GSPN BASICS - Modelling with Generalised Stochastic Petri Nets - page 97

the type of pdf for the description of the activity duration, and we can thus
only match the average observed duration. The rate of a timed transition
is the only parameter necessary for the specification of the firing delay pdf.
In a similar way:
• transition T start , which describes the activation of the two parallel
processes, is associated with a random delay, with negative exponential
pdf whose mean matches the average time for the activation operation,
• transition T par1 , which describes the first parallel process, is associ-
ated with a random delay, with negative exponential pdf whose mean
matches the average time for the execution of the first process,
• transition T par2 , which describes the second parallel process, is associ-
ated with a random delay, with negative exponential pdf whose mean
matches the average time for the process execution,
• transition T syn , which describes the process synchronization, is associ-
ated with a random delay, with negative exponential pdf whose mean
matches the average time for the synchronization to take place,
• transition T check , which describes the control of the results, is associ-
ated with a random delay, with negative exponential pdf whose mean
matches the average time for the control operation,
• transition T I/O , which describes the output of the results, is associ-
ated with a random delay, with negative exponential pdf whose mean
matches the average time for the output operation.
The situation is slightly different for the two transitions T OK , and T KO ,
that form a free-choice conflict set. Indeed, in this case the association of
the rate of the negative exponential pdf to the two transitions must account
for two aspects: the duration of the consistency check operation, and the
probability of obtaining either a positive or a negative result. If the average
duration of the consistency check is 2 time units, and the probability of a
success is 0.9, then the resulting rate associated with T OK is 0.9/2 = 0.45,
while the rate associated with T KO is 0.1/2 = 0.05. Note that the sum of
the two rates equals the inverse of the average consistency check time.
SPN models have quite nice features: their qualitative behaviour is identical
to that of their underlying PN model with no temporal specifications, so that
the reachability and structural analysis results obtained from the PN model
are valid also for the SPN model. Moreover, as we shall see in the next
chapter, SPN models are isomorphic to continuous-time Markov chains: a
class of stochastic processes for which a well-developed theory exists.