number of m issing tag s k 3 . Furth ermore, we can estimate the pr obability of having at least one tag missing and, if this p robability is above a thre shold value, we can p erform additional r eadings. This pr ocess is genera lized by d evising methods to obtain ˆ p and ˆ N fro m th ree or mo re in dependen t readings. The objective is to create a sequential decision k 2 r 1 r 2 k 2 a k 2 b k 1 k 3 Fig. 1. V enn dia gram of the possible tag sets for two reader sessions r 1 and r 2 . k 1 is the number of tags found in common in both reader sessions, k 2 a is the number of tag s only found in reader session r 1 , k 2 b only in rea der session r 2 . The set k 2 is gi ven by the sum of k 2 a and k 2 b . An unobserved number of tags, k 3 , may exist. process in which , af ter the R th reader session (ar bitration protoco l run), we calculate the probab ility of having a tag remaining an d, if this prob ability is above a thr eshold value, we carry ou t the ( R + 1) th read er session. For the general case of R > 2 rea der sessions, we propo se two classes of estimators. One class of estimato rs is em erging from the generalization of Eqn. (1). The oth er class of esti- mators is ob tained by extending a classical capture –recaptur e result by Schnabel [14], in o rder to be able to e stimate the error probability p . These estimator classes are the major contributions of this paper, along with the overall idea of sequential decision pr ocess in dealin g with the missing tag problem . I I I . S Y S T E M M O D E L The system considered consists of multiple readers and tags. Each reade r can hav e multiple reader sessions, d efined as a session in which the reader r uns its arbitration pr otocol, tryin g to r esolve the entire tag set. The outcome of a reader session is a set specifyin g the tags resolved by a read er in a reader session. Th e sets are assumed to contain n o error s, meaning that if a tag successfully backscatters a signal to the r eader without c ollision, then th e tag is p resent in a set an d th e tag is resolved. Th e r eader sessions are assumed to be coor dinated in a way that the read ers do not interfere with each other i.e. the r eader collision p roblem [15] do es not occur . Throu ghout the pap er we assume independ ent reader ses- sions, except in Sectio n VII-A, where we intro duce the corre - lated model for ev aluation of de pendent reader sessions. In a giv en session, each tag will, with probability p , be in a blind spot, i.e. no t bein g ab le to commun icate with the reader . The complete tag set S c ontains N tags and re mains un changed throug h the rea der sessions. The pr obability of error (blind spot) p is iden tical fo r each tag in each reader session. That is, for a giv en read er session and a given tag , the tag is made unread able with pro bability p , ind ependen tly of the other tag s and previous reader sessions. W e assum e indep endence acr oss the tags : the event whether a tag τ m is readab le d oes not depend on the event whether another tag τ l is readab le. On the other hand, we intro duce correlation by defin ing con ditional proba bilities that the tag τ m is read able in reader session r i +1 provided that the sam e tag τ m was readable/not readable in r i . The condition al probab ilities in the corr elated case is selec ted such that the expected number of non–reada ble tags in th e reader session r i +1 remains N p . This is ph ysically plausible, as we sh ould not be able to imp rove th e overall r eadability o f the tags set throug h a ran dom ph ysical displacem ent. I V . P R O P O S E D S O L U T I O N Four rando m variables, K 1 , K 2 a , K 2 b and K 3 , follow