**2. System model and cross-layer framework**

Consider the slotted RFID network depicted in Fig. 1 with a set R of *K* readers R = {1, . . . *K*} and a set T of *J* tags T = {1, . . . , *J*}. Each reader is provided with *M* antennas that will be used to recover, using source separation, the simultaneous transmissions of several tags. Two main processes can be distinguished in the RFID network in Fig. 1: Tag activation by the transmission of readers, also called the down-link transmission; and the backscattering response towards readers by previously activated tags, also called up-link transmission. In the down-link, the transmit power of reader *k* will be denoted by *Pr*,*<sup>k</sup>* while its probability of transmission will be denoted by *pr*,*k*. All the antennas will be assumed to transmit the same signal in the down-link. The subset of active readers at any given time will be denoted by R*t*. Tags are activated when the signal-to-interference-plus-noise ratio (SINR) given a reader transmission is above an activation threshold. The set of activated tags will be denoted by T*P*. In the up-link, the active tags proceed to transmit a backscatter signal using a randomized transmission scheme. The subset of tags that transmit a signal once they have been activated will be given by T*t*, where each tag *j* will transmit with a power level denoted by *Pt*,*j*. Details of the down- and up-link models are given in the following subsections.

#### **2.1. Tag activation process: Down-link model**

Consider that the instantaneous channel between reader *k* and tag *j* is given by the column vector **h***k*,*<sup>j</sup>* with dimensions *M* × 1, the channel experienced between reader *k* and reader *m* is given by the matrix **G***k*,*<sup>m</sup>* with dimensions *M* × *M*, and the channel experienced between tag *i* and tag *j* is given by the scalar value *ui*,*j*. The SINR experienced by tag *j* due to a transmission of reader *k* is denoted by *γk*,*j*, and it can be mathematically expressed as follows:

$$\gamma\_{k,j} = \frac{P\_{r,k} \mathbf{h}\_{k,j}^H \mathbf{h}\_{k,j}}{I\_{r\_{k,j}} + I\_{t\_j} + \sigma\_{v,j}^2}, \qquad k \in \mathcal{R}\_{t\_\prime} \tag{1}$$

**2.2. Backscattering process: Up-link model**

all *r* sources of diversity1 is given by:

receiver can be described as follows:

*lep*.

transmit power of tag *<sup>j</sup>* can be calculated as *Pt*,*<sup>j</sup>* <sup>=</sup> *<sup>β</sup>jPr*,*k*|*hkopt*,*j*<sup>|</sup>

Once a tag *j* has been activated by the transmission of a given reader, it then starts a random transmission process to prevent collisions with other active tags using a Bernoulli process with parameter *pt*,*j*, which is also the transmission probability. The backscattering factor *β<sup>j</sup>* is the fraction of the received power reused by the tag to reply to the reader. Therefore, the

An RFID Anti-Collision Algorithm Assisted by Multi-Packet Reception and Retransmission Diversity

denotes the reader that has previously activated the tag. At the reader side, source separation tools for multi-packet reception and retransmission diversity will be used. The proposed protocol consists of ensuring that the number of diversity sources is equal or larger than the number of contending tags so that the source separation technique is successful. For example, if 4 tags collide at a particular time-slot (see Fig. 2) and the reader is provided with only 2 antennas, then the system will request a retransmission from the contending tags in the following time slot. The reader will store all the signals collected during these 2 time-slots and will create a virtual MIMO system from which the signals of the contending tags can be estimated using multiuser detection. The array of stacked signals received at reader *k* across

where **H** is the stacked version of all the channels of the contending tags, **S** is the stacked version of all the signals of the contending tags, **I***r*,*<sup>k</sup>* is the collected interference created by other active readers, **E***r*,*<sup>k</sup>* is the collected leaked signal power from the transmission chain, and **V***r*,*<sup>k</sup>* is the noise term. At the reader side, a multiuser receiver such as zero forcing (ZF) or minimum mean square error (MMSE) can be implemented. For example, the zero forcing

**<sup>S</sup>** <sup>=</sup> **<sup>H</sup>** <sup>−</sup><sup>1</sup>

where **<sup>S</sup>** is the array of estimated signals of the contending tags, and **<sup>H</sup>** is the estimated channel of the contending tags. Since the resolution of a collision may take place over a random number of time slots due to the retransmission diversity scheme, then we will denote this collision resolution period as an *epoch-slot* with a length denoted by the random variable

For simplicity, it will be assumed that the performance of the multiuser receiver is described by the ability to correctly detect the presence of all the contending tags. This assumption has been used in the analysis of conventional NDMA protocols in [26]. In this assumption any detection error yields the loss of all the contending packets. Thus, it is possible to propose

> *k*,*j* **h***k*,*<sup>j</sup>*

> > *v*,*k*

*Ir*,*<sup>k</sup>* <sup>+</sup> *Pr*,*kη<sup>k</sup>* <sup>+</sup> *<sup>σ</sup>*<sup>2</sup>

<sup>1</sup> the number of diversity sources is the total number of combinations of antenna elements and retransmissions

the detection SINR of tag *<sup>j</sup>* at reader *<sup>k</sup>*, denoted by *<sup>γ</sup><sup>j</sup>*,*k*, as follows:

*<sup>γ</sup><sup>j</sup>*,*<sup>k</sup>* <sup>=</sup> *Pt*,*j***h***<sup>H</sup>*

**<sup>Y</sup>***r*,*<sup>k</sup>* = **HS** + **<sup>I</sup>***r*,*<sup>k</sup>* + **<sup>E</sup>***r*,*<sup>k</sup>* + **<sup>V</sup>***r*,*<sup>k</sup>* (3)

**Y***r*,*k*, (4)

, *j* ∈ T*<sup>t</sup>* (5)

