MODELLING FLEXIBLE MANUFACTURING SYSTEMS - Modelling with Generalised Stochastic Petri Nets - page 179

When the cells are connected, however, we have to take into account some
additional constraint. The transitions of the subnet representing cell i might
all be disabled because busyM i is empty, IB i is empty, and either BB i
contains some token but OB i−1 is empty, or OB i contains some token, but
BB i+1 is empty.
If cell i is blocked because K i tokens are in place BB i but OB i−1 is empty, it
is easy to see that OB i−1 will eventually receive a token so that enterCell i
will become enabled (and hence also transitions inM i and outM i will eventu-
ally become enabled). This can be proved by induction, showing that if OB j
is marked (for some j : 1 ≤ j < i − 1) then OB j+1 will eventually receive a
token. Indeed, due to the P-invariant above, if OB j+1 is empty, then K j+1
tokens must be distributed into places BB j+1 , IB j+1 and busyM j+1 ; since
by hypothesis OB j is marked, at least one token can eventually flow from
any of the three places listed before, to place OB j+1 . Moreover, it is always
possible to reach a marking in which OB 1 is marked due to the fact that
transition enterCell 1 can fire as soon as BB 1 contains at least one token
(the reader may easily convince him/herself that this statement is true by
playing the token game on the GSPN model).
Hence, from a marking in which BB i is marked, ∀ k : j < k ≤ i − 1,OB k are
empty and OB j is not empty, it is eventually possible to reach a marking in
which OB j+1 is marked; from here it is possible to reach a marking in which
OB j+2 is marked, and so on until a marking is reached in which OB i−1 is
marked, so that transition enterCell i may fire.
If the subnet of cell i is blocked because K i tokens are in OB i but BB i+1 is
empty, then it is possible to show that transition enterCell i+1 will eventually
become enabled. In fact, it is easy to verify that if BB j+1 is marked (for
some j : i < j ≤ n) and BB j is empty, then BB j will eventually receive
a token. Moreover, from a marking in which BB n is empty, it is always
possible to reach a marking in which BB n is marked due to the fact that
transition exitCell n can fire as soon as OB n contains at least one token.
Hence from a marking in which OB i is marked, ∀ k : i + 1 ≤ k < j,BB k
are empty and BB j is not empty, it is possible to reach a marking in which
BB j−1 is marked. From here it is possible to reach a marking in which
BB j−2 is marked, and so on, until a marking is reached in which BB i+1 is
marked, so that transition enterCell i+1 may fire.

8.2.2 Performance analysis of the Kanban system

In this section we summarize the results of the extensive performance eval-
uation study of a Kanban model presented in [ 41] . The experiments have
the objective of studying the performance and fault tolerance of a Kanban
system with five cells. The studied system is highly homogeneous: all the
cells have the same machining time (the rate of transition outM i is 4.0) and
the same number K of cards. A first set of experiments was performed to