the multinomia l d istribution, and describes the number of tags in the sets k 1 , k 2 a , k 2 b , and k 3 , respec ti vely , see Fig. 1 . The probab ility of a tag be ing read in the first reader session is (1 − p ) , and in two rea der session s (1 − p ) 2 . The p robability of a tag being read in the first but not in th e next (and vice versa) is (1 − p ) p , and not read at all is p 2 . This gives the probab ility ma ss function (pm f) Pr[ K 1 = k 1 , K 2 a = k 2 a , K 2 b = k 2 b , K 3 = k 3 ] =  N k 1 , k 2 a , k 2 b , k 3  (1 − p ) 2 k 1 ((1 − p ) p ) k 2 a + k 2 b p 2 k 3 . Lets define one more random variable, K 2 , being the sum of K 2 a and K 2 b , then, assumin g that th ey are in depende nt g i ves the pmf Pr[ K 1 = k 1 , K 2 = k 2 , K 3 = k 3 ] =  N k 1 , k 2 , k 3  (1 − p ) 2 k 1 (2(1 − p ) p ) k 2 p 2 k 3 . The expected values of the rando m variables are E [ K 1 ] = N (1 − p ) 2 E [ K 2 ] = E [ K 2 a ] + E [ K 2 b ] = 2 N (1 − p ) p E [ K 3 ] = N p 2 When measur ed v alues of k 1 and k 2 are fou nd, and by assuming tha t they are close to th eir respec ti ve e xpected value, we assume the fo llowing ap proxima tion k 1 = ˆ N (1 − ˆ p ) 2 ≈ E [ K 1 ] k 2 = 2 ˆ N (1 − ˆ p ) ˆ p ≈ E [ K 2 ] k 3 = ˆ N ˆ p 2 ≈ E [ K 3 ] (2) Based o n this, an estimate o f ˆ p can be fo und, by taking a ratio based on th e set relatio nship, na mely k 1 k 2 = ˆ N (1 − ˆ p ) 2 2 ˆ N (1 − ˆ p ) ˆ p ⇒ ˆ p = k 2 2 k 1 + k 2 . (3) Note th at the ( unknown) tag set c ardinality ˆ N is c ancelled out. Using this estimator an estimate of N can b e foun d, ba sed on the fact that E [ K 3 ] = N p 2 ≈ ˆ k 3 = ˆ N ˆ p 2 and ˆ N = k 1 + k 2 + ˆ k 3 . This yields an estimate of N for two rea der sessions k 1 + k 2 = ˆ N − ˆ k 3 = ˆ N (1 − ˆ p 2 ) ⇔ ˆ N = k 1 + k 2 1 − ˆ p 2 , (4) where ˆ p is giv en by k 1 and k 2 in Eqn. (3). When estimates of p an d N have bee n obtained, the probab ility o f missing one or more tags can b e calcu lated. As the probab ility of missing one tag in o ne r eader session is ˆ p , the p robability of not missing ˆ N tags in two reader sessions is (1 − ˆ p 2 ) ˆ N . This gives the estimate of the p robability of missing at least one tag as ˆ p M = 1 − (1 − ˆ p 2 ) ˆ N . If this p robability is large, it is likely that tag s a re left un read. It is possible to impr ove the e stimates b y makin g more than two reader sessions. T his is descr ibed in the following section . V . G E N E R A L I Z A T I O N T O M U LT I P L E R E A D E R S E S S I O N S T o provid e better estimates, the two-reader session c ase is extended to suppo rt m ore indep endent reader sessions. Th e observable sets k 1 and k 2 are extended b y de fining a vector, ¯ k = [ k 1 , . . . , k R ] T , which ho lds inform ation about how ma ny tags wer e fo und in how m any reader sessions. The first en try specify the numb er of tags fo und in R reader sessions, the second the num ber of tags fo und in R − 1 reader sessions (regardless o f w hich reader sessions) and so on. Each elemen t in ¯ k is defined by extend ing Eqn . (2) to: k i = ˆ N  R R − ( i − 1)  (1 − ˆ p ) R − ( i − 1) ˆ p i − 1 ≈ E [ K i ] , (5) where i = { 1 , 2 , ..., R } . Howev er , when extending to m ore than two reade r sessions, th ere ar e more re lationships between the sets. In the two-reader session case th e measur able sets ar e k 1 and k 2 , and the ratio k 1 /k 2 is used (see Eqn. (3)), b ut others exist, namely k 1 / ( k 1 + k 2 ) a nd k 2 / ( k 1 + k 2 ) wh ich lead to the same estimator fo r p . When th e numb er o f re ader sessions increases, then the numb er of measu rable sets and the numb er of p ossible ratios in creases, e.g . f or thr ee reader session s, the