 Research
 Open Access
Performance analysis of optimal schedulers in single channel dense radio frequency identification environments
 Javier ValesAlonso^{1}Email author,
 Francisco Javier ParradoGarcía^{1} and
 Juan J Alcaraz^{1}
https://doi.org/10.1186/16873963201311
© ValesAlonso et al.; licensee Springer. 2013
 Received: 29 November 2012
 Accepted: 22 May 2013
 Published: 17 June 2013
Abstract
Schedulers in radio frequency identification dense environments aim at distributing optimally a set of t slots between a group of m readers. In singlechannel environments, the readers within mutual interference range must transmit at different times; otherwise, interferences prevent identification of the tags. The goal is to maximize the expected number of tags successfully identified within the t slots. This problem may be formulated as a mixed integer nonlinear mathematical program, which may effectively exploit available knowledge about the number of competing tags in the reading zone of each reader. In this paper, we present this optimization problem and analyze the impact of tag estimation in the performance achieved by the scheduler. The results demonstrate that optimal solutions outperform a reference scheduler based on dividing the available slots proportionally to the number of tags in each reader. In addition, depending on the scenario load, the results reveal that there exist an optimum number of readers for the topology considered, since the total average number of identifications depend nonlinearly on the load. Finally, we study the effect of imperfect tag population knowledge on the performance achieved by the readers.
Keywords
 Frame Length
 Interference Range
 Slot Aloha
 Optimal Scheduler
 Query Packet
1 Introduction
Passive radio frequency identification (RFID) is increasingly being used to identify and trace objects in supply chains, in manufacturing process, and so forth. These environments are characterized by a large number of items with attached tags which flow on conveyor belts, inside pallets or boxes, and the like, entering and leaving facilities. In large realistic installations, several readers are commonly deployed; these are the socalled dense reader environments, comprising multiple readers within a mutual range.
In these scenarios, the rate of tags identified per reader is limited by the reader collision problems, namely:

Readertotag interferences (RTI) occur when two or more readers, irrespectively of the working frequency, transmit at the same time, overlapping their read ranges (readertotag range) and powering the same tags. For instance, in Figure 1, if readers R and R^{′} are feeding tag A simultaneously, tag is not able to produce a correct response to any of the readers.

Readertoreader interferences (RRI) occur when two or more readers working at the same frequency are in mutual range, that is, one reader that powers a tag within its readertotag range can receive stronger signals from other readers, ruining the weaker signal from the tag. For example, in Figure 1, tag B cannot be read by R if at the same time R^{′} tries to read the tag C.
Another kind of interference is tagtotag interference which is internal to the reader’s cell and is produced among tags competing to be identified by the reader. This latter type occurs even with a single reader, whereas the former ones (external) are only present with more than one reader. Indeed, the way of addressing internal and external interferences is completely different and independent. External ones are addressed by reserving (in realtime or with a preconfigured scheduling) resources to particular readers. Then, the reader uses these resources (time, frequency, power, etc.) to execute some algorithm to solve the tagtotag interference problem, as the static frame slotted ALOHA (staticFSA). Later in Section 2.1, we analyze the way how staticFSA and its derivate dynamicFSA work, since their operation is relevant to decide the readers’ scheduling.
External interferences are directly related to the readers’ output power, which delimit the interference range. For example, in Europe, output power may reach up to 2 W and guarantees a readertotag range up to 10 m, while this may cause interferences with readers up to some hundreds of meters typically, determining interference ranges:

If two or more readers are within two times the readertotag range (d_{RT}), either part or the whole reading area overlaps, preventing tag operation. Hence, both RTI and RRI interferences are present. In this case, reader operation should be allocated at different working times.

If the distance among the readers is between d_{RT} and the maximum distance determined by the RRI (d_{RR}), only RRI appears. The reader operation can be multiplexed either in frequency or in time.

If the distance among the readers is larger than the maximum RRI distance, they do not suffer interferences.
Reader operation restrictions versus d
= Freq  ≠ Freq  

= Time  d > d_{RR}  d > d_{RT} 
≠ Time  d > 0  d > 0 
Therefore, in dense reader environments, the problem is how to distribute the reading resources available among the readers to perform optimally. The main parameters involved in this problem are the following:

The number of readers, m.

The number of available frequency channels, F.

The number of timeslots available in each frequency, t.

The topology of the readers.

The implemented identification procedure in each reader (e.g., staticFSA, dynamicFSA, QueryTree protocols, etc.).

