Joint Property Estimation for Multiple RFID Tag Sets Using Snapshots of Variable Lengths

Transcription

1 Joint Proerty Estimation for Multile RFID Tag Sets Using Snashots of Variable Lengths ABSTRACT Qingjun Xiao Key Laboratory of Comuter Network and Information Integration Southeast University) Ministry of Education, PR China Radio-frequency identification RFID) technology has been widely adoted by real-world industries This aer resents a new alication for distributively deloyed RFID systems, wherein a user chooses multile tag sets at will from different satial or temoral domains, and then connects them by set oerators union, intersection and relative comlement) to form a set exression The user is allowed to query for the cardinality of an arbitrary set exression, which is called the joint roerty of multile sets We focus on the roblem of estimating the joint roerty with bounded error, which has many otential alications One of them is to allow users to check the number of tags in an arbitrary tag flow assing through a distributed RFID system For this roblem, we roose a solution with a novel design that suorts versatile snashot construction: Given the snashots of multile tag sets, although their lengths may be very different, our formulas can estimate their joint roerties, with an accuracy that can be arbitrarily set For the roosed estimator, we formally analyze its bias and variance, and also the otimal settings of rotocol arameters to minimize the time cost of taking a snashot of a tag set The simulation results show that, under redefined accuracy requirement, our solution can reduce time cost by multile folds as comared with existing works named DiffEstm and CCF, which require all tag sets must be encoded into snashots with an equal length Categories and Subject Descritors C24 [Comuter-Communication Networks]: [Distributed Systems]; C4 [Performance of Systems]: [Measurement techniques] Keywords RFID; Cardinality Estimation; Random Hashing INTRODUCTION Over the ast decade, radio-frequency identification RFID) technology has been widely used by industries such as warehouse management, logistical control, and asset tracking in Permission to make digital or hard coies of all or art of this work for ersonal or classroom use is granted without fee rovided that coies are not made or distributed for rofit or commercial advantage and that coies bear this notice and the full citation on the first age Coyrights for comonents of this work owned by others than ACM must be honored Abstracting with credit is ermitted To coy otherwise, or reublish, to ost on servers or to redistribute to lists, requires rior secific ermission and/or a fee Request ermissions from ermissions@acmorg MobiHoc 6, July 4-8, 26, Paderborn, Germany c 26 ACM ISBN /6/7 $5 DOI: htt://dxdoiorg/45/ Shigang Chen Min Chen Deartment of Comuter & Information Science & Engineering University of Florida, Gainesville, FL 326, USA {sgchen, min}@ciseufledu malls, hositals or highways [5] RFID systems can be concetually divided into two arts: RFID tags each carrying an unique ID) which are attached to hysical objects, and RFID readers, which are deloyed at laces of interest to sense the existence of tags, quickly retrieve the tag IDs, or gather the statistical information about a grou ot tags An imortant fundamental functionality of RFID system is called cardinality estimation, which is to count the number of tags in a hysical region [3 5, 7, 4, 5, 8 2] This function can be used to monitor the inventory level of a warehouse, the sales in a retail store or the oularity of a theme ark Counting the number of tags takes much less time than a full system scan that collects all tag IDs This is an imortant feature since RFID systems communicate via low-rate wireless channels and the execution time cost is the key erformance metric in system design In addition to its direct utility, tag estimation can work as a re-rocessing ste that imroves the efficiency of tag identification rocess [6, 3] More imortantly, since it does not collect any tag IDs, the anonymity of tags can be reserved, articularly in scenarios where the arty erforming the oeration such as warehouse or ort authority) does not own the tagged items Motivation The revious work mainly focuses on estimating the cardinality of a tag set within the radio range of a single reader [3 5,7,4,5,9,2], or estimating the union of tag sets near multile readers [3, 4] This aer studies the cardinality estimation roblem in a much more generalized scenario: Multile tag sets can be catured by a distributed multi-reader system at different satial or temoral domains As requested by system users, these tag sets may constitute an arbitrary set exression using the oerations of union ), intersection ) and relative comlement \) We will estimate the cardinality of this user-desired set exression, which is called a joint roerty of multile sets We use two alications to better illustrate the usefulness of this joint roerty estimation roblem Just imagine we are managing a large logistics network, where tagged roducts are shied from one location factory, warehouse, ort, or storage/retail facility) to another Assume the reader deloyed at each location takes eriodic snashots of its local set of tags and kees a series of such snashots over time When the end users want to know the quantity of goods flowing from one location to the other, we are able to address the query by estimating the cardinality of intersection between two snashots from different locations Furthermore, a more comlicated user query could be the quantity of goods traversing a routing ath comrised of multile locations We can address the query by comuting the cardinality of the intersection among two or more tag set snashots In the second alication, imagine that we are monitoring a warehouse for its inventory dynamics over time We want 5

