Towards a definition of a busy period under nonlocal description of input flows

In course of a probabilistic modeling and analysis of complex controlled queueing systems with several conflicting input flows, in a series of papers, an approach was successfully applied, one of its features being a nonlocal description of various system building blocks. In this description, some information about true arrival and leave times of customers is lost. It leads to difficulties in defining a busy period but that is one of classic performance metrics for an operating queueing system. In this paper, a controlled queuing system busy period definition is based on selecting those observation instants when queues reach zero level. A cyclic service algorithm with fixed switching times as an example using a martingale technique and effective computational formulas are obtained for the mathematical expectation of busy periods related to individual queues.