# Priority Multi-Server Queueing System with Heterogeneous Customers  
[PERSON]  
1  
[PERSON]  
1 Department of Applied Mathematics and Computer Science, Belarusian State University, 220030 Minsk, Belarus; [EMAIL_ADDRESS] (V.K. ); [EMAIL_ADDRESS] (A.D.)  
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[PERSON]  
22 Sciences and Closed Corporation Information and Networking Technologies, Institute of Control Sciences of Russian Academy, 119991 Moscow, Russia 222  
in hospital emergency departments during sorting incoming patients according to the severity o, e.g.f the injury or disease, see, e.g., [1]. In the case of unreliable systems, a priority customer may be considered as equipment failure. Priority can be also set to maximize a company's profit or increase system utilization. For instance, an online store the manager can set a high priority for customers of big spenders in order to prevent their departure to other online resources, see [2]. In some telecommunication networks, the priority of a customer is determined by its owner through a Service Level Agreement (SLA), whereby certain customers have chosen to pay more so as to get high-priority access to some high-demand resource. Another set of potential applications of the priority queueing model like considered in our paper is described in [3] as applications in customer service centers, see [4], airport security checkpoints [5], hospital emergency rooms [6], cloud computing systems [7] or processor management in certain computer operating systems [8]. There is an extensive literature on priority queues. An overview of early research works on priority queueing models can be found in monographs [9; 10; 11; 12; 13] and references therein. Mostly of these works focused on the queueing models with stationary Poisson inputs. However, the area of application of these models is currently being greatly narrowed, since flows in modern telecommunication networks do not possess memoryless property of stationary Poisson flow. They are, as a rule, correlated and heterogeneous. In the case of homogeneous customers, a good mathematical model of such flows well known in the literature as a Markovian arrival process (_MAP_), see, e.g., [14; 15; 16]. A _MAP_ is the significant generalization of the stationary Poisson process to the case of correlated bursty traffic. Queuing systems with a _MAP_ and priorities are discussed in the papers [17; 18; 19; 20; 21; 22; 23; 24; 25; 26]. Markovian arrival processes are a suitable class of stochastic processes to represent correlated traffic in case when all customers are of the same type and only the sequence of inter-arrival times is of interest. However, systems with correlated flows of heterogeneous customers are also of great interest to applications. Such flows are well modeled by a marked Markovian arrival process (_MMAP_), see [27]. Using of a _MMAP_ allows to represent flows where inter-arrival times are correlated across customer classes and to achieve models of greater accuracy. However, very few results are known about priority queue with _MMAP_. We can only refer to the papers [1; 25; 26; 28]. In [28]_MMAP\(/\)MAP\(/\)1_ queue with preemptive priorities is analysed, the moments of queue size is derived. The article [1] deals with a multi-server queue with a _MMAP_, preemptive priorities and priority upgrade. For this system, the authors found a condition for the existence of a stationary regime and the bounds for the lengths of the queues. The problem of concrete computation of the stationary distribution of the system states is not touched in that paper. This problem is considered in the recent papers [25; 26]. However, those papers deal with single server priority queues. In [25], possibility of increase of the non-preemptive priority during customer stay in the buffer is analysed. In [26], the discipline of flexible providing of non-preemptive priorities is under study. In this paper, we consider a priority \(MMAP/PH_{1,2}/N/N\) queueing system with two types of customers.