2 to know the amount of goods entering ie, the number of new tags) the warehouse, and the amount leaving ie, the number of dearted tags) between any two reference time oints Suose the warehouse has been deloyed with an RFID reader system to take eriodical snashots about the existing tags Then, a solution for this roblem could be examining the difference between two snashots taken at different time oints, which tells the information about roduct inflow and outflow within the time interval However, a key challenge is that a large warehouse inevitably needs more than one readers to achieve full coverage, and each reader can take a snashot only about its local set of tags Note that the snashots taken by different readers may not be of an identical length When a user queries for the dynamics of such a warehouse, he or she actually wants to know how the union of multile tag sets scanned by readers at different locations) fluctuates over time Such a query will roduce a comlex set exression to comute the cardinality of the set difference of two union tag sets at different time oints Problem From the above two alications, we can abstract the roblem of joint roerty estimation It is to estimate the cardinality of an arbitrary set exression that involves multile tag sets whose number is denoted by k) existing in different temoral or satial domains The rotocol designed to scan each tag set must be time-efficient, and its absolute estimation error must be ket within a redefined bound at a robability above a given threshold There are very limited rior studies on this roblem The differential estimator DiffEstm) [8] and the joint RFID estimation rotocol JREP) [7] can estimate the set exression involving only two tag sets Although the comosite counting framework CCF) [8] rovides a generalized estimator for an arbitrary exression over multile sets, it is designed based on a different, relative error model, resulting in large execution time, with unbounded worst-case time comlexity Moreover, as multile RFID readers are deloyed at different laces to scan their surroundings eriodically, the tag sets they encounter may differ significantly in sizes The biggest tag set can be many times larger than the averagesized tag sets Both the rior solutions DiffEstm and CCF encode each tag set into a data structure whose length is determined by the size of the largest ossible set or even the union of several largest tag sets for union estimation), which causes unnecessarily long rotocol execution time Our Contributions First, for the generalized joint roerty estimation roblem, we roose a solution with a novel design that suorts versatile snashot construction It adots a two-hase rotocol design between a reader and its nearby tags to construct a snashot of the tag set The length of the snashot could be but not necessary) roortional to the size of the tag set, instead of being fixed to a large worstcase value Given the snashots of any k tag sets, although their lengths may be very different, we have derived closedform formulas to estimate the joint roerties of the k sets Second, we analyze the means and variances of the estimated joint roerties comuted from the formulas We rove that the formulas roduce asymtotically unbiased results and they estimate the joint roerties with an absolute error robabilistic) bound that can be set arbitrarily We also derive formulas for determining the otimal system arameters that minimize the execution time of taking snashots, under a given accuracy constraint for joint estimation Third, we erform extensive simulations, and the results show that, by allowing the snashots to have variable lengths, the new solution significantly outerforms DiffEstm [8] and CCF [8], both of which assume a fixed length for snashots Under the same accuracy requirement, our new solution can reduce time cost by over 8% as comared with the revious 2 JOINT PROPERTY ESTIMATION In this section, we formally define the research roblem of joint roerty estimation for multile RFID tag sets Definition of Joint Proerties Suose a distributed RFID system, where tagged objects are moved from one location to another We use S, S 2,, S k to denote the tag sets catured by RFID systems at different locations or time oints They can form an arbitrary set exression as connected by the union ), intersection ) and relative comlement \) oerations The cardinality of such an exression is called a joint roerty of the k tag sets A major difficulty is that the number of ossible set exressions is really huge In order to tame the high comlexity, we start from a small grou of secial exressions We divide the union of all k sets S S 2 S k into subsets that are mutually disjoint They are called elementary subsets, and the number of elementary subsets is merely 2 k As an examle, in Fig, we illustrate the Venn diagram of three RFID tag sets S, S 2, S 3, and we divide their union into 2 3 = 7 elementary subsets Each subset is denoted by N b3 b 2 b, where b 3b 2b is a binary ranging from to that indicates whether the subset is included by S 3, S 2 or S For instance, the subset N is included by S 3 and S 2, but excluded by S In Fig, N is a secial case that corresonds to the tags not included by any sets S, S 2 or S 3 N N N S S 2 S 3 N N N N N Figure : Venn diagram of three tag sets S, S 2, S 3, and illustration of elementary subsets N b3 b 2 b We formalize the concet of elementary subset N bk b 2 b as N bk b 2 b = i k S ) i \ i k ) S i, ) b i b i = where the bit b i indicates whether the elementary subset is included or excluded by the ith set S i It is equivalent to N bk b 2 b = i k S ) i i k c S ) i, 2) b i b i = if alying the rule of relative comlement A \ B = A B c to equation ), where B c is the absolute comlement of B For a shorter notation, we relace N bk b 2 b by N x, where x is a decimal that is equal to the binary value b k b 2b Hence, the definition of elementary subset in 2) becomes N x = i k 2 i x Si) i k 2 i x= Sic), 3) where 2 i x extracts the ith bit from x by bitwise AND There are two boundary cases: N 2 k = S S 2 S k is the intersection of all sets, and N = S c S 2 c S k c = S S 2 S k ) c is the comlement of the union of all sets For an arbitrary elementary subset N x with x < 2 k, we denote its cardinality by n x, and call it a joint roerty 52

3 of the k tag sets S, S 2,, S k If the cardinalities n x x < 2 k ) of all elementary subsets are known, we can derive the cardinality of an arbitrary set exression by summing u the cardinalities of elementary subsets it includes This is because any set exression can be rewritten as the union of several elementary subsets As an examle, let the queried tag set be S 3 S 2 S ) It is equal to S 3 S 2 c S ) S 2 S c ) S 2 S ) ) By alying the distributive law, it becomes S 3 S 2 c S ) S 3 S 2 S c ) S 3 S 2 S ) By the definition of N x in 3), it equals N 5 N 6 N 7 Hence, the queried cardinality S S 2 S 3) is equal to n 5 + n 6 + n 7 The cardinality of such a set exression is called a comosite joint roerty A secial case of comosite joint roerty is the cardinality of N c, which is equal to the union of all elementary subsets N x, x < 2 k We denote such a union cardinality as n c, which is equal to x<2 k n x In a word, we emhasize on deriving the 2 k elementary joint roerties n x, and the comosite joint roerties, whose total number is huge, are left to a secondary osition Probabilistic Estimation In many ractical alications, it often does not require to know the exact value of the joint roerty n x, and an aroximated value nˆ x with desired accuracy is adequate In the following roblem definition, we require the absolute estimation error