## 1 Introduction
First, we precisely define the notion of simultaneous service of a restricted number of customers, which is described by a multidimensional Markov chain (\(MC\)), and present several examples of partial cases of such service. After that, we consider a system with the \(MAP\), a general Markovian description of the simultaneous service process for a finite number of customers, and an infinite buffer. This buffer is designed for the temporary storage of customers who arrive when service capacity is exhausted, i.e., the number of customers receiving service is equal to \(N\). It is well known that in some real-world systems, e.g., in wireless communication networks, there is no opportunity to store customers. Instead, customers have the option to temporarily leave the service area and make repeated attempts to enter service after random time intervals. This type of queueing system is called a retrial queue. For references and some known results, see the books [22; 23] and the papers [24; 25; 26; 27; 28]. Retrial queues are a more complicated research subject than queues with buffers. This statement is explained in detail in [29], where a comparison of standard and retrial queues was conducted. The second contribution of our paper is an analysis of the discipline with simultaneous service for both queueing systems with an infinite buffer and systems with customer retrials. We show how to easily obtain the generator of the multidimensional continuous-time \(MC\) describing the system with a general Markov description of customer processing and customer retrials, based on the known generator of the \(MC\) describing a similar system with an infinite buffer. As the byproduct, we illustrate why the analysis of a retrial queue is much more complicated from both theoretical and computational points of view. Also, we show that the sufficient conditions for the ergodicity of both \(M\)Cs coincide if the rate of customer retrials tends to infinity as the number of retrying customers tends to infinity. Researchers who analyze specific cases of simultaneous service for multiple customers, which can arise in the analysis of real-world systems, may find the presented results useful. The single task of a researcher is to write down three groups of matrices describing the simultaneous service process. Using the results of this paper, this researcher immediately obtains the explicit form of the generator of the Markov chain describing the behavior of the system, the sufficient condition for the ergodicity of this chain, and the manual for computation of its stationary distribution in scenarios with the existence of an infinite buffer or customers retrials (with an infinitely increasing or constant retrial rate). The outline of the results presentation is as follows. In Section 2, the Markov mathematical model of simultaneous service of a finite number of customers is described, and the necessary notation is given. Several examples of a description are presented. In Section 3, the system with the \(MAP\) arrival process, simultaneous service of a finite number of customers, and an infinite buffer is analyzed. The ergodicity condition for the multidimensional Markov process describing the behavior of this system is derived. The known algorithmic results relating to the computation of the stationary distribution of this system are presented. In Section 4, a similar system but with customer retrials is considered. A brief comparison of the generators and the multidimensional Markov chains describing the behavior of the systems with the infinite buffer and retrials is presented. The coincidence of sufficient conditions for the ergodicity of these two chains in the case of the infinitely increasing retrial rate is shown. The numerically stable algorithm for computation of the stationary distribution of the states of the system with retrials is briefly described. Required modifications in the generators of the chains analyzed in Sections 3 and 4 in the case when waiting customers are impatient are discussed in Section 5.