sets a re k 1 , k 2 and k 3 , and po ssible ratios are k 1 /k 2 , k 1 /k 3 , k 2 /k 3 , k 1 / ( k 2 + k 3 ) , k 2 / ( k 1 + k 2 ) , etc. Th erefore we do not hav e one go od ratio with equally weig hted sets, and common to almost all of the ratios is that an e xplicit e xpression for ˆ p do es not exist, and ˆ p needs there fore to be c alculated numerically . Before explaining s ome of the po ssible estimators of p , the estimator of N and the m ethod of c alculating the probab ility o f missing on e or more tags are explained. The estimator of N for R > 2 read er sessions is based on the estimator of the tag set cardinality from [8]. The percentag e of r esolved tag s is (1 − p R ) , a nd the num ber of re solved tags is the sum of k i s, th erefore the ta g set cardin ality can be generalized to ˆ N = P R i =1 k i 1 − ˆ p R The estimate o f the pr obability of missing at least one tag is extended to: ˆ p M = 1 − (1 − ˆ p R ) ˆ N . (6) This estimator is useful if an application r equires that the probab ility of on e or more missing tags shall be lower than some threshold, t 1 (e.g. t 1 = 10 − 5 ). If ˆ p M > t 1 another read er session is re quired. A new ˆ p M is estimated for each reader session until th e threshold set by the application is satisfied. As this estimated err or , ˆ p M , is b ased on estimates of p and N , it relies on these being “ good”. Therefore it can be necessary to either 1) add an artificial bias to ˆ p M , or 2) perform an extra reading af ter the criteria is satisfied. This is because ˆ p M could be lower than t 1 in some cases where it should not, as ˆ p could be underestimated . Where both the estimates o f N and p M are straightfor ward to comp ute given ˆ p , ˆ p itself is not easy to compute directly , because the estimate is fo und based o n a ratio of sums of elements in ¯ k , and the perfor mance o f the estimator de pends on choosing a go od ratio . V I . E R RO R P RO BA B I L I T Y E S T I M A T O R S An estimator of the error proba bility is define d by which elements from ¯ k are included in the ratio’ s nume rator and denomin ator respe cti vely . T wo window functio ns , φ n ( k i ) and φ d ( k i ) , are u sed to describ e wh ich elemen ts are include d. Th e ratio is then d efined as: P R i =1 φ n ( i ) k i P R i =1 φ d ( i ) k i = P R i =1 φ n ( i )  R R − ( i − 1)  (1 − ˆ p ) R − ( i − 1) ˆ p i − 1 P R i =1 φ d ( i )  R R − ( i − 1)  (1 − ˆ p ) R − ( i − 1) ˆ p i − 1 , (7) where Eqn. (5 ) is inser ted can celling ou t ˆ N . An examp le is φ n ( i ) = 1 , φ d ( i ) =  1 if i = { 2 , . . . , R } , 0 otherwise , which for two reade r sessions r esults in th e ratio ( k 1 + k 2 ) /k 2 . The estimato rs o f p p roposed here ar e defined b y their window functions. As the nu mber of re ader sessions incre ases, it beco mes more likely that elements in ¯ k be comes zero. These elements do not provide any informatio n, and are therefor e excluded by setting φ n ( i ) = φ d ( i ) = 0 wh en k i = 0 . This is u sed in the num erator window functio n for both prop osed estimators of p : φ n ( i ) =  1 if k i 6 = 0 , 0 otherwise . The difference in the e stimators is then the den ominator window f unction. Estimator 1: Remove Maximum Element The first estimato r of p is based on the simple prin ciple of removing the largest entry of ¯ k and a ll zero elemen ts in the denomin ator . This gives the win dow functio n: φ d ( i ) =  1 if k i 6 = 0 and k i 6 = max ¯ k , 0 otherwise. This estimato r is called the Rem ove Maximum Element (RME) Estimator . Estimator 2 : Remove Elemen ts Gr eater than th e Mean For the seco nd estimator of p an averaged version of ¯ k is used. As is shown in Fig . 