On the evolution of stock vectors in a deterministic integer-valued Markov system Original Research Article

For multistate systems, studies on the evolution of the stock vector have already been done under various hypothesis on the model. For example for a regular Markov chain it is known that the system evolves towards a unique limiting vector, that is a fixed point of the transition matrix and that is independent on the initial stock vector. In this paper, there are examined, under deterministic assumptions, constant size systems in which the integer-valued stock vectors are generated by a regular Markov chain. For these models, properties of the evolution of the integer-valued state sizes are proved and the limiting stock vector is examined. Sufficient conditions on the transition matrix are formulated to have, for two initial integer-valued stock vectors with lp-distance equal to 1, at each step of the trajectory of evolution integer-valued vectors of which the corresponding coordinates differ at most with one.