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| {{multiple issues|
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| {{notability|date=November 2012}}
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| {{technical|date=November 2012}}
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| {{context|date=November 2012}}
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| }}
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| ''DEVS is closed under coupling'' [[Behavior_of_Coupled_DEVS#References|[Zeigper84]]] [[Behavior_of_Coupled_DEVS#References|[ZPK00]]]. In other words, given a [[DEVS#Coupled DEVS|coupled DEVS]] model <math> N </math>, its behavior is described as an atomic DEVS model <math> M</math>. For a given coupled DEVS <math> N </math>, once we have an equivalent atomic DEVS <math> M </math>, behavior of <math> M </math> can be referred to [[Behavior of DEVS|behavior of atomic DEVS]] which is based on [[Timed Event System]].
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| Similar to [[Behavior of DEVS|behavior of atomic DEVS]], behavior of the Coupled DEVS class is described depending on definition of the total state set and its handling as follows.
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| == View1: Total States = States * Elapsed Times ==
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| Given a [[DEVS#Coupled DEVS|coupled DEVS]] model <math> N = <X,Y,D,\{M_i\},C_{xx}, C_{yx}, C_{yy}, Select></math>, its behavior is described as an atomic DEVS model <math> M = <X,Y,S,s_0,ta, \delta_{ext}, \delta_{int}, \lambda> </math>
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| where
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| * <math>X</math> and <math>Y</math> are the input event set and the output event set, respectively.
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| * <math>S=\underset{i \in D}\times Q_i</math> is the partial state set where <math>Q_i=\{(s_i,t_{ei})| s_i \in S_i, t_{ei} \in (\mathbb{T} \cap [0, ta_i(s_i)])\} </math> is the total state set of component <math> i \in D</math> (Refer to [[Behavior_of_DEVS#View_1:_total_states_=_states_*_elapsed_times|View1 of Behavior of DEVS]]), where <math> \mathbb{T}=[0,\infty)</math> is the set of non-negative real numbers.
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| * <math>s_0=\underset{i \in D}\times q_{0i}</math> is the initial state set where <math>q_{0i}=(s_{0i},0)</math> is the total initial state of component <math> i \in D </math>.
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| *<math>ta:S \rightarrow \mathbb{T}^\infty </math> is the time advance function, where <math> \mathbb{T}^\infty=[0,\infty]</math> is the set of non-negative real numbers plus infinity.Given <math>s=(\ldots, (s_{i},t_{ei}),\ldots)</math>, <center> <math> ta(s)= \min\{ ta_i(si) - t_{ei}| i \in D\}.
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| </math> </center>
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|
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|
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| *<math>\delta_{ext}:Q \times X \rightarrow S </math> is the external state function. Given a total state <math>q=(s,t_e)</math> where <math>s=(\ldots, (s_{i}, t_{ei}),\ldots), t_e \in (\mathbb{T}\cap [0,ta(s)] )</math>, and input event <math> x \in X </math>, the next state is given by <center><math> \delta_{ext}(q, x)=s'=(\ldots,(s_i', t_{ei}'), \ldots) </math><center>
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| where
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| <center> <math>
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| (s_i', t_{ei}')=
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| \begin{cases}
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| (\delta_{ext}(s_i, t_{ei}, x_i),0) & \text{if } (x, x_i) \in C_{xx}\\
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| (s_i, t_{ei}) & \text{otherwise}.