The characteristics of the traffic of tags (e.g., static tags vs. tag flow, random vs. deterministic number of tags, etc.).
Current standards (see Section 2) propose some solutions to reduce collision issues but exclusively focused on minimizing RRI. On the other hand, a number of papers (see also Section 2) deal with minimization of the RTI but without considering readertoreader interferences.
In a previous paper [1], a particular simplified problem with two readers m = 2 in readertoreader range (dual reader environment) is addressed. Besides, in our previous paper [2], the scheduling problem for singlechannel environments is firstly introduced, that is, we consider the case of any arbitrary number of readers (m) and for any particular network topology. Attending to the restrictions given above, in this case, the readers cannot transmit simultaneously if the readertoreader interferences are present, that is, if the distance between them is less than d_{RR} (note that this case also comprises readertotag interferences).
In addition to [2], in this work, we provide insight on the impact of the schedulers derived from the knowledge of the tag population associated to each reader. To the best of our knowledge, all previous optimization models (see Section 2) have largely ignored the availability of this information. This information can be effectively exploited to construct a scheduler with the goal of maximizing the number of identifications in the whole interrogation period. In this work, we assume that this information is known and show how it can be used to develop an optimal scheduler. Moreover, we analyze the improvement obtained when this information is available and the effect on the expected performance when errors occur in tag estimation.
This resource allocation problem is addressed both for static and dynamic frame length identification procedures (which are described later in Section 2.1) and that the tags remain in coverage of their corresponding reader at least during the whole period of identification (t timeslots). The goal is to maximize the expected number of identified tags in the whole network.
The rest of the paper is organized as follows: In Section 2, the most relevant research proposals are shown. Section 2.1 describes the identification procedures commonly used in RFID readers. Section 3 describes the optimization model. Section 4 shows the performance results achieved by the optimal algorithm. Section 5 deals with the analysis of the impact of tag population estimation in the scheduler. Finally, Section 6 concludes and describes future works.
2 Related work
A number of proposal for coordinating dense reader environments have been presented in the literature; most of them are based on heuristic approaches and are, thus, suboptimal by nature. A summary of these works is contained in [3]. Besides, a number of papers [4–10] propose different system models and schedulers based on the optimization of some metric, defined upon the corresponding model.
Choi and Lee [4] propose a mixed integer linear program to minimize the reader interference problem as well as other performance metrics by selecting channel, timeslots, and output power for each reader. Their strategy is based on achieving a minimal signaltointerferenceplusnoise ratio for the signal received from tags, as well as on maximizing network utilization and minimizing power consumption. However, they neglect the availability of information about the number of tags present in the reading area of each reader and the operation of the underlying reading protocols, which are major factors determining the performance of the reading process.
Kim et al. [5] propose the TPCCA algorithm based on a power control approach. It consists of controlling the reader output power optimally to reduce readertoreader collisions. Optimality criterion is related to minimize the area where interferences among readers occur.
ChuiYu et al. introduces GABPSO in [6] a scheduler based on genetic algorithm and swarm intelligence metaheuristics for singlechannel environments. These schedulers aim at minimizing the overall sum of transaction times. However, these times are provided as parameters for the scheduler and are not based on the impact of resource allocation on the reading protocols.
Deolalikar et al. derive in [7] optimal scheduling schemes for readers in RFID networks for four basic configurations. As in our work, the authors aim at maximizing the number of identification within the scheduling period(t), but they model the performance of the reading process with an approximation: the number of tags identified increases linearly up to a saturation point. From that point on, the number of identifications remains constant. As we demonstrate in Section 2.1, this approach is not realistic for different tagtotag anticollision protocol configurations (e.g., in staticFSA, there is a drop on the throughput). As in our work, only readertoreader interference (and thus singlechannel) environments are considered.
