4.4.2 Firing under time constraints

In the following subsections, you find a description of firing under time allocation in INA .

Time option Duration for transitions - Firing duration

To each transitions, a non-negative integer is assigned. A transition has to fire as soon as enough tokens are available. If such a transition t with duration d fires at the moment $\tau$, then the input tokens are removed from its pre-places at the time $\tau$, and the output tokens are available at the time $\tau+d$ on its post-places.

A transition that is being fired, i.e. when the input tokens have been removed, but the output tokens have not been created yet, cannot start a second firing procedure during this period, even if sufficiently many tokens are available at its pre-places. Such nets are executed accordingly when firing maximal steps.

In nets with firing durations, transient states can arise, i.e., states in which no transitions can fire, but out of which, as time elapses, a state results, in which actions are possible. Even the initial state can be transient, if the initial clock position is not identically zero, i.e., some transitions are firing at present.

When the initial state is transient, INA indicates the next stable state: Initial state is transient. Stabilization time = 3 is an example for the stabilization of the initial state. In general, transient states are stabilized by incrementing the clock positions, and the according change of the state: e.g., Transitional period from previous state = 2 indicates, that, since the last state, two cycles have passed in which no transitions could fire.

In case you would like to obtain an overview of all (including transient) states, use the command <R> <A> in the editor in order to insert a loop (see page in chapter 3.1). The firing duration of the new transition (loop-transition - normally numbered with the transition number 0) is to be set to 1. In each reachable state, this transition is then ready to fire; the reachability graph cannot contain dead or transient states any more.

Time option intervals for transitions - eft/lft

To each transition, an integer interval eft, lft is assigned. The clock of the transition is started at the moment its pre-places are sufficiently marked. The transition may fire as soon as the time indicated by its clock is at least equal to eft, and it has to fire when the time is equal to lft, provided it has not been disabled by the firing of other transitions. If two not necessarily different transitions share a pre-place, i.e., they are in a static conflict, and one of them fires, then the clock of the second transition is either switched off, if this transition is disabled by the firing of the first one, or reset to zero, if it remains enabled afterwards. Firing a transition does not take time.

There are states in which transitions can fire, but no transition has to fire, because their latest firing times have not been attained yet. The enabled transitions are displayed together with their possible delay, and you have to choose not only the transition to be fired but also its delay, i.e. the number of time ticks of the system clock to elapse until its firing.

Time option intervals for arcs - permeability

To each arc, which is directed from a pre-place to a transition, a time interval eft, lft is assigned . At each place, there is a clock, which indicates the time that the most recently arrived token has already waited on the place. The clock is thus set to zero at the moment a token arrives at the place, and also when a token is removed from the place. A transition can fire if all of its pre-places are sufficiently marked and the clock of each pre-place indicates a time that lies within the interval eft, lft assigned to the arc directed from this pre-place to the transition. If a transition can fire, it has to; hence such nets are executed in steps. Firing a transition does not take time. INA stabilizes transient states automatically.

Before the first firing, an overview of the time spans is displayed, after which a clock of a place is stopped: Since a token can leave a place only by an arc, if the clock of this place indicates a time between eft and lft of this arc, the clock of a place can be switched off, if the time elapsed is greater than the maximum of all lft of the arcs leaving this place. Afterwards, no more tokens can be removed from this place, so all clock positions of the place which are greater than this maximum are equivalent. INA displays these times as clock stop time.

With permeability intervals, an overview of all (including transient) states is not possible, but you can obtain some transient states by using the command <R> <A> in the editor in order to insert a loop (see page in chapter 3.1). For the interval of the arc from the new place (loop-place - normally numbered with the transition number 0) to the new transition (loop-transition - normally numbered with the transition number 0), eft is to be set to one, and lft to oo. However, the reachability graph may still contain transient states in certain cases.