108 ◾ Advances in Communications-Based Train Control Systems
In the above equations,
Ω
ϒ
()m
is the set of states that can be reached from
m
by ring a competitive exponential transition and
Ω
Γ
()
m is the set of states
that can be reached by ring the deterministic transition.
Qm()
is the innite
generator matrix, which is formed as follows: for any
n ∈Ω
, the rate from
n
to
′
∈n Ω
is given by
λ
(nn
,)
′
, and if
nm∉Ω()
, the rates out of a mark-
ing
n
are zeros.
∆∆=
′
[(
,)
]nn is the branching probability matrix, where
∆
(, ){nn Pn n
′
=
′
next markingiscurrent markingisand thedetermi
n-
|
istic transition fires}.
2. Calculate the local kernel
Et Et mn
Et
mn
mn
mn
() () (,
)(
)
,
,
,
=
[]
∈=
Ω ,where
PD
tnTt
Ym
{(),
>}
10
= =
| , as follows:
a. For state
m
when
ΓΦ
()m
=
,
Et
mn mn
m
t
,,
()=
−
δ e
Λ
(6.18)
where
δ
mn
,
is the Kronecker function dened by δ
m,n
= 1 if
mn=
and
0
otherwise.
Γ()
m is the set of deterministic transitions enabled
in state
m
, and
ΓΦ
()m
=
means that no deterministic transition is
enabled in
m
.
λ
(,
)mn
is the transition rate from marking
m
to
n
, and
Λ
Ω
m
n
mn=
∈
∑
λ(,)
.
b. For state
m
when
Γ
()
md=
with
τ
being the ring time of transition
d
,
if
nm∈Ω()
,
Et
t
t
mn
Qmt
mn
,
()
,
()
0
=
<
≥
e τ
τ
(6.19)
if
nm∉Ω()
,
Et
mn,
()
=0
(6.20)
3. Calculate the one-step transition probability matrix
PP mn
mn mn
=∈
[](, )
,,
Ω
for the EMC by the fact
PK=
(∞). erefore, we get
P
mn
,
as follows:
a. For state
m
when
ΓΦ
()m
=
,
Pt
mn
mn
m
m
m
,
()
00
(,)
0
=
=
>
Λ
Λ
Λ
λ
(6.21)
b. For state
m
when
Γ
()
md=
with
τ
being the ring time of transition
d
,
if
nm∈Ω
ϒ
()
but
nm∉Ω
Γ
()
:
Pt
mn
Qm
mn
,
()
,
()=
e
τ
(6.22)
Communication Availability in Communications-Based TCSs ◾ 109
if
nm∈Ω
Γ
()
but
nm∉Ω
ϒ
()
:
Pt
mn
mn
mm
Qm
mm
,
()
()
,
()
(,)
=
′
′
∈
′
∑
Ω
∆e
τ
(6.23)
if
nm∈Ω
Γ
()
but
nm∈Ω
ϒ
()
:
Pt
mn
mn
Qm
mn
mm
Qm
mm
,
()
,
()
()
,
()
(,)
=
+
′
′
∈
′
∑
ee
ττ
Ω
∆
(6.24)
if
nm∉Ω
Γ
()
but
nm∉Ω
ϒ
()
:
Pt
mn,
() 0
=
(6.25)
4. Calculate the steady-state probability
ξξ= ()
j
for EMC from the following
equation:
ξξξ=⋅ =
P
j
,1
(6.26)
5. Based on the kernel
Kt Kt
mn
mn
() [()]
,,
=
obtained in step 1, calculate the aver-
age time between epochs μ
m
ET
Ym= ={}
10
|
as,
a. For state
m
when
ΓΦ
()m
=
,
µ
m
m
ET Ym===
{}
1
10
|
Λ
(6.27)
b. For state
m
when
Γ
()=
md
with
τ
being the ring time of transition
d
,
µ
τ
m
n
Qmt
mn
ET Ym t===
∈
∑
∫
{}
10
()
,
0
|
Ω
ed
(6.28)
6. Based on the local kernel
Et
Et
mn
mn
() ()
,
,
=
[]
obtained in step 2, calculate the
expectation
α
mn
En=
[timespent by themarking processinstate du
ring
(0,T
1
) ∣ Y0 = m] as,
α
mn mn
PDtnTtYmtE
tt
== ==
∞∞
∫∫
{(),>} ()
10
0
,
0
|d d
(6.29)
7. Based on the parameters
ξξ= ()
j
, μ
m
, and
α
mn
, the distribution
pp
j
= ()
of
the steady-state probabilities of the MRGP, which is also the fraction of time
the marking process spends in state
j,
is given by [25]
110 ◾ Advances in Communications-Based Train Control Systems
pPDt Ym
j
t
kk
j
k
kk
k
===
→∞
∈
∈
∑
∑
lim
|{()}
0
ξα
ξµ
Ω
Ω
(6.30)
Once we get the steady-state probabilities of the MRGP
pp
j
= ()
, the system avail-
ability can be calculated by adding the probability of the states or markings where
the tokens in place
P
active
are not zero.
