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that we investigate is as follows: Given a Signal-to-
Noise-Ratio (SNR), a Burst and an amount of N data
bits to transmit in the Burst, what is the most efficient
way to allocate PDUs in the Burst so that the num-
ber of data bits that are transmitted successfully in the
Burst is maximized. In the second problem we are
given an SNR and N data bits to transmit. We look for
the optimal Burst size in relation to three performance
criterion. In the first one we look for the Burst size
that maximizes the number of successfully transmit-
ted data bits. In the second criterion we look for the
Burst size that is most efficiently used, i.e. the relation
between the number of successfully transmitted data
bits in the Burst, to the Burst size, is maximized. In
the third criterion we look for the Burst size that max-
imizes the number of successfully transmitted data
bits per transmission slot. The above problems can
arise, for example, when a MS is allocated a Burst
and wants to use it in the most efficient way, and when
the BS needs to allocate Bursts to MSs after receiving
their traffic needs. Notice that a solution to the above
problems is important in relation to a PHY layer that
is based on CTC because the Burst length determines
the FEC blocks that it contains. Next, PDUs are de-
fined over these FEC blocks. Therefore, there is a di-
rect correlation between the Burst size, the PHY layer
and the MAC layer, which influences the efficiency of
the transmissions.
The performance of IEEE 802.16e/WiMAX sys-
tems has been extensively investigated. Due to a
space limit we do not give examples for relevant stud-
ies. The interested reader can find in (So-In et al.,
2009) and (Sekercioglu et al., 2009) a very good sur-
vey on WiMAX performance. Most of the papers deal
with scheduling methods and the efficiency of trans-
port layer protocols in IEEE 802.16e systems. These
papers assume the assignment of Bursts to MSs.
However, they do not consider the issue of efficient
transmissions in the Bursts. The only works that we
are aware of, and that deal with the mutual influence
between the PHY layer budding blocks (FEC blocks)
and the MAC layer PDUs in IEEE 802.16e/WiMAX
systems are (Martikainen et al., 2008), (Alpert et al.,
2012b),(Alpert et al., 2010) and (Alpert et al., 2012a).
These papers however, handle different problems than
those in this paper.
The rest of the paper is organized as follows: In
Section 2 we answer the question of which is the most
efficient way to transmit a given amount of data bits in
a given Burst. In Section 3, given an amount of data
bits to transmit, we look for the optimal Burst sizes
according to the criterion mentioned above.
2 THE OPTIMAL
TRANSMISSION OF DATA IN A
GIVEN BURST
2.1 Problem Description
We are given an SNR, a Burst of S slots, an amount
of N data bits to transmit and the number of the PDU
overhead bits. We want to decide on the best MCS
to use in the Burst, and how to allocate PDUs in the
Burst, such that an efficiency criterion, denoted D-
Goodput, that we define next, is maximized. In Sec-
tions 2.2-2.4 we assume all the above parameters, and
also a MCS. These parameters, all together, determine
the number L of FEC blocks in the Burst, their size
F in bits and a FEC success probability p. Then, in
Sections 2.5 and 2.6 we show how to choose the best
MCS.
2.2 Definition of the D-Goodput
We are given: A Burst of L FEC blocks, every FEC
block contains F bits, every FEC block has proba-
bility p to arrive successfully at the receiver, N data
bits to transmit in the Burst and every PDU has O
overhead bits. We assume that O < F since according
to the IEEE 802.16e/WiMAX standard (IEEE, 2005),
the total length of the overheadfields in a PDU is most
likely to be smaller than one FEC block.
We want to transmit the N data bits in the Burst
such that the Data Goodput (D-Goodput) is maxi-
mized. The D-Goodput is defined as follows. Let S
be the number of data bits in the Burst, out of the N
bits, that arrive successfully at the receiver. Then: D-
Goodput =
S
N
. Notice that the D-Goodput it computed
using N in the denominator,because the Burst is given
and we want to use it to transmit to the receiver as
many data bits as possible.
Data bits are transmitted within PDUs and so we
need to decide on how many PDUs shall be allocated
in the Burst, their length and their location. We call
these decisions the division of the Burst into PDUs.
2.3 Optimal Division into PDUs
In order to find the optimal division of the Burst into
PDUs, we prove the following 5 Theorems. We omit
their proofs due to a space limit.
Theorem 1. There is an optimal division of the Burst
into PDUs such that every PDU begins and ends at
exact boundaries of FEC blocks.
Theorem 2. Assume that for a given N > L(F − O)
an optimal division of the Burst contains I PDUs. Let
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