COMPACT MODELS OF RANDOM POLLING SYSTEMS - Modelling with Generalised Stochastic Petri Nets - page 220

t a

p a T w

N

p r T a p w

S

p q

t q p s T s

t s

Figure 9.8: Abstract GSPN model obtained from that of Fig. 9.7 by elimi-
nating one immediate transition and one vanishing place

ing in models that, although easier to solve, are more di cult to understand
or to reason about.
Before applying the reduction algorithm, we can observe that in the GSPN
model of Fig. 9.7 two immediate transitions always fire in a row: the firing
of transition t q is indeed always followed by that of transition t, because t q
is enabled only if p q is marked (although the firing of t q does not modify
the marking of p q ), so that t is always enabled after the firing of t q . As
a consequence, p p is a vanishing place. Fig. 9.8 shows, with a rearranged
layout, the abstract GSPN model of Fig. 9.7 after t q and t have been merged
(the transition resulting from the combination was named t q ) and place p p
has been removed.
The net in Fig. 9.8 is covered by two P-semiflows. M(p a )+M(p q )+M(p s ) =
N is the P-invariant that represents the conservation of queues (a queue can
be either empty [p a ], or waiting for service [p q ], or in service [p s ]). The P-
invariant M(p s )+M(p w )+M(p r ) = S represents the conservation of servers
(a server may be either serving [p s ], or walking [p w ], or deciding which queue