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**:** .. // . 77. .: , 2019. .6-19. URL: https://doi.org/10.25728/ubs.2019.77.1

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** (.):** queueing system, batch arrivals, stationary distribution, probability generating functions, embedded Markov chain, renewal process, Khinchin's basic law of a stationary queue

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** (.):** This paper deals with a queuing system with general renewal arrivals, exponential service times, single service channel and infinite number of waiting positions, customers are serviced in the order of their arrival. For this queueing system, a condition for the fulfilment of the Khinchin's basic law of a stationary queue is given. The article shows that, in the case of basic law of a stationary queue for our system, the stationary distribution of the number of the customers in the system always coincides with the corresponding probability distribution of the queueing system with exponential interarrival times. In stationary case a new form of the probability generating functions of the number of clients in the system is also derived. This new form is written in terms of the probability generating functions of the tail distribution function of the number of customers per group and of the probability generating functions of a embedded discrete time homogeneous Markov chain.

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