Aggregate approximation for tandem queueing systems with production blocking

In this paper, the authors study problems arising in applications of the cross aggregation method to tandem queueing systems with production blocking, and propose two types of applications with different state descriptions. The cross aggregation method provides a nested family of approximations of stationary state probabilities of the model by imposing several different levels of assumptions on independence among nodes. Namely, in Level 1 the authors derive an approximate model by looking at one node at a time, in Level 2 by looking at two adjacent node at a time, in Level 3 by looking at three adjacent nodes at a time, and so on. The method, however, cannot be applied in a naive form to tandem queueing systems with production blocking since the state space of the system is not a product space and individual state spaces of nodes. The authors propose two ways of state description to derive a Markov chain. Using one of them, the method can be applied in Levels 2, 3 and higher, but not in Level 1. Using the other, the method can be applied in any levels of approximation after modifying the Markov chain to have a product state space, but transition rates of the modified chain become complicated. A comprehensive numerical test shows that in most cases the method provides very good approximations in Level 3 and sufficiently accurate approximations even in Level 2 for practical purposes.