2, where *kopt* = arg max*<sup>k</sup> <sup>γ</sup>k*,*<sup>j</sup>*

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53

where *Irk*,*<sup>j</sup>* <sup>=</sup> <sup>∑</sup>*m*∈R*t*,*m*�=*<sup>k</sup> Pr*,*m***h***<sup>H</sup> m*,*j* **<sup>h</sup>***m*,*<sup>j</sup>* is the interference created by other active readers, *Itj* = <sup>∑</sup>*i*∈T*t*,*i*�=*<sup>j</sup> Pt*,*i*(|*uj*,*i*<sup>|</sup> <sup>2</sup>) is the interference created by other tags, (·)*<sup>H</sup>* is the hermitian transpose operator, and *σ*<sup>2</sup> *<sup>v</sup>*,*<sup>j</sup>* is the noise. If the SINR experienced by tag *j* is above the tag sensitivity threshold *<sup>γ</sup><sup>j</sup>*, then the tag becomes activated. The probability of tag *<sup>j</sup>*, which was previously inactivated, to become activated will be given by

$$\Pr\{j \in \mathcal{T}\_P\} = \Pr\{\max\_k \gamma\_{k,j} > \tilde{\gamma}\_j\}.\tag{2}$$

**Figure 1.** Multi-tag and Multi-reader deployment scenario.

#### **2.2. Backscattering process: Up-link model**

4 Radio Frequency Identification

where *Irk*,*<sup>j</sup>* <sup>=</sup> <sup>∑</sup>*m*∈R*t*,*m*�=*<sup>k</sup> Pr*,*m***h***<sup>H</sup>*

<sup>∑</sup>*i*∈T*t*,*i*�=*<sup>j</sup> Pt*,*i*(|*uj*,*i*<sup>|</sup>

operator, and *σ*<sup>2</sup>

**2.1. Tag activation process: Down-link model**

Consider that the instantaneous channel between reader *k* and tag *j* is given by the column vector **h***k*,*<sup>j</sup>* with dimensions *M* × 1, the channel experienced between reader *k* and reader *m* is given by the matrix **G***k*,*<sup>m</sup>* with dimensions *M* × *M*, and the channel experienced between tag *i* and tag *j* is given by the scalar value *ui*,*j*. The SINR experienced by tag *j* due to a transmission

of reader *k* is denoted by *γk*,*j*, and it can be mathematically expressed as follows:

*k*,*j* **h***k*,*<sup>j</sup>*

*v*,*j*

threshold *<sup>γ</sup><sup>j</sup>*, then the tag becomes activated. The probability of tag *<sup>j</sup>*, which was previously

Pr{*<sup>j</sup>* ∈ T*P*} = Pr{max

<sup>2</sup>) is the interference created by other tags, (·)*<sup>H</sup>* is the hermitian transpose

*<sup>v</sup>*,*<sup>j</sup>* is the noise. If the SINR experienced by tag *j* is above the tag sensitivity

, *k* ∈ R*t*, (1)

*<sup>k</sup> <sup>γ</sup>k*,*<sup>j</sup>* <sup>&</sup>gt; *<sup>γ</sup><sup>j</sup>*}. (2)

**<sup>h</sup>***m*,*<sup>j</sup>* is the interference created by other active readers, *Itj* =

*Irk*,*<sup>j</sup>* <sup>+</sup> *Itj* <sup>+</sup> *<sup>σ</sup>*<sup>2</sup>

*<sup>γ</sup>k*,*<sup>j</sup>* <sup>=</sup> *Pr*,*k***h***<sup>H</sup>*

*m*,*j*

inactivated, to become activated will be given by

**Figure 1.** Multi-tag and Multi-reader deployment scenario.

Once a tag *j* has been activated by the transmission of a given reader, it then starts a random transmission process to prevent collisions with other active tags using a Bernoulli process with parameter *pt*,*j*, which is also the transmission probability. The backscattering factor *β<sup>j</sup>* is the fraction of the received power reused by the tag to reply to the reader. Therefore, the transmit power of tag *<sup>j</sup>* can be calculated as *Pt*,*<sup>j</sup>* <sup>=</sup> *<sup>β</sup>jPr*,*k*|*hkopt*,*j*<sup>|</sup> 2, where *kopt* = arg max*<sup>k</sup> <sup>γ</sup>k*,*<sup>j</sup>* denotes the reader that has previously activated the tag. At the reader side, source separation tools for multi-packet reception and retransmission diversity will be used. The proposed protocol consists of ensuring that the number of diversity sources is equal or larger than the number of contending tags so that the source separation technique is successful. For example, if 4 tags collide at a particular time-slot (see Fig. 2) and the reader is provided with only 2 antennas, then the system will request a retransmission from the contending tags in the following time slot. The reader will store all the signals collected during these 2 time-slots and will create a virtual MIMO system from which the signals of the contending tags can be estimated using multiuser detection. The array of stacked signals received at reader *k* across all *r* sources of diversity<sup>1</sup> is given by:

$$\mathbf{Y}\_{r,k} = \mathbf{H}\mathbf{S} + \mathbf{I}\_{r,k} + \mathbf{E}\_{r,k} + \mathbf{V}\_{r,k} \tag{3}$$

where **H** is the stacked version of all the channels of the contending tags, **S** is the stacked version of all the signals of the contending tags, **I***r*,*<sup>k</sup>* is the collected interference created by other active readers, **E***r*,*<sup>k</sup>* is the collected leaked signal power from the transmission chain, and **V***r*,*<sup>k</sup>* is the noise term. At the reader side, a multiuser receiver such as zero forcing (ZF) or minimum mean square error (MMSE) can be implemented. For example, the zero forcing receiver can be described as follows:

$$
\hat{\mathbf{S}} = \hat{\mathbf{H}}^{-1} \mathbf{Y}\_{r,k\prime} \tag{4}
$$

where **<sup>S</sup>** is the array of estimated signals of the contending tags, and **<sup>H</sup>** is the estimated channel of the contending tags. Since the resolution of a collision may take place over a random number of time slots due to the retransmission diversity scheme, then we will denote this collision resolution period as an *epoch-slot* with a length denoted by the random variable *lep*.

