# Stalling in Queuing Systems with Heterogeneous Channels
## 1 Introduction  
Queueing systems (QSs) with different capacity channels are often used in technology and business. Usually, the theory of discrete Markov processes is used for QS modelling, and the probabilities of steady states are calculated from a linear equations system; here, the coefficients are equal to the rates of transition, while the order of the system is equal to the number of states [1]. If all channels of the linear system have the same capacity, this system has an analytical solution with a complexity that does not depend on the number of states. However, in general, this system of equations for the state probabilities of a QS with several heterogeneous channels can depend on the number of states, which might be significant, and it can be unstable. The queueing strategies in QSs with two or several different channels have been analysed by many authors, see, for example [2; 3; 4; 5; 6; 7; 8; 9; 10; 11]. Computer programs are developed to determine the optimal allocation of storage spaces among three heterogeneous servers, by calculating the bounds of sizes of finite sources for different traffic intensities [12]. Particular cases of the steady-state behaviour of a discrete-time queueing problem with S heterogeneous groups with a discipline FCFS and a limited waiting space were analysed in [13; 14]; a set of heterogeneous and exponential servers were discussed in [15]. [PERSON] and [PERSON] [16], using deterministic (sample-path) analysis, generalized and extended the fundamental properties of systems with \"stationary deterministic flows\". They proved the renewal-reward theorem and established a relationship that shows that the \"operational analysis\" definition of average service times--when considered as the observation period \(t\rightarrow\infty\)--coincides with the standard definition of average service times for all stable queueing systems. Moreover, the equilibrium in a single-server queueing system with retrials and strategic timing of the customers was analysed in [17]. The systems modelled by M/M/n queues with heterogeneous servers and non-informed customers show that thereis a value of arrival rate; below this, the slow server should not be used, and above which, it should be used [18]. A multi-server controllable queueing system with heterogeneous servers was considered in [19]; here, several monotonicity properties of optimal policies for such a system were proved. [PERSON] and [PERSON] studied [7] the queuing system with K heterogeneous servers using a heuristic approach. [PERSON] and [PERSON] [20,21] showed that, in an M/M/K queue, the optimal threshold levels may also depend on the states of slower servers; however, this influence is negligible. In [22], a finite-source queueing system serving one class of customers and consisting of heterogeneous servers with unequal service intensities and of one common queue were investigated. The authors of [23] provided an analysis of the dependence of the convergence of experimental results on the type of distribution used by the system parameters in heterogeneous queueing resource system with an unlimited number. In [24], a multi-server infinite buffer queueing system with additional servers (assistants) providing help to the main servers when they encountered problems was analysed through multi-dimensional Markov chains. The authors of [25] analysed heterogeneous queues where servers differ not only in service rates but also in operating costs; they presented a simple heuristic solution. The proposed that the solutions for systems of heterogeneous services can also be applied in specific tasks of computer networks and multiprocessor systems in addition to solving social tasks and others. Heterogeneous server problems occur in multiprocessor systems, which combine CPU and GPU processors [26, 27, 28, 29, 30]. Expressions of the queue length distributions of the GPS and individual traffic flows were presented in [31].