<s>
SP-DEVS	B-Application
abbreviating	O
"	O
Schedule-Preserving	O
Discrete	B-Application
Event	I-Application
System	I-Application
Specification	I-Application
"	O
is	O
a	O
formalism	O
for	O
modeling	O
and	O
analyzing	O
discrete	O
event	O
systems	O
in	O
both	O
simulation	O
and	O
verification	O
ways	O
.	O
</s>
<s>
SP-DEVS	B-Application
also	O
provides	O
modular	O
and	O
hierarchical	O
modeling	O
features	O
which	O
have	O
been	O
inherited	O
from	O
the	O
Classic	O
DEVS	B-Application
.	O
</s>
<s>
SP-DEVS	B-Application
has	O
been	O
designed	O
to	O
support	O
verification	O
analysis	O
of	O
its	O
networks	O
by	O
guaranteeing	O
to	O
obtain	O
a	O
finite-vertex	O
reachability	O
graph	O
of	O
the	O
original	O
networks	O
,	O
which	O
had	O
been	O
an	O
open	O
problem	O
of	O
DEVS	B-Application
formalism	O
for	O
roughly	O
30	O
years	O
.	O
</s>
<s>
To	O
get	O
such	O
a	O
reachability	O
graph	O
of	O
its	O
networks	O
,	O
SP-DEVS	B-Application
has	O
been	O
imposed	O
the	O
three	O
restrictions	O
:	O
</s>
<s>
Thus	O
,	O
SP-DEVS	B-Application
is	O
a	O
sub-class	O
of	O
both	O
DEVS	B-Application
and	O
FD-DEVS	B-Application
.	O
</s>
<s>
These	O
three	O
restrictions	O
lead	O
that	O
SP-DEVS	B-Application
class	O
is	O
closed	O
under	O
coupling	O
even	O
though	O
the	O
number	O
of	O
states	O
are	O
finite	O
.	O
</s>
<s>
This	O
property	O
enables	O
a	O
finite-vertex	O
graph-based	O
verification	O
for	O
some	O
qualitative	O
properties	O
and	O
quantitative	O
property	O
,	O
even	O
with	O
SP-DEVS	B-Application
coupled	O
models	O
.	O
</s>
<s>
The	O
above	O
controller	O
for	O
crosswalk	O
lights	O
can	O
be	O
modeled	O
by	O
an	O
atomic	O
SP-DEVS	B-Application
model	O
.	O
</s>
<s>
To	O
captured	O
the	O
dynamics	O
of	O
an	O
atomic	O
SP-DEVS	B-Application
,	O
we	O
need	O
to	O
introduce	O
two	O
variables	O
associated	O
to	O
time	O
.	O
</s>
<s>
shows	O
a	O
state	O
trajectory	O
associated	O
with	O
an	O
event	O
segment	O
of	O
the	O
SP-DEVS	B-Application
model	O
shown	O
in	O
Fig	O
.	O
</s>
<s>
3	O
shows	O
the	O
time	O
flow	O
of	O
the	O
elapsed	O
time	O
which	O
is	O
the	O
only	O
one	O
continuous	O
variable	O
in	O
SP-DEVS	B-Application
.	O
</s>
<s>
One	O
interesting	O
feature	O
of	O
SF-DEVS	O
might	O
be	O
the	O
preservation	O
of	O
schedule	O
the	O
restriction	O
(	O
3	O
)	O
of	O
SP-DEVS	B-Application
which	O
is	O
drawn	O
at	O
time	O
47	O
in	O
Fig	O
.	O
</s>
<s>
in	O
a	O
SP-DEVS	B-Application
model	O
.	O
</s>
<s>
The	O
property	O
of	O
non-negative	O
rational-valued	O
lifespans	O
which	O
are	O
not	O
changed	O
by	O
input	O
events	O
along	O
with	O
finite	O
numbers	O
of	O
states	O
and	O
events	O
guarantees	O
that	O
the	O
behavior	O
of	O
SP-DEVS	B-Application
networks	O
can	O
be	O
abstracted	O
as	O
an	O
equivalent	O
finite-vertex	O
reachability	O
graph	O
by	O
abstracting	O
the	O
infinitely-many	O
values	O
of	O
the	O
elapsed	O
times	O
.	O
</s>
<s>
To	O
abstract	O
the	O
infinitely-many	O
cases	O
of	O
elapsed	O
times	O
for	O
