Graphics Programs Reference
In-Depth Information
t
(q)
p
(q)
T
(q)
r0
w0
w0
t
(q)
p
(q)
T
(q)
r1
w1
w1
p
(q)
r
t
(q)
p
(q)
T
(q)
r2
w2
w2
t
(q)
p
(q)
T
(q)
r3
w3
w3
Figure 9.3: The subnet modelling the choice of the next queue and the server
movement
After the next queue has been selected, the server walks to the chosen queue.
The timed transitions T
(q)
wp
, p = 0, 1, 2, 3 model the walk times to reach queue
p; therefore, in a marking M they are assigned a rate equal to M(p
(q)
wp
)ω,
where ω
−1
is the mean walk time required to move from one queue to the
next.
9.2.1
The first complete random polling model
Fig.
9.4
shows a GSPN model describing a multiserver random polling sys-
tem with four queues. It is clearly composed of four replicas of the queue
submodel, connected through the submodels describing the queue selection
and the server walks. The initial marking (S tokens in place p
(0)
p
and one
token in places p
(q
a
) defines the number of servers in the system, as well as
their initial position and the initial states of the individual queues.
The characteristics of the timed and immediate transitions in the GSPN
The GSPN model in Fig.
9.4
comprises four ECSs of immediate transitions
(counting only the ECSs that include more than one transition, and that
may thus originate a conflict), corresponding to the choice of the next queue
to be visited at the output of the four queues.
Five P-semiflows cover all the GSPN places. Four of them cover the triplets
of places p
(q
a
, p
(q)
, p
(q)
, with q = 0, 1, 2, 3.
The resulting P-invariants are
q
s
M(p
(q
a
) + M(p
(q)
) + M(p
(q)
This guarantees that places p
(q
a
, p
(q)
) = 1.
,
q
s
q
Search WWH ::
Custom Search