For simplicity, it will be assumed that the performance of the multiuser receiver is described by the ability to correctly detect the presence of all the contending tags. This assumption has been used in the analysis of conventional NDMA protocols in [26]. In this assumption any detection error yields the loss of all the contending packets. Thus, it is possible to propose the detection SINR of tag *<sup>j</sup>* at reader *<sup>k</sup>*, denoted by *<sup>γ</sup><sup>j</sup>*,*k*, as follows:

$$\widehat{\gamma}\_{j,k} = \frac{P\_{t,j} \mathbf{h}\_{k,j}^H \mathbf{h}\_{k,j}}{\widehat{I}\_{r,k} + P\_{r,k} \eta\_k + \widehat{\sigma}\_{v,k}^2}, \qquad j \in \mathcal{T}\_t \tag{5}$$

<sup>1</sup> the number of diversity sources is the total number of combinations of antenna elements and retransmissions

where *Ir*,*<sup>k</sup>* <sup>=</sup> <sup>∑</sup>*m*�=*<sup>k</sup>* tr(**G***<sup>H</sup> <sup>k</sup>*,*m***G***k*,*m*) is the interference created by other active readers, tr(·) is the trace operator, *<sup>η</sup><sup>k</sup>* is the power ratio leaked from the down-link chain, and *<sup>σ</sup>*<sup>2</sup> *<sup>v</sup>*,*<sup>k</sup>* is the noise. Note that tag-to-tag interference is not considered as an independent orthogonal training signal for each tag is used in each transmission for purposes of tag detection and channel estimation, which is also used in the original NDMA protocol in [26]. Thus, tag *j* can be detected by reader *k* if the received SINR is above a threshold denoted by *γ*ˇ *<sup>k</sup>*. The set of detected tags by reader *k* will be denoted by T*D*,*k*, thus the probability of tag *j* being in T*D*,*<sup>k</sup>* will be given by

$$\Pr\{j \in \mathcal{T}\_{D,k}\} = \Pr\{\hat{\gamma}\_{j,k} > \check{\gamma}\_k\}.\tag{6}$$

 

 

**3.1. Network state information and tag activation model**

N (*n*) and given that the tag was previously inactivated as follows:

*Qj*|T*P*(*n*) <sup>=</sup> ∑

follows:

*Qj*|N (*n*) <sup>=</sup> Pr{*<sup>j</sup>* ∈ T*P*(*<sup>n</sup>* <sup>+</sup> <sup>1</sup>)|N (*n*), *<sup>j</sup>* �∈ T*P*(*n*)} <sup>=</sup> Pr{max

set of active tags T*P*(*n*) by averaging over all values of N (*n*) where T*t*(*n*) ∈ T*P*(*n*):

N (*n*);T*t*(*n*)∈T*P*(*n*)

 

An RFID Anti-Collision Algorithm Assisted by Multi-Packet Reception and Retransmission Diversity

 

 

N (*n*) = {R*t*(*n*), T*t*(*n*)}. (10)

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55

*<sup>k</sup> <sup>γ</sup>k*,*j*(*n*) <sup>&</sup>gt; *<sup>γ</sup><sup>j</sup>*}. (11)

Pr{N (*n*)}*Qj*|N (*n*) (12)

**Figure 2.** Example of the operation of the proposed protocol with multi-packet reception and retransmission diversity.

The network state information can be defined as all the parameters that completely describe the network at any given time. In our case, the network state information N (*n*) at epoch-slot *n* is defined as the collection of the sets of active readers R*t*(*n*) and contending tags T*t*(*n*):

Once the network state information has been defined, we can define the probability of tag *j* being activated in slot *n* conditional on a given realization of the network state information

For convenience in the analysis, let us rewrite this tag activation probability in terms of the

where Pr{N (*n*)} is the probability of occurrence of the network state information N (*n*). This term can be calculated by considering all the combinations of active tags and readers as

The set of correctly detected tags across all the readers will be simply given by T*D*, where T*<sup>D</sup>* = ∪*k*T*D*,*k*. Since this detection process is prone to errors, we will use in this paper the same assumption used in the original paper for NDMA in [26] where tags are only correctly received at the reader side if all the contending tags are correctly detected and none of the remaining silent tags is incorrectly detected as active (i.e., false alarm). This means that correct tag reception for tag *j* only occurs when:

$$\Pr\{j \in \mathcal{T}\_{\mathbb{R}}\} = \Pr\{\mathcal{T}\_{\mathbb{D}} = \mathcal{T}\_{\mathbb{t}}\}, \quad \text{where} \quad j \in \mathcal{T}\_{\mathbb{t}} \tag{7}$$

where T*<sup>R</sup>* is the set of tags correctly received at the reader side. A tag that has transmitted to the reader side can be correctly detected with probability *PD*, which can be defined as:

$$P\_D = \Pr\{j \in \mathcal{T}\_D | j \in \mathcal{T}\_l\} = \sum\_k \Pr\{\hat{\gamma}\_{j,k} > \hat{\gamma}\_k | j \in \mathcal{T}\_l\},\tag{8}$$

and which can be read as the probability that tag *j* is correctly detected as active given it has transmitted a signal. Similarly, the probability of false alarm is given by:

$$P\_{\mathcal{F}} = \Pr\{j \in \mathcal{T}\_{\mathcal{D}} | j \notin \mathcal{T}\_{\mathcal{t}}\} = \sum\_{k} \Pr\{\hat{\gamma}\_{j,k} > \hat{\gamma}\_{k} | j \notin \mathcal{T}\_{\mathcal{t}}\},\tag{9}$$

which can be read as the probability that tag *j* is incorrectly detected as active when it has transmitted no signal at all.

#### **3. Performance metrics and Markov model**

The main performance metric to be used in this chapter is the average tag throughput, which can be defined as the long term ratio of correct tag readings to the total number of time-slots used in the measurement. Before providing an expression for this metric, it is first necessary to define the network state information, as well as the tag activation and tag reception probability models, and the definition of the Markov model for the dynamic analysis of an RFID network.

<sup>54</sup> Radio Frequency Identification from System to Applications An RFID Anti-Collision Algorithm Assisted by Multi-Packet Reception and Retransmission Diversity 7 An RFID Anti-Collision Algorithm Assisted by Multi-Packet Reception and Retransmission Diversity http://dx.doi.org/10.5772/54069 55

**Figure 2.** Example of the operation of the proposed protocol with multi-packet reception and retransmission diversity.

