Decomposable Semi-regenerative Processes: and Their Applications

A very known notion of regeneration means that the "future" of a stochastic process became independent from its "past" in some random times, which is usually times of some state (regeneration state) destination. Presence of regeneration times allows to represent appropriate process as independent functional elements, regeneration cycles, investigate its characteristics in terms of them at separate regeneration cycles and proof some asimptotic theorem about this type of processes. If there are several such regeneration states this notion is generalised up to notion of semi- regeneration. In this case the process could be represented as a Markov chain of its cycles. Next step of generalization consists in discovering of some embedded regeneration times that allows to construct some hierarchical structure for the processes, possessing this property. This processes are named as decomposable semi-regenerative processes. The methods of these processes applied then for the investigation of several models: M/GI/1 queueing system, M_r/GI_r/1 priority queueing system, GI/GI/1 queueing system, polling system and reliability of complex hierarchical system.