GSPN BASICS - Modelling with Generalised Stochastic Petri Nets - page 108

p 1

T 1 t 2

Figure 5.4: A conflict set comprising a timed and an immediate transition

1. Consider a free-choice conflict set comprising an immediate and a
timed transition, and assume for a moment that priority does not
exist. The race policy makes the immediate transition always win,
except for the case in which a zero delay is sampled from the nega-
tive exponential pdf. Although the probability of selecting the value
zero is null, some problem may arise when the conflict set is enabled
infinitely often in a finite time interval. For example, in the case of
Fig. 5.4, the timed and the immediate transitions are always enabled,
because the firing of the immediate transition t 2 does not alter the PN
marking. The situation is changed only when the timed transition T 1
fires. This happens with probability one in time zero, possibly after
an infinite number of firings of the immediate transition. To avoid
these (sometimes strange) limiting behaviours, the priority of imme-
diate over timed transitions was introduced in the GSPN definition.
This makes the timed transition T 1 in Fig. 5.4 never enabled.

2. The mentioned memoryless property of the negative exponential pdf,
at any time instant makes the residual time until a timer associated
with a transition expires statistically equivalent to the originally sam-
pled timer reading. Thus, whether a new timer value is set at every
change of marking, or at every instant a transition becomes enabled
after disabling, or after firing, makes no difference from the point of
view of the probabilistic metrics of the GSPN.

3. Since the probability that a sample extracted from a negative expo-
nential pdf takes a specific value x equals zero, the probability of two
timers expiring at the same time is null. Indeed, given the value sam-
pled by the first timer, the probability that the second one samples
the same value is zero.