nˆ x n x to be bounded by a redefined range ±θ at a robability of at least δ, which is called the θ, δ) model Besides n x, we also consider to kee the absolute estimation error of n c bounded by ±θ, which can act as a reresentative of comosite joint roerty Definition Joint Proerty Estimation Problem) For joint roerties n c or n x x < 2 k ), the joint estimation roblem is to find an algorithm for generating estimations ˆn c and nx ˆ They should satisfy the accuracy constraint: P rob{n c θ ˆn c nc + θ} δ P rob{n x θ nˆ x n x + θ} δ, where n c ±θ and n x ±θ are the confidence interval of estimations ˆn c and nx, ˆ resectively, and δ is the confidence level 4) An alternative way of secifying the estimation accuracy is based on a relative error bound ɛ, ): P rob{n c ɛ) ˆn c nc + ɛ)} δ P rob{n x ɛ) nˆ x n x + ɛ)} δ According to this model, the robabilities for the relative errors ˆc nc n n c and nx nx ˆ n x to fall into the range ±ɛ are at least δ This relative error model has been adoted by the revious work [8,8], with a time comlexity of O ln ), where J ɛ 2 J δ is Jaccard similarity the ratio of the intersection size of all tag sets to the union size of all sets However, could be J very large, since the intersection size can be small or even zero while the union size is very large, which may be the routine case in ractice, instead of being rare For the alications in the introduction, there can be two warehouses with few roducts moved between them, or there can be times when few roducts are moved in or out of a warehouse In both cases, is very large or even tends to infinite J In conclusion, the relative error model, as a remanent from the earlier literature on cardinality estimation of a single tag set, is no longer suitable for joint roerty estimation of multile sets Therefore, this aer adots the absolute error model in 4) The revious rotocols named DiffEstm [8] and CCF [8] are not designed under this model 5) 3 ALOHA-BASED RFID PROTOCOL This section introduces a standardized RFID communication rotocol based on slotted ALOHA, which can be used to take a snashot of a tag set without collecting any tag IDs 3 ALOHA Communication Protocol A reader communicates with the tags in its radio range, using the following slotted ALOHA rotocol, which is artially comliant with the EPCglobal RFID standard [] Initially, the reader broadcasts a Query command to start an ALOHA frame with m time slots and using R as random seed Uon hearing the command, a tag selects a time slot in a seudorandom fashion by a hash function Hid R) mod m, where m is the frame length, id is the tag ID, and is bitwise XOR that mixes tag id and random seed R Then, the reader transmits the m time slots one by one At the boundary of each two adjacent slots, it broadcasts a QueryRe command to terminate the current slot and start the next As the slot index grows, the tags whose generated hash values are consistent with the current slot index will send out their resonses By executing the above framed-slotted ALOHA rotocol, from the ersective of the reader, the resonses of all tags distribute uniformly in the frame with m time slots Furthermore, a samling mechanism can be incororated into the ALOHA frame as a field in the frame header Due to the samling, only fraction < ) of tags resond in the frame and the rest of them just kee silent 32 Emty/Busy Time Slots A time slot is said to be emty if it contains no tag relies, or busy if there are at least one tags resonses Busy slots can be further classified into singleton slots containing exactly one tag resonse) and collision slots containing at least two tag resonses) According to the EPCglobal RFID standard [], each tag resonse needs to contain a 6-bits random number in order to differentiate singleton slots from collision ones or 96 bits for reorting a tag ID) By contrast, if it only needs to distinguish emty slots from busy ones, transmitting merely one bit is sufficient for indicating the resence of a tag in the slot, which may reduce by multile folds the average time cost er slot Symbolically, we use to denote emty state and to denote busy state We will design rotocols that only require the RFID reader to differentiate between emty slots and busy ones From the ersective of reader, an ALOHA frame is equivalent to an array of bits recording the / states of the sequence of time slots, which is called a bitma that encodes the current set of tags [5] Afterwards, the three terms tag set snashot, frame and bitma will be used interchangeably We assume that the RFID reader at each location invokes the ALOHA rotocol eriodically to take a snashot of the set of tags currently in its radio range For the tag sets S, S 2,, S k in different satial and temoral domains, let B, B 2,, B k be the bitmas that encode them, resectively 4 A BASELINE SOLUTION In this section, for imlementing the joint roerty estimation, we resent a baseline solution that uses the bitmas B, B 2,, B k This rotocol imlicitly assumes that all these bitmas must have an equal length and use a common samling robability, because it needs to aly the bitwise OR oeration to any subset of these bitmas This baseline rotocol is doomed to have oor time efficiency, which is exlained as follows For a small tag set, if 53

4 the samling robability is very small, too few or even no tag will be samled for the snashot construction Hence, the samling robability has to be reasonably large, as deicted by the first bitma in Fig 2, where 2 tags are recorded with 4% samling robability However, for a large set, a significant samling robability will cause all bits to be set as ones as illustrated by the second bitma in Fig 2), unless the bitma length is sufficiently large see the third bitma of Fig 2) Now because the same large length has to be alied to all bitmas, it becomes a great waste for small tag sets the fourth bitma of Fig 2) Since each bit takes one time slot to determine, a large bitma length imlies a long time for taking a snashot, even for a very small tag set 2 tags, 4% samling 2 tags, 4% samling 2 tags, 4% samling large bitma for a small set Figure 2: Inefficiency roblem of baseline rotocol when handling small tag sets and large tag sets Therefore, the baseline rotocol will waste execution time, if the tag sets it handles dramatically differ in sizes But still we need it to comare with ours when conducting simulation studies So we describe its estimation equations as follows Union Cardinality Estimation For the frames B, B 2,, B k, an imortant roerty is that, if a tag is samled to rely, it will resond in the same time slot among all frames This is because the tag uses the same hash function Hid R) mod m for slot selection in different frames, where m is the common length of all frames and R is the