Graphics Programs Reference
In-Depth Information
the theory of discrete-state stochastic processes in continuous time can be
applied for the evaluation of the performance of the real system described
with the timed PN model.
As we note in the appendix on stochastic processes, discrete-state stochastic
processes in continuous time may be very di cult to characterize from a
probabilistic viewpoint and virtually impossible to analyse. Their charac-
terization and analysis become instead reasonably simple in some special
cases that are often used just because of their mathematical tractability.
The simplicity in the characterization and analysis of discrete-state stochas-
tic processes in continuous time can be obtained by eliminating (or at least
drastically reducing) the amount of memory in the dynamics of the process.
This is the reason for the frequent adoption of the negative exponential
probability density function (pdf) for the specification of random delays.
Indeed, the negative exponential is the only continuous pdf that enjoys the
memoryless property, i.e., the only continuous pdf for which the residual
delay after an arbitary interval has the same pdf as the original delay (for
more details see Appendix A).
In the case of PN models with timed transitions adopting the race policy,
a random delay with negative exponential pdf can be associated with each
transition. This guarantees that the qualitative behaviour of the resulting
timed PN model is the same as the qualitative behaviour of the PN model
with no temporal specification. The timed PN models that are obtained
with such an approach are known by the name Stochastic PNs (SPNs). This
modelling paradigm was independently proposed by G.Florin and S.Natkin
[ 27] and by M.Molloy [51] .
Similar ideas appeared also in the works of
F.Symons [66] .
An SPN model is thus a PN model augmented with a set of rates (one for
each transition in the model). The rates are su cient for the characteriza-
tion of the pdf of the transition delays (remember that the only parameter
of the negative exponential pdf is its rate, obtained as the inverse of the
mean).
The use of negative exponential pdfs for the specification of the tempo-
ral characteristics in a PN model has some interesting effects that deserve
comment. Let us consider the dynamic evolution of an SPN model. As in
Chapter 3, for the description we can assume that each timed transition
possesses a timer. The timer is set to a value that is sampled from the
negative exponential pdf associated with the transition, when the transition
becomes enabled for the first time after firing (thus we are assuming an age
memory policy). During all time intervals in which the transition is enabled,
the timer is decremented. For the time being, let us assume that the speed
at which the timer is decremented is constant in all markings. Transitions
fire when their timer reading goes down to zero.
With this interpretation, each timed transition can be used to model the
execution of an activity in a distributed environment; all activities execute in
 
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