98 ◾ Advances in Communications-Based Train Control Systems
that when one SA on the train is at the coverage edge, the other one is still in the
coverage of another AP. We name this system as the data communication system
with redundancy and no backup link.
e second proposed system with redundancy conguration is shown in
Figure6.3. In this system, only one AP with directional antenna is used in each
location. Two directional antennas, i.e., head antenna and tail antenna, are con-
nected with independent SA at each end of the train. All the SAs and APs in the
system have the same SSID. Compared to the rst proposed system, the AP space
in the second system is halved to make sure that any SA on the train is on the
coverage of two APs. ere is one active link and backup link for both the head
SA and the tail SA. e corresponding backup link will become active when the
active link fails. e communication will be interrupted only if all four links fail.
We name this system as the data communication system with redundancy and
backup link.
We take the data communication system with redundancy and no backup
link (as shown in Figure6.2) as an example to illustrate how to implement it in
practice. ere are normally two active links between the train and the ground.
In real systems, we connect each end of the active link with a network adapter.
Coverage of AP21
Coverage of AP11
Coverage of AP12
Coverage of AP22
Ground equipment
NA1 NA2
NA1
Train equipment
NA2
Backbone network 1
Backbone network 2
AP11 AP21 AP12 AP22
SA1 SA2 SA1 SA2
SA1 SA2
SA1 SA2
Figure6.2 First proposed data communication system with redundancy and no
backup link.
Communication Availability in Communications-Based TCSs ◾ 99
As shown in Figure6.2, there are two network adapters for both the train and the
ground equipment. A high-level protocol needs to be designed between the train
and the ground to keep the data from one active link and delete the redundant
data from the other active link. Communication will not be interrupted if only
one of the active links fails. e availability of these systems will be analyzed in
Section 6.4.
6.4 Data Communication System Availability Analysis
System availability is the probability that the system is operating at a specied time
[19]. It is primarily used in repairable systems, where brief interruptions in service
can be tolerated.
e WLAN-based data communication system is modeled as a CTMC [8]. We
dene the link state as
si
jk=
(,
,)
, where
i
is the number of active links,
j
is the
number of failed links caused by deep channel fading, and
k
is the number of failed
links caused by hando.
According to CTMC theory [8], let
Pt
s
()
be the unconditional probability of
the CTMC being in state
s
at time
t
. en the row vector,
Pt Pt
tP
t
s
() (),,..., ()
=
[(
)]
12
P
(6.1)
represents the transient state probability vector of the CTMC. Given
P(0
)
, the
behavior of the CTMC can be described by the following Kolmogorov dierential
equation:
d
dt
Pt Pt Q() ()
=×
(6.2)
Coverage of AP1
Coverage of AP2
Coverage of AP3
Coverage of AP4
Backbone network
AP1
SA1 SA2
AP2
AP3
AP4
Figure6.3 Second proposed data communication system with redundancy and
backup link.
100 ◾ Advances in Communications-Based Train Control Systems
where:
P(0)
represents the initial probability vector (at time
t = 0
) of the CTMC
Q
is the innitesimal generator matrix
Qq
ss
=
′
[]
represents the transition rate from state
s
to state
′
s
e diagonal elements are
qq
ss ss ss
=−∑
′
≠
′
e CTMC model for data communication system with basic conguration is
shown in Figure6.4. In this model, state
(1
,0,0
)
is the only service state where the
only existing link is active,
λ
1 is the deep channel fading rate,
λ2
is the hando
rate, μ1 is the fading recovery rate, and μ2 is the hando recovery rate.
e innitesimal generator matrix for this model is
Q
1=
(1 2)
12
11
0
202
−+
−
−
λλ λλ
µµ
µµ
(6.3)
With the same link state
si
j
=
(, , k
)
, the CTMC model for the rst proposed data
communication system with redundancy and no backup link is shown in Figure6.5.