1, the two subsets k 2 a and k 2 b are a dded tog ether into k 2 . Instead o f u sing th is sum, a new vector is defined , ¯ k ′ , con taining estimates o f these sub sets. The estimate of th e subset is th e average o f the entries in ¯ k , with regard to the number of sub sets per en try in ¯ k . This is defined a s: ¯ k ′ = " k 1  R R  , k 2  R R − 1  , . . . , k R  R 1  # T . The second estimator of p is nam ed th e Remove Elements Greater than the Mean (REGM) Estimato r . The den ominator window f unction is φ d ( i ) =  1 if k ′ i 6 = 0 and k ′ i < m ¯ k ′ , 0 otherwise , where m ¯ k ′ is the sample mean of the no nzero elemen ts in ¯ k ′ . This e stimator r emoves all no nzero elemen ts and all elemen ts greater th an m ¯ k ′ . The S chnabel Estimato r W e pro pose to use the simple capture–re capture mo del, which provides an estimate of N . When the reade r sessions are assumed to be indepen dent, and as th e tag s ar e assumed to be in a closed p opulation , the tag card inality estimation can b e assumed to be a simple captu re–recap ture experimen t. When the nu mber of r eader sessions, R , is two, the Linco ln-Peterson method provides a maximu m likelihood estimate [16], wher e the tag set card inality is fou nd as ˆ N LP = n 1 n 2 m 2 , where n 1 is the numb er of tags found in the first reader session, n 2 is the n umber of tags fou nd in the second , an d m 2 is the number of re-found tags in the second reader session. F or more than two reader sessions, the Sch nabel meth od from [14] can be used , which is a weighted av erage over a series o f Linco ln- Peterson estimates ˆ N S = P R i n i M i P R i m i , (8) where M i is the tota l n umber of tags fo und in the ( i − 1) th reader session. N ote th at the two equation s are equal f or R = 2 , as M 1 = m 1 = 0 and M 2 = n 1 . The method do es not m ake an in termediate estimate of p , but finds an estimate o f N d irectly . T o c ompare them and to make an estimate o f the erro r pro bability p M , an estimator for p , ˆ p S , is derived. An estimate of the pro bability of suc cess for the i th reader session is n i ˆ N , and the estimator is fou nd, b y av eraging over the err ors, ˆ p S = 1 R R X i =1  1 − n i ˆ N S  , which is th e sample mean o f the error pr obabilities found in all read er sessions. This is u sed for compar ison and for calculation of ˆ p M as with th e oth er estima tors. V I I . A N A LY T I C A L E V A L U A T I O N The analy tical work is ma de for two re ader sessions, as then an explicit estimate of p can be foun d. The estimator fo r p is a function of the observations k 1 and k 2 , denoted g ( k 1 , k 2 ) : g ( k 1 , k 2 ) = ˆ p =    1 if k 1 = 0 and k 2 = 0 , 0 if k 1 > 0 and k 2 = 0 , k 2 2 k 1 + k 2 otherwise, (9) which fo llows from Eqn . (3), but with two special cases where either no tags ar e found o r all tags ar e found in both r eader sessions. Its exp ected value is giv en in the f ollowing Lemma. Lemma 1 Let the estimate o f p be d efined as in Eqn. (9), then the expected v alue of ˆ p for known N an d p is E [ g ( k 1 , k 2 ) | N , p ] = 2 N ( p − p 2 N ) 2 N − 1 + p 2 N . Pr oo f: See App endix fo r the proo f. The above result shows that the estimator is bia sed, but as N incr eases an d p de creases, then the bias can b e neglected. The bias can in princip le be r emoved, as it arises due to the definition of the estimator in the marginal cases. A ppropr iate choices of th e marginal cases can make it u nbiased. The lower limit f or N is, if th e expected error made is allowed to be e.g. 1 % and p ≤ 0 . 9 , E [ g ( k 1 , k 2 ) | N ≥ 46 , p ≤ 0 . 9] − p < 0 . 01 , that is, if the m aximum assumed error p robability is p = 0 . 