5.4 Mapping from DEVS to DCharts

DEVS models can also be transformed into DCharts. Because of the closure under coupling of DEVS, any Coupled DEVS can be replaced by an Atomic DEVS that has the same behavior. It is not necessary to consider coupled DEVS in proving the mapping from DEVS to DCharts.

The DEVS formalism discussed here is real-time DEVS, which use the real time instead of virtual time as global time. The time unit is default to second.

Theorem 3   DEVS models can be transformed into DCharts that have exactly the same behavior.

Proof Different parts of an Atomic DEVS are mapped to DCharts constructs as following:

• The state set $S$ is mapped to a single variable $v$ of a DCharts model, whose state hierarchy has only one default state $s$. The state space of $v$ is a superset of $S$. This variable can always be found. It can be of a primitive type such as integer or string, a list which contains multiple elements, or any other types supported by the action language. This variable is used to keep track of the model state.

• The time advance function $ta$ and internal transition function $\delta_{int}$ are transformed into transitions with $after$ events. Each of such DCharts transitions is a self-loop on the state $s$. It uses $after(t)$ as its event, where $t$ is the $ta$ of a DEVS state. The guard of the transition checks the current state of the variable $v$, and tests if the model is in the old state of the DEVS model. In the output of this transition, $v$ is modified according to the $\delta_{int}$ of the DEVS model.

• The external transition function $\delta_{ext}$ is transformed into transitions with the same event names. This transformation is similar to the transformation between $\delta_{int}$ and DCharts transitions.

  The elapsed time of an external transition can be computed with the primitive action that allows access to the time since the simulation or execution starts. Suppose the time when the last state is entered is $t_last$. It is obtained with the time action in the enter actions of the state. The time when the external event is received is denoted by $t_event$. This time can be obtained in the guard (since the time action is side-effect-free) or the output of the transition. Then the elapsed time is equal to $t_event-t_last$, which can be known in the guard and the output.

• The $X$ (input set) and $Y$ (output set) of the DEVS model is ignored, since DCharts do not require to explicitly declare them.

• The output function $\lambda$ is transformed into action code in the output of transitions. It produces the same output as the DEVS model, according to the current state of the model.

  The events sent in the output of a DEVS transition are different from the events broadcast in a statecharts, because the first kind of events are explicitly sent to an output port. Fortunately, this kind of events are equivalent to the out-going messages in DCharts, which are textually represented as a port name and an event name separated by a dot.

$\Box$