On tandem stochastic networks with time-deteriorating product quality

We consider an n-site tandem stochastic production network where each product moves sequentially through the sites, and the product's quality deteriorates with its sojourn time in the system. At each site the product goes through two stages: the first stage is a processing operation with a generally-distributed random duration. This operation either does or does not conclude successfully; in the latter case, the operation is repeated immediately. Once the processing operation concludes successfully, the product goes through an inspection stage lasting a generally-distributed random duration. At the end of the inspection the product's state is determined as follows: either (i) it requires additional processing and moves forward to the next site; or (ii) it is found ‘good’ and exits the network with quality value depending on its total sojourn time in the system; or (iii) it is declared ‘failed’, discarded, and exits the network with zero quality value. Two scenarios are analysed: (i) a new product enters the system only after the preceding product has exited and (ii) the network is a tandem Jackson-type system. For each scenario, we construct both time-dependent and quality-dependent performance measures. In the case where the sites can be arranged in an arbitrary order, we derive easy to implement optimal index-type policies of ordering the sites so as to maximise the quality rate of the production network.