9 , then the m inimum num ber of tags should be N = 46 to satisfy the error requirement. The estimate of N is shown to be un biased in th e following. Lemma 2 Let the estimate of t he tag set car dinality be defined as in Eq n. (4), then, fo r known N a nd p , the estimate of N is unbiased , that is E [ ˆ N | N , p ] = N . Pr oo f: Fro m Eq n. (4) it follows that E [ ˆ N | N , p ] = X k 1 ,k 2 k 1 + k 2 1 − p 2 Pr[ K 1 = k 1 , K 2 = k 2 ] , and by in serting the multino mial distribution, and th e proba - bilities for e ach set: E [ ˆ N | N , p ] = N X k 1 =0 N − k 1 X k 2 =0 k 1 + k 2 1 − p 2  N k 1 , k 2 , N − k 1 − k 2  · (1 − p ) 2 k 1 (2(1 − p ) p ) k 2 p 2( N − k 1 − k 2 ) . This can be split into two sums, an d by th e expectatio n o f a multinomia l d istribution: E [ ˆ N | N , p ] = 1 1 − p 2 ( E [ K 1 ] + E [ K 2 ]) = N 1 − p 2  (1 − p ) 2 + 2(1 − p ) p  = N . This result ensures, given a good estimate of the error p roba- bility , that the tag set estimator p roduces an unb iased tag set cardinality estimate. For the m ethod to work, the tag sets f ound in each reader session h av e to b e ind ependen t, as shown in e.g . the examp les in Sectio n I. T o in vestigate what happ ens if th e read er sessions are dependen t, the estimators are tested in scenario s with depend ent reader sessions. Th e following section e xplains how the dep endency is mo delled, u sing a co rrelation c oefficient to specify the co rrelation b etween read er sessions. A. Model for Dependent Reader Sessions So far it has been assumed that the reade r sessions ar e indepen dent, but wh at if this does n ot h old? In th e fo llowing, a m ethod is introduc ed to de fine the correlation for tag τ m between the reade r sessions r i and r i +1 . For two reader sessions, de fine the Bernoulli r andom variable X 1 signifying the outcome of one tag in the first re ader session, and X 2 the outcome in the secon d re ader session, then X 1 =  1 w .p. p , 0 w .p. 1 − p , X 2 =  1 w .p. p q + (1 − p ) r , 0 otherwise, where p is the pro bability of a tag not being read in the first reader session, q is the probab ility that it is not rea d in the second r eader session e ither , and r is the pro bability of a tag not bein g read in the seco nd, but in th e first. This gives the relations: Pr[ X 1 = 1] = p Pr[ X 1 = 0] = 1 − p Pr[ X 2 = 1 | X 1 = 1] = q Pr[ X 2 = 0 | X 1 = 1] = 1 − q Pr[ X 2 = 1 | X 1 = 0] = r Pr[ X 2 = 0 | X 1 = 0] = 1 − r It is assumed tha t the expec ted error pr obability remain s the same b etween reader sessions, b ecause of the ra ndom p hysical displacement of the tags. Theref ore E [ X 1 ] ≡ E [ X 2 ] , an d pq + (1 − p ) r = p , where r and q for ms the bo und r < p < q because an erro r in th e first reader session incre ases th e pr obability of error in the seco nd. T o specify the level of correlation , the correlation coefficient is used, tha t is ρ = Cov ( X 1 , X 2 ) σ X 1 σ X 2 = q − p 1 − p , where 0 ≤ ρ ≤ 1 . This yields the correlated probab ilities q and r with respect to p an d ρ as q = ρ (1 − p ) + p, r = p (1 − q ) 1 − p . (10) This is u sed to show how th e pr esented app roach to solve the missing tag problem is affected if the reader sessions are not indepen dent. Th e results are shown in the following section. V I I I . S I M U L A T I O N E V A L U A T I O N T o ev aluate the estimators against eac h o ther , and to a ssert that th ey perform as expected, simulatio ns have been car ried out. Th e true n umber of tags is set to N = 500 an d each result is a veraged over 10 00 experime nts. A. Indep endent Reader Sessions N umber of r eader ses s ions ( R ) Estim ate d err or pr obabili ty ( ˆ p ) 2 4 6 8 10 12 14 16 18 20 0.19 0.2 0.21 0.22 0.23 0.24 0.25 0.26 RME Estimator REGM Estimator Schnabel Fig. 2. Si mulated estimation of p vs. number of reader sessions for p = 0 . 