## 1 Introduction  
Queueing theory is a subfield of probability theory within mathematics. It was founded just over 100 years ago, by the publication of works and successful practical application of the Danish mathematician, statistician, and engineer [PERSON] (1878-1929). In his honor, the system solved by M/M/1 and the unit of communication activity bears [PERSON]'s name. The effective practical application of the new queueing theory has given a strong impetus for it is further development. At the same time, the tasks posed by the practice require the solutions of non-Markovian and non-stationary queueing systems with a permanent imperative according to the standard set by [PERSON]--a transparent analytical solution. The M/G/1 model was solved in 1930 by the Austrian-French engineer and mathematician, [PERSON] (1892-1981) [1]. Our contemporary, Sir [PERSON], in 1961 [2], published an approximate formula for the mean waiting time in a G/G/1 queue. Although approximate, it is considered to be a very precise formula, especially for a system operating close to saturation. Of course, the most important contributions to the development of queueing theory must include contributions by [PERSON] (1918-2007) and another contemporary [PERSON] (born 1928). The solutions of non-stationary queueing systems begin with the works by [PERSON] (1903-1987) [3], which emerged almost simultaneously with the first general solutions of the non-Markov queueing systems in 1930 [1]. Nevertheless, in spite of the great effort, intensive development of the world economy and the requirements of practice have far exceeded the achievements of queueing system researchers. The emergence and design of the development of computers in the 1980s and 1990s resulted in the prognosis that each queueing system could be solved by the use of simulations. This attitude has somewhat discouraged further efforts in the analytical approach of queueing theory and was consistently described by [PERSON] in the set and reasoned antithesis [4]. His absolutely correct assessment of the necessity of an analytical approach and positive development prognosis was confirmed by [PERSON], [PERSON], and [PERSON], especially for non-stationary time-dependent queueing systems [5]. The analytical approach to solving the queueing system remains an imperative. This imperative does not exist in itself, but is encouraged by the practical application of the queueing system. The theory of the Hidden Markov Model (HMM) was founded by [PERSON] and [PERSON] in the mid-1960s [6]. It is based on the simple idea that in the observed system, the Markov processes are a priori, and that the existence of hidden Markov states can be confirmed statistically, representing the criterion for optimizing the maximum probability. HMM is classified into methods of artificial intelligence. It is characterized by an intense development [7] and capacity for effectively solving a wide range of real tasks: character recognition [8], medicine [9], helping blind people [10], infrastructure management [11], driver fatigue detection [12], etc. Basically, HMM is related to queueing theory. It also shares the same analytical problems in the principles of the functioning of a state in a non-Markovian domain [13]. The application of the idea of hidden states from HMM in queueing systems has an initially modest, but extremely important, realized methodological idea in the birth-death process [14]. The synthesis of HMM and queueing theory leads to the idea of abstract hidden states of Markov (HMS) in the first-order transition probability distribution among the hidden states with a possible catalytic function. This idea is not new, and HMS have already been developed for some higher-order Markov models [15]. For non-Markov queueing systems, the HMS function is contrary to convolution. It is based on the process of the decomposition of exponential random variables to the sum of exponential distributions.