The paper studies a queueing system with periodic intensity of conflicting input flows, reflecting, in particular, the functioning of the traffic flow control system in the class of cyclic algorithms with a fixed signal durations and daily fluctuations in the traffic flow intensity. The process of conflicting flows control with homogeneous customers and periodic intensity is considered in detail. Sequences of random variables are constructed that describe the processes occurring in the system. Recurrence relations are derived that connect these quantities. The mathematical model of the system is a countable Markov chain. Transition probabilities for this Markov chain are calculated. An ordering of essential states is introduced, revealing the block structure of the transition probability matrix. An algorithm for calculating a stationary distribution is proposed, based on the well-known chain censoring method. The restriction of the Markov chain to the class of essential communicating states is considered, recurrence relations are established for generating functions for the distribution of the queue length, the state of the service device and the time of day. Using the iterative-majorant approach, necessary and sufficient conditions for the existence of a stationary distribution for queue lengths are studied.