Verbal Description of Dynamics

Suppose that $A$ is an atomic DEVS such that $A=<X, Y, S, s_0, \tau, \delta_x, \delta_y>$ , $s$ is the current state of $A$ , and $p=(s, t_s, t_e) \in P$ . Then the possible discrete state transitions are:

1. If an external input $x$ comes in, $A$ executes $\delta_x(p,x)=(s',b)$ where $b \in \{0,1\}$ and the lifespan and the elapsed time with $s'$ can change or be preserved as follows.
   1. update $t_s = \tau(s')$ and $t_e = 0$ if $b=1$ ;
   2. keep $t_s$ and $t_e$ preserved if $b=0$ .

2. If no external input comes in, then when the elapsed time reaches the lifetime, $A$ executes $\delta_y(s)=(y,s')$ and update $t_s = \tau(s')$ and $t_e = 0$ .

Notice that the elapsed time $t_e$ increases linearly over time so it is a continuous variable whose time derivative is constant 1. However, the lifetime $t_s$ is not changing continuously but it is determined discretely at the time of executing either $\delta_x$ or $\delta_y$ .

In other word, suppose that there is $p=(s, t_s, t_e) \in P$ at time $t_l \in T$ , there is another $p'=(s',t_s',t_e') \in P$ at time $t_u \in T$ and $t_l \le t_u$ . If there is no event between $t_l$ and $t_u$ , it implies that the only difference between $p$ and $p'$ is that their elapsed time such that $(s'=s) \wedge (t_s' = t_s) \wedge (t_e'=t_e+t_u-t_l)$ .

Example 2.1 (Ping-Pong Player)   Figure 1.2 shows an atomic DEVS model for a ping-pong player. This model has an input event ``?receive'' and an output event ``!send''. And it has two states: ``Send'' and ``Wait''. Once the player gets into ``Send'', it will generates ``!send'' and backs to ``Wait'' after the sending time which is an random variant in the uniform probability distribution function (pdf) of [0.1, 1.2]. When staying at ``Wait'' and if it gets ``?receive'', it changes into ``Send'' again.

Formally we can rewrite this player as $M_{Player}=<X,Y,S,s_0,\tau,\delta_x,\delta_y>$ where $X$ ={?receive}; $Y$ ={!send};$S$ ={Send, Wait}; $s_0$ =Send; $\tau$ (Send)$\in$ [0.1, 1.2], $\tau$ (Wait)=$\infty$ ; $\delta_x(s,t_s,t_e,x)=\delta_x$ (Send,$\infty$ ,[0,$t_s$ ],?receive) =(Send,1),
$\delta_x(s,t_s,t_e,x)=\delta_x$ (Send,[0.1, 1.2],[0,$t_s$ ],?receive)=(Send,0);
$\delta_y(s)=\delta_y$ (Send)=(!send,Wait); $\square$

Figure 1.2: State Transition Diagram of Ping-Pong Player