random seed Out of the k bitmas, we can arbitrarily select c bitmas and combine them by bitwise OR, which is denoted as B i B i2 B ic with i < i 2 < i c k Because a tag resonds at the same slot in all these frames, when calculating the bitwise OR, its dulicated resonses in different frames will be filtered automatically Therefore, the OR of c bitmas B i B i2 B ic is equivalent to the bitma encoding of the union of c tag sets S i S i2 S ic [5] For the cardinality of such a union set, a good estimator is to use the fraction of zero bits in the corresonding OR bitma [5] Secifically, let z be the fraction of zero bits in an OR bitma Then, the corresonding union cardinality can be estimated as m logz) /, where m is the number of bits in the OR bitma, and is the samling robability Joint Proerty Estimation Since the cardinality of any union set is known, we can easily derive the joint roerty n 2 k = S S 2 S k, which is the cardinality of the intersection of all the k tag sets According to the wellknown rincile of inclusion and exclusion, n 2 k is equal to S S 2 S k = S i S i S i2 i k + i <i 2 <i 3 k i <i 2 k S i S i2 S i3 + + ) k S S 2 S k, 6) where the union cardinalities S i, S i S i2, S i S i2 S i3,, S S 2 S k all have unbiased estimators as stated Interestingly, 6) can be extended to estimating any joint roerty n x with x < 2 k By the definition of N x in 3), n x = i k 2 i x Si i k 2 i x Si ) i k 2 i x= Si ) The first term i k Si is the intersection of multile 2 i x tag sets and hence can be estimated using 6) The second term is the intersection of i k 2 i x Si and i k Si, 2 i x= where the latter can be treated as a single tag set encoded into the OR bitma i k Bi Hence, the second term 2 i x= can be regarded as the intersection of multile sets with a union set, and hence can also be estimated by 6) Because this baseline rotocol deends on the INClusion- EXClusion rincile in 6) for estimating the intersection of multile sets from their unions, we call it INC-EXC for short 5 ADAPTIVE ESTIMATION PROTOCOL We resents our Joint RFID Estimation Protocol named M-JREP to derive joint roerties of Multile tag sets, even when they are encoded into bitmas of different lengths Naturally, it is desirable to let each bitma have a different length, deending on the cardinality of the tag set it encodes This insires us to develo a new algorithm of combining k bitmas with variable lengths The real difficulty is not at how to combine k bitmas; there are simle ways to combine them The real difficulty comes after the combination how to erform analysis on the information combined from nonuniformly sized snashots, how to use that information for joint roerty estimation, and most imortantly, how to ensure the satisfaction of accuracy requirements in 4) These are the tasks that have not been fulfilled in the literature Our M-JREP rotocol is comrised of two comonents: an online encoding comonent for comressing each tag set into a bitma, and an offline analysis comonent for estimating the joint roerties of multile tag sets, using their bitmabased snashots Whenever a bitma encoding of a tag set is collected by the online hase, the RFID reader offloads it immediately to a central server for long-term storage and for offline rocessing Such an asymmetric design will ush most comlexity to the offline comonent at the server side, while keeing the online comonent at the side of reader and tags for raw data collection) as efficient as ossible We will describe the online comonent in the first subsection, and then the offline comonent in the subsequent subsection 5 Online Encoding of a Tag Set We use a two-hase rotocol to encode tag set S i into a bitma with length roortional to the set size s i The first hase generates an estimation for the number of tags s i with coarse accuracy, and the second hase uses the coarse estimation ŝ i to configure ALOHA frame length and rescan the tag set It in fact is a common ractice for RFID researchers to use such a two-hase rotocol for estimating the cardinality of a single tag set [3], which will be exlained later in section 9 for related works In following, we describe the two-hase rotocol with more details Firstly, the RFID reader that covers the tag set S i invokes an existing rotocol to generate a coarse estimation of the set size s i Because only a single tag set is handled in this hase, the estimation accuracy is secified by the ɛ, δ) model in 5), in which the robability for the relative estimation error to fall within the range ±ɛ is at least δ Since the accuracy is rather coarse, we often configure ɛ = 2% and δ = 5% To attain the goal, many existing rotocols can be alied, such as LoF [4], GMLE [7], PET [2] and ZOE [9] In order to attain the re-defined estimation accuracy of tag set size, the needed number of time slots for scanning the tag set is O logs ɛ max)) log ) for LoF, or 2 δ O log logs ɛ max)) log ) for PET, where 2 smax is an uer δ bound for the size of any tag set, according to a recent survey 54

5 study [3] It is clear that the time exense is roortional to the logarithm or even the log-logarithm of the tag set size Hence, the time cost of the first hase is negligibly small when its accuracy requirement is coarse For instance, when the relative estimation error ɛ is 2% and δ is 5%, the time cost of the LoF algorithm [4] is 32 logs max) Secondly, because the size of tag set has been coarsely estimated by the first hase as ŝ i, the reader can re-scan the tag set by an ALOHA frame B i whose length m i is linearly roortional to ŝ i Then, the frame length m i satisfies m i = min,] { 2 log 2 ŝi ρ ) }, 7) where ρ is the load factor of the frame Note that the frame length m i is exlicitly configured to an integral ower of two Then, for any two frames, the length of the longer frame is always integral multile of the length of the shorter frame Equation 7) can be aroximated as the rimary time cost of the online comonent for tag set encoding Later in section 7, we will rove that an otimized value of the load factor ρ that minimizes 7) while satisfying the re-defined accuracy constraint of joint roerty estimation in 4) is ρ = 2 k max θ 2 k maxs maxz δ 2 + ) 5 ), 8) where θ is the bound of absolute estimation error, Z δ is the δ quantile of standard Gaussian distribution eg, 2 Z 5 96), s max is the uer bound of the cardinality of a tag set, k max is the largest number of tag sets that may involve in any user query, and is the samling robability Since s max, k max, θ and Z δ are all fixed values, 8) can be regarded a function of only one variable Let be the otimal samling robability that maximizes 8), which in turn will minimize the time cost of encoding a tag set in 7) It is clear that the value of only deends on s max, k max, θ and δ Hence, is