The study of MohsenianRad et al. [8] is the work more closely related to ours. The authors design two optimizationbased distributed channel selection and randomized interrogation algorithms for dense RFID systems: FDFA (which is fully distributed and achieves a local optimum) and SDFA (semidistributed and reach to the global optimum). In addition, the authors realistically assume that the reader may operate asynchronously. Similarly to our work, they consider a FDMA/TDMA scheduler, where the medium access control layer of the readers complies with EPCglobal Class1 Gen2 standard (therefore, a reader may allocate a number of interrogation frames within its allocated time). In this work, the authors focus on the probability that a reader starts an interrogation interval without experiencing either readertoreader or readertotag collisions. The goal is to achieve maxmin fairness in the network; as a result, the processing load is evenly distributed among all readers. However, this paper does not consider the knowledge about the number of contending tags in range of each reader. This information allows us to formulate the optimization problem in terms of reading efficiency (maximizing the number of tags identified in the overall time period). An additional contribution of [8] is to develop a protocol to construct the topology (i.e., readertoreader and readertotag constraints) by exchanging some messages in three control channels. This protocol may be implemented in other schedulers (like ours) to determine the network topology in real time.
Tanaka and Sasase [9] also determine an interference model which they apply later to formulate constraints in a binary integer linear program aimed at maximizing the ratio of total time where readers can successfully communicate with the tags and total interrogation time of the readers. As in our model, the goal is selecting suitable timeslot and channels for each reader. They also propose two heuristics (one distributed and one centralized) to solve the allocation problem efficiently.
Seo and Lee [10] propose a FDM/TDM scheduler (RAGA) based on a readertoreader interference model, which seeks to maximize a utility function depending upon the operating time slots. This problem is solved using a genetic algorithm metaheuristic.
As many of the previous works, neither in [9] nor in [10] the reading protocol or the current load (unidentified tags) of each cell is considered. Summarizing, to the best of our knowledge, all previous optimization models ignore the availability of information about the number of tags within the range of each reader. This information can be very effectively exploited to construct a scheduler with the goal of maximizing the number of identifications in the interrogation period. Besides, most previous works assume a model view from the physical layer perspective and are usually aimed at minimizing interference. This view has notable limitations since tag identification performance, and thus scheduling, heavily depends on the underlying tagtotag anticollision protocol, as discussed in the next Section.
2.1 Tag identification procedure
 1.
static frame length FSA (staticFSA). The reader starts the identification process with an identification frame by sending a Query packet with information about the frame length (k slots) to the tags. The frame length is kept unchanged during the whole identification process. At each frame, each unidentified tag selects a slot at random from among the k slots to send its identifier to the reader. FSA achieves reasonably good performance at the cost of requiring a central node (the reader) to manage slot and frame synchronization. FSA has been implemented in many commercial products and has been standardized in the ISO/IEC 180006C [11], ISO/IEC 180007 [12], and EPCGlobal Class1 Gen2 (EPCC1G2) standards [13].
 2.
dynamic frame length FSA (dynamicFSA). When the tags outnumber the available slots, the identification time increases considerably due to frequent tagtotag collisions. On the other hand, if the slots outnumber the tags, many slots will be empty in the frame, which also leads to long identification times. DynamicFSA protocols were conceived to address this problem. They are similar to staticFSA, but the number of slots per frame is variable. In other words, parameter k may change from frame to frame in the Query packet to adjust the frame length. DynamicFSA operation is optimal in terms of reading throughput (rate of identified tags per slot) when the frame length equals the number of contenders [14]. Therefore, to maximize throughput, the reader should ideally know the actual number of competing tags and allocate that number of slots to the next frame. Different dynamicFSA algorithms have been proposed to estimate the number of competing nodes based on the collected statistical information. The most relevant ones have been studied in depth in our previous papers [15, 16].
In the next Section, both algorithms (staticFSA and dynamicFSA) are considered in order to propose an optimal slot distribution for the single channel environment. In the case of staticFSA, the frame length is k for all readers; in the case of dynamicFSA, we are assuming that each reader j actually knows the number of competing nodes at frame i (n_{j,i}) and that the reader is adjusting k_{j,i} = n_{j,i} if the number of the remaining available slots is greater than n_{j,i}.
3 Optimal time distribution
Recall from the introduction that a densereader environment with the limitation of a single frequency channel F = 1, m readers, and t slots available in the channel is assumed. In addition, for each reader j = 1, …, m, let us denote