2 . The r esults of the estimate of p a re shown in Fig. 2 . It shows that th e RME E stimator is not perfo rming as good as the other s. This is beca use the maximum element that is rem oved m ay con tain almost all the tags and there by all the inform ation. By rem oving it, th e estimator makes a bad estimate. The pro blem decrea ses, as th e number of reader sessions increase as th e tags are sprea d out in more sets. Because of the fluctuations fo r the RME Estimato r in its estimate of p , it is n ot co nsidered further and is not inclu ded in any of the following figures. Num be r o f rea de r sess io ns ( R ) Esti m a ted numbe r o f ta g s ( ˆ N ) 2 4 6 8 10 12 14 16 18 20 9995 9996 9997 9998 9999 10000 10001 10002 10003 10004 10005 REGM Estimator Schnabel Number of resolved tags Fig. 3. Simulate d estimation of N vs. number of reader sessions for p = 0 . 2 . N umber of r eader ses s ions ( R ) MSE (( ˆ N − N ) 2 ) 2 4 6 8 10 12 14 16 18 20 10 −20 10 −15 10 −10 10 −5 10 0 10 5 REGM Estimator Schnabel Fig. 4. Simulated MSE of N vs. number of reader sessions for p = 0 . 2 . The tag set cardinality is estimated in Fig. 3. The estimate giv en by the two estimators is similar, but the REGM Esti- mator con verges faster to the true number of tags. This can be seen in Fig. 4, wher e the mean-sq uare err or of N is given, showing that the Schnab el Estimator co n verges to zero mo re slowly than the REGM E stimator . N umber of r eader ses s ions ( R ) Estim ate d probabil ity of mi ss i ng one tag ( ˆ p 1 ) 10 − 5 p = 0 . 1 p = 0 . 2 2 4 6 8 10 12 14 16 18 20 10 −12 10 −10 10 −8 10 −6 10 −4 10 −2 10 0 REGM Estimator Schnabel True p 1 Fig. 5. Simulat ed estimation of p M vs. number of rea der sessions for p = 0 . 1 and p = 0 . 2 . An example threshold is at 10 − 5 . The estimate of p M is the most imp ortant estimate, as it shows how many r eader sessions are need ed to b e certain, with hig h probab ility , that all tags are resolved. Results are in Fig. 5 f or p = 0 . 1 a nd p = 0 . 2 . It can be seen that bo th estimators are close to the tru e p M calculated using Eqn . (6) using true p an d N as if they were known a prior i. Th erefore, if th e er ror pr obability is p = 0 . 1 , then the sequ ential decision process determines to stop after R = 8 reader sessions, and for p = 0 . 2 it is R = 1 2 , if the allowed thresho ld is 1 0 − 5 . The p = 0 . 2 case can be comp ared with Fig . 3, wher e it is seen, that a ll tag s a re found in app roximately 8 reader sessions. B. Depend ent Read er Sessions In th e following the estimators are tested in scenarios where the ind ependen ce assumption does not hold. For the simulations it is cho sen to use ρ = 0 . 1 and ρ = 0 . 3 , to demonstrate the effect of correlated reader sessions. The correlated error pro babilities are fo und using Eqn. (10), in which the co rrelation co efficient ρ is a par ameter . N umber of r eader ses s ions ( R ) Estim ate d err or pr obabili ty ( ˆ p ) ρ = 0 . 1 ρ = 0 . 3 ρ = 0 . 1 ρ = 0 . 3 2 4 6 8 10 12 14 16 18 20 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 REGM Estimator Schnabel Fig. 6. Simulate d estimati on of p vs. number of reader sessions with correla tion between the re ader sessions. p = 0 . 2 and t he cor relati on coeffici ent is ρ = 0 . 1 and ρ = 0 . 