In this paper, we present the results of a study of a priority multi-server queuing system with heterogeneous customers arriving according to a marked Markovian arrival process (MMAP), phase-type service times (PH), and a queue with finite capacity. Priority traffic classes differ in PH distributions of the service time and the probability of joining the queue, which depends on the current length of the queue. If the queue is full, the customer does not enter the system. An analytical model has been developed and studied for a particular case of a queueing system with two priority classes. We present an algorithm for calculating stationary probabilities of the system state, loss probabilities, the average number of customers in the queue, and other performance characteristics for this particular case. For the general case with K priority classes, a new method for assessing the performance characteristics of complex priority systems has been developed, based on a combination of machine learning and simulation methods. We demonstrate the high efficiency of the new method by providing numerical examples.

This paper presents a study of fork–join systems. The fork–join system breaks down each customer into numerous tasks and processes them on separate servers. Once all tasks are finished, the customer is considered completed. This design enables the efficient handling of customers. The customers enter the system in a MAP flow. This helps create a more realistic and flexible representation of how customers arrive. It is important for modeling various real-life scenarios. Customers are divided into $K\ge 2$ tasks and assigned to different subsystems. The number of tasks matches the number of subsystems. Each subsystem has a server that processes tasks, and a buffer that temporarily stores tasks waiting to be processed. The service time of a task by the k-th server follows a PH (phase-type) distribution with an irreducible representation (${\beta }_k$, $S_k$), $1\le k\le K$. An analytical solution was derived for the case of $K=2$ when the input MAP flow and service time follow a PH distribution. We have efficient algorithms to calculate the stationary distribution and performance characteristics of the fork–join system for this case. In general cases, this paper suggests using a combination of Monte Carlo and machine learning methods to study the performance of fork–join systems. In this paper, we present the results of our numerical experiments.

Рассматривается многофазная система массового обслуживания с входящим коррелированным МАР-потоком, РН-распределением времени обслуживания и ограничен­ным размером буфера на фазах системы. Приведен краткий исторический обзор по анализу моделей таких систем и методов их исследования. На основании проведенного обзора обосно­­вана новизна постановки задачи, рассматриваемой в статье, методов ее решения и результатов. Дано описание алгоритма точного расчета характеристик производительности многофазных систем малой размерности и оценки сложности этого алгоритма. Для исследо­вания многофазных систем большой размерности предложен подход, основанный на комби­на­ции методов имитационного моделирования и машинного обучения. Приве­дены результа­ты численного анализа, подтвердившие эффективность применения методов машинного обучения для оценки характеристик производительности тандемных систем, адекватно опи­сы­вающих функционирование широкополосных беспроводных сетей. This paper considers tandem queuing systems with limited buffer sizes in each phase. The system handles an incoming correlated MAP flow and the service time obeys a PH-distribution. Models of such systems and methods for their investigation are briefly reviewed from the historical perspective. According to the review, the problem statement presented below, the methods proposed for solving this problem, and the corresponding results are novel. An accurate algorithm for calculating the performance characteristics of low-dimensional tandem queuing systems is described, including an estimate of the algorithm’s complexity. An approach using both machine learning and simulation modeling is suggested for the investigation of high-dimensional tandem queuing systems. Numerical analysis results are provided to show the effectiveness of machine learning methods for estimating the performance of tandem queuing systems.