#### **3.1. Network state information and tag activation model**

6 Radio Frequency Identification

where *Ir*,*<sup>k</sup>* <sup>=</sup> <sup>∑</sup>*m*�=*<sup>k</sup>* tr(**G***<sup>H</sup>*

transmitted no signal at all.

analysis of an RFID network.

correct tag reception for tag *j* only occurs when:

will be given by

*<sup>k</sup>*,*m***G***k*,*m*) is the interference created by other active readers, tr(·) is the

Pr{*<sup>j</sup>* ∈ T*D*,*k*} = Pr{*<sup>γ</sup><sup>j</sup>*,*<sup>k</sup>* > *<sup>γ</sup>*<sup>ˇ</sup> *<sup>k</sup>*}. (6)

Pr{*<sup>j</sup>* ∈ T*R*} = Pr{T*<sup>D</sup>* = T*t*}, where *<sup>j</sup>* ∈ T*t*, (7)

Pr{*<sup>γ</sup><sup>j</sup>*,*<sup>k</sup>* > *<sup>γ</sup>*<sup>ˇ</sup> *<sup>k</sup>*|*<sup>j</sup>* ∈ T*t*}, (8)

Pr{*<sup>γ</sup><sup>j</sup>*,*<sup>k</sup>* > *<sup>γ</sup>*<sup>ˇ</sup> *<sup>k</sup>*|*<sup>j</sup>* �∈ T*t*}, (9)

*<sup>v</sup>*,*<sup>k</sup>* is the noise.

trace operator, *<sup>η</sup><sup>k</sup>* is the power ratio leaked from the down-link chain, and *<sup>σ</sup>*<sup>2</sup>

Note that tag-to-tag interference is not considered as an independent orthogonal training signal for each tag is used in each transmission for purposes of tag detection and channel estimation, which is also used in the original NDMA protocol in [26]. Thus, tag *j* can be detected by reader *k* if the received SINR is above a threshold denoted by *γ*ˇ *<sup>k</sup>*. The set of detected tags by reader *k* will be denoted by T*D*,*k*, thus the probability of tag *j* being in T*D*,*<sup>k</sup>*

The set of correctly detected tags across all the readers will be simply given by T*D*, where T*<sup>D</sup>* = ∪*k*T*D*,*k*. Since this detection process is prone to errors, we will use in this paper the same assumption used in the original paper for NDMA in [26] where tags are only correctly received at the reader side if all the contending tags are correctly detected and none of the remaining silent tags is incorrectly detected as active (i.e., false alarm). This means that

where T*<sup>R</sup>* is the set of tags correctly received at the reader side. A tag that has transmitted to the reader side can be correctly detected with probability *PD*, which can be defined as:

and which can be read as the probability that tag *j* is correctly detected as active given it has

which can be read as the probability that tag *j* is incorrectly detected as active when it has

The main performance metric to be used in this chapter is the average tag throughput, which can be defined as the long term ratio of correct tag readings to the total number of time-slots used in the measurement. Before providing an expression for this metric, it is first necessary to define the network state information, as well as the tag activation and tag reception probability models, and the definition of the Markov model for the dynamic

*k*

*k*

*PD* = Pr{*<sup>j</sup>* ∈ T*D*|*<sup>j</sup>* ∈ T*t*} = ∑

transmitted a signal. Similarly, the probability of false alarm is given by:

*PF* = Pr{*<sup>j</sup>* ∈ T*D*|*<sup>j</sup>* �∈ T*t*} = ∑

**3. Performance metrics and Markov model**

The network state information can be defined as all the parameters that completely describe the network at any given time. In our case, the network state information N (*n*) at epoch-slot *n* is defined as the collection of the sets of active readers R*t*(*n*) and contending tags T*t*(*n*):

$$\mathcal{N}(n) = \{\mathcal{R}\_l(n), \mathcal{T}\_l(n)\}. \tag{10}$$

Once the network state information has been defined, we can define the probability of tag *j* being activated in slot *n* conditional on a given realization of the network state information N (*n*) and given that the tag was previously inactivated as follows:

$$Q\_{\hat{\jmath}|\mathcal{N}(n)} = \Pr\{\hat{\jmath} \in \mathcal{T}\_{\mathcal{P}}(n+1)|\mathcal{N}(n), \hat{\jmath} \notin \mathcal{T}\_{\mathcal{P}}(n)\} = \Pr\{\max\_{\hat{k}} \gamma\_{k,\hat{\jmath}}(n) > \tilde{\gamma}\_{\hat{\jmath}}\}.\tag{11}$$

For convenience in the analysis, let us rewrite this tag activation probability in terms of the set of active tags T*P*(*n*) by averaging over all values of N (*n*) where T*t*(*n*) ∈ T*P*(*n*):

$$Q\_{\vec{j}|\mathcal{T}\_{\mathbb{F}}(n)} = \sum\_{\mathcal{N}(n): \mathcal{T}\_{\mathbb{F}}(n) \in \mathcal{T}\_{\mathbb{F}}(n)} \text{Pr}\{\mathcal{N}(n)\} Q\_{\vec{j}|\mathcal{N}(n)} \tag{12}$$

where Pr{N (*n*)} is the probability of occurrence of the network state information N (*n*). This term can be calculated by considering all the combinations of active tags and readers as follows:

$$\Pr\{\mathcal{N}(n)\} = \prod\_{k \in \mathcal{R}\_l} p\_{r,k} \prod\_{m \notin \mathcal{R}\_l} \overline{p}\_{r,m} \prod\_{j \in \mathcal{T}\_l} p\_{t,j} \prod\_{i \notin \mathcal{T}\_l} \overline{p}\_{t,i} \tag{13}$$

*qj*|N (*n*) <sup>=</sup> Pr{*<sup>j</sup>* ∈ T*R*(*<sup>n</sup>* <sup>+</sup> <sup>1</sup>)} <sup>=</sup> Pr{T*<sup>D</sup>* <sup>=</sup> <sup>T</sup>*t*}, where *<sup>j</sup>* ∈ T*<sup>t</sup>* (16)

An RFID Anti-Collision Algorithm Assisted by Multi-Packet Reception and Retransmission Diversity

Pr{N (*n*)}*qj*|N (*n*) (17)

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57

Pr{T*P*}*qj*|T*<sup>P</sup>* . (18)

Pr{T*P*}|T*P*|. (20)

, (19)

It is also convenient to re-write this reception probability in terms of the set of active tags

The tag throughput per resolution period can be finally calculated by adding all the contributions over the calculated probability space Pr{T*P*} using the Markov model

Now, the throughput per time-slot can be calculated as the ratio of the throughput per

 |T*D*| *M*

where |·| is the set cardinality operator and ⌈·⌉ is the ceil integer operator. As a measure of stability we will use the average number of activated tags, which can be calculated as follows:

T*P*

A high number of activated tags means that stability is compromised, while a relatively low