(0,1,0) (0,0,1)
(1,0,0)
λ1
μ1
λ2
μ2
Figure 6.4 CTMC model for the data communication system with basic
conguration.
λ1
μ1
λ2
μ2
λ2
μ2
λ1
μ1
λ1
μ1
(0,1,1)
(0,2,0)(1,0,1)
(1,1,0)
(2,0,0)
Figure6.5 CTMC model for the rst proposed data communication system with
redundancy and no backup link.
Communication Availability in Communications-Based TCSs ◾ 101
In this model, (2,0,0), (1,0,1), and (1,1,0) are service states where at least one link
is active. As explained in Section 6.2, state (0,0,2) does not exist because the head
SA and the tail SA will not hando at the same time. States (0,1,1) and (0,2,0) are
out-of-service states for lack of active links.
e innitesimal generator matrix for this model is
Q2 =
−+
−+
−++
−
(1 2) 12
00
2(12)0
10
10(1 21)2 1
01 2(1
λλ λλ
µλ
λλ
µλλµ
λλ
µµµ++
−
µ
µµ
2) 0
00
10
1
(6.4)
Next, we extend the link state
si
jk=
(,
,)
to
shijk
=
(,,,
)
, where
h
is the total
number of active links and backup links, μ
3
is the rate of successful transitions
from backup links to active links. e CTMC model for the second proposed data
communication system with redundancy and backup link is shown in Figure6.6.
In this model,
(2
,0,1,1
)
,
(1
,0,2,1
)
, and
(0,0,4,0)
are out-of-service states where
no active link exists. All the other states are service states that have at least one
activelink.
e innitesimal generator matrix for this model is
Q
A
A
A
A
3
10 0
130
10 1
00 1
00 20
0000
0002
00 0
00 00
00 20
0
1
2
3
43
=
λ
µµ
µλ
µ
λ
λ
µ
λ
000 0
00 01
0
0
0
0
0000
0200
0002
0000
0200
00 10
00 01
0100
λ
µ
µ
µ
µ
µ
µ
µ
1100
00 0
000
0000
0
0
0
1
5
6
7
A
A
A
−
µ
(6.5)
where:
A
1
(1
2)=− +λλ
A
2
(1
32
)
=− ++µµ λ
102 ◾ Advances in Communications-Based Train Control Systems
A
3
(1
21)=− ++λλ µ
A
4
(1
32
)
=− ++µµ λ
A
5
(1 2)
=− +µµ
A
6
(1
2)=− +µµ
A
7
(1
2)=− +µµ
In order to derive the system availability, let π
n
be the steady-state probability of
state
n
for the three CTMCs. π
n
shall satisfy the following equations [8]:
(, ,..., )(0,0,...,0)
12
ππ π
N
Q
×=
(6.6)
n
N
n
=1
1
∑
=π
(6.7)
where
n
is the number of states for the three CTMCs.
e steady-state probability can be obtained by resolving the previous two
equations, and the system availability for each of the three systems can be derived
from the steady-state probability as follows:
A
wW
w
=
∈
∑
π
(6.8)
where
W
is the aggregation of service states.
e system unavailability
UA
is then derived as
UA =1
−
A
(6.9)
μ1
λ1
λ1
λ1
λ1
μ3 μ3
μ1
μ
1
μ1
μ1
λ1
μ
2
μ2
μ2
μ2
μ2
μ2
λ2
λ2
λ2
λ2
λ2
λ2
μ1
μ
1
λ1
λ1
μ1
μ
1
(4,2,0,0)
(3,1,0,1)
(3,1,1,1) (3,2,1,0) (2,1,2,0) (2,2,2,0)
(1,1,2,1)
(1,1,3,0)
(0,0,3,1)(2,1,1,1)(2,0,1,1) (1,0,2,1)
(0,0,4,0)
Figure 6.6 CTMC model for the rst proposed data communication system with
redundancy and backup link.
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