172 RFID Handbook: Applications, Technology, Security, and Privacy
Theorem 9.2
The average slot delay of optimal framed slotted ALOHA protocol approximates to e3m.
Proof
When the probability that m tags transmit to a slot is p, the successful transmission
probability of a tag, S, is given by,
S ?? mp(1 p)m1: (9:12)
Due to the concavity of the equation, we can ?¬?nd the optimal condition through the ?¬?rst
derivative with respect to p as follows:
ds
dp ?? m(1 p)m1 m(m 1)p(1 p)m2 ?? 0: (9:13)
Using this, the optimal condition is given by p ?? 1m
. When frame size is L, p is represented
as p ?? 1L
. Therefore, the relationship of L and m when it is in the optimal condition is L??m.
Under this condition, the (n??1)th frame size Ln??1 in adaptive framed slotted ALOHA is
denoted as the following relationship:
Ln??1 ?? m X n
i??0
Li S*, (9:14)
where S* is the optimal utilization of a frame as follows:
S* ?? lim
m!1
P(X ?? 1jL ?? m) ?? lim
m!1
1
1
L n1
??
1
e
: (9:15)
To know the asymptotic property of frame size, n is taken to in?¬?nity, then:
lim
n!1
Ln??1 ?? lim
n!1
m X n
i??0
Li S* !: (9:16)
Intuitively, as time goes by, the number of unidenti?¬?ed tags will decrease. By the optimal
condition, frame size will decrease as well. Hence, lim
n!1
Ln??1 converges to zero, and ?¬?nally
we get the following relationship:
X1
i??0
Li ?? e m: (9:17)
Two theorems in this section allow us to measure the exact performance of tag anticollision
protocols.
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