re-determined for a system once these arameters are set, and the best ρ is also re-determined 52 Offline Estimation of Joint Proerties of Multile Tag Sets In this subsection, we resent an offline analysis algorithm that derives the joint roerties n c and n x, x < 2 k, of k tag sets, using the bitmas B, B 2,, B k Without loss of generality, we assume the bitmas are sorted by their lengths in non-descending order that satisfies m m 2 m k Although all these bitmas are assumed by equation 7) to have the same load factor ρ, our offline algorithm to describe later can in fact work well if each bitma B i has its own load factor ρ i We will rove in Section 6 that, only when ρ = ρ 2 = = ρ k = ρ, can we minimize the rotocol time cost Exanded Bitwise OR We introduce two bitwise oerations which will be used later In the binary reresentation of a value y, let loy) be the location of the lowest-order -bit, and let hiy) be the location of the highest-order -bit For examle, if the binary reresentation of y is, then we have loy) = and hiy) = 7 A boundary case is that y is equal to zero In this case, we define hiy) = and loy) = For the bitmas, we introduce an auxiliary bitma called exanded OR to combine them, which is noted by OR bk b 2 b OR bk b 2 b = i k ExandB i, m hibk b 2 b )) b i The subscrit b k b 2b indicates for each bitma whether it is involved in the exanded OR: If b i is one, then the bitma B i is involved Among the chosen bitmas, the length of the longest bitma is m hibk b 2 b ), because m m 2 m k and hib k b 2b ) is the osition of the highest -bit in binary b k b 2b The function ExandB i, m hibk b 2 b )) increases the length of B i to the largest bitma length m hibk b 2 b ) by self relication, such that all bitmas after exansion have an equal length and can be combined by bitwise OR Figure 3 illustrates an examle of alying exanded OR to three bitmas B, B 2 and B 3 Among them, B 3 is the longest We relicate B for one time and B 2 for three times, such that after exansion, all bitmas are of the same length and the bitwise OR oeration can be used to combine them B B 2 B 3 OR Relicate Bitwise OR of B 3 and exanded B 2 and B Figure 3: Exanded OR of three bitmas B, B 2, B 3, whose lengths are 4, 8, 6, resectively For simlicity, we relace OR bk b 2 b by a shorter notation OR y, where y is a decimal equal to b k b 2b Therefore, OR y = i k 2 i y ExandBi, ) 9) It is always feasible to exand the bitma B i to have the same length with the longest bitma B hiy), because both their lengths m i and are the owers of two by equation 7), and the ratio /m i is definitely an integer Exected Zero Fraction of OR y We analyze the exected fraction of zero bits in the bitma OR y Let us focus on just one bit in OR y, and we have the following roerty Proerty Probability of a Tag Assigning a Bit) Considering an arbitrary bit in bitma OR y and an arbitrary tag from elementary subset N x, when the bitwise AND of x and y is non-zero, the robability of the tag assigning the bit to one is ; when x y is zero, the robability is zero m lox y) Proof Please check out the Aendix A According to Proerty, a tag has the chance to assign a bit of OR y to one, only when it is in a subset N x that satisfies x y Because in such a subset N x, any tag assigns the jth bit of OR y at a robability of, the robability m lox y) that all tags in N x do not assign this bit is ) nx m lox y) Let X y j) be the event that the jth bit in OR y remains zero The occurrence of the event needs all tags in any N x with x y do not assign this bit Thus, its robability is P rob{x y j) } = x y ) x<2 nx ) k m lox y) Let z y be the fraction of bits in OR y that remains zeros: z y = j< j) X y, ) where is the number of bits in bitma OR y see 9)), and j) X is the indicator function of the event X y j), whose y value is one if the event occurs and is zero otherwise Since the zero fraction z y is the arithmetic mean of a large number of indeendent variables, by the central limit theorem, z y aroximates a Gaussian distribution Its exected value is Ez y) = E X j) y ) = j< E X j) y ) 55

6 Clearly, E j) X ) = P rob{x y j) } Hence, alying ), y Ez y) = j< P rob{x y j) } = P rob{x y j) } = x y ) x<2 nx k m lox y) Alying the aroximation m )n m ) m m n which works when m and m are both large, we have Ez y) x y x<2 k ) m = ) x y x<2 k hiy) m lox y) n x m lox y) n x Using the sign function sgn which equals to, or when its inut arameter is ositive, zero or negative), we have Ez y) ) x<2 k sgnx y) m lox y) n x 2) Estimator of Joint Proerty n x Using the fraction of zero bits in OR y with y < 2 k, we are able to estimate each joint roerty n x with x < 2 k We will show later that this essentially is a fully determined linear system, which uts 2 k constraints over 2 k unknown variables By ), we know that the variance of z y is inversely roortional to the number of observations Hence, when the number of slots in the frame OR y is sufficiently large, we can aroximate Ez y) by z y Then, 2) becomes z y ) x<2 k sgnx y) n m x lox y) We will rove later that such an aroximation indeed roduces unbiased estimators Taking the logarithm of both sides, logz y) / log ) m m hiy) x<2k sgnx y) hiy) n m lox y) x Alying the aroximation log ) for large m, m m logz y) m x<2 sgnx y) hiy) n k m lox y) x If we define the measurement of the number of tags in OR y as û y = logz y), 3) the above equation can be simlified as m x<2 sgnx y) hiy) n k m x û lox y) y 4) Alying 3) to each OR y bitma, we can collect a vector of measurements û = [ û, û 2,, û y,, û 2 k ] T If utting together the sizes of all elementary subsets, we can obtain the vector of unknowns: n = [ n, n 2,, n x,, n 2 k ] T With û and n roerly defined, 4) can be rewritten as M n û, 5) where M is the coefficient matrix defined in 6) Its element uses y as row index and x as column index, x, y < 2 k [ ] M = sgnx y) 6) m lox y) For elements of M, if x y is zero, then sgnx y) is zero, m and the exression sgnx y) hiy) is treated as zero m lox y) In Aendix B of the extended version [2], we rove that the coefficient matrix M is non-singular and rovide a recursive formula for calculating M Hence, we can solve the equation system and obtain an estimator of joint roerties ˆn = M û 7) Estimator of Comosite Joint Proerty n c Among the numerous comosite joint roerties, the largest and the most imortant roerty is n c the number of tags in the union of all the sets We estimate it as ˆn c = x<2 nˆ x, k the sum of all elements in the vector ˆn After simlification, ˆn c = û 2 k m2 m m mj+ mj m j û m3 m2 m 2 û 3 û 2 j m k m k m k u 2 k 8) 6 THEORETICAL ANALYSIS In this section, we analyze the bias and variance of the M-JREP estimators It is easy to rove that the estimators nˆ x in 7) and ˆn c in 8) are asymtotically