## 1 Introduction  
Single-server queueing systems, where customers receive service one by one, have received significant attention in the existing queueing literature. However, impressive progress in service provision technologies in various real-world systems, particularly telecommunication systems and networks, has made it important to investigate queueing systems that can provide service to many customers simultaneously. The simplest case of such systems is the standard multi-server queueing system, where the system resource is divided into a fixed number, say, \(N\), of parts (servers), each of which can provide service to customers independently of the other servers. This type of system has also been well investigated in the literature since the pioneering work of [PERSON] in the early 1900s. A shortcoming in the analysis of multi-server queueing systems compared to single-server queueing systems is that single-server systems are well studied in the case of _arbitrary_ distributions of service times and even mutually dependent service times, while for multi-server systems, except for the model without a buffer ([PERSON] loss model), exact analyses of multi-server queues are only implemented when the service time has an exponential distribution or, at most, an essential generalization such as the phase-type (\(PH\)) distribution. For definitions and properties, see, e.g., [1, 2]. Further progress in technology has led to the possibility and the necessity of analyzing other, more effective mechanisms of resource sharing for the simultaneous service of several customers, like processor sharing (\(PS\)), see, e.g., [3; 4]. This mechanism assumes, in the simplest settings, that all customers receive service simultaneously at a rate inversely proportional to the number of customers. The classical \(PS\) mechanism is not suitable for many real-world systems, e.g., telecommunication systems, where service at a very low rate may not be acceptable to customers. Therefore, the number of simultaneously serviced customers must be restricted. Thus, the discipline of limited processor sharing (\(LPS\)) was introduced. This discipline assumes that no more than \(N\) customers can receive service at the same time. The number \(N\) is sometimes called the multi-programming, concurrency level, or multiplicity. For details, see, e.g., [5; 6; 7; 8; 9]. The mixed-service discipline (for details see e.g., [10; 11]) assumes that a limited number of customers (corresponding to the so-called non-elastic traffic) receive service at a constant rate, as in the standard multi-server system, while another finite group of customers is processed using the \(LPS\) discipline. As another example of systems in which up to a limited number of customers may receive service simultaneously, the so-called semi-open queueing networks can be mentioned (see, e.g., [12; 13; 14; 15]), in which only a finite number of customers can be processed in the inner network. Therefore, the range of queueing models with different mechanisms for the simultaneous service of customers is quite wide. The first contribution of this paper is the presentation of a framework for the uniform consideration of queueing systems with existing and future service disciplines. Besides the service discipline, the performance characteristics of any queueing model essentially depend on the arrival process. Here, we assume a general Markov arrival process (\(MAP\)), which is a more general process than the stationary Poisson process and the renewal process with a \(PH\) distribution of inter-arrival times. It is known that the \(MAP\) is suitable for describing arrival processes with correlated inter-arrival times and times with large variance, which are typical of certain information flows in telecommunication systems and contact centers. For more information about the \(MAP\), see, e.g., [16; 17; 18; 19; 20; 21].