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Special Issue "Advances in Queueing Theory"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Dynamical Systems".

Deadline for manuscript submissions: 15 March 2023 | Viewed by 4033

Special Issue Editors

Prof. Dr. Anatoly Nazarov
E-Mail Website
Guest Editor
Institute of Applied Mathematics and Computer Science, Tomsk State University, 36 Lenin Ave., 634050 Tomsk, Russia
Interests: queueing theory; applied probability
Prof. Dr. Alexander Dudin
E-Mail Website
Guest Editor
1. Department of Applied Mathematics and Computer Science, Belarusian State University, 4, Nezavisimosti Ave., 220030 Minsk, Belarus
2. Applied Mathematics and Communications Technology Institute, Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St., 117198 Moscow, Russia
Interests: queueing theory; applied probability

Special Issue Information

Dear Colleagues,

The purpose of this Special Issue is to gather a collection of articles devoted to recent studies in the field of Queueing Theory. The topic includes theoretical studies in queueing theory, their application in practice for real systems and processes, and related fields that correspond to stochastic modeling.

Prof. Dr. Anatoly Nazarov
Prof. Dr. Alexander Dudin
Guest Editors

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 1800 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

Published Papers (7 papers)

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Research

Article
Analysis of Multi-Server Priority Queueing System with Hysteresis Strategy of Server Reservation and Retrials
Mathematics 2022, 10(20), 3747; https://doi.org/10.3390/math10203747 - 12 Oct 2022
Viewed by 296
Abstract
A multi-server queueing system with two types of requests and preemptive priority of one type is considered as a model of a cell of a cognitive radio system under practical suggestions about the arrival flows. A hysteresis type strategy for server reservation is [...] Read more.
A multi-server queueing system with two types of requests and preemptive priority of one type is considered as a model of a cell of a cognitive radio system under practical suggestions about the arrival flows. A hysteresis type strategy for server reservation is suggested to mitigate the effect of interruption of service of low priority requests. Under the arbitrarily fixed values of the sets of the thresholds defining this strategy, the behavior of the system is described by a level-dependent multi-dimensional Markov chain. Formulas for computation of values of performance characteristics of the system are derived. Numerical examples illustrating the dependence of the main performance characteristics on the thresholds defining the strategy of control and the numerical solution of the problem of the optimal choice of the thresholds are reported. Full article
(This article belongs to the Special Issue Advances in Queueing Theory)
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Article
Optimization of Open Queuing Networks with Batch Services
Mathematics 2022, 10(16), 3027; https://doi.org/10.3390/math10163027 - 22 Aug 2022
Viewed by 457
Abstract
In this paper, open queuing networks with Poisson arrivals and single-server infinite buffer queues are considered. Unlike traditional queuing models, customers are served (with exponential service time) in batches, so that the nodes are non-work-conserving. The main contribution of this work is the [...] Read more.
In this paper, open queuing networks with Poisson arrivals and single-server infinite buffer queues are considered. Unlike traditional queuing models, customers are served (with exponential service time) in batches, so that the nodes are non-work-conserving. The main contribution of this work is the design of an efficient algorithm to find the batch sizes which minimize the average response time of the network. As preliminary steps at the basis of the proposed algorithm, an analytical expression of the average sojourn time in each node is derived, and it is shown that this function, depending on the batch size, has a single minimum. The goodness of the proposed algorithm and analytical formula were verified through a discrete-event simulation for an open network with a non-tree structure. Full article
(This article belongs to the Special Issue Advances in Queueing Theory)
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Article
Tail Asymptotics for a Retrial Queue with Bernoulli Schedule
Mathematics 2022, 10(15), 2799; https://doi.org/10.3390/math10152799 - 07 Aug 2022
Cited by 1 | Viewed by 369
Abstract
In this paper, we study the asymptotic behaviour of the tail probability of the number of customers in the steady-state M/G/1 retrial queue with Bernoulli schedule, under the assumption that the service time distribution has a regularly varying tail. [...] Read more.
In this paper, we study the asymptotic behaviour of the tail probability of the number of customers in the steady-state M/G/1 retrial queue with Bernoulli schedule, under the assumption that the service time distribution has a regularly varying tail. Detailed tail asymptotic properties are obtained for the conditional probability of the number of customers in the (priority) queue and orbit, respectively, in terms of the recently proposed exhaustive stochastic decomposition approach. Numerical examples are presented to show the impacts of system parameters on the tail asymptotic probabilities. Full article
(This article belongs to the Special Issue Advances in Queueing Theory)
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Article
Pseudo Steady-State Period in Non-Stationary Infinite-Server Queue with State Dependent Arrival Intensity
Mathematics 2022, 10(15), 2661; https://doi.org/10.3390/math10152661 - 28 Jul 2022
Cited by 1 | Viewed by 411
Abstract
An infinite-server queueing model with state-dependent arrival process and exponential distribution of service time is analyzed. It is assumed that the difference between the value of the arrival rate and total service rate becomes positive starting from a certain value of the number [...] Read more.
An infinite-server queueing model with state-dependent arrival process and exponential distribution of service time is analyzed. It is assumed that the difference between the value of the arrival rate and total service rate becomes positive starting from a certain value of the number of customers in the system. In this paper, time until reaching this value by the number of customers in the system is called the pseudo steady-state period (PSSP). Distribution of duration of PSSP, its raw moments and its simple approximation under a certain scaling of the number of customers in the system are analyzed. Novelty of the considered problem consists of an arbitrary dependence of the rate of customer arrival on the current number of customers in the system and analysis of time until reaching from below a certain level by the number of customers in the system. The relevant existing papers focus on the analysis of time interval since exceeding a certain level until the number of customers goes down to this level (congestion period). Our main contribution consists of the derivation of a simple approximation of the considered time distribution by the exponential distribution. Numerical examples are presented, which confirm good quality of the proposed approximation. Full article
