Combined M/G/1-G/M/1 type structured chains: a simple algorithmic solution and applications

We consider combined M/G/1-G/M/1 type Markov chains with block-structured transition. It is assumed that all the blocks are generated with rational generating matrices. We provide an algorithmic approach to find the stationary probability distribution based on well-known/concepts in linear system theory. These chains arise in the (correlated) G/G/1 queueing systems and known structured Markov chains such as canonical and non-canonical M/G/1 and G/M/1 types which frequently arising in teletraffic analysis of computer and communications networks are special cases. We provide a truncation-free algorithmic solution in a simple geometric form of the the stationary probability vector of the chain taking full advantage of the rational generating matrices