Product-forms from a CAT and DOG
Peter G. Harrison · ACM SIGMETRICS Performance Evaluation Review · 2002
The equilibrium state space probabilities of a stationary Markov chain can be obtained immediately from its reversed process. There are two main steps in the derivation of product-form solutions for multi-dimensional Markov chains using this approach. First, the reversed process must be determined. This is achieved for a wide class of cooperating processes using a compound agent theorem (CAT), a compositional result from Markovian Process Algebra (MPA). Secondly, a path to each state must be found from some specified reference state. This is usually obtained in a simple way by considering the components of the state in order of dimension, e.g. in a dimension-ordered graphical (DOG) representation. In this note, the main results for reversing a stationary compound Markov process, under appropriate conditions, are given and applied to deriving product-forms. No balance equations are solved.