unbiased, due to the rigid rocess by which they are derived We lace the detailed roof of asymtotic unbiasedness in Aendix C of the extended version [2] In following, we focus on analyzing their variances, which determine their estimation errors 6 Probabilistic Distribution of Measurements z y Because the zero fraction z y of bitma OR y is the inut of M-JREP estimator, we need to first analyze its mean and variance By ), the zero ratio z y is the arithmetic mean of indeendent variables When the number of variables is large, according to the central limit theorem, z y aroximates a Gaussian distribution The mean value of z y is given in 2), and can be further simlified as Ez y) e ωy, where ω y is defined below and its hysical meaning is the load factor of OR y ω y = x<2 k sgnx y) n x m lox y) 9) Note that the symbol ω y is different from ρ i, which later will be used to denote the load factor of the frame B i We derive in Aendix D [2] that the covariance of zero ratio z y of OR y and zero ratio z y2 of OR y2 is aroximately Covz y, z y2 ) e ω y y ω y,y 2 ) e ωy +ωy 2 ) minm hiy ),m hiy 2 ) ), 2) where ω y,y 2 is density of common tags of OR y and OR y2 ωy,y 2 = minm hiy ), m hiy2 )) x y n x x y 2 2) m lox y ) m lox y 2 ) 62 Variance of Cardinality Measurements û y We have defined the measurement of the number of tags û y in the OR bitma OR y in 3) We will analyze the covariance of cardinality measurement uˆ y of bitma OR y and cardinality measurement uˆ y2 of bitma OR y2 Cov uˆ y, uˆ y2 ) = Cov m hiy ) logz y ), m hiy 2 ) logz y2 ) ) Proved in Aendix E [2], the above equation aroximates Cov uˆ y, uˆ y2 ) m hiy ) m hiy 2 ) Covz 2 e ωy +ωy 2 ) y, z y2 ) By substituting Covz y, z y2 ) with its aroximation in 2), Cov uˆ y, uˆ y2 ) maxm 2 hiy ), m hiy2 )) e ω y +ω y2 ω y y ) ωy,y 2 ) ), 22) where ω y can be found in 9), and ωy,y 2 is defined in 2) When y = y 2 = y, the above covariance becomes V arû y) V arû y) m 2 hiy) e ω y + 2 )) m 2 n x 23) lox y) x y 56

7 63 Estimation Variance of Joint Proerty The covariance Cov nˆ x, nˆ x2 ) of any two joint roerty estimations nˆ x and nˆ x2, x, x 2 < 2 k, can form a matrix Covˆn, ˆn) According to the roof in Aendix F of [2], Covˆn, ˆn) = M Covû, û) M ) T, 24) where Covû, û) is a covariance matrix whose entry is in 22) In following, we analyze the estimation variance of the comosite joint roerty n c Equation 8) is rewritten as ˆn c = û + m 2 m û + û 3) + m 3 û 3 + û 7) + m j m j m 2 u 2 j + û 2 j ) + m k m k To simlify this equation, we define ˆd j as ˆd j = m j m j u 2 k + û 2 k ) 25) u 2 j + û 2 j, with 2 j k 26) We have roved in Aendix C of [2] that ˆd j is an unbiased estimation of d j = S j \ S S 2 S j ) Alying 26) to 25), we have ˆn c = û + 2 j k ˆd j 27) Then, V ar ˆn c ) = V arû + 2 j k ˆd j) In Aendix G of the extended version [2], we rove that Cov ˆd i, ˆd j) for any i, j values with 2 i < j, and Covû, ˆd i) Hence, V ar ˆn c ) V arû) + 2 j k V ar ˆd j), 28) where V arû y) is given in 23) By definition of ˆd j in 26), V ar ˆd j) = m j 2 m 2 V ar u j 2 j ) + V arû 2 j ) 2 m j m j Cov u 2 j, û 2 j ) By 22), Cov u 2 j, û 2 j ) m j m j V ar u 2 j ) Then, V ar ˆd j) m j 2 m 2 V ar u j 2 j ) + V arû 2 j ) 29) Alying the above equation of V ar ˆd j) to 28), we have V ar ˆn c ) V arû ) + V arû2 j ) m j 2 m 2 V ar u 2 j k j 2 j )) = V arû 2 k ) j<k m j+ 2 mj 2 m j 2 V arû 2 j ) 3) 7 PROTOCOL PARAMETERS In this section, we otimize the arameters of M-JREP rotocol, under the accuracy constraints of joint roerty estimation in 4) There are many system arameters But the size of largest tag sets s max is a henomenon of hysical world, and is beyond the control of RFID system The accuracy model θ, δ) and the number of tag sets k totally deends on the demand of users Hence, only two arameters are controllable and should be otimized, ie, the load factor ρ i of the frame B i and the common samling robability In the first subsection, we investigate the aroriate configuration for the load factor ρ i for frame B i In the second subsection, we study how to otimize the samling robability to minimize the execution time or the size of frame B i) 7 Configuration of Load Factors Equation 4) requires that the robability for the absolute estimation errors of joint roerties n c and n x to fall within ±θ is at least δ We roved before that both nˆ x and ˆn are asymtotically unbiased estimations and they aroximate Gaussian distributions Hence, Eq 4) can be translated to V ar ˆn c ) θ / Z δ) 2 and V ar nˆ x) θ / Z δ ) 2, 3) where Z δ is δ quantile of standard Gaussian distribution 2 Proerty 2 Variance Uer Bounds) The tight uer bound of V ar nˆ x), x < 2 k, and V ar ˆn c ) is V arû 2 k ) V ar nˆ x), V ar ˆd j) V ar ˆn c ) V arû 2 k ) 32) Meanwhile, V arû 2 k ) is tightly uer bounded by V arû 2 k ) 2 s k ρk e i k ρ i + 2 i k ρi)) 33) Proof From 3), V ar ˆn c ) V arû 2 k ) For roof of other arts, see Aendix H of the extended version [2] By the above roerty, V ar ˆn c ) and V ar nx) ˆ are tightly uer bounded by 33) Hence, the two constraints in 3) can be tightened as 2 s k ρk e i k ρ i + 2 i k ρi)) θ / Z δ ) 2, 34) which guarantees that 3) is satisfied even in the worst case In our system design, we shall configure ρ = ρ = ρ 2 = ρ k as a system-wide otimal load factor, which will be exlained at the end of this subsection Then, 34) becomes 2 s kρ e kρ + 2 kρ) ) θ / Z δ ) 2 35) In this subsection, we focus on the otimization of the load factor ρ, and kee the samling robability temorally fixed, whose otimization is ostoned to the next subsection Equation 35) has no exlicit solution for ρ due to the existence of exonential term e kρ Hence, we aly the Taylor series e x + x + x2 + x3 + Ox 4 ) to 35) Then, 2! 3! s kρ + kρ + kρ) 2 + kρ)3 + 2 kρ) ) θ Z 2 δ ρ 2k ks k θ 2 Z δ 2 + ) 5 ) Assume the size of any tag set s i is at most s max, and the number of tag sets k involved in any query is at most k max In the worst case s k = s max and k = k max, we must ensure ρ 2k max θ 2 k maxs max Z 2 + ) 5 ) 36) δ Since ρ = ρ i = s i m i, it is inversely roortional to the frame length m i, which measures the rotocol time cost for encoding the tag set S i Thus, we configure the target load factor as large as ossible under the constraint, and obtain Eq 8) We justify our choice of setting ρ = ρ = ρ 2 = ρ k The left side of 34) is an increasing function in each ρ i If we allow these load factors to be unequal and still set their values to be as small as ossible, then some of them will be greater than the right side of 36) and others will be small Because S, S 2,, S k are arbitrary tag sets under consideration, it means that some tag sets will be encoded with their load factors greater than the right side of 36) and some other will have smaller load factors Let S, S 2,, S k be the k tag sets with load factors greater than the right side of 36) We should be able to erform joint estimation on any k encoded tag sets without violating the accuracy requirement However, if we erform joint estimation on S, S 2,, S k, because their load factors are larger than 36), the constraint of 34) will not hold 72 Otimization of Samling Probability Because ρ = ρ i = s i m i, we have m i = s i ρ Recall that the value of m i must be a ower of two to suort the exanded OR of multile ALOHA frames Hence, m i = 2 log 2 s i ) ρ We want to choose the otimal samling robability that 57