n_{j}, the tags unidentified in the range of the reader j

t_{j}, the number of slots assigned to reader j.
Let us remark that the methods used in dynamicFSA tagtotag anticollision protocols to determine the number of contenders can be directly applied in our case to estimate n_{j} in realtime (see [15] and [16] for details). Besides, topological dependencies among readers are defined by an m × m matrix $A=\left({a}_{{\text{jj}}^{\prime}}\right)$, the elements of which are 1 if reader j and j^{′} cannot operate at the same time, and 0 otherwise.
where ${I}_{{t}_{\mathrm{j}}}$ is 1 if t_{j} is greater than 0, and 0 otherwise.
Constraint (3) expresses a basic limiting condition on the values assigned to the number of assigned slots. The key in our problem formulation is constraint (4) which establishes local conditions to regulate the spatial reuse of the resources in our network. This condition states that the number of slots assigned to a reader j plus those assigned to its neighbors can not surpass the number of available slots. ${I}_{{t}_{\mathrm{j}}}$ is included since readers without slots assigned should be considered as disconnected, and no constraints have to be applied to that particular readers.
The former constraint guarantees that enough slots are available for each node in each neighborhood (set of nodes bonded with topological constraints, i.e., ${a}_{{\mathrm{j}}^{\prime}\mathrm{j}}=1$) to obey with the limit of t slots among all neighbors. Note that it does not guarantee that these slots can be allocated consecutively. However, this is not an issue since tags do not proceed with the next slot until a QueryRep packet arrives from the reader. Hence, even if the slots are not consecutively allocated, the tags perceive continuity and the identification can be performed seamlessly.
We must remark that this set of constraints produces feasible solutions regardless of the considered topology. However, in some cases (as we will show in the next section), the constraint is too strict and may lead to suboptimal solutions since space reutilization is limited. If the network graph has a large density (i.e., the number of edges is close to the maximal number of edges), the results provided by solving problem (1) will be close to the optimal solution with maximal space resource reutilization. Whereas, for sparse network graphs, the space reutilization will be small. The first kind of scenario will likely occur (due to the large readertoreader interference range) in facilities with nonscreened readers; thus, the solutions found will be realistic.
3.1 φ(n_{ j },t_{ j }) computation for staticFSA
Finally, in order to solve the optimization problem, the expected number of identifications φ(n,t) must be computed. The next sections deal with its computation both for staticFSA and for dynamicFSA.
In this case, the reading process for each reader j consists of several consecutive reading frames of length k until all the t_{j} reading slots are eventually exhausted. It is assumed that t_{j} = k a, being a a positive integer or zero. Given the last condition, and since expectation is a linear operator, φ(n_{j},t_{j}) can be computed as the sum of the average number of tags identified in the first frame (φ(n_{j},k)) plus those identified in the remainder process (φ(n_{j}η,t_{j}k)), where η denotes the random number of tags identified in the first frame.
3.2 φ(n_{ j },t_{ j }) computation for dynamicFSA
In this second case, the reading process for each reader j also consists of several reading frames but of variable length k_{j,1},k_{j,2},…, until all the t_{j} reading slots are exhausted. Besides, denote the number of contenders in each frame as n_{j,1},n_{j,2},…. Since the dynamicFSA operation is used (see Section 2.1), the reader seeks to maximize reading throughput and allocates the optimal number of slots in each frame. That is, as much slots as the number of contending tags (k_{j,i} = n_{j,i}). This is possible while ${n}_{j,i}<{t}_{\mathrm{j}}\sum _{c=1}^{i1}{k}_{j,c}$, that is, if the remainder number of slots is greater that the number of contenders. Otherwise, we assume that a last frame is allocated with all the remaining slots (k_{j,i} = ${t}_{\mathrm{j}}\sum _{c=1}^{i1}{k}_{j,i}$).
4 Results
Besides, the following parameters have been considered:

t = 512,

n tags to be identified at each reader, from 1 to 100 tags,

m = 2, 4, 6, 8, and 10,

and for staticFSA k = 16 and 64.
Our optimization algorithm has been implemented using the General Algebraic Modeling System, a highlevel modeling system for mathematical programming and optimization, and AlphaECP, a MINLP (MixedInteger NonLinear Programming) solver based on the extended cutting plane method. It allowed us to define our optimization problem directly from the mathematical description provided in Section 3.
Fullmesh scenario
Number of tags  Φ  R1  R2  R3  R4 

10  40.000  128  128  128  128 
20  80.000  126  115  155  116 
30  119.999  128  128  128  128 
40  159.427  128  128  128  128 
50  186.355  128  128  128  128 
60  189.195  94  140  139  139 
70  189.113  110  70  166  166 
80  188.949  80  162  190  80 
90  188.992  0  212  89  211 
100  188.905  157  157  99  99 
Star scenario
Number of tags  Φ  R1  R2  R3  R4 

