Behavior of coupled DEVS

Given a coupled DEVS model ${\displaystyle N=<X,Y,D,\{M_{i}\},C_{xx},C_{yx},C_{yy},Select>}$, its behavior is described as an atomic DEVS model ${\displaystyle M=<X,Y,S,s_{0},ta,\delta _{ext},\delta _{int},\lambda >}$

where

• ${\displaystyle X}$ and ${\displaystyle Y}$ are the input event set and the output event set, respectively.
• ${\displaystyle S={\underset {i\in D}{\times }}Q_{i}}$ is the partial state set where ${\displaystyle Q_{i}=\{(s_{i},t_{ei})|s_{i}\in S_{i},t_{ei}\in (\mathbb {T} \cap [0,ta_{i}(s_{i})])\}}$ is the total state set of component ${\displaystyle i\in D}$ (Refer to View1 of Behavior of DEVS), where ${\displaystyle \mathbb {T} =[0,\infty )}$ is the set of non-negative real numbers.
• ${\displaystyle s_{0}={\underset {i\in D}{\times }}q_{0i}}$ is the initial state set where ${\displaystyle q_{0i}=(s_{0i},0)}$ is the total initial state of component ${\displaystyle i\in D}$.
• ${\displaystyle ta:S\rightarrow \mathbb {T} ^{\infty }}$ is the time advance function, where ${\displaystyle \mathbb {T} ^{\infty }=[0,\infty ]}$ is the set of non-negative real numbers plus infinity. Given ${\displaystyle s=(\ldots ,(s_{i},t_{ei}),\ldots )}$,
  ${\displaystyle ta(s)=\min\{ta_{i}(si)-t_{ei}|i\in D\}.}$

• ${\displaystyle \delta _{ext}:Q\times X\rightarrow S}$ is the external state function. Given a total state ${\displaystyle q=(s,t_{e})}$ where ${\displaystyle s=(\ldots ,(s_{i},t_{ei}),\ldots ),t_{e}\in (\mathbb {T} \cap [0,ta(s)])}$, and input event ${\displaystyle x\in X}$, the next state is given by
  ${\displaystyle \delta _{ext}(q,x)=s'=(\ldots ,(s_{i}',t_{ei}'),\ldots )}$

where

${\displaystyle (s_{i}',t_{ei}')={\begin{cases}(\delta _{ext}(s_{i},t_{ei},x_{i}),0)&{\text{if }}(x,x_{i})\in C_{xx}\\(s_{i},t_{ei})&{\text{otherwise}}.\end{cases}}}$

Given the partial state ${\displaystyle s=(\ldots ,(s_{i},t_{ei}),\ldots )\in S}$, let ${\displaystyle IMM(s)=\{i\in D|ta_{i}(s_{i})=ta(s)\}}$ denote the set of imminent components. The firing component ${\displaystyle i^{*}\in D}$ which triggers the internal state transition and an output event is determined by

${\displaystyle i^{*}=Select(IMM(s)).}$

• ${\displaystyle \delta _{int}:S\rightarrow S}$ is the internal state function. Given a partial state ${\displaystyle s=(\ldots ,(s_{i},t_{ei}),\ldots )}$, the next state is given by
  ${\displaystyle \delta _{int}(s)=s'=(\ldots ,(s_{i}',t_{ei}'),\ldots )}$

where

${\displaystyle (s_{i}',t_{ei}')={\begin{cases}(\delta _{int}(s_{i}),0)&{\text{if }}i=i^{*}\\(\delta _{ext}(s_{i},t_{ei},x_{i}),0)&{\text{if }}(\lambda _{i^{*}}(s_{i^{*}}),x_{i})\in C_{yx}\\(s_{i},t_{ei})&{\text{otherwise}}.\end{cases}}}$

• ${\displaystyle \lambda :S\rightarrow Y^{\phi }}$ is the output function. Given a partial state ${\displaystyle s=(\ldots ,(s_{i},t_{ei}),\ldots )}$,
  ${\displaystyle \lambda (s)={\begin{cases}\phi &{\text{if }}\lambda _{i^{*}}(s_{i^{*}})=\phi \\C_{yy}(\lambda _{i^{*}}(s_{i^{*}}))&{\text{otherwise}}.\end{cases}}}$

Given a coupled DEVS model ${\displaystyle N=<X,Y,D,\{M_{i}\},C_{xx},C_{yx},C_{yy},Select>}$, its behavior is described as an atomic DEVS model ${\displaystyle M=<X,Y,S,s_{0},ta,\delta _{ext},\delta _{int},\lambda >}$

where

• ${\displaystyle X}$ and ${\displaystyle Y}$ are the input event set and the output event set, respectively.
• ${\displaystyle S={\underset {i\in D}{\times }}Q_{i}}$ is the partial state set where ${\displaystyle Q_{i}=\{(s_{i},t_{si},t_{ei})|s_{i}\in S_{i},t_{si}\in \mathbb {T} ^{\infty },t_{ei}\in (\mathbb {T} \cap [0,t_{si}])\}}$ is the total state set of component ${\displaystyle i\in D}$ (Refer to View2 of Behavior of DEVS).
• ${\displaystyle s_{0}={\underset {i\in D}{\times }}q_{0i}}$ is the initial state set where ${\displaystyle q_{0i}=(s_{0i},ta_{i}(s_{0i}),0)}$ is the total initial state of component ${\displaystyle i\in D}$.
• ${\displaystyle ta:S\rightarrow \mathbb {T} ^{\infty }}$ is the time advance function. Given ${\displaystyle s=(\ldots ,(s_{i},t_{si},t_{ei}),\ldots )}$,
  ${\displaystyle ta(s)=\min\{t_{si}-t_{ei}|i\in D\}.}$

