COMPACT MODELS OF RANDOM POLLING SYSTEMS - Modelling with Generalised Stochastic Petri Nets - page 215

GSPN model featuring the remarkable property of being parametric with
respect to the number of queues as well.
First of all, note that the total number of customers being served in the
system equals the number of queues with a customer in service, and the
total number of waiting customers equals the number of queues in which
there is a customer waiting.
The latter observation may appear too trivial to be made, but it has far-
reaching consequences, since it permits the global state of the system to be
identified as the sum of the individual states of the queues.
The equally likely selection of the next queue to be visited allows the prob-
ability of providing service at the next queue to be written as:
N q

N = C q

(9.1)

N

where N q and C q are the number of queues with waiting customers, and the
total number of waiting customers in the system, respectively.
Similarly, the probability of being unable to provide service at the next
queue because a service is in progress can be written as:
N s

N = C s

(9.2)

N

where N s and C s are the number of queues with customers being served,
and the total number of customers being served in the system, respectively.
Finally, the probability of being unable to provide service at the next queue
because the buffer is empty can be written as:
N a

N = C a

(9.3)

N

where N a and C a are the number of queues with empty buffer and the total
number of empty buffers in the system, respectively.

9.4.1 The abstract GSPN model

The abstract GSPN model of the symmetric multiserver random polling
system with N queues is depicted in Fig. 9.7. The characteristics of the
timed and immediate transitions in this GSPN model are summarized in
Tables 9.6 and 9.7, respectively.
The global system state is defined by three quantities: the number of queues
with the buffer empty (N a ), the number of queues with a waiting customer
(N q ), and the number of walking servers. Note that the number of queues
being attended by a server (N s ) is simply the complement to N of the sum
of the first two quantities (N s = N − N a − N q ), which also equals the number
of busy servers.
These aggregate quantities are represented in the GSPN by the markings
of places p a ,p q ,p w and p s .

Tokens in place p a represent queues with the