6.6 Numerical Results and Discussions
In this section, we rst present the numerical parameters. Numerical results are
presented to show the soundness of our availability models. Finally, the availability
improvement of the proposed systems is presented.
6.6.1 System Parameters
In CBTC systems, APs are usually deployed to make their coverage areas overlap
with each other in order to decrease the shadow zone. In our numerical examples,
the distance between two successive APs is
l
= 200
m
for the rst proposed data
communication system with redundancy and no backup link, then the distance is
l
=100
m
for the second proposed data communication system with redundancy
and backup link. e hando rate
λ2
is determined by the train velocity. Given the
distance
l
between two successive APs and the train velocity
v
, the average time
between two successive handos is
lv
/
, which gives
(1
/2)=(/
)λ
lv.
Similarly, the hando end rate μ
2
is determined by the hando time. Given
the average hando time
T
h
, it can be calculated as
µ2/=
lT
h
. Other parameters we
used for the CTMC model are shown in Table6.1.
For the DSPN model parameters, according to [22], the average transition time
for an exponentially distributed transition is the reciprocal of the transition rate.
erefore, all the exponentially distributed transition times in our DSPN models
Table 6.1 Parameters Used in Numerical Examples
Notation Denition Value
λ1
Channel fading rate 0.01s
–1
λ2
Handoff rate Determined by train
velocity
μ1
Channel fading
recovery rate
0.2s
–1
μ3
Backup to active rate 0.2s
–1
Communication Availability in Communications-Based TCSs ◾ 111
such as
T
fading
and
T
fading recovery_
can be calculated from the parameters in the CTMC
models described above. And for the deterministic transitions
T
handoff
, as it indi-
cates the average time between two successive handos, it can be calculated as
Tl
v
fhandof
=
/
.
6.6.2 Model Soundness
Given all the parameters of the CTMC and DSPN models, Figure6.10 illustrates
the unavailability in the two proposed models for dierent congurations. As we
can observe from the gure, the unavailability dierence between the two models
changes with model parameters, and the dierence is not signicant in most cases.
Compared to the DSPN model, the unavailability error in the CTMC model is due
to the fact that the time between two successive handos is assumed to be exponen-
tially distributed in the CTMC model, which is not very accurate in real systems.
6.6.3 Availability Improvement
e unavailability for the three data communication systems when the train veloc-
ity changes from 20 to 100 km/h is shown in Figure6.11. For comparison, we
0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19
0.20
10
−4
10
−3
10
−2
10
−1
10
0
Average handoff time (seconds)
Unavailability
Basic configuration (CTMC model)
Redundency configuration without backup link (CTMC model)
Redundency configuration with backup link (CTMC model)
Basic configuration (DSPN model)
Redundency configuration without backup link (DSPN model)
Redundency configuration with backup link (DSPN model)
Figure 6.10 Comparison of CTMC and DSPN model solutions for different
redundancy congurations.
112 ◾ Advances in Communications-Based Train Control Systems
calculate the unavailability of the system with semi-redundancy studied in [5],
where only one SA is installed on the train. As shown in the gure, the system
unavailability increases with the train velocity. is is because the larger the veloc-
ity, the more frequently the hando occurs, which increases the hando rate.
Consequently, the unavailability of all the three systems is increased. Particularly,
the unavailability of the existing system with basic conguration is more than 5%
with dierent train speeds. Such a high unavailability would not be acceptable in
practice. By contrast, the proposed data communication systems can decrease the
unavailability below 1% with a wide range of the train velocity. From Figure6.11,
we can also observe that the second proposed data communication system with
redundancy and backup link has better availability than the rst one, although
handos occur more frequently in the second system. is is because the prob-
ability of successful transitions from a backup link to an active link is much higher
than the probability of deep channel fading. When a hando happens at one of
the active links, the corresponding backup link will become active before the other
active link encounters deep channel fading. erefore, the system will become
unavailable only when all the four links fail. Our proposed redundancy congura-
tions always give better availability performance compared to the semi-redundancy
scheme as well.
20 30 40 50 60 70 80 90
100
Train speed (km/h)
Basic configuration
Semi−redundancy configuration
Redundency configuration without backup link
Redundency configuration with backup link
10
−4
10
−3
10
−2
10
−1
10
−5
Unavailability
Figure 6.11 Unavailability of the three WLAN-based data communication
systems.
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