<s>
DEVS	B-Application
abbreviating	O
Discrete	B-Application
Event	I-Application
System	I-Application
Specification	I-Application
is	O
a	O
modular	O
and	O
hierarchical	O
formalism	O
for	O
modeling	O
and	O
analyzing	O
general	O
systems	O
that	O
can	O
be	O
discrete	O
event	O
systems	O
which	O
might	O
be	O
described	O
by	O
state	B-Application
transition	I-Application
tables	I-Application
,	O
and	O
continuous	O
state	O
systems	O
which	O
might	O
be	O
described	O
by	O
differential	O
equations	O
,	O
and	O
hybrid	O
continuous	O
state	O
and	O
discrete	O
event	O
systems	O
.	O
</s>
<s>
DEVS	B-Application
is	O
a	O
timed	B-Application
event	I-Application
system	I-Application
.	O
</s>
<s>
DEVS	B-Application
abbreviating	O
Discrete	B-Application
Event	I-Application
System	I-Application
Specification	I-Application
is	O
a	O
modular	O
and	O
hierarchical	O
formalism	O
for	O
modeling	O
and	O
analyzing	O
general	O
systems	O
that	O
can	O
be	O
discrete	O
event	O
systems	O
which	O
might	O
be	O
described	O
by	O
state	B-Application
transition	I-Application
tables	I-Application
,	O
and	O
continuous	O
state	O
systems	O
which	O
might	O
be	O
described	O
by	O
differential	O
equations	O
,	O
and	O
hybrid	O
...	O
</s>
<s>
DEVS	B-Application
is	O
a	O
formalism	O
for	O
modeling	O
and	O
analysis	O
of	O
discrete	O
event	O
systems	O
(	O
DESs	O
)	O
.	O
</s>
<s>
The	O
DEVS	B-Application
formalism	O
was	O
invented	O
by	O
Bernard	O
P	O
.	O
Zeigler	O
,	O
who	O
is	O
emeritus	O
professor	O
at	O
the	O
University	O
of	O
Arizona	O
.	O
</s>
<s>
DEVS	B-Application
was	O
introduced	O
to	O
the	O
public	O
in	O
Zeigler	O
's	O
first	O
book	O
,	O
,	O
in	O
1976	O
,	O
while	O
Zeigler	O
was	O
an	O
associate	O
professor	O
at	O
University	O
of	O
Michigan	O
.	O
</s>
<s>
DEVS	B-Application
can	O
be	O
seen	O
as	O
an	O
extension	O
of	O
the	O
Moore	B-General_Concept
machine	I-General_Concept
formalism	O
,	O
which	O
is	O
a	O
finite	B-Architecture
state	I-Architecture
automaton	I-Architecture
where	O
the	O
outputs	O
are	O
determined	O
by	O
the	O
current	O
state	O
alone	O
(	O
and	O
do	O
not	O
depend	O
directly	O
on	O
the	O
input	O
)	O
.	O
</s>
<s>
Since	O
the	O
lifespan	O
of	O
each	O
state	O
is	O
a	O
real	O
number	O
(	O
more	O
precisely	O
,	O
non-negative	O
real	O
)	O
or	O
infinity	O
,	O
it	O
is	O
distinguished	O
from	O
discrete	O
time	O
systems	O
,	O
sequential	O
machines	O
,	O
and	O
Moore	B-General_Concept
machines	I-General_Concept
,	O
in	O
which	O
time	O
is	O
determined	O
by	O
a	O
tick	O
time	O
multiplied	O
by	O
non-negative	O
integers	O
.	O
</s>
<s>
The	O
state	B-Application
transition	I-Application
and	O
output	O
functions	O
of	O
DEVS	B-Application
can	O
also	O
be	O
