**p2ke error 1**, p] = + p2N. between the reader sessions ri and ri+1. For two reader 2N − 1 sessions, define the Bernoulli random variable X1 signifying Proof: See Appendix for the proof. the outcome of one tag in the first reader session,

*p2ke error 1*,

**p2ke error 1**, and X2 the The above result shows that the estimator is biased, but as outcome in the second reader session, then N increases and

**p2ke error 1**decreases, then the bias can be neglected. The bias can in principle be removed, as it arises due to the 1 w.p,

**p2ke error 1**. p, 1 w.p. pq + (1 − p)r,

*p2ke error 1*X1 = X2 = definition of the estimator in the marginal cases. Appropriate 0 w.p. 1 − p,

**p2ke error 1**0 otherwise, choices of the marginal cases can make it unbiased. where p is the probability of a tag not being read in the first The lower limit for N is, if the expected error made is reader session,

*p2ke error 1*, q is the probability that it is not read in the allowed to be e.g. 1% and p ≤ 0.9,

**p2ke error 1**second reader session either,

*p2ke error 1*, and r is the probability of a tag E[g(k1k2 ) X1 = 1] = 1 − q The estimate of N is shown to be unbiased in the following,

*p2ke error 1*. Pr[X2 = 1 N,

*p2ke error 1*, p] = · ρ= =

*p2ke error 1*, 1 − p2 k1k2N − k1 − k2 σX1 σX2 1−p k1 =0 k2 =0

**p2ke error 1**(1 − p) (2(1 − p)p)k2 p2(N −k1 −k2 ). where 0 ≤ ρ ≤ 1. This yields the correlated probabilities q and r with respect to p and ρ as This can be split into two sums, and by the expectation of a multinomial distribution:

*p2ke error 1*p(1 − q)

**p2ke error 1**q = ρ(1 − p) + p,

**p2ke error 1**,

*p2ke error 1*r=

*p2ke error 1*.

**p2ke error 1**(10) 1 1−p

**P2ke error 1**N,

**p2ke error 1**, p] = (E[K1 ] + E[K2 ]) 1 − p2 This is used to show how the presented approach to solve the N missing tag problem is affected if the reader sessions are not (1 − p)2 + 2(1 − p)p = N.

**p2ke error 1**= 2 1−p independent. The results are shown in the following section.

**p2ke error 1**

*p2ke error 1*

*p2ke error 1*5

*p2ke error 1*10 VIII,

**p2ke error 1**. S IMULATION E VALUATION

*p2ke error 1*REGM Estimator Schnabel To evaluate the estimators against each other, and to assert that they perform as expected,

**p2ke error 1**, simulations have been carried 10

**p2ke error 1**

*p2ke error 1*

**p2ke error 1**0 out,

*p2ke error 1*. The true number of tags is set to Counter strike vp4 terrorist missions = 500 and each result is averaged over 1000 experiments.

*p2ke error 1*smtp error code 553 MSE ((N̂ −N )2 )

*p2ke error 1*−5

*p2ke error 1*10 A. Independent Reader Sessions

**p2ke error 1**

**p2ke error 1**−10

**p2ke error 1**

**p2ke error 1**10 0.26 RME Estimator REGM Estimator 0.25 Schnabel −15 10 Estimated error probability (p̂)

*p2ke error 1*0.24 terrorist takedown 2 patch icq error 1460 −20

*p2ke error 1*10 2 4 6 8 10 12 14 16 18 20 0.23 Number of reader sessions (R)

**p2ke error 1**Fig. 4,

*p2ke error 1*. Simulated MSE of N vs. number of reader sessions for p = 0.2. 0.22

*p2ke error 1*

*p2ke error 1*0.21 The tag set cardinality is estimated in Fig,

*p2ke error 1*,

**p2ke error 1**. 3. The estimate nvop error ae_not_found 0.2 given by the two estimators is similar, but the REGM Esti- mator converges faster to the true number of tags. This can 0.19

**p2ke error 1**

*p2ke error 1*be seen in Fig,

*p2ke error 1*. 4, where the mean-square error of N is given, 2 4 6 8 10 12 14 16 18 20 Number of reader sessions (R) showing that the Schnabel Estimator converges to zero more

**p2ke error 1**slowly than the REGM Estimator. Fig. 2,

**p2ke error 1**. Simulated estimation of p vs. number of reader sessions for p = 0.2,

*p2ke error 1*.

**p2ke error 1**0

**p2ke error 1**The results of the estimate of p are shown in Fig. 2,

*p2ke error 1*. This is because the maximum element that is removed may contain almost all the tags and thereby all

*p2ke error 1*−4 the information,

**p2ke error 1**. By removing it, the estimator makes a bad

**p2ke error 1**ole error 800 referstorange excel 10

*p2ke error 1*10−5 estimate. The problem decreases,

*p2ke error 1*, as the number of reader sessions increase as the tags are spread out in more sets,

*p2ke error 1*.

