Behavior of DEVS

The behavior of a given DEVS model is a set of sequences of timed events including null events, called event segments, which make the model move from one state to another within a set of legal states. To define it this way, the concept of a set of illegal state as well a set of legal states needs to be introduced.

In addition, since the behavior of a given DEVS model needs to define how the state transition change both when time is passed by and when an event occurs, it has been described by a much general formalism, called general system [ZPK00]. In this article, we use a sub-class of General System formalism, called timed event system instead.

Depending on how the total state and the external state transition function of a DEVS model are defined, there are two ways to define the behavior of a DEVS model using Timed Event System. Since the behavior of a coupled DEVS model is defined as an atomic DEVS model, the behavior of coupled DEVS class is also defined by timed event system.

View 1: total states = states * elapsed times
Suppose that a DEVS model, $$\mathcal{M}=$$ has


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

Then the DEVS model, $$\mathcal{M} $$ is a Timed Event System $$\mathcal{G}= $$ where \times Q$$ is defined for two different cases: $$ q \in Q_N $$ and $$ q \in Q_A $$. For a non-accepting state $$ q \in Q_N $$, there is no change together with any even segment $$\omega \in \Omega_{Z,[t_l,t_u]}$$ so $$(q,\omega,q) \in \Delta.$$
 * The event set $$Z=X \cup Y^\phi$$.
 * The state set $$Q=Q_A \cup Q_N$$ where $$ Q_N=\{\bar{s} \not \in S \}$$.
 * The set of initial states $$ \,Q_0 = \{(s_0,0)\}$$.
 * The set of accepting states $$ Q_A = \mathcal{M}.Q.$$
 * The set of state trajectories $$ \Delta \subseteq Q \times \Omega_{Z,[t_l,t_u]}

For a total state $$ q=(s,t_e) \in Q_A$$ at time $$ t \in \mathbb{T} $$ and an event segment $$ \omega \in \Omega_{Z,[t_l,t_u]}$$ as follows.

If unit event segment $$ \omega$$ is the null event segment, i.e. $$ \omega=\epsilon_{[t, t+dt]}$$ $$\, (q, \omega, (s, t_e+dt)) \in \Delta.$$

If unit event segment $$ \omega$$ is a timed event $$ \omega=(x, t)$$ where the event is an input event $$ x \in X$$, $$ (q, \omega, (\delta_{ext}(q,x), 0)) \in \Delta. $$

If unit event segment $$ \omega$$ is a timed event $$ \omega=(y, t)$$ where the event is an output event or the unobservable event $$ y \in Y^\phi$$, $$ \begin{cases} (q, \omega,(\delta_{int}(s), 0)) \in \Delta& \textrm{if } ~ t_e = ta(s), y = \lambda(s)\\ (q, \omega, \bar{s})                     & \textrm{otherwise}. \end{cases} $$

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

View 2: total states = states * lifespans * elapsed times
Suppose that a DEVS model, $$\mathcal{M}=$$ has


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

Then the DEVS $$Q=\mathcal{D}$$ is a timed event system $$\mathcal{G}= $$ where \times Q$$ is depending on two cases: $$q \in Q_N $$ and $$q \in Q_A $$. For a non-accepting state $$ q \in Q_N $$, there is no changes together with any segment $$\omega \in \Omega_{Z,[t_l,t_u]}$$ so $$(q,\omega,q) \in \Delta.$$
 * The event set $$Z=X \cup Y^\phi$$.
 * The state set $$Q=Q_A \cup Q_N $$ where $$ Q_N=\{ \bar{s} \not \in S \}$$.
 * The set of initial states$$ \,Q_0 = \{(s_0,ta(s_0),0)\}$$.
 * The set of acceptance states $$ Q_A = \mathcal{M}.Q$$.
 * The set of state trajectories $$ \Delta \subseteq Q \times \Omega_{Z,[t_l,t_u]}

For a total state $$ q=(s,t_s, t_e) \in Q_A$$ at time $$ t \in \mathbb{T} $$ and an event segment $$ \omega \in \Omega_{Z,[t_l,t_u]}$$ as follows.

If unit event segment $$ \omega$$ is the null event segment, i.e. $$ \omega=\epsilon_{[t, t+dt]}$$ $$ (q, \omega, (s, t_s, t_e+dt)) \in \Delta.$$

If unit event segment $$ \omega$$ is a timed event $$ \omega=(x, t)$$ where the event is an input event $$ x \in X$$, $$ \begin{cases} (q, \omega, (s', ta(s'), 0))\in \Delta& \textrm{if } ~\delta_{ext}(s,t_s,t_e,x)=(s',1),\\ (q, \omega, (s', t_s, t_e))\in \Delta& \textrm{otherwise, i.e. } ~\delta_{ext}(s,t_s,t_e,x)=(s',0). \end{cases} $$

If unit event segment $$ \omega$$ is a timed event $$ \omega=(y, t)$$ where the event is an output event or the unobservable event $$ y \in Y^\phi$$, $$ \begin{cases} (q, \omega, (s', ta(s'),0)) \in \Delta& \textrm{if } ~t_e = t_s, y = \lambda(s), \delta_{int}(s)=s',\\ (q, \omega, \bar{s} )\in \Delta& \textrm{otherwise}. \end{cases} $$

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

Features of View1
View1 has been introduced by Zeigler [Zeigler84] in which given a total state $$ q=(s,t_e) \in Q $$ and $$\, ta(s)=\sigma $$ where $$ \sigma $$ is the remaining time [Zeigler84] [ZPK00]. In other words, the set of partial states is indeed $$S=\{(d,\sigma)| d \in S', \sigma \in \mathbb{T}^\infty \} $$ where $$ S'$$ is a state set.

When a DEVS model receives an input event $$ x \in X$$, View1 resets the elapsed time $$ t_e $$ by zero, if the DEVS model needs to ignore $$ x $$ in terms of the lifespan control, modellers have to update the remaining time $$\, \sigma = \sigma - t_e$$ in the external state transition function $$ \delta_{ext} $$ that is the responsibility of the modellers.

Since the number of possible values of $$ \sigma $$ is the same as the number of possible input events coming to the DEVS model, that is unlimited. As a result, the number of states $$ s=(d, \sigma) \in S $$ is also unlimited that is the reason why View2 has been proposed.

If we don't care the finite-vertex reachability graph of a DEVS model, View1 has an advantage of simplicity for treating the elapsed time $$ t_e=0$$ every time any input event arrives into the DEVS model. But disadvantage might be modelers of DEVS should know how to manage $$\sigma$$ as above, which is not explicitly explained in $$\delta_{ext} $$ itself but in $$\Delta$$.

Features of View2
View2 has been introduced by Hwang and Zeigler[HZ06][HZ07] in which given a total state $$ q=(s, t_s, t_e) \in Q $$, the remaining time, $$ \sigma$$ is computed as

$$\, \sigma = t_s - t_e. $$

When a DEVS model receives an input event $$ x \in X$$, View2 resets the elapsed time $$ t_e $$ by zero only if $$ \delta_{ext}(q,x)=(s',1)$$. If the DEVS model needs to ignore $$ x $$ in terms of the lifespan control, modellers can use $$ \delta_{ext}(q,x)=(s',0) $$. Unlike View1, since the remaining time $$ \sigma $$ is not component of $$ S $$ in nature, if the number of states, i.e. $$ |S| $$ is finite, we can draw a finite-vertex (as well as edge) state-transition diagram [HZ06][HZ07]. As a result, we can abstract behavior of such a DEVS-class network, for example SP-DEVS and FD-DEVS, as a finite-vertex graph, called reachability graph [HZ06][HZ07].