8 minimizes the rotocol execution time by keeing the frame length m i as small as ossible Hence, we have the formula for the frame length defined by section 5-A as 7) and quoted here: m i = min,] {2 log 2 ŝi ) ρ }, where the load factor ρ is determined by 8), and the samling robability is hidden inside 8) and needs to be otimized The otimal that can minimize m i deends on the re-determined arameters s max, k max, θ and δ We can numerically comute from 7) the otimal samling robability that minimizes m i 8 SIMULATION STUDIES We evaluate the erformance of our M-JREP rotocol by simulations The most related work include CCF Comosite Counting Framework [8]), which is based on the relative error model, and DiffEstm Differential Estimator [8]), which is also based on the relative error model and can only handle two tag sets Please check Section 2 for discussion on the absolute error model and the relative error model In this section, we will comare our M-JREP rotocol with CCF and INC-EXC Note that INC-EXC rotocol in Section 4 degrades to DiffEstm [8], when it handles only two tag sets We will consider two erformance metrics First, given the same accuracy requirement defined in 4), we comare the execution times of all the three rotocols For M-JREP, its execution time is measured as the number of time slots it takes the reader to encode a tag set into a bitma, including the frame length m i and other slots needed to give an initial rough estimation of tag set size s i We adot GMLE [7] to generate an initial estimate with a 95% confidence interval of ±2% error The time cost of GMLE hence is aroximately 544 Z 2 5 / slots [7] Second, when the rotocols are subject to the same average execution time, we comare their robabilities of meeting a given error bound ±θ The robability is measured as the number of joint estimations that meet the error bound divided by the total number of joint estimations erformed in the simulation When resenting simulation results, we only show the robability of successfully bounding the estimation error of union cardinality n c, and omit the bounding robability of n x, since by Proerty 2, V ar nˆ x) V ar ˆn c ) and the bounding robability of n x is always larger than n c The system model is a distributed RFID system of multile locations, where each reader eriodically takes a snashot of its local set of tags, whose number ranges from to 5,, with s max = 5, The average cardinality of a tag set is s avg =,, which reflects that the normal business flow of tagged objects is smaller than the worstcase number that the system is designed to handle The size of each tag set will be taken from a Gaussian distribution N, 2 2 ) truncated by the range, s max] For the accuracy requirement, we configure δ = 5% and θ = 8 by default We will erform simulation studies with other values of system arameters δ, θ, s max and s avg as well 8 Protocol Execution Time to Achieve the Same Estimation Accuracy We comare the average time cost of the three rotocols, which are forced to meet the same accuracy constraint A critical arameter that affects rotocol erformance is k max, the largest number of tag sets involved in a query We erform simulations with k max set to the value 2, 4, 6 or 8 Before simulation studies, we exlain how to theoretically configure the arameters of M-JREP, ie, comute the value of load factor ρ from 8) and the otimal samling robability from 7) For different k max values, we show the corresonding ρ and in Table We find that in most circumstances with reasonable high accuracy, the otimal samling robability should be configured close to one in order to avoid samling error Table : Parameter Settings for M-JREP Protocol Number of Sets k max Otimal Samling Probability Theoretical Value of Load factor ρ Emirical Value of Load factor ρ The theoretical values of ρ are set conservatively on the third row of Table ) to guarantee that the accuracy constraint is satisfied even in the worst case Alternatively, their values can be set emirically through simulations for normal situations In our simulation, we first comute the initial value of ρ from 8) and then erform bi-section search to increase it as large as ossible such that the resulting value of m i will still satisfy the accuracy requirement Consequently, on the last row of the above table, when k max equals 2, 4, 6 or 8, the load factor ρ is emirically configured to 39, 68, 35 or 3, resectively It shows that, to suort the user queries that involve more tag sets, the load factor ρ must decrease, which is consistent with equation 8) In the second row of Table 2, we resent the average execution time of M-JREP in simulations using the emirical arameters in Table ) When INC-EXC and CCF realize the same estimation accuracy, their execution times are shown in the third and fourth rows of Table 2, resectively Because they are not designed for absolute error bound, there is no formula to comute their frame length or the number of hash values stored With s max = 5,, we use exhaustive search by simulation to find their minimum time cost that can meet the error bound Table 2 shows that the frame length used by INC-EXC is at least 5% larger than that of M-JREP, and the time cost of CCF is even higher This is because EXC-INC or CCF) has to adot a large frame length or store a large amount of tag hash values) to tolerate the worst case of estimating joint roerties for k max tag sets whose cardinalities range between 45, and 5, This exensive time cost is fixed even when encoding small tag sets whose average size is only about, The time costs of CCF could get even worse than the results shown in the last row of Table 2, if it is alied to another worst scenario of estimating the intersection of multile sets, which is emty Table 2: Comarison of Time Cost Among Protocols Number of Sets k max Time Cost of M-JREP,274 2,72 4,234 77,44 Time Cost of INC-EXC 5,92 24,725 23,2 384,66 Time Cost of CCF for Union 42,244 68,976 38,96 675,94 To give a straightforward imression on the time costs of M-JREP, INC-EXC and CCF, we configure k max to 4 and show their comarison results in Fig 4, where the horizontal axis is the size of a tag set, which varies from to 5,, and the vertical axis is the number of time slots needed or hash values stored for CCF) to take a snashot of the tag set Due to the nature of their designs, INC-EXC uses a constant frame length of 24,725 slots, and CCF uses constant time cost of recording 68,976 hash values The frame length of M-JREP is variable It is small when the tag set is small For examle, for a set of, tags, the number of time slots needed by M-JREP is 2 log 2,/68) +48 = 6, 532, only 3% of what is needed by INC-EXC The average time cost of M-JREP is 2,72 shown by the solid horizontal line 58

