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Source: DEVS

DEVS, abbreviating Discrete Event System Specification, is a modular and hierarchical formalism for modeling and analyzing general systems that can be discrete event systems which might be described by state transition tables, and continuous state systems which might be described by differential equations, and hybrid continuous state and discrete event systems. DEVS is a timed event system.

History

DEVS

Formalism

DEVS

= Atomic DEVS

DEVS

}

such that

X

=
{
?

receive

}

Y

=
{
!

send

}

S

=
{
(
d
,
σ
)

|

d
∈
{

Wait

,

Send

}
,
σ
∈

T

∞

}

s

0

=
(

Send

,
0.1
)

t
a
(
s
)

=
σ

for all

s
∈
S

δ

e
x
t

(
(
(

Wait

,
σ
)
,

t

e

)
,
?

receive

)

=
(

Send

,
0.1
)

δ

i
n
t

(

Send

,
σ
)

=
(

Wait

,
∞
)

δ

i
n
t

(

Wait

,
σ
)

=
(

Send

,
0.1
)

λ
(

Send

,
σ
)

=
!

send

λ
(

Wait

,
σ
)

=
ϕ

{\displaystyle {\begin{aligned}X&=\{?{\textit {receive}}\}\\Y&=\{!{\textit {send}}\}\\S&=\{(d,\sigma )|d\in \{{\textit {Wait}},{\textit {Send}}\},\sigma \in \mathbb {T} ^{\infty }\}\\s_{0}&=({\textit {Send}},0.1)\\ta(s)&=\sigma {\text{ for all }}s\in S\\\delta _{ext}((({\textit {Wait}},\sigma ),t_{e}),?{\textit {receive}})&=({\textit {Send}},0.1)\\\delta _{int}({\textit {Send}},\sigma )&=({\textit {Wait}},\infty )\\\delta _{int}({\textit {Wait}},\sigma )&=({\textit {Send}},0.1)\\\lambda ({\textit {Send}},\sigma )&=!{\textit {send}}\\\lambda ({\textit {Wait}},\sigma )&=\phi \end{aligned}}}

Both Player A and Player B are atomic DEVS models.

Behavior of atomic DEVS

Simply speaking, there are two cases that an atomic DEVS model

M

{\displaystyle M}

can change its state

s
∈
S

{\displaystyle s\in S}

: (1) when an external input

x
∈
X

{\displaystyle x\in X}

comes into the system

M

{\displaystyle M}

; (2) when the elapsed time

t

e

{\displaystyle t_{e}}

reaches the lifespan of

s

{\displaystyle s}

which is defined by

t
a
(
s
)

{\displaystyle ta(s)}

. (At the same time of (2),

M

{\displaystyle M}

generates an output

y
∈
Y

{\displaystyle y\in Y}

which is defined by

λ
(
s
)

{\displaystyle \lambda (s)}

.) .
For formal behavior description of given an Atomic DEVS model, refer to the page Behavior of DEVS. Computer algorithms to implement the behavior of a given Atomic DEVS model are available at Simulation Algorithms for Atomic DEVS.

= Coupled DEVS

=
The coupled DEVS defines which sub-components belong to it and how they are connected with each other. A coupled DEVS model is defined as an 8-tuple

where

X

{\displaystyle X}

is the set of input events;

Y

{\displaystyle Y}

is the set of output events;

D

{\displaystyle D}

is the name set of sub-components;

{

M

i

}

{\displaystyle \{M_{i}\}}

is the set of sub-components where for each

i
∈
D
,

M

i

{\displaystyle i\in D,M_{i}}

can be either an atomic DEVS model or a coupled DEVS model.

C

x
x

⊆
X
×

⋃

i
∈
D

X

i

{\displaystyle C_{xx}\subseteq X\times \bigcup _{i\in D}X_{i}}

is the set of external input couplings;

C

y
x

⊆

⋃

i
∈
D

Y

i

×

⋃

i
∈
D

X

i

{\displaystyle C_{yx}\subseteq \bigcup _{i\in D}Y_{i}\times \bigcup _{i\in D}X_{i}}

is the set of internal couplings;

C

y
y

:

⋃

i
∈
D

Y

i

→

Y

ϕ

{\displaystyle C_{yy}:\bigcup _{i\in D}Y_{i}\rightarrow Y^{\phi }}

is the external output coupling function;

S
e
l
e
c
t
:

2

D

→
D

{\displaystyle Select:2^{D}\rightarrow D}

is the tie-breaking function which defines how to select the event from the set of simultaneous events;

The coupled DEVS model for Ping-Pong Game
The ping-pong game of Fig. 1 can be modeled as a coupled DEVS model

N
=<
X
,
Y
,
D
,
{

M

i

}
,

C

x
x

,

C

y
x

,

C

y
y

,
S
e
l
e
c
t
>

{\displaystyle N=}

where

X
=
{
}

{\displaystyle X=\{\}}

;

