Here m is the maximal retransmission number
and m
’
is the maximal backoff time number. In
addition,
and
'
max min
[ ] 2 [ ]
m
CW i CW i
.
Let
be the probability of i-th class to send data
frames within a given time slot.
is the
probability to send information in the website within
the time interval.
is the detection probability of
busy channel for the i-th traffic class during the
backoff time cycle. The
indicates as the conflict
probability in sending data when the backoff timer
is countering to 0.
The next step transition probability in the
Markov chain mode is formulated as following:
,
,
,,
,0 ,0
{ , , , , 1} 1 , [0, 2], [0, ]
{ , , , , } , [0, 1], [0, ]
{ , , , 1,0} / , [0, 1], [1, ]
{ ,0, , ,0} (1 ) / , [0, 1], [0,
it i j
it i j
io i j i j
io i i
p i j k i j k p k W j m
p i j k i j k p k W j m
p i j k i j p W k W j m
p i k i j p W k W j
,0 ,0
1]
{ ,0, , ,0} 1/ , [0, 1]
ii
m
p i k i m W k W
(3)
This equation displays five possible conditions.
denotes the probability of the countdown timer
in the current traffic state. It keeps in certain level of
countdown probability while not detecting the data
flow in the network channel.
is the original
probability of countdown timer, which begins to
gradually decrease in the uniform interval of idle
time. From 3-subequation, the original probability
of the i+1-th countdown timer is randomly selected
in the range of
when the collision is
happen at the i-th transition state. It presents in the
above description of 4-subequation. After the data
frame is successfully transformed, the probability of
countdown timer for new frame is randomly
selected in the range of
. If the network
channel reaches the maximum count downing
number, all traffic flames will be withdrawn. New
frames retransmit in the transmission probability of
. Based on the internal scheduling algorithm
of the EDCA machine in the traffic channel, the
delivering probability of the packets is described as
formulas:
2
0 1 0 1
0
(1 ) ... (1 )
N
bN
j
(4)
The probability formulas of Markov chain model
is discussed in the previous formulas. It is shown
that the traffic flow in the countdown is idle. The
is the probability that the other different sites
of networks does not send to the traffic flow in this
website. The probability
presents that
1
1
1 (1 ) (1 )
0,
N
n
Pt
it
bh
h h i
(5)
A primary traffic service successfully transmits
the network packet when the timer is counting down
to 0. The unsuccessful transmission probability is
1
1
0
1 (1 ) (1 )
i
bn
io h
h
Pt
(6)
Suppose that
,,
lim { ( , ) , ( , ) }
i j k
t
b P s i t j b i t k
, it denotes the
distributed steady state of the Markov chain model.
Thus,
, ,0 ,0,0
, [0, ]
j
i j io i
b p b j m
(7)
And
1
,
, [1, 1]
,
, ,0
1
,
,,
,0
,0,0
Wk
ij
b k W
ij
ij
Wp
b
i j it
i j k
j
p b k
io
i
(8)
The stationary condition is satisfied by
1
,
1
,,
0
0
W
ij
m
b
i j k
j
k
(9)
Therefore, the calculation of
denotes as
formulas (10),
,0,0
1 ' 1
'1
0
1
' 1 1
'
2(1 2 )(1 )(1 )
2(1 )(1 2 )(1 ) (1 (2 ) )(1 ) (1 )(1 2 )
,'
2(1 2 )(1 )(1 )
2(1 )(
2(1 2 )(1 )(1 )
(1 2 )(2 1)( )
i
io io it
mm
m
io it i io io io
io io
io io it
m
io
io io it
mm
m
io
io io
b
ppp
p p W p p p
mm
ppp
ppp
pW
pp
p
pp
+
1
0
1
1 2 )(1 ) (1 (2 ) )(1 )
2(1 2 )(1 )(1 )
, '
(1 )(1 2 )
m
io it i io io
io io it
m
io
io
p p W p p
ppp
mm
p
p
-
(10)
Let the i-th class of traffic sending data for a
given time slot of the probability (
) presents in the
formula (11)
, ,0 ,0,0
, [0, ]
j
i j io i
b p b j m
(11)
1
, , ,0,0
0
1
1
m
m
io
i i j k i
j
io
p
bb
p
(12)
WSEAS TRANSACTIONS on COMMUNICATIONS
Ben-Bin Chen, Hua-Ching Chen, Dong-Hui Guo