• ${\displaystyle \delta _{ext}:Q\times X\rightarrow S\times \{0,1\}}$ is the external state function. Given a total state ${\displaystyle q=(s,t_{s},t_{e})}$ where ${\displaystyle s=(\ldots ,(s_{i},t_{si},t_{ei}),\ldots ),t_{s}\in \mathbb {T} ^{\infty },t_{e}\in (\mathbb {T} \cap [0,t_{s}])}$, and input event ${\displaystyle x\in X}$, the next state is given by
  ${\displaystyle \delta _{ext}(q,x)=((\ldots ,(s_{i}',t_{si}',t_{ei}'),\ldots ),b)}$

where

${\displaystyle (s_{i}',t_{si}',t_{ei}')={\begin{cases}(s_{i}',ta_{i}(s_{i}'),0)&{\text{if }}(x,x_{i})\in C_{xx},\delta _{ext}(s_{i},t_{si},t_{ei},x_{i})=(s_{i}',1)\\(s_{i}',t_{si},t_{ei})&{\text{if }}(x,x_{i})\in C_{xx},\delta _{ext}(s_{i},t_{si},t_{ei},x_{i})=(s_{i}',0)\\(s_{i},t_{si},t_{ei})&{\text{otherwise}}\end{cases}}}$

and

${\displaystyle b={\begin{cases}1&{\text{if }}\exists i\in D:(x,x_{i})\in C_{xx},\delta _{ext}(s_{i},t_{si},t_{ei},x_{i})=(s_{i}',1)\\0&{\text{otherwise}}.\end{cases}}}$

Given the partial state ${\displaystyle s=(\ldots ,(s_{i},t_{si},t_{ei}),\ldots )\in S}$, let ${\displaystyle IMM(s)=\{i\in D|t_{si}-t_{ei}=ta(s)\}}$ denote the set of imminent components. The firing component ${\displaystyle i^{*}\in D}$ which triggers the internal state transition and an output event is determined by

${\displaystyle i^{*}=Select(IMM(s)).}$

• ${\displaystyle \delta _{int}:S\rightarrow S}$ is the internal state function. Given a partial state ${\displaystyle s=(\ldots ,(s_{i},t_{si},t_{ei}),\ldots )}$, the next state is given by
  ${\displaystyle \delta _{int}(s)=s'=(\ldots ,(s_{i}',t_{si}',t_{ei}'),\ldots )}$

where

${\displaystyle (s_{i}',t_{si}',t_{ei}')={\begin{cases}(s_{i}',ta_{i}(s_{i}'),0)&{\text{if }}i=i^{*},\delta _{int}(s_{i})=s_{i}',\\(s_{i}',ta_{i}(s_{i}'),0)&{\text{if }}(\lambda _{i^{*}}(s_{i^{*}}),x_{i})\in C_{yx},\delta _{ext}(s_{i},t_{si},t_{ei},x_{i})=(s',1)\\(s_{i}',t_{si},t_{ei})&{\text{if }}(\lambda _{i^{*}}(s_{i^{*}}),x_{i})\in C_{yx},\delta _{ext}(s_{i},t_{si},t_{ei},x_{i})=(s',0)\\(s_{i},t_{si},t_{ei})&{\text{otherwise}}.\end{cases}}}$

• ${\displaystyle \lambda :S\rightarrow Y^{\phi }}$ is the output function. Given a partial state ${\displaystyle s=(\ldots ,(s_{i},t_{si},t_{ei}),\ldots )}$,
  ${\displaystyle \lambda (s)={\begin{cases}\phi &{\text{if }}\lambda _{i^{*}}(s_{i^{*}})=\phi \\C_{yy}(\lambda _{i^{*}}(s_{i^{*}}))&{\text{otherwise}}.\end{cases}}}$

Since in a coupled DEVS model with non-empty sub-components, i.e., ${\displaystyle |D|>0}$, the number of clocks which trace their elapsed times are multiple, so time passage of the model is noticeable.

For View1

Given a total state ${\displaystyle q=(s,t_{e})\in Q}$ where ${\displaystyle s=(\ldots ,(s_{i},t_{ei}),\ldots )}$

If unit event segment ${\displaystyle \omega }$ is the null event segment, i.e. ${\displaystyle \omega =\epsilon _{[t,t+dt]}}$, the state trajectory in terms of Timed Event System is

${\displaystyle \Delta (q,\omega )=((\ldots ,(s_{i},t_{ei}+dt),\ldots ),t_{e}+dt).}$

For View2

Given a total state ${\displaystyle q=(s,t_{s},t_{e})\in Q}$ where ${\displaystyle s=(\ldots ,(s_{i},t_{si},t_{ei}),\ldots )}$

If unit event segment ${\displaystyle \omega }$ is the null event segment, i.e. ${\displaystyle \omega =\epsilon _{[t,t+dt]}}$, the state trajectory in terms of Timed Event System is

${\displaystyle \Delta (q,\omega )=((\ldots ,(s_{i},t_{si},t_{ei}+dt),\ldots ),t_{s},t_{e}+dt).}$

1. The behavior of a couple DEVS network whose all sub-components are deterministic DEVS models can be non-deterministic if ${\displaystyle Select(IMM(s))}$ is non-deterministic.

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

• [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.