Graphics Programs Reference

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destination queue makes it possible to pay it before the selection of the next

queue.

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

places, p
w0
,p
w1
,p
w2
, and p
w3
. In the decision subnet in the GSPN model in

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.

9.3.2

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

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

modified.

9.4

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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