Graphics Programs Reference
In-Depth Information
Table 9.3: Characteristics of the timed transitions in the first GSPN model
of a multiserver random polling system (p,q = 0, 1, 2, 3)
transition
rate
semantics
T
(q)
λ
single-server
a
T
(q)
µ
single-server
s
T
(q)
ω
multiple-server
wp
Table 9.4: Characteristics of the immediate transitions in the first GSPN
model of a multiserver random polling system (p,q = 0, 1, 2, 3)
transition
weight
priority ECS
t
(q)
1
2
q + 1
s
t
(q)
1
1
q + 5
w
t
(q)
1
3
q + 9
rp
p
(q
s
, with q = 0, 1, 2, 3 are 1-bounded, as was expected, since the number of
customers either waiting or being served at any queue cannot exceed one.
The fifth P-semiflow covers places p
(q
p
, p
(q)
, p
(q)
, p
(q)
w0
, p
(q)
w1
, p
(q)
w2
, p
(q)
w3
, with
q = 0, 1, 2, 3. The token count of the resulting P-invariant is S, since the
P-invariant refers to the conservation of the number of servers in the system.
As a result, the GSPN model is bounded, and thus the number of reachable
states is finite. A number of T-semiflows (192) cover all transitions. The
structural characteristics of the GSPN fulfil the necessary condition given in
Chapter 2 for the model to be live and reversible. The reachability analysis
shows that the GSPN model indeed is live and reversible.
Note that the GSPN model in Fig.
9.4
does not account for the system
symmetries. Indeed, with such a PN structure, we might allow the queue
parameters to be different for different queues, and the choice of the next
queue to depend on both the queue that the server is leaving, and the one
it is going to visit next. The price that has to be paid for this generality
is a high number of reachable states: it can thus happen that in the case
of large values of N, the complexity of the solution exceeds the capacity of
existing software packages.
The second column of Table
9.5
shows the number of tangible and vanishing
states of the model presented in Fig.
9.4
in the case of two servers (S = 2),
for an increasing number of queues: only the entries for models with up
to six queues were computed, because larger values of N lead to memory
s
r
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