stochastic	O
.	O
</s>
<s>
Zeigler	O
proposed	O
a	O
hierarchical	O
algorithm	O
for	O
DEVS	B-Application
model	O
simulation	O
in	O
1984	O
 [ Zeigler84 ] 	O
which	O
was	O
published	O
in	O
Simulation	O
journal	O
in	O
1987	O
.	O
</s>
<s>
Since	O
then	O
,	O
many	O
extended	O
formalism	O
from	O
DEVS	B-Application
have	O
been	O
introduced	O
with	O
their	O
own	O
purposes	O
:	O
DESS/DEVS	O
for	O
combined	O
continuous	O
and	O
discrete	O
event	O
systems	O
,	O
P-DEVS	O
for	O
parallel	O
DESs	O
,	O
G-DEVS	O
for	O
piecewise	O
continuous	O
state	O
trajectory	O
modeling	O
of	O
DESs	O
,	O
RT-DEVS	O
for	O
realtime	O
DESs	O
,	O
Cell-DEVS	O
for	O
cellular	O
DESs	O
,	O
Fuzzy-DEVS	O
for	O
fuzzy	O
DESs	O
,	O
Dynamic	O
Structuring	O
DEVS	B-Application
for	O
DESs	O
changing	O
their	O
coupling	O
structures	O
dynamically	O
,	O
and	O
so	O
on	O
.	O
</s>
<s>
In	O
addition	O
to	O
its	O
extensions	O
,	O
there	O
are	O
some	O
subclasses	O
such	O
as	O
SP-DEVS	B-Application
and	O
FD-DEVS	B-Application
have	O
been	O
researched	O
for	O
achieving	O
decidability	O
of	O
system	O
properties	O
.	O
</s>
<s>
DEVS	B-Application
defines	O
system	O
behavior	O
as	O
well	O
as	O
system	O
structure	O
.	O
</s>
<s>
System	O
behavior	O
in	O
DEVS	B-Application
formalism	O
is	O
described	O
using	O
input	O
and	O
output	O
events	O
as	O
well	O
as	O
states	O
.	O
</s>
<s>
In	O
the	O
classic	O
DEVS	B-Application
formalism	O
,	O
Atomic	O
DEVS	B-Application
captures	O
the	O
system	O
behavior	O
,	O
while	O
Coupled	O
DEVS	B-Application
describes	O
the	O
structure	O
of	O
system	O
.	O
</s>
<s>
The	O
following	O
formal	O
definition	O
is	O
for	O
Classic	O
DEVS	B-Application
 [ ZKP00 ] 	O
.	O
</s>
<s>
For	O
more	O
how	O
to	O
understand	O
this	O
function	O
,	O
refer	O
to	O
the	O
article	O
,	O
Behavior	B-Application
of	I-Application
DEVS	I-Application
.	O
</s>
<s>
The	O
atomic	O
DEVS	B-Application
model	O
for	O
player	O
A	O
of	O
Fig	O
.	O
</s>
<s>
Both	O
Player	O
A	O
and	O
Player	O
B	O
are	O
atomic	O
DEVS	B-Application
models	O
.	O
</s>
<s>
Simply	O
speaking	O
,	O
there	O
are	O
two	O
cases	O
that	O
an	O
atomic	O
DEVS	B-Application
model	O
can	O
change	O
its	O
state	O
:	O
(	O
1	O
)	O
when	O
an	O
external	O
input	O
comes	O
into	O
the	O
system	O
;	O
(	O
2	O
)	O
when	O
the	O
elapsed	O
time	O
reaches	O
the	O
lifespan	O
of	O
which	O
is	O
defined	O
by	O
.	O
</s>
<s>
For	O
formal	O
behavior	O
description	O
of	O
given	O
an	O
Atomic	O
DEVS	B-Application
model	O
,	O
refer	O
to	O
the	O
page	O
Behavior	B-Application
of	I-Application
DEVS	I-Application
.	O
</s>
<s>
Computer	O
algorithms	O
to	O
implement	O
the	O
behavior	O
of	O
a	O
given	O
Atomic	O
DEVS	B-Application
model	O
are	O
available	O
at	O