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| \end{cases}
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| </math></center>
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| Given the partial state <math>s=(\ldots,(s_i, t_{ei}),\ldots) \in S </math>, let <math> IMM(s)=\{i \in D| ta_i(s_i) = ta(s) \} </math> denote ''the set of imminent components''. The ''firing component'' <math> i^* \in D </math> which triggers the internal state transition and an output event is determined by <center> <math> i^* = Select(IMM(s)).</math></center>
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| *<math>\delta_{int}:S \rightarrow S </math> is the internal state function. Given a partial state <math> s=(\ldots, (s_{i}, t_{ei}),\ldots)</math>, the next state is given by <center><math> \delta_{int}(s)=s'=(\ldots,(s_i', t_{ei}'), \ldots) </math><center>
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| where
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| <center> <math>
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| (s_i', t_{ei}')=
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| \begin{cases}
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| (\delta_{int}(s_i),0) & \text{if } i = i^*\\
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| (\delta_{ext}(s_i, t_{ei}, x_i),0) & \text{if } (\lambda_{i^*}(s_{i^*}), x_i) \in C_{yx}\\
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| (s_i, t_{ei}) & \text{otherwise}.
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| \end{cases}
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| </math></center>
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| *<math>\lambda:S \rightarrow Y^\phi </math> is the output function. Given a partial state <math> s=(\ldots, (s_{i}, t_{ei}),\ldots)</math>, <center><math> \lambda(s)=
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| \begin{cases}
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| \phi &\text{if } \lambda_{i^*}(s_{i^*})=\phi \\
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| C_{yy}(\lambda_{i^*}(s_{i^*})) &\text{otherwise}.
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| \end{cases}
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| </math><center> | |
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| == View2: Total States = States * Lifespan * Elapsed Times ==
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| Given a [[DEVS#Coupled DEVS|coupled DEVS]] model <math> N = <X,Y,D,\{M_i\},C_{xx}, C_{yx}, C_{yy}, Select></math>, its behavior is described as an atomic DEVS model <math> M = <X,Y,S,s_0,ta, \delta_{ext}, \delta_{int}, \lambda> </math>
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| where
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| * <math>X</math> and <math>Y</math> are the input event set and the output event set, respectively.
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| * <math>S=\underset{i \in D}\times Q_i</math> is the partial state set where <math>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}])\} </math> is the total state set of component <math> i \in D</math> (Refer to [[Behavior_of_DEVS#View_2:_total_states_=_states_*_lifespans_*_elapsed_times|View2 of Behavior of DEVS]]).
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| * <math>s_0=\underset{i \in D}\times q_{0i}</math> is the initial state set where <math>q_{0i}=(s_{0i},ta_i(s_{0i}),0)</math> is the total initial state of component <math> i \in D </math>.
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| *<math>ta:S \rightarrow \mathbb{T}^\infty </math> is the time advance function. Given <math>s=(\ldots, (s_{i},t_{si},t_{ei}),\ldots)</math>, <center> <math> ta(s)= \min\{ t_{si} - t_{ei}| i \in D\}.
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| </math> </center>
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|
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|
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| *<math>\delta_{ext}:Q \times X \rightarrow S \times \{0, 1\} </math> is the external state function. Given a total state <math>q=(s,t_s,t_e)</math> where <math>s=(\ldots, (s_{i}, t_{si},t_{ei}),\ldots), t_s \in \mathbb{T}^\infty, t_e \in (\mathbb{T}\cap [0,t_s] )</math>, and input event <math> x \in X </math>, the next state is given by <center><math> \delta_{ext}(q, x)=((\ldots,(s_i', t_{si}', t_{ei}'), \ldots),b) </math><center>
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| where
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| <center> <math>
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| (s_i', t_{si}', t_{ei}')=
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| \begin{cases}
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| (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)\\
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| (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)\\
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| (s_i, t_{ei}) & \text{otherwise}
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| \end{cases}
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| </math></center>
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| and
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| <center> <math>
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| b=
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| \begin{cases}
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| 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)\\
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| 0 & \text{otherwise}.