The parameters to be optimized are the vector of reader transmission probabilities **<sup>p</sup>***<sup>r</sup>* <sup>=</sup>

of transmission probabilities of the active tags **<sup>p</sup>***<sup>t</sup>* = [*pt*,1,... *pt*,*J*]. The objective of the optimization is the total throughput, so the optimization problem with transmit power

*<sup>T</sup>*, the vector of reader transmit powers **<sup>P</sup>***<sup>r</sup>* = [*Pr*,1,... *Pr*,*K*] and the vector

{**P***r*, **<sup>p</sup>***t*, **<sup>p</sup>***r*}*opt* <sup>=</sup> arg max {**P***r*,**p***t*,**p***r*}∑*Tj* s.t. **<sup>P</sup>***<sup>r</sup>* <sup>&</sup>lt; **<sup>P</sup>***r*,0 (21)

+ Pr{T*<sup>D</sup>* = <sup>∅</sup>}

N (*n*);T*t*(*n*)∈T*P*(*n*)

presented in previous subsections. This can be mathematically expressed as:

*Sj* = ∑ T*P*,*j*∈T*<sup>P</sup>*

resolution period to the average number of time-slots per resolution period:

*Tj* <sup>=</sup> *Sj* ∑T*<sup>D</sup>* Pr{T*D*}

*<sup>E</sup>*[|T*P*|] = ∑

number indicates that the algorithm is more stable.

**4. Optimization and results**

constraint can thus be written as follows:

[*pr*,1,... *pr*,*K*]

T*P*(*n*) by averaging over all values of N (*n*) where T*t*(*n*) ∈ T*P*(*n*):

**3.4. Tag throughput and stability**

*qj*|T*P*(*n*) <sup>=</sup> ∑

where (·) = 1 − (·). This concludes the definition of the tag activation probability and the network state information.

#### **3.2. Markov model**

In order to define the Markov model for dynamic analysis of the system, let us now calculate the probability of having a set of active tags T*P*(*<sup>n</sup>* + <sup>1</sup>) in epoch-slot *<sup>n</sup>* + 1 conditional on having the set of active tags T*P*(*n*) during the previous epoch-slot. This transition probability must consider all the combinations of tags that either enter (i.e., they are activated in epoch slot *n*) or leave the set of active tags (i.e., they transmit in epoch slot *n*). This can be expressed as follows:

$$\Pr\{\mathcal{T}\_{\mathcal{P}}(n+1)|\mathcal{T}\_{\mathcal{P}}(n)\} = \prod\_{j \in \mathcal{T}\_{\mathcal{P}}(n), j \notin \mathcal{T}\_{\mathcal{P}}(n+1)} p\_{t,j} \prod\_{i \notin \mathcal{T}\_{\mathcal{P}}(n), i \in \mathcal{T}\_{\mathcal{P}}(n+1)} Q\_{i|\mathcal{T}\_{\mathcal{P}}(n)} \prod\_{l \notin \mathcal{T}\_{\mathcal{P}}(n), l \notin \mathcal{T}(n+1)} \overline{Q}\_{l|\mathcal{T}\_{\mathcal{P}}(n)}$$

$$\times \prod\_{\substack{w \in \mathcal{T}\_\mathbb{P}(n), w \in \mathcal{T}\_\mathbb{P}(n+1)}} \overline{p}\_{t,w}.\tag{14}$$

Let us now arrange the probability of occurrence of all the possible sets of activated tags Pr{T*P*} into a one-dimensional vector given by **<sup>s</sup>** = [*s*0,...*sJJ* ] *<sup>T</sup>*, where (·)*<sup>T</sup>* is the transpose operator (see Fig. 3). This means that we are mapping the asymmetrical states into a linear state vector where each element represents the probability of occurrence of one different state Pr{T*P*}. In the example given in Fig. 3 we have only two possible tags, where the first system state is given by both tags being active, the second state with only tag 1 as active, the third state with only tag 2 as active, and the fourth state with both tags inactive. Once these states are mapped into the state vector **s**, the transition probabilities between such states (Pr{T*P*(*<sup>n</sup>* + <sup>1</sup>)|T*P*(*n*)) can also be mapped into a matrix **<sup>M</sup>**, which defines the Markov model for state transition probabilities (see Fig. 3). The *i*, *j* entry of the matrix **M** denotes the transition probability between state *i* and state *j*. The vector of state probabilities can thus be obtained by solving the following characteristic equation:

$$\mathbf{s} = \mathbf{M} \mathbf{s},\tag{15}$$

by using standard eigenvalue analysis or iterative schemes. Each one of the calculated terms of the vector **s** can be mapped back to the original probability space Pr{T*P*}, which can then be used to calculate relevant performance metrics.

#### **3.3. Tag detection model**

Before calculating the tag throughput, first we must define the correct reception probability of tag *j* at the reader side conditional on the network state information N (*n*) as follows:

$$\mathfrak{q}\_{j|\mathcal{N}(n)} = \Pr\{j \in \mathcal{T}\_{\mathbb{R}}(n+1)\} = \Pr\{\mathcal{T}\_{\mathbb{D}} = \mathcal{T}\_{\mathbb{I}}\}, \quad \text{where} \quad j \in \mathcal{T}\_{\mathbb{I}} \tag{16}$$

It is also convenient to re-write this reception probability in terms of the set of active tags T*P*(*n*) by averaging over all values of N (*n*) where T*t*(*n*) ∈ T*P*(*n*):

$$q\_{\vec{j}|\mathcal{T}\_{\mathbb{P}}(n)} = \sum\_{\mathcal{N}(n): \mathcal{T}\_{\mathbb{P}}(n) \in \mathcal{T}\_{\mathbb{P}}(n)} \text{Pr}\{\mathcal{N}(n)\} q\_{\vec{j}|\mathcal{N}(n)} \tag{17}$$

#### **3.4. Tag throughput and stability**

8 Radio Frequency Identification

network state information.

Pr{T*P*(*<sup>n</sup>* + <sup>1</sup>)|T*P*(*n*)} = ∏

**3.2. Markov model**

as follows:

Pr{N (*n*)} = ∏

*j*∈T*P*(*n*),*j*�∈T*P*(*n*+1)

Pr{T*P*} into a one-dimensional vector given by **<sup>s</sup>** = [*s*0,...*sJJ* ]

obtained by solving the following characteristic equation:

be used to calculate relevant performance metrics.