Simulation	B-Algorithm
Algorithms	I-Algorithm
for	I-Algorithm
Atomic	I-Algorithm
DEVS	I-Algorithm
.	O
</s>
<s>
The	O
coupled	O
DEVS	B-Application
defines	O
which	O
sub-components	O
belong	O
to	O
it	O
and	O
how	O
they	O
are	O
connected	O
with	O
each	O
other	O
.	O
</s>
<s>
is	O
the	O
set	O
of	O
sub-components	O
where	O
for	O
each	O
can	O
be	O
either	O
an	O
atomic	O
DEVS	B-Application
model	O
or	O
a	O
coupled	O
DEVS	B-Application
model	O
.	O
</s>
<s>
1	O
can	O
be	O
modeled	O
as	O
a	O
coupled	O
DEVS	B-Application
model	O
where	O
;;;	O
is	O
described	O
as	O
above	O
;	O
;	O
;	O
and	O
.	O
</s>
<s>
Simply	O
speaking	O
,	O
like	O
the	O
behavior	O
of	O
the	O
atomic	O
DEVS	B-Application
class	O
,	O
a	O
coupled	O
DEVS	B-Application
model	O
changes	O
its	O
components	O
 '	O
states	O
(	O
1	O
)	O
when	O
an	O
external	O
event	O
comes	O
into	O
;	O
(	O
2	O
)	O
when	O
one	O
of	O
components	O
where	O
executes	O
its	O
internal	O
state	B-Application
transition	I-Application
and	O
generates	O
its	O
output	O
.	O
</s>
<s>
For	O
formal	O
definition	O
of	O
behavior	O
of	O
the	O
coupled	O
DEVS	B-Application
,	O
you	O
can	O
refer	O
to	O
Behavior	B-Application
of	I-Application
Coupled	I-Application
DEVS	I-Application
.	O
</s>
<s>
Computer	O
algorithms	O
to	O
implement	O
the	O
behavior	O
of	O
a	O
given	O
coupled	O
DEVS	B-Application
mode	O
are	O
available	O
at	O
Simulation	B-Algorithm
Algorithms	I-Algorithm
for	I-Algorithm
Coupled	I-Algorithm
DEVS	I-Algorithm
.	O
</s>
<s>
The	O
simulation	O
algorithm	O
of	O
DEVS	B-Application
models	O
considers	O
two	O
issues	O
:	O
time	O
synchronization	O
and	O
message	O
propagation	O
.	O
</s>
<s>
Time	O
synchronization	O
of	O
DEVS	B-Application
is	O
to	O
control	O
all	O
models	O
to	O
have	O
the	O
identical	O
current	O
time	O
.	O
</s>
<s>
However	O
,	O
for	O
an	O
efficient	O
execution	O
,	O
the	O
algorithm	O
makes	O
the	O
current	O
time	O
jump	O
to	O
the	O
most	O
urgent	O
time	O
when	O
an	O
event	O
is	O
scheduled	O
to	O
execute	O
its	O
internal	O
state	B-Application
transition	I-Application
as	O
well	O
as	O
its	O
output	O
generation	O
.	O
</s>
<s>
Message	O
propagation	O
is	O
to	O
transmit	O
a	O
triggering	O
message	O
which	O
can	O
be	O
either	O
an	O
input	O
or	O
output	O
event	O
along	O
the	O
associated	O
couplings	O
which	O
are	O
defined	O
in	O
a	O
coupled	O
DEVS	B-Application
model	O
.	O
</s>
<s>
For	O
more	O
detailed	O
information	O
,	O
the	O
reader	O
can	O
refer	O
to	O
Simulation	B-Algorithm
Algorithms	I-Algorithm
for	I-Algorithm
Atomic	I-Algorithm
DEVS	I-Algorithm
and	O
Simulation	B-Algorithm
Algorithms	I-Algorithm
for	I-Algorithm
Coupled	I-Algorithm
DEVS	I-Algorithm
.	O
</s>
<s>
By	O
introducing	O
