COMPACT MODELS OF RANDOM POLLING SYSTEMS - Modelling with Generalised Stochastic Petri Nets - page 219

This is because no coherence is guaranteed between the system state at the
moment of the choice, and the system state at the end of the walk.
More subtle error situations can be easily devised. For example, a server
may choose a queue with a customer waiting with probability 1/N, if the
number of tokens in place p q at the time of the choice is equal to one; but
it may be the case that when the server terminates its walk, there is more
than one token in p q (there are several queues with waiting customers in the
system) so that the choice probability is incorrect.
The reason why this problem was not present in the GSPN model in Fig. 9.6
is that there queues had an identity, and servers chose according to this
identity. The fact that the state of the chosen queue could change was
irrelevant, because of the fact that the choice was not based on the queue
state. Now, queues are identified only by their local state, but this local
state changes in time, so that choices can only be based on the instantaneous
situation.

9.4.3 Number of states and limitations of the model

The reduction in the state space size with respect to the models in which
queues are individually described, is shown in Table 9.5, in the case of two
servers (S = 2), for an increasing number of queues.
The differences in the number of states become drastic for larger values of
N, due to a linear growth with N, which can be expected by the elimina-
tion of the combinatorial growth of the possible symmetric combinations of
individual queue states.
It is also interesting to note that the resulting model is parametric in both
the number of queues and the number of servers. The parametrization is a
result of the exploitation of the system symmetries.

9.5 From GSPNs to SPNs

The abstractions discussed so far have always been due to the ingenuity of
the modeller. In Chapter 6 we briefly illustrated the rules for the struc-
tural reduction of GSPN models, aiming at the elimination of all immediate
transitions from a GSPN, thus producing an equivalent SPN. We now apply
those rules to the reduction of the abstract GSPN model of Fig. 9.7. It is
worth mentioning at this point that in this particular case the reduction
process not only eliminates all immediate transitions, and consequently in-
duces a (consistent) reduction in the number of states, but also produces
a model that is much simpler than the initial one, thus allowing a deeper
understanding of the intrinsic system behaviour. Unfortunately, this is not
always the case: in general, the elimination of immediate transitions is paid
with the introduction of a large number of timed transitions, usually result-