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Advances in Queueing Theory and Applications
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Advances in Queueing Theory and Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "D1: Probability and Statistics".

Deadline for manuscript submissions: 20 November 2025 | Viewed by 4618

Special Issue Editor

Prof. Dr. Alexander Dudin

E-Mail Website
Guest Editor
Department of Applied Mathematics and Computer Science, Belarusian State University, 4, Nezavisimosti Ave., 220030 Minsk, Belarus
Interests: queueing theory; applied probability
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

We are delighted to announce the forthcoming opening of the third volume of our Special Issue, "Advances in Queueing Theory and Applications". Building upon the resounding success of the previous two editions, we present a renewed opportunity for you to contribute to this Special Issue.

Our aim is to gather a comprehensive collection of articles and reviews that focus on recent studies in the field of Queueing Theory. Topics of interest include theoretical advancements in Queueing Theory, practical applications for real systems and processes, and related fields that involve stochastic modeling.

We look forward to your valuable contributions to this Special Issue.

Prof. Dr. Alexander Dudin
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • queueing theory
  • stochastic modeling
  • applied probability

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Related Special Issue

Published Papers (5 papers)

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Research

20 pages, 580 KiB  
Article
Analysis of BMAP/PH/N-Type Queueing System with Flexible Retrials Admission Control
by Sergei A. Dudin, Olga S. Dudina, Azam A. Imomov and Dmitry Y. Kopats
Mathematics 2025, 13(9), 1434; https://doi.org/10.3390/math13091434 - 27 Apr 2025
Viewed by 68
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Advances in Queueing Theory and Applications)
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17 pages, 448 KiB  
Article
Equilibrium Strategies for Overtaking-Free Queueing Networks under Partial Information
by David Barbato, Alberto Cesaro and Bernardo D’Auria
Mathematics 2024, 12(19), 2987; https://doi.org/10.3390/math12192987 - 25 Sep 2024
Viewed by 579
Abstract
We investigate the equilibrium strategies for customers arriving at overtaking-free queueing networks and receiving partial information about the system’s state. In an overtaking-free network, customers cannot be overtaken by others arriving after them. We assume that customer arrivals follow a Poisson process and [...] Read more.
We investigate the equilibrium strategies for customers arriving at overtaking-free queueing networks and receiving partial information about the system’s state. In an overtaking-free network, customers cannot be overtaken by others arriving after them. We assume that customer arrivals follow a Poisson process and that service times at any queue are independent and exponentially distributed. Upon arrival, the received partial information is the total number of customers already in the network; however, the distribution of these among the queues is left unknown. Adding rewards for being served and costs for waiting, we analyze the economic behavior of this system, looking for equilibrium threshold strategies. The overtaking-free characteristic allows for coupling of its dynamics with those of corresponding closed Jackson networks, for which an algorithm to compute the expected sojourn times is known. We exploit this feature to compute the profit function and prove the existence of equilibrium threshold strategies. We also illustrate the results by analyzing and comparing two simple network structures. Full article
(This article belongs to the Special Issue Advances in Queueing Theory and Applications)
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27 pages, 1590 KiB  
Article
Sojourn Time Analysis of a Single-Server Queue with Single- and Batch-Service Customers
by Yusei Koyama, Ayane Nakamura and Tuan Phung-Duc
Mathematics 2024, 12(18), 2820; https://doi.org/10.3390/math12182820 - 11 Sep 2024
Cited by 1 | Viewed by 1136
Abstract
There are various types of sharing economy services, such as ride-sharing and shared-taxi rides. Motivated by these services, we consider a single-server queue in which customers probabilistically select the type of service, that is, the single service or batch service, or other services [...] Read more.