3 . The e stimated error p robabilities are shown in Fig. 6, where it can be seen , that the estimators are affected by the correlated reader sessions. The Schnabel Estimator co n verges to the co rrect error pr obability , where the REGM Estimato r conv erges to som e o ther error probability , d epending on the correlation . N umber of r eader ses s ions ( R ) Estim ate d number of tag s ( ˆ N ) 2 4 6 8 10 12 14 16 18 20 9300 9400 9500 9600 9700 9800 9900 10000 10100 REGM Estimator Schnabel Number of resolved tags Fig. 7. Simulate d estimation of N vs. number of reader sessions with correla tion between the re ader sessions. p = 0 . 2 and t he cor relati on coeffici ent is ρ = 0 . 1 for the upper value s, and ρ = 0 . 3 for the lower va lues. Even th ough the error pro bability estimates for the REGM Estimator conver ges to wro ng values of ˆ p , it p erforms better than the Sch nabel Estimator whe n estimating the tag set cardinality . This is seen in Fig. 7, wher e the REGM Estimato r never p rovides an estimate lo wer then the actual n umber of resolved tag s. Both estimato rs c on verges slower to wards the true N , b ecause of the co rrelation between the r eader sessions. N umber of r eader ses s ions ( R ) Estim ate d probabil ity of mi ss i ng one tag ( ˆ p 1 ) 10 − 5 p = 0 . 1 p = 0 . 2 2 4 6 8 10 12 14 16 18 20 10 −12 10 −10 10 −8 10 −6 10 −4 10 −2 10 0 REGM Estimator Schnabel True p 1 Fig. 8. Simulate d estimation of p M vs. number of reader sessions with correla tion betwee n the reader sessions. p = 0 . 1 and p = 0 . 2 , and the correla tion coef ficient is ρ = 0 . 3 . An exampl e threshold is at 10 − 5 . The estimate o f the proba bility of co mpletely missing one tag is sho wn in Fig. 8, where it can be seen, that the correlation affects the p erforman ce of th e estimate. Therefore if the estimator is used as is, the estimate is wrong. Th e ideas fo r a so lution to this, proposed in Section V, is to make some estimation margin, e.g. two ad ditional reader sessions, so th at more reader sessions th an strictly necessary is u sed, to be certain the pro bability of missing one o r mo re tag s is below the c hosen threshold. I X . C O N C L U S I O N In this p aper two different me thods for obtainin g th e error probab ility estimate and the tag set ca rdinality estimate are propo sed. The first method, named the REGM Estimator, is based on the assumption that it is possible to obtain statistically indepen dent, un correlated reader sessions. First this estimator is intr oduced and explain ed with two read er sessions, after which it is extende d to the general case. Then a method is devised to calculate the n umber o f r equired re ader sessions to g uarantee, with some proba bility , that no tag s a re missing . The secon d estimator is based on the Schn abel method, kn own from cap ture–recap ture liter ature, wh ich is extended to also provide estimates of the error p robability and th e pr obability that tag s are missing. It is shown that th e REGM Estimator for the error pr obabil- ity for two read er sessions is biased, but that the bias be comes insignificant wh en the number o f tag s increases and the err or probab ility decr eases. Also, it is shown that the estimate of the tag set ca rdinality is u nbiased in th e case of two read er sessions. For the estimators to work it is impor tant that th e assumption of independen t, uncorrelated reader sessions holds. T o show how the e stimators b ehave when the read er sessions are correlated, a mo del is devised fo r u se in the simulations. Simulations are perfo rmed, which show th at th e tag set cardinality e stimator using th e estimated error pro bability from the REGM Estimator converges towards the c orrect value faster th an th e Schn abel Estimator . Th ey also show that more read er sessions d ecreases the probab ility of a missing tag, indicatin g that