10  40.000  78  74  76  77 
20  80.000  126  115  155  116 
30  119.999  128  128  128  128 
40  159.427  128  128  128  128 
50  186.355  128  128  128  128 
60  189.195  94  140  139  139 
70  210.000  0  512  512  512 
80  240.000  0  512  512  512 
90  270.000  0  512  512  512 
100  300.000  0  512  512  512 
Line scenario
Number of tags  Φ  R1  R2  R3  R4 

10  40.000  85  86  165  221 
20  80.000  118  118  124  248 
30  120.000  153  153  171  171 
40  160.000  173  170  169  173 
50  199.913  174  169  169  174 
60  236.153  176  168  168  176 
70  254.481  188  162  162  188 
80  263.693  215  172  125  215 
90  273.451  242  180  90  242 
100  287.462  256  256  0  453 
In addition, note that the results obtained for the star scenario (Table 3) can be improved. For example, for n = 30, after assigning 128 slots to R1, it would be possible to assign 384 to all remainder readers, which will provide a solution better than that obtained by solving problem (1). As discussed in Section 3, this is caused by the strict resource reutilization obtained by applying the set of constraints given by Equation (4). This problem does not appear for networks characterized by a dense graph, as the fullmesh scenario.
Another important result shown in these figures is the existence of saturation points in the system. That is, in some cases, the throughput does not increase when the load is increased. For dynamicFSA, in all cases, the throughput never decreases; this is caused by the flexibility of dynamicFSA to adapt to different loads. For staticFSA k = 64, the effect is almost similar to the dynamicFSA case, except in the fullmesh scenario where the throughput slightly decreases when the tag is beyond n = 60. However, for all staticFSA k = 16 cases, the effect of the load in the throughput is dramatic, with a throughput minima and a step decreasing performance. This is of considerable importance, since staticFSA k = 16 is the default configuration of many readers in the market, and this configuration leads to poor collective performance.
5 Tag estimation impact on scheduler performance
 (1)
Quantify the improvement achieved in the scheduler when tag instant population estimation is available.
 (2)
Quantify the impact of tag population estimation errors on the performance achieved by the scheduler.
As stated in Section 2, previous works do not assume knowledge about the tag population and are mostly based on minimizing interferences. To establish a comparison between our model and a reference model that do not use population information, at least, we must focus on the same performance metric, i.e., the expected number of identifications (which can also be viewed as throughput).
Although our reference model does not use information about the instant population, it is rational to assume at least a coarse knowledge of the environment, typically the average number of competing tags. This allows the designer to configure the system for a standard case. Note that if this information is unavailable, the designer should guess somehow a configuration, and the performance would be lower than in the reference model.
Henceforth, let us assume that our reference model is based on the availability of information about the average tag population and that the designer is able to select the optimal scheduler configuration for this case (e.g., by solving problem (1)).
For simplicity, let us denote by $\overrightarrow{n}$ the mdimensional vector (n_{1},…,n_{ m }), and by ${\mathrm{\Phi}}_{\overrightarrow{n}}\left(\overrightarrow{{n}^{\prime}}\right)$ the expected number of identifications when the optimal solution to problem (1) with tag estimation parameter $\overrightarrow{n}$ is applied to the actual population $\overrightarrow{{n}^{\prime}}$. Besides, let $\overrightarrow{{n}^{\ast}}$ denote the mdimensional vector, where the j th component is the average number of tags in reader j.
By solving problem (1), both optimal slot assignments can be computed. Let us denote $\hat{t}$ and $\hat{{t}^{\ast}}$ to the optimal assignments for tag populations $\overrightarrow{n}$ and $\overrightarrow{{n}^{\ast}}$, respectively, and ${\hat{t}}_{j}$ and ${\hat{{t}^{\ast}}}_{j}$ the slots assigned to particular reader j th.
where φ(n,t) is computed directly with formulas (9) and (10) for staticFSA and dynamicFSA, respectively.
Note that we use a perfect knowledge of the average number of tags; therefore, we are assuming the leastfavorable comparison case for our scheduler versus the reference model.
5.1 Numerical examples
Fullmesh scenario
Number of readers  DynamicFSA  StaticFSA,  StaticFSA, 

k = 64  k = 16  
2  0.0273  0  0.0889 
4  0.0120  0.0008  0.2891 
6  0.0423  0.0050  0.4831 
8  0.0791  0.0086  0.6024 
10  0.0947  0.0086  0.6731 
Star scenario
Number of readers  DynamicFSA  StaticFSA,  StaticFSA, 