each	O
components	O
of	O
SP-DEVS	B-Application
networks	O
,	O
a	O
time-abstraction	O
method	O
,	O
called	O
the	O
time-line	O
abstraction	O
has	O
been	O
introduced	O
 [ Hwang05 ]  ,  [ HCZF07 ] 	O
in	O
which	O
the	O
orders	O
and	O
relative	O
difference	O
of	O
schedules	O
are	O
preserved	O
.	O
</s>
<s>
By	O
using	O
the	O
time-line	O
abstraction	O
technique	O
,	O
the	O
behavior	O
of	O
any	O
SP-DEVS	B-Application
network	O
can	O
be	O
abstracted	O
as	O
a	O
reachability	O
graph	O
whose	O
numbers	O
of	O
vertices	O
and	O
edges	O
are	O
finite	O
.	O
</s>
<s>
As	O
a	O
qualitative	O
property	O
,	O
safety	O
of	O
a	O
SP-DEVS	B-Application
network	O
is	O
decidable	O
by	O
(	O
1	O
)	O
generating	O
the	O
finite-vertex	O
reachability	O
graph	O
of	O
the	O
given	O
network	O
and	O
(	O
2	O
)	O
checking	O
whether	O
some	O
bad	O
states	O
are	O
reachable	O
or	O
not	O
 [ Hwang05 ] 	O
.	O
</s>
<s>
As	O
a	O
qualitative	O
property	O
,	O
liveness	O
of	O
a	O
SP-DEVS	B-Application
network	O
is	O
decidable	O
by	O
(	O
1	O
)	O
generating	O
the	O
finite-vertex	O
reachability	O
graph	O
(	O
RG	O
)	O
of	O
the	O
given	O
network	O
,	O
(	O
2	O
)	O
from	O
RG	O
,	O
generating	O
kernel	O
directed	O
acyclic	O
graph	O
(	O
KDAG	O
)	O
in	O
which	O
a	O
vertex	O
is	O
strongly	O
connected	O
component	O
,	O
and	O
(	O
3	O
)	O
checking	O
if	O
a	O
vertex	O
of	O
KDAG	O
contains	O
a	O
state	O
transition	O
cycle	O
which	O
contains	O
a	O
set	O
of	O
liveness	O
states[Hwang05]	O
.	O
</s>
<s>
As	O
a	O
quantitative	O
property	O
,	O
minimum	O
and	O
maximum	O
processing	O
time	O
bounds	O
from	O
two	O
events	O
in	O
SP-DEVS	B-Application
networks	O
can	O
be	O
computed	O
by	O
(	O
1	O
)	O
generating	O
the	O
finite-vertex	O
reachability	O
graph	O
and	O
(	O
2.a	O
)	O
by	O
finding	O
the	O
shortest	O
paths	O
for	O
the	O
minimum	O
processing	O
time	O
bound	O
and	O
(	O
2.b	O
)	O
by	O
finding	O
the	O
longest	O
paths	O
(	O
if	O
available	O
)	O
for	O
the	O
maximum	O
processing	O
time	O
bound	O
 [ HCZF07 ] 	O
.	O
</s>
<s>
Let	O
a	O
total	O
state	O
of	O
a	O
SP-DEVS	B-Application
model	O
be	O
passive	O
if	O
;	O
otherwise	O
,	O
it	O
be	O
active	O
.	O
</s>
<s>
One	O
of	O
known	O
SP-DEVS	B-Application
'	O
s	O
limitation	O
is	O
a	O
phenomenon	O
that	O
"	O
once	O
an	O
SP-DEVS	B-Application
model	O
becomes	O
passive	O
,	O
it	O
never	O
returns	O
to	O
become	O
active	O
(	O
OPNA	O
)	O
"	O
.	O
</s>
<s>
3(b )	O
are	O
not	O
SP-DEVS	B-Application
because	O
the	O
total	O
state	O
associated	O
with	O
"	O
idle	O
"	O
(	O
I	O
)	O
,	O
is	O
passive	O
but	O
it	O
moves	O
to	O
an	O
active	O
state	O
,	O
"	O
toast	O
"	O
(	O
T	O
)	O
whose	O
toating	O
time	O
is	O
20	O
seconds	O
or	O
40	O
seconds	O
.	O
</s>
<s>
3(b )	O
is	O
FD-DEVS	B-Application
.	O
</s>
<s>
There	O
is	O
an	O
open	O
source	O
library	O
,	O
called	O
DEVS#	O
at	O
,	O
that	O
supports	O
some	O
algorithms	O
for	O
finding	O
safeness	O
and	O
liveness	O
as	O
well	O
as	O
Min/Max	O
processing	O
time	O
bounds	O
.	O
</s>