Y
=
{
}

{\displaystyle Y=\{\}}

;

D
=
{
A
,
B
}

{\displaystyle D=\{A,B\}}

;

M

A

and

M

B

{\displaystyle M_{A}{\text{ and }}M_{B}}

is described as above;

C

x
x

=
{
}

{\displaystyle C_{xx}=\{\}}

;

C

y
x

=
{
(
A
.
!
s
e
n
d
,
B
.
?
r
e
c
e
i
v
e
)
,
(
B
.
!
s
e
n
d
,
A
.
?
r
e
c
e
i
v
e
)
}

{\displaystyle C_{yx}=\{(A.!send,B.?receive),(B.!send,A.?receive)\}}

; and

C

y
y

(
A
.
!
s
e
n
d
)
=
ϕ
,

C

y
y

(
B
.
!
s
e
n
d
)
=
ϕ

{\displaystyle C_{yy}(A.!send)=\phi ,C_{yy}(B.!send)=\phi }

.

Behavior of coupled DEVS

Simply speaking, like the behavior of the atomic DEVS class, a coupled DEVS model

N

{\displaystyle N}

changes its components' states (1) when an external event

x
∈
X

{\displaystyle x\in X}

comes into

N

{\displaystyle N}

; (2) when one of components

M

i

{\displaystyle M_{i}}

where

i
∈
D

{\displaystyle i\in D}

executes its internal state transition and generates its output

y

i

∈

Y

i

{\displaystyle y_{i}\in Y_{i}}

. In both cases (1) and (2), a triggering event is transmitted to all influencees which are defined by coupling sets

C

x
x

,

C

y
x

,

{\displaystyle C_{xx},C_{yx},}

and

C

y
y

{\displaystyle C_{yy}}

.
For formal definition of behavior of the coupled DEVS, you can refer to Behavior of Coupled DEVS. Computer algorithms to implement the behavior of a given coupled DEVS mode are available at Simulation Algorithms for Coupled DEVS.

Analysis methods

= Simulation for discrete event systems

=
The simulation algorithm of DEVS models considers two issues: time synchronization and message propagation. Time synchronization of DEVS is to control all models to have the identical current time. However, for an efficient execution, the algorithm makes the current time jump to the most urgent time when an event is scheduled to execute its internal state transition as well as its output generation. Message propagation is to transmit a triggering message which can be either an input or output event along the associated couplings which are defined in a coupled DEVS model. For more detailed information, the reader can refer to Simulation Algorithms for Atomic DEVS and Simulation Algorithms for Coupled DEVS.

= Simulation for continuous state systems

=
By introducing a quantization method which abstracts a continuous segment as a piecewise const segment, DEVS can simulate behaviors of continuous state systems which are described by networks of differential algebraic equations. This research has been initiated by Zeigler in 1990s and many properties have been clarified by Prof. Kofman in 2000s and Dr. Nutaro. In 2006, Prof. Cellier who is the author of Continuous System Modeling[Cellier91], and Prof. Kofman wrote a text book, Continuous System Simulation[CK06] in which Chapters 11 and 12 cover how DEVS simulates continuous state systems. Dr. Nutaro's book [Nutaro10], covers the discrete event simulation of continuous state systems too.

= Verification for discrete event systems

=
As an alternative analysis method against the sampling-based simulation method, an exhaustive generating behavior approach, generally called verification has been applied for analysis of DEVS models. It is proven that infinite states of a given DEVS model (especially a coupled DEVS model ) can be abstracted by behaviorally isomorphic finite structure, called a reachability graph when the given DEVS model is a sub-class of DEVS such as Schedule-Preserving DEVS (SP-DEVS), Finite & Deterministic DEVS (FD-DEVS) [HZ09], and Finite & Real-time DEVS (FRT-DEVS) [Hwang12]. As a result, based on the reachability graph, (1) dead-lock and live-lock freeness as qualitative properties are decidable with SP-DEVS [Hwang05], FD-DEVS [HZ06], and FRT-DEVS [Hwang12]; and (2) min/max processing time bounds as a quantitative property are decidable with SP-DEVS so far by 2012.

Variations of DEVS

= Extensions (superclassing)

=

Numerous extensions of the classic DEVS formalism have been developed in the last decades.
Among them formalisms which allow to have changing model structures while the simulation time evolves.
G-DEVS [Giambiasi01][Zacharewicz08], Parallel DEVS, Dynamic Structuring DEVS, Cell-DEVS [Wainer09], dynDEVS, Fuzzy-DEVS, GK-DEVS, ml-DEVS, Symbolic DEVS, Real-Time DEVS, rho-DEVS

= Restrictions (subclassing)

=
There are some sub-classes known as Schedule-Preserving DEVS (SP-DEVS) and Finite and Deterministic DEVS (FD-DEVS) which were designated to support verification analysis.
SP-DEVS and FD-DEVS whose expressiveness are E(SP-DEVS)

⊂

{\displaystyle \subset }

E(FD-DEVS)

⊂

{\displaystyle \subset }

E(DEVS) where E(formalism) denotes the expressiveness of formalism.