*p2ke error 1*−6 lotus domino error 2360 10 Because of the fluctuations for the RME Estimator in its p = 0.2 estimate of p,

**p2ke error 1**, it is not considered further and is not included

**p2ke error 1**−8 error code 1073741515 0x80131700

*p2ke error 1*10 p = 0.1 in any of the following figures.

**p2ke error 1**5,

*p2ke error 1*. Simulated estimation of pM vs. number of reader sessions for p = 0.1 Estimated number of tags (N̂ ) and p = 0.2,

**p2ke error 1**. An example threshold is at 10−5. 10001 10000 The estimate of pM is the most important estimate, as it 9999

**p2ke error 1**shows how many reader sessions are needed to be certain, 9998

*p2ke error 1*with high probability,

*p2ke error 1*,

*p2ke error 1*all tags are resolved,

**p2ke error 1**. Results are

*p2ke error 1*in Fig,

**p2ke error 1**. 5 for p = 0.1 and p = 0.2. It can be seen that both 9997

**p2ke error 1**estimators are close to the true pM calculated using Eqn. (6)

*p2ke error 1*9996

*p2ke error 1*sql error 10054 using true p and N as if they were known a priori. Therefore, 9995 if the error probability is p = 0.1,

**p2ke error 1**, then the sequential decision 2 4

*p2ke error 1*6 8 10 12 14 16 18 20 Number of reader sessions (R) process determines to stop after R = 8 reader sessions, and

*p2ke error 1*for p = 0.2 it is R = 12, if the allowed threshold is 10−5. The Fig. 3,

*p2ke error 1*. Simulated estimation of N vs,

*p2ke error 1*. number of reader sessions for p = 0.2. p = 0.2 case can be compared with Fig. 3,

*p2ke error 1*,

*p2ke error 1*, where it is seen,

**p2ke error 1**,

**p2ke error 1**

*p2ke error 1*that all tags are found in approximately 8 reader sessions,

*p2ke error 1*. B. Dependent Reader Sessions

*p2ke error 1*

*p2ke error 1*than the Schnabel Estimator when estimating the tag set In the following the estimators are tested in scenarios cardinality. This is seen in Fig. 7, where the REGM Estimator where the independence assumption does not hold,

*p2ke error 1*. For the never provides an estimate lower then the actual number of simulations it is chosen to use ρ = 0.1 and ρ = 0.3,

**p2ke error 1**,

*p2ke error 1*resolved tags. Both estimators

*p2ke error 1*slower towards the to demonstrate the effect of correlated reader sessions. The true Nbecause of the correlation between the reader sessions. correlated error probabilities are found using Eqn. (10), in 0 10 which the correlation coefficient ρ is a parameter.

*p2ke error 1*

*p2ke error 1*REGM Estimator

**p2ke error 1**

**p2ke error 1**Schnabel

**p2ke error 1**

**p2ke error 1**

*p2ke error 1*True p Estimated probability of missing one tag (p̂1 )

**p2ke error 1**−2 1 0.2 10 ρ = 0.1 0.19 −4

**p2ke error 1**

*p2ke error 1*10

*p2ke error 1*ρ = 0.1 10−5 Estimated error probability (p̂) 0.18

*p2ke error 1*−6 kerio error 1714 10 p = 0.2 0.17 −8 ρ = 0.3 10 0.16 ρ = 0.3

**p2ke error 1**

**p2ke error 1**

**p2ke error 1**p = 0.1

**p2ke error 1**−10

*p2ke error 1*

**p2ke error 1**10 0.15

*p2ke error 1*

*p2ke error 1*−12

*p2ke error 1*

*p2ke error 1*10 0.14

*p2ke error 1*2 4 6 8

*p2ke error 1*10 12 14 16 18 20

**p2ke error 1**

*p2ke error 1*Number of reader sessions (R)

**p2ke error 1**REGM Estimator Schnabel

**p2ke error 1**0.13 Fig. 8. Simulated estimation of pM vs. number of reader sessions with 2 4 6 8 10 12 14 16 18 20 Number of reader sessions (R) correlation between the reader sessions,

**p2ke error 1**. p = 0.1 and p = 0.2, and the

**p2ke error 1**correlation coefficient is ρ = 0.3,

**p2ke error 1**. An example threshold is at 10−5. Fig. 6,

**p2ke error 1**. Simulated estimation of p vs,

*p2ke error 1*. number of reader sessions with correlation between the reader sessions,

**p2ke error 1**. p = 0.2 and the correlation coefficient The estimate of the probability of completely missing one is ρ = 0.1 and ρ = 0.3.

