Analysis of BMAP / PH / N -Type Queueing System with Flexible Retrials Admission Control

Sergei A. Dudin (), Olga S. Dudina, Azam A. Imomov and Dmitry Y. Kopats
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Sergei A. Dudin: Department of Applied Mathematics and Computer Science, Belarusian State University, 4 Nezavisimosti Ave., 220030 Minsk, Belarus
Olga S. Dudina: Department of Applied Mathematics and Computer Science, Belarusian State University, 4 Nezavisimosti Ave., 220030 Minsk, Belarus
Azam A. Imomov: Physics and Mathematical Faculty, Karshi State University, 17 Kuchabag Str., Karshi City 180100, Uzbekistan
Dmitry Y. Kopats: Faculty of Mathematics and Computer Science, Yanka Kupala State University of Grodno, 22 Orzheshko Str., 230027 Grodno, Belarus

Mathematics, 2025, vol. 13, issue 9, 1-20

Abstract: This research examines a multi-server retrial queueing system with a batch Markov arrival process and a phase-type service time distribution. The system’s distinguishing feature is its ability to control the admission of retrial customers. An attempt by a customer to retry is successful only if the number of busy servers does not exceed certain threshold values, which may depend on the state of the fundamental process of the primary customer’s arrival. Impatient retrying customers may abandon the system without obtaining service. A group of primary customers that arrives while the number of available servers is fewer than the group size is either entirely rejected or occupies all available servers, while the remainder of the group transitions to the orbit. The system’s behavior, under a defined set of thresholds, is characterized by a multidimensional Markov chain classified as asymptotically quasi-Toeplitz. This enables the acquisition of the ergodicity condition and the computation of the steady-state distribution of the Markov chain and the system’s performance measures. The presented numerical examples demonstrate the impact of threshold value variation. An example of solving an optimization problem is presented. The importance of the account of the batch arrivals is shown.

Keywords: batch Markov arrival process; phase-type service time distribution; impatience; retrial admission control; asymptotically quasi-Toeplitz Markov chain (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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