k = 64  k = 16  
2  0.0273  0  0.0889 
4  0.1592  0.1702  0.2636 
6  0.0509  0.0023  0.1940 
8  0.0417  0.0005  0.1756 
10  0.0333  0.0000  0.1683 
Line scenario
Number of readers  DynamicFSA  StaticFSA,  StaticFSA, 

k = 64  k = 16  
2  0.0266  0  0.0817 
4  0.0331  0.0144  0.2114 
6  0.0049  0.0018  0.2105 
8  0.0298  0.0050  0.2073 
10  0.0101  0.0013  0.2056 
The results clearly depend on the scenario and on the tagtotag anticollision protocols. Improvement ranges from nearly 0% in many staticFSA k=64 cases, while it may reach up to 67% for staticFSA, k=16 in the fullmesh scenario. For dynamicFSA, the improvement is between 2.7% and 21.14%, depending on the particular scenario. Let us remark again that this comparison is performed against the average tags identified when the optimal configuration is computed using as information the mean number of competing tags. Therefore, this is the minimum improvement ratio: nonoptimal schedulers (as the reference heuristic used in Section 4) will obtain worse results.
Fullmesh scenario
Number of readers  DynamicFSA  StaticFSA,  StaticFSA, 

k = 64  k = 16  
2  0.0284  0  0.0527 
4  0.0064  0.0007  0.2334 
6  0.0361  0.0040  0.4107 
8  0.0580  0.0101  0.5515 
10  0.0747  0.0070  0.5712 
Star scenario
Number of readers  DynamicFSA  StaticFSA,  StaticFSA, 

k = 64  k = 16  
2  0.0275  0  0.0704 
4  0.1541  0.1750  0.2087 
6  0.0543  0.0017  0.2123 
8  0.0406  0.0002  0.1878 
10  0.0302  0.0000  0.1925 
Line scenario
Number of readers  DynamicFSA  StaticFSA,  StaticFSA, 

k = 64  k = 16  
2  0.0442  0.0006  0.0023 
4  0.0324  0.0212  0.0991 
6  0.0003  0.0019  0.1169 
8  0.0279  0.0071  0.0815 
10  0.0118  0.0012  0.0942 
Again, the results heavily depend on the configuration, but in most cases, even assuming an error in the tag number estimation, they show a positive feedback using the estimation. In some cases, in the fullmesh and line scenarios, there is a negative impact, but almost negligible. Therefore, we can conclude that even assuming errors, the utilization of tag estimators is worth to be considered.
6 Conclusions
This work introduced a novel optimal scheduler for a particular dense reader environment composed by m readers which must share a single frequency channel. The scheduler proposed exceeds in performance to heuristic algorithms, improving the average number of tags identified in an RFID facility. Besides, the effect of the reading protocols has also been studied in depth, concluding that a dynamic FSA algorithm excels static frame length ones. Indeed, the impact of using knowledge about tag population in the scheduler has been analyzed. It has been concluded that even assuming errors in the estimation, our scheduler is able to obtain a higher performance than a reference model, where the average population is perfectly known.
As future works, we aim at extending our model to multichannel scenarios, developing a model that allow full resource reutilization and further analyze RFID realistic scenarios to propose optimal configuration strategies.
Appendix
Computation of P(an,t)
To compute the probability P(an,t), we apply the technique in [17], where the authors formulate probabilistic transforms for urn models that convert the dependent random variables describing urn occupancies (slot occupancies in our case) into independent random variables. Due to the independence of random variables in the transform domain, it is simpler to compute the statistics of interest, and afterwards the transform is inverted to get the desired result.
Let us denote P(an,t) as the probability of interest and P(λ,t,i) its transformation, with λ as parameter meaningful in the transform domain only. Indeed, there is no dependence on the number of balls (tags), n, in the transform domain.
and extracting the coefficient of λ^{ n }for the appropriate n value, n = j + i, we obtain the result in Equation (8).
Declarations
Acknowledgements
This work has been supported by project CALM TEC201021405C02 which is funded by the Spanish Ministerio de Innovación y Ciencia. It has been developed within the framework of ‘Programa de Ayudas a Grupos de Excelencia de la Región de Murcia’, funded by Fundación Seneca, Agencia de Ciencia y Tecnología de la Región de Murcia (Plan Regional de Ciencia y Tecnología 2007/2010). We are also indebted to Javier FernándezNogueira for his help in MINLP optimization.
Authors’ Affiliations
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