Behavior of DEVS

Suppose that a DEVS model, ${\displaystyle {\mathcal {M}}=<X,Y,S,s_{0},ta,\delta _{ext},\delta _{int},\lambda >}$ has

1. the external state transition ${\displaystyle \delta _{ext}:Q\times X\rightarrow S}$.
2. the total state set ${\displaystyle Q=\{(s,t_{e})|s\in S,t_{e}\in (\mathbb {T} \cap [0,ta(s)])\}}$ where ${\displaystyle t_{e}}$ denotes elapsed time since last event and ${\displaystyle \mathbb {T} =[0,\infty )}$ denotes the set of non-negative real numbers, and

Then the DEVS model, ${\displaystyle {\mathcal {M}}}$ is a Timed Event System ${\displaystyle {\mathcal {G}}=<Z,Q,Q_{0},Q_{A},\Delta >}$ where

• The event set ${\displaystyle Z=X\cup Y^{\phi }}$.
• The state set ${\displaystyle Q=Q_{A}\cup Q_{N}}$ where ${\displaystyle Q_{N}=\{{\bar {s}}\not \in S\}}$.
• The set of initial states ${\displaystyle \,Q_{0}=\{(s_{0},0)\}}$.
• The set of accepting states ${\displaystyle Q_{A}={\mathcal {M}}.Q.}$
• The set of state trajectories ${\displaystyle \Delta \subseteq Q\times \Omega _{Z,[t_{l},t_{u}]}\times Q}$ is defined for two different cases: ${\displaystyle q\in Q_{N}}$ and ${\displaystyle q\in Q_{A}}$. For a non-accepting state ${\displaystyle q\in Q_{N}}$, there is no change together with any even segment ${\displaystyle \omega \in \Omega _{Z,[t_{l},t_{u}]}}$ so ${\displaystyle (q,\omega ,q)\in \Delta .}$

For a total state ${\displaystyle q=(s,t_{e})\in Q_{A}}$ at time ${\displaystyle t\in \mathbb {T} }$ and an event segment ${\displaystyle \omega \in \Omega _{Z,[t_{l},t_{u}]}}$ as follows.

If unit event segment ${\displaystyle \omega }$ is the null event segment, i.e. ${\displaystyle \omega =\epsilon _{[t,t+dt]}}$

${\displaystyle \,(q,\omega ,(s,t_{e}+dt))\in \Delta .}$

If unit event segment ${\displaystyle \omega }$ is a timed event ${\displaystyle \omega =(x,t)}$ where the event is an input event ${\displaystyle x\in X}$,

${\displaystyle (q,\omega ,(\delta _{ext}(q,x),0))\in \Delta .}$

If unit event segment ${\displaystyle \omega }$ is a timed event ${\displaystyle \omega =(y,t)}$ where the event is an output event or the unobservable event ${\displaystyle y\in Y^{\phi }}$,

${\displaystyle {\begin{cases}(q,\omega ,(\delta _{int}(s),0))\in \Delta &{\textrm {if}}~t_{e}=ta(s),y=\lambda (s)\\(q,\omega ,{\bar {s}})&{\textrm {otherwise}}.\end{cases}}}$

Computer algorithms to simulate this view of behavior are available at Simulation Algorithms for Atomic DEVS.

Suppose that a DEVS model, ${\displaystyle {\mathcal {M}}=<X,Y,S,s_{0},ta,\delta _{ext},\delta _{int},\lambda >}$ has

1. the total state set ${\displaystyle Q=\{(s,t_{s},t_{e})|s\in S,t_{s}\in \mathbb {T} ^{\infty },t_{e}\in (\mathbb {T} \cap [0,t_{s}])\}}$ where ${\displaystyle t_{s}}$ denotes lifespan of state ${\displaystyle s}$, ${\displaystyle t_{e}}$ denotes elapsed time since last ${\displaystyle t_{s}}$update, and ${\displaystyle \mathbb {T} ^{\infty }=[0,\infty )\cup \{\infty \}}$ denotes the set of non-negative real numbers plus infinity,
2. the external state transition is ${\displaystyle \delta _{ext}:Q\times X\rightarrow S\times \{0,1\}}$.

