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In this section, we analyze the performance of tag anticollision protocols. All four tree- based protocols (binary tree, query tree, ABS, and AQS) use binary search tree as their searching method to identify tags. Therefore, we choose binary tree protocol as the representative of tree-based protocols. On the other hand, adaptive framed slotted ALOHA is mainly researched among the probabilistic tag anticollision protocols. Hence, we compare the performance of adaptive framed slotted ALOHA with the binary tree protocol.

9.5.1 Average Slot Delay Analysis

In general, the performance of anticollision protocols is represented by average slot delay. Average slot delay can be defined as the expected number of slots consumed for identify- ingmtags. In most existing protocols to date which are related with RFID tag, anticollision aimed to minimize average slot delay or maximize average slot throughput. In similar vein, we will present the performance of tag anticollision protocols in slot delay in analytic manner, dividing them into binary tree protocol and adaptive framed slotted ALOHA.

TABLE 9.1

Notations

Symbol Description

F Frame size

Ntag Estimated number of tags

S Number of readable slots

C Number of collided slots

I Number of idle slots

SEXP(F,Ntag) Expected value of readable slot given frame size and the number of tags

CEXP(F,Ntag) Expected value of collided slot given frame size and the number of tags

Theorem 9.1

The average slot delay of binary tree protocol approximates to 2.8853m. Proof

The proof of this theorem follows an analysis presented in Ref. [20], which is based on a research that investigated on tree-based multiple access channel [5]. The average slot delay

T(m), which is the expected number of consumed slots until identifyingmtags, in binary tree protocol is as follows:

T(m)¼C(m)þI(m)þm, (9:5) whereC(m) andI(m) denote the average number of collided slots and idle slots, respect- ively. Themin the equation accounts for themslots corresponding to readable slots. Since the random number generator in a tag for binary tree protocol follows uniform distribu- tion, the probability thatkout ofmtags try to reply at levelLin the tree is given by the binomial distribution as following:

P(X¼kjm,L)¼ m

k p

k_ð1pÞmk_{, (9:6)}

wherep¼1L

2. Using this, we get the probabilities that a slot at levelLof the tree is readable,

idle, or collision:

Pread(X¼1jm,L)¼(1p)m, (9:7)

Pidle(X¼0jm,L)¼mp(1p)m1, (9:8)

Pcoll(X2jm,L)¼1Pread(X¼1jm,L)Pidle(X¼0jm,L)

¼1(1p)mmp(1p)m1: (9:9)

For slots are visited only when their parent experiences collisions, we can write the average slot delay by the summation of the expected number of nodes whose parents are collided slots in all levels:

tTS(m)¼ X1 L¼0 X 2L₁ i¼0 Pcoll(X2jm,L1) ¼1þ2X 1 L¼0 2LPcoll(X2jm,L): (9:10)

Substituting from Equation 9.9 gives _tTS(m)¼1þ2X1

L¼0

2L1(1p)mmp(1p)m12:885m: (9:11)

In case of framed slotted ALOHA, not like tree-based protocols, there can be diverse variant protocols according to which frame adaptation algorithm is used. However, since the optimal condition of slotted ALOHA channel with given number of nodes has been revealed in the other work [9], we will assume an optimal framed slotted ALOHA protocol.

Theorem 9.2

The average slot delay of optimal framed slotted ALOHA protocol approximates toe3m. Proof

When the probability that m tags transmit to a slot is p, the successful transmission probability of a tag,S, is given by,

S¼mp(1p)m1: (9:12)

Due to the concavity of the equation, we canfind the optimal condition through thefirst derivative with respect topas follows:

ds

dp¼m(1p)

m1_m(m1)p(1p)m2_{¼0: (9:13)}

Using this, the optimal condition is given byp¼1

m. When frame size isL,pis represented

asp¼1

L. Therefore, the relationship ofLandmwhen it is in the optimal condition isL¼m.

Under this condition, the (nþ1)th frame sizeLnþ1in adaptive framed slotted ALOHA is denoted as the following relationship:

Lnþ1 ¼m

Xn

i¼0

LiS*, (9:14)

whereS* is the optimal utilization of a frame as follows:

S*¼ lim m!1P(X¼1jL¼m)¼mlim!1 1 1 L _n1 ¼1 e: (9:15)

To know the asymptotic property of frame size,nis taken to infinity, then:

lim n!1Lnþ1¼nlim!1 m Xn i¼0 LiS* ! : (9:16)

Intuitively, as time goes by, the number of unidentified tags will decrease. By the optimal condition, frame size will decrease as well. Hence, lim

n!1Lnþ1converges to zero, andfinally

we get the following relationship:

X1

i¼0

Li¼em: (9:17)

Two theorems in this section allow us to measure the exact performance of tag anticollision protocols. As a result, it seems that adaptive framed slotted ALOHA protocol is little better than binary tree protocol. However, as this result gives us only asymptotic property, it is needed to confirm that in reality. We conducted a Monte Carlo simulation to make sure what we have analyzed. Figure 9.8 depicts the measurement of the average slot delay of the two protocols. The adaptive framed slotted ALOHA protocol with DFSA is assumed as the optimal one in our simulation. The difference between two protocols is not obvious when the number of tags is low, but it clearly increases as the number of tags increases. The reason for this observation is that they converge to the asymptotic average delay values as

the number of tags is getting bigger. Nevertheless, we cannot determine which protocol is the best in reality because, in the RFID standards, the frame sizes of probabilistic protocols are formatted to the powers of two, not integer values. Of course, the protocols with the powers of two frames will show the same average slot delay asymptotically, but its converging speed will be fairly degraded.