Algorithmic approach for the design of Markovian queueing network with multiple closed chains

This paper mainly deals with design of an algorithm in which the customer transitions are characterized by more than one closed Markov chain. Generating functions are implemented to derive closed form of solutions and product form solution with the parameters such as stability, normalizations constant and marginal distributions. For such a system with `N´ servers and `L´ chains, the solutions are considerably more complicated than those for the systems with one sub-chain only. Hence the result is generalized to a queueing network in which the customer routing transitions are characterized by a Markov chain decomposable into multiple sub-chains. Networks with closed subchains are introduced as a limiting case of suitable chosen open network. The several aggregate states and their marginal distributions are introduced. An algorithmic approach is implemented from the generating function representation for the general class of Networks. Based on the algorithmic approach it is proved that how open and closed sub-chain interact with each other in such system.