*p2ke error 1*tag is shown in Fig. 8, where it can be seen, that the correlation

**p2ke error 1**

*p2ke error 1*affects the performance of the estimate,

*p2ke error 1*. Therefore if the The estimated error probabilities are shown in Fig. 6,

*p2ke error 1*,

**p2ke error 1**estimator is used

**p2ke error 1**is, the estimate is wrong. The ideas for where it can be seen,

**p2ke error 1**, that the estimators are affected by the

*p2ke error 1*a solution to this, proposed in Section V, is to make some correlated reader sessions. The Schnabel Estimator converges

*p2ke error 1*

**p2ke error 1**estimation margin, e.g,

**p2ke error 1**. two additional reader sessions, so that to the correct error probability,

**p2ke error 1**, where the REGM Estimator adobe flash error #1009 more reader sessions than strictly necessary is used, to be converges to some other error probability, depending on the

*p2ke error 1*certain the probability of missing one or more tags

**p2ke error 1**below correlation.

*p2ke error 1*the chosen threshold. 10100

**p2ke error 1**IX,

**p2ke error 1**. C ONCLUSION

**p2ke error 1**10000 In this paper two different methods for obtaining the error

**p2ke error 1**probability estimate and the tag set cardinality estimate are 9900

*p2ke error 1*

**p2ke error 1**

*p2ke error 1*proposed. The first method, named the REGM Estimator, is Estimated number of tags (N̂ )

**p2ke error 1**9800

**p2ke error 1**

**p2ke error 1**

*p2ke error 1*based on the assumption that it is possible to obtain statistically

**p2ke error 1**independent, uncorrelated reader sessions,

**p2ke error 1**. First this estimator 9700 is introduced and explained with two reader sessions,

*p2ke error 1*, after which it is extended to the general case. Then a method is 9600 devised to calculate the number of required reader sessions

**p2ke error 1**9500

*p2ke error 1*

**p2ke error 1**to guarantee, with some probability, that no tags are missing. The second estimator is based on the Schnabel method, known 9400 REGM Estimator from capture–recapture literature,

**p2ke error 1**, which is extended to also

**p2ke error 1**Schnabel

*p2ke error 1*Number of resolved tags provide estimates of the error probability and the probability 9300 2 4 6 8 10 12 14 16 18 20 that tags are missing. Number of reader sessions (R)

*p2ke error 1*It is shown that the REGM Estimator for the error probabil- Fig. 7. Simulated estimation of N vs,

*p2ke error 1*. number of reader sessions with ity for two reader sessions is biased,

*p2ke error 1*, but that the

**p2ke error 1**becomes correlation between the reader sessions. p = 0.2 and the correlation coefficient insignificant when the number of tags increases and the error is ρ = 0.1 for the upper values, and ρ = 0.3 for the lower values. probability decreases. Also, it is shown that the estimate of

**p2ke error 1**

*p2ke error 1*

**p2ke error 1**the tag set cardinality is unbiased in the case of two reader Even though the error probability estimates for the REGM sessions. For the estimators to work it is important that the Estimator converges to wrong values of p̂, it performs better assumption of independent, uncorrelated reader sessions holds,

**p2ke error 1**. To show how the estimators behave when the reader sessions [15] D,

*p2ke error 1*. Engels and S. Sarma,

**p2ke error 1**, “The Reader Collision Problem,” Systems, Man are correlated, a model is devised for use in the simulations,

**p2ke error 1**. and Cybernetics,

*p2ke error 1*, 2002 IEEE International Conference on, vol. 3,

*p2ke error 1*, p. 6 pp.,

*p2ke error 1*, Oct. 2002. Simulations are performed,

**p2ke error 1**, which show that the tag set [16] P. Yip,

**p2ke error 1**, “A Martingale Estimating Equation for a Capture-Recapture cardinality estimator using the estimated error probability Experiment in Discrete Time,” Biometrics, vol,

*p2ke error 1*,

**p2ke error 1**. 47, no. 3, pp,

*p2ke error 1*. 1081– from the REGM Estimator converges towards the correct

*p2ke error 1*1088, 1991,

*p2ke error 1*. value faster than the Schnabel Estimator,

*p2ke error 1*. They also show that A PPENDIX more reader sessions decreases the probability of a missing tag,

**p2ke error 1**, indicating that the proposed method for estimating the The following is the proof of Lemma 1. probability of missing a tag is working. Experiments with Proof: The estimator is defined as in Eqn. (9) and the dependent reader sessions show that the estimation of the tag expected value E[g(K1Oki 5850 231 fatal error ) N,

*p2ke error 1*, p] = N.

*p2ke error 1*pq + (1 − p)r = p, Proof: From Eqn,

*p2ke error 1*. (4) it follows that X k1 + k2 where r and q forms the bound r < p < q because an error E[N̂ X1 = 1] = q Pr[X2 = 0

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