COMPACT MODELS OF RANDOM POLLING SYSTEMS - Modelling with Generalised Stochastic Petri Nets - page 217

buffer empty (M(p a ) = N a ), tokens in place p q represent queues with a
waiting customer (M(p q ) = N q ), and tokens in place p w represent walking
servers. Tokens in place p s model queues being served, as well as busy
servers (M(p s ) = N s ).
Consequently, T a is a timed transition with rate M(p a )λ, modelling the su-
perposition of the interrupted Poisson arrival processes, T s has rate M(p s )µ,
and models the M(p s ) parallel service activities, and T w has rate M(p w )ω
to model the server walks that are currently in progress.
Place p r is the place where the decision about the queue to be visited next
is made: the choice is implemented by firing one of the three conflicting
immediate transitions t a , t q , or t s (all with priority 3), corresponding to the
choice of a queue with an empty buffer, of a queue with a waiting customer,
and of a queue with a service in progress, respectively.
The initial parametric marking of a GSPN model of a multiserver random
polling system with N queues and S servers has N tokens in place p a and
S tokens in place p r .
The rationale of the model is as follows. When a server terminates a service
at a queue (firing of T s ), it must execute a walk (firing of T w ) before choosing
the next queue to visit. In this abstract model, queues are not modelled
individually, so that instead of explicitly modelling the choice of the next
queue to be visited, we model the causes and the consequences of the choice
(this is the only relevant information from the point of view of the underlying
stochastic model). A server chooses a queue with a waiting customer with
the probability given in ( 9.1) , which can be rewritten as M(p q )/N.
This choice is conditioned on the existence of at least one queue with a
waiting customer, and it is modelled in the GSPN by immediate transition
t q , whose weight (probability) is M(p q )/N. As a consequence of this choice
a customer will be served; therefore the number of waiting customers M(p q )
is decremented by one, while the number of queues being served M(p s ) is
incremented by one (by the firing of the immediate transition t with priority
2), and a new service can start (timed transition T s is enabled).
Alternatively, a server may choose a queue where the buffer is empty with
the probability given in ( 9.3) , which can be rewritten as M(p a )/N, or it may
choose a queue at which a service is in progress with the probability given
in ( 9.2) , which can be rewitten as M(p s )/N.
The first choice is modelled by transition t a , while the second is modelled
by transition t s . Again, both choices are conditioned on the presence of at
least one queue in the required situation, i.e., on the presence of at least one
token in the appropriate place, and this explains the presence of test arcs
from t a to p a and from t s to p s (as well as from t q to p q ).
The consequence of both choices is that the server must execute another
walk (return to place p w to execute T w ), and then repeat the choice of the
next queue to be visited (coming back to the decision place p r ).
The GSPN model in Fig. 9.7 comprises only one non-trivial ECS of im-