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**:** 2019
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**:** 80
**:** .. // . 80. .: , 2019. .20-39. DOI: https://doi.org/10.25728/ubs.2019.80.2

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** (.):** queueing system, batch arrivals, stationary distribution, probability generating functions, Fibonacci numbers, binomial coefficients, sums of binomial coefficients, Fibonacci sequence, Generalized Fibonacci polynomials, Binet's Fibonacci number formula, Generalized Fibonacci numbers, Binet form, generating function, generating function for a Fibonacci numbers, geometric distribution, Pisot number

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** (.):** This paper deals with a queuing system with Poisson arrivals, exponential service times, single service channel and infinite number of waiting positions, customers are serviced in the order of their arrival. The requests arrives in groups and the number of requests in a group is one or two. For this queueing system be found in algebraic form the steady-state probabilities for the number of customers in the system. A probability mass function of this distribution can be defined by polynomials like polynomials Fibonacci. The geometric distribution is a special case of this distribution. Fibonacci numbers can be expressed in terms of the polynomials like polynomials Fibonacci. Consequently our distribution expressed in terms of this polynomials under certain conditions can be written in terms of Fibonacci numbers. Using the Binet formula is shown that in some cases the found distribution is asymptotically geometric distribution. In this paper it is shown that the Bernoulli numbers can be expressed as an elementary double sum of binomial coefficients. Changing the order in that double sum and summing one of them get a formula for Fibonacci numbers which Catalan developed or Lucas formula for Fibonacci numbers.

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