**3.3. Tag detection model**

*k*∈R*<sup>t</sup>*

*pr*,*<sup>k</sup>* ∏*m*�∈R*<sup>t</sup>*

where (·) = 1 − (·). This concludes the definition of the tag activation probability and the

In order to define the Markov model for dynamic analysis of the system, let us now calculate the probability of having a set of active tags T*P*(*<sup>n</sup>* + <sup>1</sup>) in epoch-slot *<sup>n</sup>* + 1 conditional on having the set of active tags T*P*(*n*) during the previous epoch-slot. This transition probability must consider all the combinations of tags that either enter (i.e., they are activated in epoch slot *n*) or leave the set of active tags (i.e., they transmit in epoch slot *n*). This can be expressed

*pt*,*<sup>j</sup>* ∏

*w*∈T*P*(*n*),*w*∈T*P*(*n*+1)

Let us now arrange the probability of occurrence of all the possible sets of activated tags

operator (see Fig. 3). This means that we are mapping the asymmetrical states into a linear state vector where each element represents the probability of occurrence of one different state Pr{T*P*}. In the example given in Fig. 3 we have only two possible tags, where the first system state is given by both tags being active, the second state with only tag 1 as active, the third state with only tag 2 as active, and the fourth state with both tags inactive. Once these states are mapped into the state vector **s**, the transition probabilities between such states (Pr{T*P*(*<sup>n</sup>* + <sup>1</sup>)|T*P*(*n*)) can also be mapped into a matrix **<sup>M</sup>**, which defines the Markov model for state transition probabilities (see Fig. 3). The *i*, *j* entry of the matrix **M** denotes the transition probability between state *i* and state *j*. The vector of state probabilities can thus be

by using standard eigenvalue analysis or iterative schemes. Each one of the calculated terms of the vector **s** can be mapped back to the original probability space Pr{T*P*}, which can then

Before calculating the tag throughput, first we must define the correct reception probability of tag *j* at the reader side conditional on the network state information N (*n*) as follows:

× ∏

*i*�∈T*P*(*n*),*i*∈T*P*(*n*+1)

*pr*,*<sup>m</sup>* ∏ *j*∈T*<sup>t</sup>* *pt*,*<sup>j</sup>* ∏ *i*�∈T*<sup>t</sup>*

*Qi*|T*P*(*n*) ∏

**s** = **Ms**, (15)

*l*�∈T*P*(*n*),*l*�∈T (*n*+1)

*<sup>T</sup>*, where (·)*<sup>T</sup>* is the transpose

*pt*,*w*. (14)

*Ql*|T*P*(*n*)

*pt*,*<sup>i</sup>* (13)

The tag throughput per resolution period can be finally calculated by adding all the contributions over the calculated probability space Pr{T*P*} using the Markov model presented in previous subsections. This can be mathematically expressed as:

$$S\_{\hat{\jmath}} = \sum\_{\mathcal{T}\_{\mathcal{P}}, \hat{\jmath} \in \mathcal{T}\_{\mathcal{P}}} \Pr\{\mathcal{T}\_{\mathcal{P}}\} q\_{\hat{\jmath}|\mathcal{T}\_{\mathcal{P}}}.\tag{18}$$

Now, the throughput per time-slot can be calculated as the ratio of the throughput per resolution period to the average number of time-slots per resolution period:

$$T\_{\dot{j}} = \frac{S\_{\dot{j}}}{\sum\_{\mathcal{T}\_D} \Pr\{\mathcal{T}\_D\} \left\lceil \frac{|\mathcal{T}\_D|}{M} \right\rceil + \Pr\{\mathcal{T}\_D = \mathcal{Q}\}},\tag{19}$$

where |·| is the set cardinality operator and ⌈·⌉ is the ceil integer operator. As a measure of stability we will use the average number of activated tags, which can be calculated as follows:

$$E[|\mathcal{T}\_P|] = \sum\_{\mathcal{T}\_P} \Pr\{\mathcal{T}\_P\} |\mathcal{T}\_P|. \tag{20}$$

A high number of activated tags means that stability is compromised, while a relatively low number indicates that the algorithm is more stable.

#### **4. Optimization and results**

The parameters to be optimized are the vector of reader transmission probabilities **<sup>p</sup>***<sup>r</sup>* <sup>=</sup> [*pr*,1,... *pr*,*K*] *<sup>T</sup>*, the vector of reader transmit powers **<sup>P</sup>***<sup>r</sup>* = [*Pr*,1,... *Pr*,*K*] and the vector of transmission probabilities of the active tags **<sup>p</sup>***<sup>t</sup>* = [*pt*,1,... *pt*,*J*]. The objective of the optimization is the total throughput, so the optimization problem with transmit power constraint can thus be written as follows:

$$\{\mathbf{P}\_{r\prime}\mathbf{p}\_{l\prime}\mathbf{p}\_r\}\_{opt} = \arg\max\_{\{\mathbf{P}\_r\mathbf{p}\_l, \mathbf{p}\_r\}} \sum T\_j \quad \text{s.t.} \quad \mathbf{P}\_{r\prime} < \mathbf{P}\_{r,0} \tag{21}$$

**Figure 3.** Example of the Markov model for a two-tag system.

Since the explicit optimization of the expressions is difficult to achieve, particularly when considering the Markov model proposed in the previous section, in this section we will simplify the optimization problem by applying the previous concepts to an ALOHA protocol implemented at the reader side. This means that tags can be only activated when the readers' transmissions are collision-free. Power levels will be fixed, and the maximum throughput performance will be investigated by simply plotting the surface versus the reader and tag transmission probabilities. At the tag side we will consider the following three options: a conventional ALOHA protocol without MPR, ALOHA with multi-packet reception (simply tagged ALOHA MPR), and the proposed scheme with retransmission diversity and multi-packet reception (tagged NDMA MPR).

**Figure 4.** Throughput (*T*) vs. reader and tag transmissions probabilities (*pr* and *pt*) of a symmetrical ALOHA protocol for reader

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**Figure 5.** Throughput (*T*) vs. reader and tag transmissions probabilities (*pr* and *pt*) of a symmetrical ALOHA protocol for reader

and tag anti-collision assuming interference between readers and tags.

and tag anti-collision assuming no interference between readers and tags.

