GSPN REPRESENTATION OF PHASE-TYPE DISTRIBUTIONS - Modelling with Generalised Stochastic Petri Nets - page 161

A complete treatment of how to include PH distributions in GSPN models
can be found in [21, 2, 13] . In [ 21] the expansion is performed at the CTMC
level with the use of an algorithm that maps the reachability graph of the
GSPN into the expanded state space of the CTMC. With this approach,
the timed transitions of the GSPN model are annotated with attributes
that specify the characteristics of their distributions and of their memory
policies. These attributes are used by the expansion algorithm to obtain the
desired extended CTMC. IN [13] a methodology is proposed to perform the
expansion at the net level. With this approach, timed transitions with PH
distributions and with specified memory policies are substituted by proper
subnets. After the substitution takes place, the resulting model is a standard
(but expanded) GSPN that can be analyzed with usual techniques. In this
topic we consider only this second method and we limit our discussion to
simply highlighting the fundamental aspects of the problem by using our
example. For this purpose we assume that the CPU service time for the
customers of the Central Server System has one of the simplest possible PH
distributions, namely an Erlang-3. The discussion of this simple example
will be su cient to illustrate a technique that can be applied to embed into
GSPN models any PH distribution.

7.2.1 Erlang distributions

Erlang distributions correspond to series of exponential stages all with the
same mean. A random variable x with Erlang-k distribution can be in-
terpreted as a sum of k independent and identically distributed random
variables. Each of these components has a negative exponential distribu-
tion with parameter kµ. Under this assumption, the expected value of the
Erlang-k variable is E[x] = 1/µ and its variance is V AR[x] = 1/(kµ 2 ).
Timed transitions with Erlang firing time distributions can be easily repre-
sented with GSPN models in which each stage is represented by an exponen-
tially timed transition. As usual, the single-server policy can be enforced by
using a control place to insure that at most one stage be active at any given
time. Considering as an example an Erlang distribution with three expo-
nential stages, Fig. 7.5 depicts one of the possible GSPN representations of
such a distribution in which transitions T 1 , T 2 , and T 3 implement the three
exponential stages and have all the same firing rate.
This straightforward representation of an Erlang distribution is, however,
not suited for conflicting timed transitions in which the race policy requires
the firing action to be instantaneous after a generally distributed amount of
time is elapsed. Consider for instance the GSPN system of Fig. 7.6 in which
a token in place p in enables two conflicting transitions. The observation that
the “horizontal” subnet provides an Erlang-3 delay could lead to the super-
ficial impression that this is a correct implementation of a conflict between a
transition with negative exponential firing time distribution (transition T 4 )