9 Protocol Time Cost x 4 ) M-JREP Average Time Cost x 4 Cardinality of a Tag Set M-JREP INC-EXC CCF for Union Figure 4: Protocol execution time under the arameter settings: k max = 4, s max = 5, θ = 8, δ = 5% 82 Estimation Accuracy under the Same Execution Time We comare the estimation accuracy of INC-EXC and M- JREP, when giving them the same execution time, which is configured by the second row of Table 2 Here, we omit the results of CCF for sace, which are worse than INC-EXC When resenting simulation results, a difficulty is that the estimation accuracy is strongly affected by the sizes of tag sets involved and their ways of overlaing It is imossible to resent the simulation results of all the cases, and hence we focus on only two of them The first is an extreme case that deals with k max large sets from 45, to 5,) which are slightly overlaed The second case can be regarded as a normal case that handles two large sets and k max 2 small sets whose sizes randomly distribute between and 5, The simulation results of the extreme case are shown in Table 3, and the results of the normal case is in Table 4 In both tables, our M-JREP rotocol erforms well, because its robability of bounding absolute estimation error with θ is always above δ = 95% In contrast, the accuracy of INC- EXC is non-satisfactory for the normal case in Table 4, and severely degrades when handling the extreme case in Table 3 Table 3: Accuracy when Handling k max Large Sets Number of Sets k max Bounding Probability of M-JREP 95% 95% 95% 95% Bounding Probability of INC-EXC 78% 96% 84% 288% Table 4: Accuracy Comarison when Handling Two Large Tag Sets and k max 2 Small Tag Sets Number of Sets k max Bounding Probability of M-JREP 95% 96% 966% 996% Bounding Probability of INC-EXC 78% 54% 866% 99% 9 RELATED WORK Much existing RFID work concentrates on how to collect efficiently the IDs of a grou of tags, which is called tag identification Since the tags communicate with a reader through wireless medium, inevitably collisions will haen when multile tags resond to the same reader simultaneously Collision arbitration rotocols mainly fall into two categories, ie, tree-based rotocols [3], and slotted ALOHA rotocols [6] The de-facto RFID standard, EPCglobal CG2, is a variant of the slotted ALOHA rotocol [] Another branch of RFID research investigates how to accurately estimate the cardinality of a tag set at low time cost without any ID collection To minimize the time cost, a lethora of rotocols have been develoed, including unified robabilistic estimator [5], lottery frame rotocol [4], generalized maximum likelihood estimation [7], first non-emty slot based estimation [4], robabilistic estimating tree [2], average run based tag estimation [5], and zero-one estimator [9] An imortant recent study has roosed a twohase rotocol named SRC s [3], which uses the first hase to swiftly make a rough estimation of the tag cardinality, and the second hase based on ALOHA frame for achieving better accuracy Our M-JREP also adots a two-hase rotocol for efficiently encoding a tag set into a bitma A recent trend is to extend the tag counting roblem from a single set to multile sets Some researchers focus on two tag sets scanned by a reader at different time oints, and estimate the cardinalities of their intersection/differences [9, 6, 8] Such information can hel detect missing tags which exist in the revious tag set, but no longer in the current set), remaining tags existing in both sets), and new tags oosite to missing tags) Another work named CCF is able to estimate the cardinality of an arbitrary set exression [8] It assumes that each tag set is encoded into a sketch named k-min hash values [2] and the configured value of k must be the same for the sketches of all tag sets However, the aforementioned revious studies on multileset counting roblem are limited from three ersectives First, most of them are not designed to handle a general set exression, excet the work in [8] Second, all of them secifies the accuracy requirement by the relative error model Unfortunately, when the quantity to estimate aroaches zero, their time cost to meet the accuracy requirement skyrockets to infinity see Section 2 for detailed discussion) The correct choice is to use instead the absolute error model Third, revious work requires that all tag sets must be comressed into data structures called snashots) with the same length, such that multile snashots can be merged easily to estimate the union of multile sets [8,9,6,8] However, these snashots may not have an equal length, esecially when the tag sets they encode dramatically differ in sizes, which is commonly seen in real-world scenarios A very recent work [7] addresses this third roblem by allowing the bitma-based snashots of tag sets to have adatively different lengths, in order to imrove the rotocol time efficiency But it is still inadequate in that it only deals with the joint cardinality estimation of two tag sets In many alications, it is required to estimate the cardinality of a general set exression that may involve an arbitrary number of tag sets Our aer can solve this roblem efficiently CONCLUSION In this aer, we have formulated a roblem called joint roerty estimation, in which the cardinality of an arbitrary set exression involving multile tag sets from different satial or temoral domains) is estimated with bounded absolute error We roose a rotocol named M-JREP with a novel design that allows multile tag sets to be encoded into bitmas with varied lengths It rovides a new method called exanded OR to combine the multile bitmas, and it designs formulas to exloit the combined information, estimate the cardinalities of all elementary subsets, and finally calculate the cardinality of the desired set exression We have analyzed the bias and variance of M-JREP, and also the otimal setting of its rotocol arameters under redefined accuracy requirements We have erformed extensive simulation studies The results show that our rotocol can reduce the execution time by multile folds as comared with INC- EXC and CCF rotocols, which require all tag sets must be encoded into length-consistent snashots 59