Two scenarios are considered: one in which tags and readers operate in the same channel, thereby interfering with each other, and the second scenario where readers and tags operate in a synchronized manner in different channels, which eliminates the probability of collision between them. For convenience, let us consider in first instance that all tags and readers experience channel and queuing states that are statistically identical (symmetrical system). A tag activation probability of *q* = 0.7 and a tag detection probability at the reader side of *Q* = 0.95 have been used in the theoretical calculations. A probability of false alarm for the NDMA protocol has been set to *PF* = 0.01. The results have been calculated with *<sup>J</sup>* = 15 tags and *K* = 5 readers.

Fig. 4 shows the results of average throughput *<sup>T</sup>* <sup>=</sup> <sup>∑</sup>*<sup>j</sup> Tj* versus various values of reader and tag transmission probability *pt* and *pr* for a conventional ALOHA protocol without multi-packet reception (*M* = 1) and without retransmission diversity considering full interference between readers an tags. Fig. 5 shows the results of average throughput *T* versus various values of reader and tag transmission probability *pt* and *pr* for a conventional ALOHA protocol without multi-packet reception (*M* = 1) and without retransmission diversity considering no interference between readers an tags. Note how the throughput shape is considerably affected by the interference assumption between readers and tags. Fig. 6 shows the results of average number of tags versus various values of reader and tag transmission probability *pt* and *pr*. Fig. 7 shows the results of average throughput *T*

<sup>58</sup> Radio Frequency Identification from System to Applications An RFID Anti-Collision Algorithm Assisted by Multi-Packet Reception and Retransmission Diversity 11 An RFID Anti-Collision Algorithm Assisted by Multi-Packet Reception and Retransmission Diversity http://dx.doi.org/10.5772/54069 59

10 Radio Frequency Identification

and *K* = 5 readers.

Tag 1 Tag 2

Transition probability between state 3 and state 1 (M31)

**Markov chain** 

Active Active Active Inactive Inactive Active Inactive Inactive

diversity and multi-packet reception (tagged NDMA MPR).

**Figure 3.** Example of the Markov model for a two-tag system.

State

**s=[** s1 s2 s3 s4] Transition probabilities: Matrix**<sup>M</sup>**

Since the explicit optimization of the expressions is difficult to achieve, particularly when considering the Markov model proposed in the previous section, in this section we will simplify the optimization problem by applying the previous concepts to an ALOHA protocol implemented at the reader side. This means that tags can be only activated when the readers' transmissions are collision-free. Power levels will be fixed, and the maximum throughput performance will be investigated by simply plotting the surface versus the reader and tag transmission probabilities. At the tag side we will consider the following three options: a conventional ALOHA protocol without MPR, ALOHA with multi-packet reception (simply tagged ALOHA MPR), and the proposed scheme with retransmission

Two scenarios are considered: one in which tags and readers operate in the same channel, thereby interfering with each other, and the second scenario where readers and tags operate in a synchronized manner in different channels, which eliminates the probability of collision between them. For convenience, let us consider in first instance that all tags and readers experience channel and queuing states that are statistically identical (symmetrical system). A tag activation probability of *q* = 0.7 and a tag detection probability at the reader side of *Q* = 0.95 have been used in the theoretical calculations. A probability of false alarm for the NDMA protocol has been set to *PF* = 0.01. The results have been calculated with *<sup>J</sup>* = 15 tags

Fig. 4 shows the results of average throughput *<sup>T</sup>* <sup>=</sup> <sup>∑</sup>*<sup>j</sup> Tj* versus various values of reader and tag transmission probability *pt* and *pr* for a conventional ALOHA protocol without multi-packet reception (*M* = 1) and without retransmission diversity considering full interference between readers an tags. Fig. 5 shows the results of average throughput *T* versus various values of reader and tag transmission probability *pt* and *pr* for a conventional ALOHA protocol without multi-packet reception (*M* = 1) and without retransmission diversity considering no interference between readers an tags. Note how the throughput shape is considerably affected by the interference assumption between readers and tags. Fig. 6 shows the results of average number of tags versus various values of reader and tag transmission probability *pt* and *pr*. Fig. 7 shows the results of average throughput *T*

1 2 3 4

M31

**1 2 3 4** 

State probability vector **s** 

M14

M41

Characteristic equation

**s**=**Ms**

**Figure 4.** Throughput (*T*) vs. reader and tag transmissions probabilities (*pr* and *pt*) of a symmetrical ALOHA protocol for reader and tag anti-collision assuming interference between readers and tags.

**Figure 5.** Throughput (*T*) vs. reader and tag transmissions probabilities (*pr* and *pt*) of a symmetrical ALOHA protocol for reader and tag anti-collision assuming no interference between readers and tags.

**Figure 6.** Average number of active tags vs. reader and tag transmissions probabilities (*pr* and *pt*).

**Figure 8.** Throughput (*T*) vs. reader and tag transmissions probabilities (*pr* and *pt*) of a symmetrical ALOHA MPR protocol for

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**Figure 9.** Throughput (*T*) vs. reader and tag transmissions probabilities (*pr* and *pt*) of a symmetrical NDMA MPR protocol for

of reader and tag transmission probability *pt* and *pr* for an NDMA MPR protocol considering interference between readers an tags, while Fig. 10 shows the results without considering interference between readers an tags. The results in Fig.9 and 10 show that the proposed NDMA MPR solution considerably outperforms its ALOHA counterparts in both scenarios:

Let us now address an asymmetrical scenario. For this purpose consider that the tag/reader space is divided into two different sets of readers and three different sets of tags. Readers and tags are working in different channels. The first and second sets of tags can only be reached by the first and second sets of readers, respectively. The third set of tags can be reached by both sets of readers. All tags have the same transmission probability *pt* as well as

reader and tag anti-collision assuming no interference between readers and tags.

reader and tag anti-collision assuming interference between readers and tags.

with or without interference between readers and tags.

