GSPN REPRESENTATION OF PHASE-TYPE DISTRIBUTIONS - Modelling with Generalised Stochastic Petri Nets - page 162

p in T 1 p 1 T 2 p 2 T 3 p out

p control

Figure 7.5: The GSPN representation of an Erlang-3 server

p in T 1 p 1 T 2 p 2 T 3 p out

p control

p 3 T 4

p 4

Figure 7.6: The GSPN model of a conflict between Erlang-3 and exponential
servers

and a subnet representing an activity with an Erlang-3 distribution. The
model of Fig. 7.6 instead solves the conflict by making transition T 1 race
against transition T 4 . In this way only the first stage of the Erlang dis-
tribution takes part in the race that is won by the whole subnet whenever
transition T 1 fires before transition T 4 , thus removing the token from place
p in .
In order to overcome this problem, an alternative representation of PH
distributions was proposed in [ 13] . According to this proposal, the subnet
corresponding to the PH distribution comprises an input place that, upon
the enabling of the subnet, is kept marked until the completion of the last
stage of the distribution. For the case of the Erlang-3 distribution considered
in Fig. 7.5, the new GSPN representation is depicted in Fig. 7.7.
When place p in becomes marked and the whole subnet is idle (place p control
is marked), two token s 2 are immediately deposited in place p 1 and the first
delay stage starts; at the same time the token in place p control is withdrawn

2 The multiplicity of the arcs between t 1 and p 1 , and between p 3 and T 3 , depends on the
degree of the Erlang distribution that is being modelled; Erlang- k distributions require a
k− 1 multiplicity.