Graphics Programs Reference
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destination queue makes it possible to pay it before the selection of the next
Again, the feature that we are introducing in the model has nothing to
do with the actual system operations. It is instead an abstraction that
is permitted by the symmetry in the model and that has the advantage of
reducing the size of the state space of the model. Moreover, the performance
parameters that we declared of interest are not affected by the inversion.
In fact, this abstraction is equivalent to a lumping of the state space of
the underlying CTMC. To see this, note that the decision subnet of the
GSPN model in Fig. 9.5 consists of one vanishing place p r , and four tangible
places, p w0 ,p w1 ,p w2 , and p w3 . In the decision subnet in the GSPN model in
Fig. 9.6, the four tangible places are substituted by one tangible place p w .
All markings that in the GSPN in Fig. 9.5 contain the same sum of tokens
in the four places p w0 ,p w1 ,p w2 , and p w3 , are grouped into a single marking
of the GSPN model in Fig. 9.6.
The GSPN model in Fig. 9.6 comprises only one ECS of immediate transi-
tions, that describes the selection of the next queue, like in the GSPN model
of Fig. 9.5. Five P-semiflows cover all the GSPN places, similarly to what
happened in the GSPN of Fig. 9.5. As a result, the GSPN model is bounded.
A number of T-semiflows cover all transitions, and the GSPN model is live
and reversible.
Number of states and limitations of the model
The second, third, and fourth columns in Table 9.5 show the cardinality
of the tangible and vanishing state spaces for the three GSPN models in
Figs. 9.4, 9.5, and 9.6, in the case of two servers (S = 2) and for an in-
creasing number of queues. Although in the three cases the growth of the
total number of states with N is always combinatorial, the exploitation of
symmetries yields very significant reductions, particularly for larger values
of N (for N = 6 the reduction is more than one order of magnitude).
Note that all the models we have presented so far are parametric in S, the
number of servers, but not in N, the number of queues in the system; indeed,
in order to change the values of S we only need to change the initial marking,
while in order to change the value of N, the structure of the net must be
Abstracting from the Queue Identifiers
Since the system is completely symmetric, i.e., since all queues can be char-
acterized with the same parameters, and the selection of the next queue is
made on the basis of a uniform probability distribution, it is possible to pro-
duce a very abstract model in which queues are not modelled individually.
This is a leap in the process towards abstraction, which produces a compact
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