There are various types of sharing economy services, such as ride-sharing and shared-taxi rides. Motivated by these services, we consider a single-server queue in which customers probabilistically select the type of service, that is, the single service or batch service, or other services (e.g., train). In the proposed model, which is denoted by the M+M(K)/M/1 queue, we assume that the arrival process of all the customers follows a Poisson distribution, the batch size is constant, and the common service time (for the single- and batch-service customers) follows an exponential distribution. In this model, the derivation of the sojourn time distribution is challenging because the sojourn time of a batch-service customer is not determined upon arrival but depends on single customers who arrive later. This results in a two-dimensional recursion, which is not generally solvable, but we made it possible by utilizing a special structure of our model. We present an analysis using a quasi-birth-and-death process, deriving the exact and approximated sojourn time distributions (for the single-service customers, batch-service customers, and all the customers). Through numerical experiments, we demonstrate that the approximated sojourn time distribution is sufficiently accurate compared to the exact sojourn time distributions. We also present a reasonable approximation for the distribution of the total number of customers in the system, which would be challenging with a direct-conventional method. Furthermore, we presented an accurate approximation method for a more general model where the service time of single-service customers and that of batch-service customers follow two distinct distributions, based on our original model. Full article
(This article belongs to the Special Issue Advances in Queueing Theory and Applications)
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31 pages, 449 KiB  
Article
Explicit Solutions for Coupled Parallel Queues
by Herwig Bruneel and Arnaud Devos
Mathematics 2024, 12(15), 2345; https://doi.org/10.3390/math12152345 - 26 Jul 2024
Cited by 2 | Viewed by 904
Abstract
We consider a system of two coupled parallel queues with infinite waiting rooms. The time setting is discrete. In either queue, the service of a customer requires exactly one discrete time slot. Arrivals of new customers occur independently from slot to slot, [...] Read more.
We consider a system of two coupled parallel queues with infinite waiting rooms. The time setting is discrete. In either queue, the service of a customer requires exactly one discrete time slot. Arrivals of new customers occur independently from slot to slot, but the numbers of arrivals into both queues within a slot may be mutually dependent. Their joint probability generating function (pgf) is indicated as A(z1,z2) and characterizes the whole model. In general, determining the steady-state joint probability mass function (pmf) u(m,n),m,n≥0 or the corresponding joint pgf U(z1,z2) of the numbers of customers present in both queues is a formidable task. Only for very specific choices of the arrival pgf A(z1,z2) are explicit results known. In this paper, we identify a multi-parameter, generic class of arrival pgfs A(z1,z2), for which we can explicitly determine the system-content pgf U(z1,z2). We find that, for arrival pgfs of this class, U(z1,z2) has a denominator that is a product, say r1(z1)r2(z2), of two univariate functions. This property allows a straightforward inversion of U(z1,z2), resulting in a pmf u(m,n) which can be expressed as a finite linear combination of bivariate geometric terms. We observe that our generic model encompasses most of the previously known results as special cases. Full article
(This article belongs to the Special Issue Advances in Queueing Theory and Applications)
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16 pages, 313 KiB  
Article
Stability of Queueing Systems with Impatience, Balking and Non-Persistence of Customers
by Alexander N. Dudin, Sergey A. Dudin, Valentina I. Klimenok and Olga S. Dudina
Mathematics 2024, 12(14), 2214; https://doi.org/10.3390/math12142214 - 15 Jul 2024
Cited by 2 | Viewed by 1265
Abstract
The operation of many queueing systems is adequately described by the structured multidimensional continuous-time Markov chains. The most well-studied classes of such chains are level-independent Quasi-Birth-and-Death processes, GI/M/1 type and M/G/1 type Markov chains, [...] Read more.
The operation of many queueing systems is adequately described by the structured multidimensional continuous-time Markov chains. The most well-studied classes of such chains are level-independent Quasi-Birth-and-Death processes, GI/M/1 type and M/G/1 type Markov chains, generators of which have the block tri-diagonal, lower- and upper-Hessenberg structure, respectively. All these classes assume that the matrices of transition rates are quasi-Toeplitz. This property greatly simplifies their analysis but makes them inappropriate for the study of many important systems, e.g., retrial queues with a retrial rate depending on the number of customers in orbit, queues with impatient customers, etc. The importance of such systems attracts significant interest to their analysis. However, in the literature, there is a methodological gap relating to the ergodicity condition of the corresponding Markov chains. To fulfill this gap and facilitate the analysis of a wide range of such systems, we show that under non-restrictive assumptions, the following hold true: (i) if the customers can balk or are impatient or non-persistent, then the Markov chain describing the behavior of the system belongs to the class of asymptotically quasi-Toeplitz Markov chains; (ii) this chain is ergodic; (iii) known algorithms can be applied for the calculation of the stationary distribution of the corresponding queueing system. Full article
(This article belongs to the Special Issue Advances in Queueing Theory and Applications)
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