GSPN REPRESENTATION OF PHASE-TYPE DISTRIBUTIONS - Modelling with Generalised Stochastic Petri Nets - page 160

before the interruption is lost and that the interrupted job should start anew
when the CPU becomes available again after its repair. The subnet added
to the original Central Server Model and depicted in Fig. 7.3 can be in-
terpreted according to both the phenomena that we want to model, while
the specification of the way of treating interrupted work must be added as
an “attribute” to transition CPU specifying whether it behaves according
to an age or an enabling memory policy, respectively (see Section 3.4 for a
discussion of this topic).
When the firing time distributions are negative exponential, the distinction
between the two policies is inessential, due to the memoryless property of
the distribution (see Chapter 6) and the model of Fig. 7.3 can be used
to study both phenomena without the need for any additional specification
1 . When instead the firing time distributions are not negative exponential,
considerable differences exist between the two cases.
The objective of the methodology discussed in this chapter is to allow treat-
ing these cases with GSPN models. In the next section, we will show how
predefined subnets can replace the exponential transitions taking automat-
ically into account all the details that need to be considered for the correct
implementation of the different types of distributions and of memory poli-
cies.

7.2 Phase-Type Distributions

The possibility of including timed transitions with general firing time distri-
butions in GSPN models is provided by the phase expansion [ 55] that allows
the behaviour of a random variable with general distribution to be approxi-
mated by means of a combination of negative exponential random variables
with appropriate parameters. These distributions are called Phase-Type
(PH) distributions. This means that an activity that lasts for a generally
distributed random amount of time can be modelled by breaking it down
into a set of stages (phases), all lasting for exponentially distributed peri-
ods of time. Considering that the state of an activity represented by an
exponentially distributed random variable is completely captured by the in-
dication that the activity is under way, because of the memoryless property
of this distribution (see Appendix A), in the case of the phase expansion,
the same state is characterized by the additional indication of the index of
the stage that is currently active. As a consequence, a multidimensional
CTMC can be used to describe the evolution of any activity represented
with PH distributions, and an overall CTMC (although more complex) can
still be recognized as the probabilistic model underlying SPNs and GSPNs
including timed transitions with PH distributed firing times.

1 This is indeed the case, provided that we don't need to implement a preemptive-
repeat-identical execution policy