COMPACT MODELS OF RANDOM POLLING SYSTEMS - Modelling with Generalised Stochastic Petri Nets - page 225

p a

N

p a

N

T a

T a

p q

T w p s T s

T w p w T s

p w

p p

S

(b)

(a)

Figure 9.11: Two further simplifications of the SPN model

pendency on S is embedded into the arc weights and transition rates. This
embedding bears some resemblance to the embedding of the model charac-
teristics into appropriate functions, typical of high-level SPNs [36] . Second,
note that servers have disappeared from the model; the server utilization
policy is not obvious, as it is not obvious that servers can perform several
walks in a row. Third, note that we ended up with three transitions and
three places arranged in a circle; that two of the transitions have infinite-
server rates, while the rate of the third one is an infinite-server multiplied
by a function that accounts for the number of servers in the system and the
number of tokens in the transition output place.
Finally, a few words are needed to comment on the numerical results in
Table 9.5, which report the cardinality of the tangible and vanishing state
spaces of the various GSPN models presented so far, in the case of two
servers (S = 2), for an increasing number of queues.
While a linear growth with N was already achieved with the model in
Fig. 9.7, that already exhibits a tangible state space cardinality equal to
3N, the reduction of immediate transitions allows the elimination vanishing
markings, so that the cardinality of the state space of the final model equals
3N. This is extraordinarily less than the number of states of the original
simple GSPN model we started with!