**Figure 7.** Throughput (*T*) vs. reader and tag transmissions probabilities (*pr* and *pt*) of a symmetrical ALOHA MPR protocol for reader and tag anti-collision assuming interference between readers and tags.

versus various values of reader and tag transmission probability *pt* and *pr* for a conventional ALOHA protocol with multi-packet reception (*M* = 2) and without retransmission diversity considering no interference between readers an tags. Note that the maximum throughput has been considerably improved over the conventional ALOHA protocol without MPR capabilities in Fig. 4. Similarly, Fig. 8 shows the results of average throughput *T* versus various values of reader and tag transmission probability *pt* and *pr* for a conventional ALOHA protocol with multi-packet reception (*M* = 2) and without retransmission diversity by considering no interference between readers an tags. The improvement over the algorithm without MPR in fig. 5 is considerable from almost 0.2 tags/timeslot in the case of ALOHA, to almost 0.4 tags/time-slot in the case of ALOHA MPR, and up to 0.7 tags/time-slot in the case of NDMA MPR. Fig. 9 shows the results of average throughput *T* versus various values <sup>60</sup> Radio Frequency Identification from System to Applications An RFID Anti-Collision Algorithm Assisted by Multi-Packet Reception and Retransmission Diversity 13 An RFID Anti-Collision Algorithm Assisted by Multi-Packet Reception and Retransmission Diversity http://dx.doi.org/10.5772/54069 61

12 Radio Frequency Identification

**Figure 6.** Average number of active tags vs. reader and tag transmissions probabilities (*pr* and *pt*).

reader and tag anti-collision assuming interference between readers and tags.

**Figure 7.** Throughput (*T*) vs. reader and tag transmissions probabilities (*pr* and *pt*) of a symmetrical ALOHA MPR protocol for

versus various values of reader and tag transmission probability *pt* and *pr* for a conventional ALOHA protocol with multi-packet reception (*M* = 2) and without retransmission diversity considering no interference between readers an tags. Note that the maximum throughput has been considerably improved over the conventional ALOHA protocol without MPR capabilities in Fig. 4. Similarly, Fig. 8 shows the results of average throughput *T* versus various values of reader and tag transmission probability *pt* and *pr* for a conventional ALOHA protocol with multi-packet reception (*M* = 2) and without retransmission diversity by considering no interference between readers an tags. The improvement over the algorithm without MPR in fig. 5 is considerable from almost 0.2 tags/timeslot in the case of ALOHA, to almost 0.4 tags/time-slot in the case of ALOHA MPR, and up to 0.7 tags/time-slot in the case of NDMA MPR. Fig. 9 shows the results of average throughput *T* versus various values

**Figure 8.** Throughput (*T*) vs. reader and tag transmissions probabilities (*pr* and *pt*) of a symmetrical ALOHA MPR protocol for reader and tag anti-collision assuming no interference between readers and tags.

**Figure 9.** Throughput (*T*) vs. reader and tag transmissions probabilities (*pr* and *pt*) of a symmetrical NDMA MPR protocol for reader and tag anti-collision assuming interference between readers and tags.

of reader and tag transmission probability *pt* and *pr* for an NDMA MPR protocol considering interference between readers an tags, while Fig. 10 shows the results without considering interference between readers an tags. The results in Fig.9 and 10 show that the proposed NDMA MPR solution considerably outperforms its ALOHA counterparts in both scenarios: with or without interference between readers and tags.

Let us now address an asymmetrical scenario. For this purpose consider that the tag/reader space is divided into two different sets of readers and three different sets of tags. Readers and tags are working in different channels. The first and second sets of tags can only be reached by the first and second sets of readers, respectively. The third set of tags can be reached by both sets of readers. All tags have the same transmission probability *pt* as well as

**Figure 10.** Throughput (*T*) vs. reader and tag transmissions probabilities (*pr* and *pt*) of a symmetrical NDMA MPR protocol for reader and tag anti-collision assuming no interference between readers and tags.

**Figure 12.** Throughput (*T*) vs. reader and tag transmissions probabilities (*pr* and *pt*) of an asymmetrical NDMA protocol for

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This chapter presented a novel algorithm for passive RFID anti-collision based on the concepts of multi-packet reception and retransmission diversity. In addition, the design of the algorithm has been based on a new design paradigm called cross-layer design, where physical and medium access control layers are jointly designed, and where reader and tag anti-collision components are also jointly considered. The proposed Markov model is a new approach for the modeling of RFID networks, as it captures both the activation process given by the operation of readers sending requests to tags, and the tag detection process that results from tags randomly transmitting their information back to the readers that previously activated them. The results for tag throughput showed considerable improvement over conventional ALOHA solutions that have been implemented in current deployments and commercial platforms for RFID. This opens an interesting area for the design of advanced

2 Center for TeleInfrastruktur, Department of Electronic Systems, Aalborg University,

[1] Ahmed, N.; Kumar, R.; French R.S. & Ramachandran, U.; (2007). RF2ID: A Reliable Middleware Framework for RFID Deployment. *IEEE International Parallel and Distributed*

random access protocols for future RFID systems and for the internet of things.

1 Instituto de Telecomunicações, Campus Universitário, Aveiro, Portugal

Ramiro Sámano-Robles1, Neeli Prasad2 and Atílio Gameiro1

*processing Symposium*, March 2007, pp. 1-10.

reader and tag anti-collision without interference between readers and tags.

**5. Conclusions**

**Author details**

Aalborg, Denmark

**References**

**Figure 11.** Throughput (*T*) vs. reader and tag transmissions probabilities (*pr* and *pt*) of an asymmetrical ALOHA protocol for reader and tag anti-collision without interference between readers and tags.

all readers transmit with the same parameter *pr*. A tag activation probability of *q* = 0.7 and a tag detection probability at the reader side of *Q* = 0.95 have been used in the theoretical calculations. A probability of false alarm for the NDMA protocol has been set to *PF* = 0.01. The results of Fig. 11 and Fig. 12 have been obtained using three groups of tags with *<sup>J</sup>*<sup>1</sup> = 3,*J*<sup>2</sup> = 5 and *<sup>J</sup>*<sup>3</sup> = 7 tags, and two groups of readers with *<sup>K</sup>*<sup>1</sup> = 5 and *<sup>K</sup>*<sup>2</sup> = 10 readers. While Fig. 11 shows the results of an ALOHA protocol without MPR capabilities (*M* = 1), Fig. 12 shows the results of the proposed NDMA protocol with *M* = 1. In both cases, the readers and tags are assumed to transmit in different channels, thereby avoiding interference between their transmissions. It can be observed the significant gain provided by the NDMA protocol for all values of *pt* and *pr*.

**Figure 12.** Throughput (*T*) vs. reader and tag transmissions probabilities (*pr* and *pt*) of an asymmetrical NDMA protocol for reader and tag anti-collision without interference between readers and tags.
