PETRI NETS AND THEIR PROPERTIES - Modelling with Generalised Stochastic Petri Nets - page 29

t

p

t 1 t 2

p 1 p 2

Figure 2.8: Modelling conflict

•∃ p ∈ • t : M(p) < (k + 1) · I(p,t)

We therefore indicate with ED(t,M) the number of times that transition t
is enabled in marking M. We indicate with E(M) the multiset of transitions
enabled in marking M. We denote by t ∈ E(M) the condition “t enabled in
M”, and with T 0 ⊆ E(M), where T 0 ⊆ T, a set of transitions enabled in M.
In general, a conflict is any situation in which the firing of a transition
reduces the enabling degree of another transition (note that we disregard
conflicts of transitions with themselves). If the enabling degree is reduced
to zero, then the firing of a transition has disabled another transition that
was previously enabled.
If we go back to the example of Fig. 2.1, then we see that t 2 and t 3 are in
conflict and in any marking where place p 2 is marked with only one token,
the firing of t 2 also causes the disabling of t 3 . This conflict represents the
choice made by a process when it decides to require either a read or a write
access to the database. Also transitions t 4 and t 5 form a conflict, which is
however less obvious. Indeed, t 4 and t 5 share an input place (p 5 ), but this is
not the reason why the firing of t 4 disables t 5 (it is however the reason why
the firing of t 5 disables t 4 ). Transition t 4 only tests the number of tokens
in p 5 (note the output arc from t 4 to p 5 ). The reason why the firing of t 4
disables t 5 is the existence of the inhibitor arc from p 6 to t 5 . Since p 6 is an
output of t 4 , the firing of t 4 marks p 6 , and t 5 becomes disabled. In general,
a test for zero on an output place of a transition is a condition for conflict.
It is worth observing that in our readers & writers example conflicts are
always symmetric: whenever the firing of t decreases the enabling degree