Then the DEVS ${\displaystyle Q={\mathcal {D}}}$ is a timed event system ${\displaystyle {\mathcal {G}}=<Z,Q,Q_{0},Q_{A},\Delta >}$ where

• The event set ${\displaystyle Z=X\cup Y^{\phi }}$.
• The state set ${\displaystyle Q=Q_{A}\cup Q_{N}}$ where ${\displaystyle Q_{N}=\{{\bar {s}}\not \in S\}}$.
• The set of initial states${\displaystyle \,Q_{0}=\{(s_{0},ta(s_{0}),0)\}}$.
• The set of acceptance states ${\displaystyle Q_{A}={\mathcal {M}}.Q}$.
• The set of state trajectories ${\displaystyle \Delta \subseteq Q\times \Omega _{Z,[t_{l},t_{u}]}\times Q}$ is depending on two cases: ${\displaystyle q\in Q_{N}}$ and ${\displaystyle q\in Q_{A}}$. For a non-accepting state ${\displaystyle q\in Q_{N}}$, there is no changes together with any segment ${\displaystyle \omega \in \Omega _{Z,[t_{l},t_{u}]}}$ so ${\displaystyle (q,\omega ,q)\in \Delta .}$

For a total state ${\displaystyle q=(s,t_{s},t_{e})\in Q_{A}}$ at time ${\displaystyle t\in \mathbb {T} }$ and an event segment ${\displaystyle \omega \in \Omega _{Z,[t_{l},t_{u}]}}$ as follows.

If unit event segment ${\displaystyle \omega }$ is the null event segment, i.e. ${\displaystyle \omega =\epsilon _{[t,t+dt]}}$

${\displaystyle (q,\omega ,(s,t_{s},t_{e}+dt))\in \Delta .}$

If unit event segment ${\displaystyle \omega }$ is a timed event ${\displaystyle \omega =(x,t)}$ where the event is an input event ${\displaystyle x\in X}$,

${\displaystyle {\begin{cases}(q,\omega ,(s',ta(s'),0))\in \Delta &{\textrm {if}}~\delta _{ext}(s,t_{s},t_{e},x)=(s',1),\\(q,\omega ,(s',t_{s},t_{e}))\in \Delta &{\textrm {otherwise,i.e.}}~\delta _{ext}(s,t_{s},t_{e},x)=(s',0).\end{cases}}}$

If unit event segment ${\displaystyle \omega }$ is a timed event ${\displaystyle \omega =(y,t)}$ where the event is an output event or the unobservable event ${\displaystyle y\in Y^{\phi }}$,

${\displaystyle {\begin{cases}(q,\omega ,(s',ta(s'),0))\in \Delta &{\textrm {if}}~t_{e}=t_{s},y=\lambda (s),\delta _{int}(s)=s',\\(q,\omega ,{\bar {s}})\in \Delta &{\textrm {otherwise}}.\end{cases}}}$

Computer algorithms to simulate this view of behavior are available at Simulation Algorithms for Atomic DEVS.

• DEVS
• Behavior of Coupled DEVS
• Simulation Algorithms for Atomic DEVS
• Simulation Algorithms for Coupled DEVS

• [Zeigler76] Bernard Zeigler (1976). Theory of Modeling and Simulation (first ed.). Wiley Interscience, New York.
• [Zeigler84] Bernard Zeigler (1984). Multifacetted Modeling and Discrete Event Simulation. Academic Press, London; Orlando. ISBN 978-0-12-778450-2.
• [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.
• [HZ06] M. H. Hwang and Bernard Zeigler, ``A Reachable Graph of Finite and Deterministic DEVS Networks``, Proceedings of 2006 DEVS Symposium, pp48-56, Huntsville, Alabama, USA, (Available at https://web.archive.org/web/20120726134045/http://www.acims.arizona.edu/ and http://moonho.hwang.googlepages.com/publications)
• [HZ07] M.H. Hwang and Bernard Zeigler, ``Reachability Graph of Finite & Deterministic DEVS``, IEEE Transactions on Automation Science and Engineering, Volume 6, Issue 3, 2009, pp. 454–467, http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=5153598&arnumber=5071137&count=19&index=7