a	O
quantization	O
method	O
which	O
abstracts	O
a	O
continuous	O
segment	O
as	O
a	O
piecewise	O
const	O
segment	O
,	O
DEVS	B-Application
can	O
simulate	O
behaviors	O
of	O
continuous	O
state	O
systems	O
which	O
are	O
described	O
by	O
networks	O
of	O
differential	O
algebraic	O
equations	O
.	O
</s>
<s>
In	O
2006	O
,	O
Prof	O
.	O
Cellier	O
who	O
is	O
the	O
author	O
of	O
Continuous	O
System	O
Modeling[Cellier91],	O
and	O
Prof	O
.	O
Kofman	O
wrote	O
a	O
text	O
book	O
,	O
Continuous	O
System	O
Simulation[CK06]	O
in	O
which	O
Chapters	O
11	O
and	O
12	O
cover	O
how	O
DEVS	B-Application
simulates	O
continuous	O
state	O
systems	O
.	O
</s>
<s>
As	O
an	O
alternative	O
analysis	O
method	O
against	O
the	O
sampling-based	O
simulation	O
method	O
,	O
an	O
exhaustive	O
generating	O
behavior	O
approach	O
,	O
generally	O
called	O
verification	O
has	O
been	O
applied	O
for	O
analysis	O
of	O
DEVS	B-Application
models	O
.	O
</s>
<s>
It	O
is	O
proven	O
that	O
infinite	O
states	O
of	O
a	O
given	O
DEVS	B-Application
model	O
(	O
especially	O
a	O
coupled	O
DEVS	B-Application
model	O
)	O
can	O
be	O
abstracted	O
by	O
behaviorally	O
isomorphic	O
finite	O
structure	O
,	O
called	O
a	O
reachability	O
graph	O
when	O
the	O
given	O
DEVS	B-Application
model	O
is	O
a	O
sub-class	O
of	O
DEVS	B-Application
such	O
as	O
Schedule-Preserving	O
DEVS	B-Application
(	O
SP-DEVS	B-Application
)	O
,	O
Finite	O
&	O
Deterministic	O
DEVS	B-Application
(	O
FD-DEVS	B-Application
)	O
[HZ09],	O
and	O
Finite	O
&	O
Real-time	O
DEVS	B-Application
(	O
FRT-DEVS	O
)	O
 [ Hwang12 ] 	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
based	O
on	O
the	O
rechability	O
graph	O
,	O
(	O
1	O
)	O
dead-lock	O
and	O
live-lock	O
freeness	O
as	O
qualitative	O
properties	O
are	O
decidable	O
with	O
SP-DEVS	B-Application
[Hwang05],	O
FD-DEVS	B-Application
[HZ06],	O
and	O
FRT-DEVS	O
[Hwang12];	O
and	O
(	O
2	O
)	O
min/max	O
processing	O
time	O
bounds	O
as	O
a	O
quantitative	O
property	O
are	O
decidable	O
with	O
SP-DEVS	B-Application
so	O
far	O
by	O
2012	O
.	O
</s>
<s>
Numerous	O
extensions	O
of	O
the	O
classic	O
DEVS	B-Application
formalism	O
have	O
been	O
developed	O
in	O
the	O
last	O
decades	O
.	O
</s>
<s>
There	O
are	O
some	O
sub-classes	O
known	O
as	O
Schedule-Preserving	O
DEVS	B-Application
(	O
SP-DEVS	B-Application
)	O
and	O
Finite	O
and	O
Deterministic	O
DEVS	B-Application
(	O
FD-DEVS	B-Application
)	O
which	O
were	O
designated	O
to	O
support	O
verification	O
analysis	O
.	O
</s>
<s>
SP-DEVS	B-Application
and	O
FD-DEVS	B-Application
whose	O
expressiveness	O
are	O
E(SP-DEVS )	O
E(FD-DEVS )	O
E(DEVS )	O
where	O
E(formalism )	O
denotes	O
the	O
expressiveness	O
of	O
formalism	O
.	O
</s>