= DEVS-related articles

=
Event segment
Timed event system
Verifiable sub-classes of DEVS: SP-DEVS, FD-DEVS
Behavior of atomic DEVS
Behavior of coupled DEVS
Simulation algorithms for atomic DEVS
Simulation algorithms for coupled DEVS

= Other formalisms

=
Automata theory: a formal method for state transition systems
Finite-state machine: a state transition machine with finite sets of events and states
Petri nets: a graphical representation of state and transition relations
Markov chain: a stochastic process in which the future will be determined by the current state
Specification and Description Language: SDL, a formal complete and unambiguous language to graphically represent simulation models.

Footnotes

References

[Cellier91] Francois E. Cellier (1991). Continuous System Modeling (first ed.). Springer. ISBN 978-0-387-97502-3.
[CK06] Francois E. Cellier; Ernesto Kofman (2006). Continuous System Simulation (first ed.). Springer. ISBN 978-0-387-26102-7.
[Giambiasi01] Giambiasi N., Escude B. Ghosh S. “Generalized Discrete Event Simulation of Dynamic Systems”, in: Issue 4 of SCS Transactions: Recent Advances in DEVS Methodology-part II, Vol. 18, pp. 216–229, dec 2001
[Hwang05] M.H. Hwang, "Tutorial: Verification of Real-time System Based on Schedule-Preserved DEVS", Proceedings of 2005 DEVS Symposium, San Diego, Apr. 2-8, 2005, ISBN 1-56555-293-8,
[HZ06] M.H. Hwang and B. P. Zeigler, "A Modular Verification Framework using Finite and Deterministic DEVS", Proceedings of 2006 DEVS Symposium, pp. 57–65, Huntsville, Alabama, USA,
[HZ09] M.H. Hwang and B.P. Zeigler, "Reachability Graph of Finite and Deterministic DEVS Networks", IEEE Transactions on Automation Science and Engineering, Volume 6, Issue 3, 2009, pp. 454–467,
[Hwang12] M.H. Hwang, "Qualitative verification of finite and real-time DEVS networks", Proceedings of the 2012 Symposium on Theory of Modeling and Simulation - DEVS Integrative M&S Symposium, Article No. 43,
[Mittal13] Saurabh Mittal; Jose L. Risco Martin (2013). Netcentric System of Systems Engineering with DEVS Unified Process (first ed.). CRC Press. ISBN 978-1439827062.
[Nutaro10] James Nutaro (2010). Building Software for Simulation: Theory, Algorithms, and Applications in C++ (first ed.). Wiley. ISBN 978-0-470-41469-9.
[Sarjoughian09] Hessam S. Sarjoughian; Vignesh Elamvazhuthi (2009). "CoSMoS: A Visual Environment for Component-Based Modeling, Experimental Design, and Simulation". Proceedings of the International Conference on Simulation Tools and Techniques. {{cite journal}}: Cite journal requires |journal= (help)
[Wainer09] Gabriel A. Wainer (2009). Discrete-Event Modeling and Simulation: A Practitioner's Approach (first ed.). CRC Press. ISBN 978-1-4200-5336-4.
[Wainer10] Gabriel A. Wainer and Pieter Mosterman Eds. (2010). Discrete-Event Modeling and Simulation: Theory and Applications (first ed.). CRC Press. ISBN 978-1-4200-7233-4.
[Zacharewicz08] Gregory Zacharewicz, Claudia Frydman, and Norbert Giambiasi (2008) G-DEVS/HLA Environment for Distributed Simulations of Workflows, SIMULATION May 2008 84: 197–213, doi:10.1177/0037549708092833.
[Zeiger68] Bernard Zeigler (1968). On the Feedback Complexity of Automata (Ph.D. Thesis ed.). University of Michigan.
[Zeigler76] Bernard Zeigler (1976). Theory of Modeling and Simulation (first ed.). Wiley Interscience, New York. ISBN 0-12-778455-1.
[Zeigler84] Bernard Zeigler (1984). Multifacetted Modeling and Discrete Event Simulation. Academic Press, London; Orlando. ISBN 978-0-12-778450-2.
[Zeigler87] Bernard Zeigler (1987). "Hierarchical, modular discrete-event modelling in an object-oriented environment". Simulation. 49 (5): 219–230. doi:10.1177/003754978704900506. S2CID 62648626.
[ZKP00] Bernard Zeigler; Tag Gon Kim; Herbert Praehofer (2000). Theory of Modeling and Simulation (second ed.). Academic Press, New York. ISBN 978-0-12-778455-7.