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| \end{cases}
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| </math></center>
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| Given the partial state <math>s=(\ldots,(s_i, t_{si}, t_{ei}),\ldots) \in S </math>, let <math> IMM(s)=\{i \in D| t_{si} - t_{ei} = ta(s) \} </math> denote ''the set of imminent components''. The ''firing component'' <math> i^* \in D </math> which triggers the internal state transition and an output event is determined by <center> <math> i^* = Select(IMM(s)).</math></center>
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| *<math>\delta_{int}:S \rightarrow S </math> is the internal state function. Given a partial state <math> s=(\ldots, (s_{i},t_{si}, t_{ei}),\ldots)</math>, the next state is given by <center><math> \delta_{int}(s)=s'=(\ldots,(s_i', t_{si}', t_{ei}'), \ldots) </math><center>
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| where
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| <center> <math>
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| (s_i', t_{si}', t_{ei}')=
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| \begin{cases}
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| (s_i', ta_i(s_i'),0) & \text{if } i = i^*,\delta_{int}(s_i)=s_i',\\
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| (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)\\
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| (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)\\
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| (s_i, t_{si}, t_{ei}) & \text{otherwise}.
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| \end{cases}
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| </math></center>
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| *<math>\lambda:S \rightarrow Y^\phi </math> is the output function. Given a partial state <math> s=(\ldots, (s_{i}, t_{si}, t_{ei}),\ldots)</math>, <center><math> \lambda(s)=
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| \begin{cases}
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| \phi &\text{if } \lambda_{i^*}(s_{i^*})=\phi \\
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| C_{yy}(\lambda_{i^*}(s_{i^*})) &\text{otherwise}.
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| \end{cases}
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| </math><center>
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| == Time Passage ==
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| Since in a coupled DEVS model with non-empty sub-components, i.e., <math> |D|>0</math>, the number of clocks which trace their elapsed times are multiple, so time passage of the model is noticeable.
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| ;For View1
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| Given a total state <math> q=(s,t_e) \in Q </math> where <math> s = (\ldots,(s_i, t_{ei}),\ldots) </math>
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| If [[Event Segment#Unit event segment|unit event segment]] <math> \omega</math> is the [[Event Segment#Null event segment|null event segment]], i.e. <math> \omega=\epsilon_{[t, t+dt]}</math>, the state trajectory in terms of [[Timed Event System]] is
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| <center> <math> \Delta(q, \omega)=((\ldots,(s_i, t_{ei}+dt),\ldots), t_e+dt).</math> </center>
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| ; For View2
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| Given a total state <math> q=(s,t_s,t_e) \in Q </math> where <math> s = (\ldots,(s_i, t_{si}, t_{ei}),\ldots) </math>
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| If [[Event Segment#Unit event segment|unit event segment]] <math> \omega</math> is the [[Event Segment#Null event segment|null event segment]], i.e. <math> \omega=\epsilon_{[t, t+dt]}</math>, the state trajectory in terms of [[Timed Event System]] is
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| <center> <math> \Delta(q, \omega)=((\ldots,(s_i,t_{si}, t_{ei}+dt),\ldots), t_{s}, t_e+dt).</math> </center>
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| == Remarks ==
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| # The behavior of a couple DEVS network whose all sub-components are [[DEVS#Deterministic_DEVS_and_Non-deterministic_DEVS|''deterministic DEVS'']] models can be ''non-deterministic'' if <math> Select(IMM(s))</math> is ''non-deterministic''.
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| ==See also==
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| *[[DEVS]]
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| *[[Behavior of DEVS|Behavior of Atomic DEVS]]
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| *[[Simulation Algorithms for Coupled DEVS]]
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| *[[Simulation Algorithms for Atomic DEVS]]
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| == References ==
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| * [Zeigler84] {{cite book|author = Bernard Zeigler | year = 1984| title = Multifacetted Modeling and Discrete Event Simulation | publisher = Academic Press, London; Orlando | isbn = 978-0-12-778450-2 }}
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| * [ZKP00] {{cite book|author = Bernard Zeigler, Tag Gon Kim, Herbert Praehofer| year = 2000| title = Theory of Modeling and Simulation| publisher = Academic Press, New York | isbn= 978-0-12-778455-7 |edition=second}}
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| {{DEFAULTSORT:Behavior Of Coupled Devs}}
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| [[Category:Automata theory]]
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| [[Category:Formal specification languages]]
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