PETRI NETS AND THEIR PROPERTIES - Modelling with Generalised Stochastic Petri Nets - page 21

p 1 p 2

n m

t

k

p 3

Figure 2.2: The enabling and firing rules

respectively. Several processes may access simultaneously the database for
reading, as long as no write is under way. Indeed, the enabling condition of
t 4 requires a token in input place p 5 (that represents the condition “no write
is in progress”). Observe that the firing of t 4 does not change the condition
“no write is in progress”, since it does not alter the content of p 5 . A process
may get exclusive access for a write whenever no read is under way and no
other write is in progress. The “and” of the two conditions is obtained by
the enabling condition of t 5 that requires zero tokens in inhibitor place p 6
(whose marking represents the number of processes currently performing a
read access), and one token in input place p 5 (representing the condition
“no write is in progress”). Observe also that the firing of t 5 changes the
condition “no write is in progress” by removing the token from p 5 . The
condition is re-established at the end of the write access, when transition t 7
fires by putting a token back to place p 5 .
The natural extension of the concept of transition firing, is the firing of
a transition sequence (or execution sequence). A transition sequenc e 1 σ =
t (1) , ··· ,t (k) can fire starting from marking M if and only if there exists a se-
quence of markings M = M (1) , ··· ,M (k+1) = M 0 such that ∀ i = (1, ··· ,k),M (i) [t (i) i M (i+1) .
We denote by M[σ i M 0 the firing of a transition sequence, and we say that
M 0 is reachable from M.
An important final consideration is that the enabling and firing rules for a
generic transition t are “local”: indeed, only the numbers of tokens in the
input and inhibitor places of t, and the weights of the arcs connected to t
need to be considered to establish whether t can fire and to compute the
change of marking induced by the firing of t. This justifies the assertion

1 We write t (1) rather than t 1 because we want to indicate the first transition in the
sequence, that may not be the one named t 1 .