the p roposed metho d for estimating the probab ility o f missing a tag is working. Exp eriments with depend ent reader sessions show that the estimation of th e tag set cardinality requ ires more rea der sessions to be precise, but that th e REGM E stimator’ s estimate of th e tag set cardinality still con verges faster than the o ne based on the Schnabel method. Howe ver both estimator s und erestimate the probabil- ity of missing on e or more tag s, resulting in a p ossibility o f prematur e term ination o f th e algorith m. T o co unter this, some estimation margin shou ld be used when the re ader sessions are depend ent, and the an alysis o f this margin will be investigated in further work. Another interesting venue for future work is to in vestigate the cases whe n the reading er rors h av e corre lations across the tags in th e same reader session. The futu re work should in clude ev aluation o f the proposed metho ds by using more detailed physical mod els for the tag reading e rrors. R E F E R E N C E S [1] R. Angel es, “RFID T echnolo gies: Supply-Chain Applications and Im- plementa tion Issues, ” Information Systems Manage ment , vol. 22, no. 1, pp. 51–65, 2005. [2] A. Juels, D. Molnar , and D. W agne r , “Security and Pri v acy Iss ues in E - passports, ” Security and Privacy for Emerging Are as in Communicat ions Network s, 2005. SecureC omm 2005. First Internatio nal Confere nce on , pp. 74–88, Sept. 2005. [3] D. Molnar and D. 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IEE E 10th Internatio nal Symposium on , pp. 18–22, Aug. 2008. [14] Z. E. Schnabel, “The Estimation of T otal Fish Population of a Lake , ” The A merican Mathemati cal Monthly , vol. 45, no. 6, pp. 348–352, 1938. [15] D. Engels and S. Sarma, “The Reader Collision P roblem, ” Systems, Man and Cybernetics, 2002 IEEE International Confer ence on , vol. 3, p. 6 pp., Oct. 2002. [16] P . Y ip, “A Martingale Estimating Equation for a Capture-Rec apture Experiment in Discrete Time, ” Biometrics , vol. 47, no. 3, pp. 1081– 1088, 1991. A P P E N D I X The following is the pro of of Lemma 1. Pr oo f: The estimator is defined as in Eqn. (9) and the expected value E [ g ( K 1 , K 2 ) | N , p ] is E [ g ( K 1 , K 2 )] = X k 1 ,k 2 g ( k 1 , k 2 ) Pr[ K 1 = k 1 , K 2 = k 2 ] = N X k 1 =0 N − k 1 X k 2 =1 k 2 2 k 1 + k 2 Pr[ K 1 = k 1 , K 2 = k 2 ]+ Pr[ K 1 = 0 , K 2 = 0] . W e in sert th e multinomia l distribution w ith the prob abilities for each set, E [ g ( K 1 , K 2 )] = N X k 1 =0 N − k 1 X k 2 =1 k 2 2 k 1 + k 2  N k 1 , k 2 , N − k 1 − k 2  · (1 − p ) 2 k 1 (2(1 − p ) p ) k 2 p 2( N − k 1 − k 2 ) + p 2 N = p 2 N N X k 1 =0 N − k 1 X k 2 =1 2 k 2 k 2 2 k 1 + k 2 ·  N k 1 , k 2 , N − k 1 − k 2   1 − p p  2 k 1 + k 2 + p 2 N . W e defin e a function h , which is all but the two p 2 N , and we differentiate it w ith respe ct to p , dh dp = N X k 1 =0 N − k 1 X k 2 =0 2 k 2 k 2  N k 1 , k 2 , N − k 1 − k 2  ·  1 − p p  2 k 1 + k 2 − 1  − 1 p 2  = − 1 (1 − p ) p 2 N +1 · N X k 1 =0 N − k 1 X k 2 =0 k 2  N k 1 , k 2 , N − k 1 − k 2  · (1 − p ) 2 k 1 (2(1 − p ) p ) k 2 p 2( N − k 1 − k 2 ) = − 1 (1 − p ) p 2 N +1 E [ K 2 ] = − 1 (1 − p ) p 2 N +1 N 2(1 − p ) p = − N 2 p 2 N . This fu nction is integrated and merged with the part no t differentiated, this giv es h = Z − 2 N p 2 N d p = 2 N p 1 − 2 N 2 N − 1 + c E [ g ( K 1 , K 2 )] = p 2 N  2 N p 1 − 2 N 2 N − 1 + c  + p 2 N . By inserting k nown p , and solving with respect to the integral coefficient c , c is found to − 2 N 2 N − 1 , and the expected value is E [ g ( K 1 , K 2 )] = p 2 N  2 N ( p 1 − 2 N − 1) 2 N − 1  + p 2 N = 2 N ( p − p 2 N ) 2 N − 1 + p 2 N , which is ap proximately p fo r large N .