In a GKCS, the number of entities in the various queues of each linkage station [J.sub.i,i + 1] are related by:
In a GKCS, the number of customers in each queue of a given stage i is bounded by the two parameters [K.sub.i] and [S.sub.i] according to relations (5) to (10), which give upper bounds for the WIP, the number of finished parts and the number of demands within a given stage i:
The inequalities given by property 2 bring to light some differences between the KCS and the GKCS. First, recall that in the KCS, for each stage i, the WIP, the number of finished parts as well as the number of demands of each stage i, are all bounded by the number of stage i kanbans, [K.sub.i].
Therefore, the GKCS with parameters [S.sub.i] = [K.sub.i] has the same behaviour as the KCS.
Note that relation (12) means that if there are more than [K.sub.i] [greater than] [S.sub.i] + [K.sub.i + 1] kanbans in stage i, then there are [K.sub.i] - [S.sub.i] - [K.sub.i + 1] kanbans that always stay in linkage station [J.sub.i, i + 1], without any impact on the behaviour of the GKCS. Thus, as already pointed out in and in, there is no need to put more kanbans in stage i than [S.sub.i] + [K.sub.i + 1], i.e., the number of stage i kanbans has to satisfy:
The steady state performance measures of particular interest in a GKCS are the following: the average work in process and the average number of finished parts for each stage, the proportion of back-ordered demands, the average number of back-ordered demands and the average waiting time of a back-ordered demand.
The queueing network of a GKCS can be viewed as a multi-class closed queueing network by considering each type of kanbans as one class of customers.
Example 1 is a single-stage GKCS and Example 2 is a 3-stage GKCS.
Consider a single-stage GKCS, consisting of four machines in tandem with all the service times being exponentially distributed, with the same mean equal to 1.
Consider a GKCS composed of three identical stages in tandem.
Consider now the case of the non-saturated GKCS with back-ordered demands.
In this section, we presented an analytical method for evaluating the performance of a GKCS. It appears to be rapid and very accurate.