(This article belongs to the Special Issue Advances in Queueing Theory)
Article
Double Sources Queuing-Inventory System with Hybrid Replenishment Policy
Mathematics 2022, 10(14), 2423; https://doi.org/10.3390/math10142423 - 11 Jul 2022
Viewed by 473
Abstract
A hybrid replenishment policy in double sources queuing-inventory system is proposed. If the inventory level drops to the reorder point s, then a regular order of the fixed volume Q = Ss is generated to a slow and cheap source, where [...] Read more.
A hybrid replenishment policy in double sources queuing-inventory system is proposed. If the inventory level drops to the reorder point s, then a regular order of the fixed volume Q = Ss is generated to a slow and cheap source, where S denotes the maximum size of the system’s warehouse. If the inventory level falls below a certain threshold value r, where r < s, then the system instantly cancels the regular order and generates an emergency order to a fast and expensive source where the replenishment quantity should be able to bring the inventory level back to S at the replenishment epoch. In addition to consuming customers, the system also receives destructive customers that do not require inventory but destroy them. The stability condition for the system under study is found, steady-state probabilities are calculated, and formulas for finding performance measures are proposed. The problem of minimizing the total cost of the system under the proposed hybrid replenishment policy is solved by choosing the appropriate values of the order point and the threshold value. Full article
(This article belongs to the Special Issue Advances in Queueing Theory)
Article
A Multi-Type Queueing Inventory System—A Model for Selection and Allocation of Spectra
Mathematics 2022, 10(5), 714; https://doi.org/10.3390/math10050714 - 24 Feb 2022
Cited by 3 | Viewed by 698
Abstract
The model discussed in this paper provides an efficient mechanism for the selection and allocation of available limited spectra for transmission of heterogeneous data in a network. The data packets (customers), belonging to different classes, arrive according to a batch marked the Markovian [...] Read more.
The model discussed in this paper provides an efficient mechanism for the selection and allocation of available limited spectra for transmission of heterogeneous data in a network. The data packets (customers), belonging to different classes, arrive according to a batch marked the Markovian arrival process (BMMAP). The inventory considered is of multi-type (different types of channels becoming available) and are generated according to a marked Markovian arrival process (MMAP). The number of distinct types of inventory and that of the customers are the same. Arriving customers are allowed to wait in finite buffers of each category which are reserved for distinct classes of customers except for the most general class, which is provided with an infinite waiting space. The number of servers also equals the number of distinct types of inventory. When items of a particular type arrive in the inventory, the service starts, providing the buffer of customers of the corresponding class is non-empty. The service can be viewed as a selection process with Coxian distributed service times. The system is analyzed using the matrix analytic method and performance measures are obtained. The model is illustrated with suitable numerical examples. Full article
(This article belongs to the Special Issue Advances in Queueing Theory)
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Article
Retrial BMAP/PH/N Queueing System with a Threshold-Dependent Inter-Retrial Time Distribution
Mathematics 2022, 10(2), 269; https://doi.org/10.3390/math10020269 - 16 Jan 2022
Cited by 2 | Viewed by 535
Abstract
In this paper, we study a multi-server queueing system with retrials and an infinite orbit. The arrival of primary customers is described by a batch Markovian arrival process (BMAP), and the service times have a phase-type ( [...] Read more.
In this paper, we study a multi-server queueing system with retrials and an infinite orbit. The arrival of primary customers is described by a batch Markovian arrival process (BMAP), and the service times have a phase-type (PH) distribution. Previously, in the literature, such a system was mainly considered under the strict assumption that the intervals between the repeated attempts from the orbit have an exponential distribution. Only a few publications dealt with retrial queueing systems with non-exponential inter-retrial times. These publications assumed either the rate of retrials is constant regardless of the number of customers in the orbit or this rate is constant when the number of orbital customers exceeds a certain threshold. Such assumptions essentially simplify the mathematical analysis of the system, but do not reflect the nature of the majority of real-life retrial processes. The main feature of the model under study is that we considered the classical retrial strategy under which the retrial rate is proportional to the number of orbital customers. However, in this case, the assumption of the non-exponential distribution of inter-retrial times leads to insurmountable computational difficulties. To overcome these difficulties, we supposed that inter-retrial times have a phase-type distribution if the number of customers in the orbit is less than or equal to some non-negative integer (threshold) and have an exponential distribution in the contrary case. By appropriately choosing the threshold, one can obtain a sufficiently accurate approximation of the system with a PH distribution of the inter-retrial times. Thus, the model under study takes into account the realistic nature of the retrial process and, at the same time, does not resort to restrictions such as a constant retrial rate or to rough truncation methods often applied to the analysis of retrial queueing systems with an infinite orbit. We describe the behavior of the system by a multi-dimensional Markov chain, derive the stability condition, and calculate the steady-state distribution and the main performance indicators of the system. We made sure numerically that there was a reasonable value of the threshold under which our model can be served as a good approximation of the BMAP/PH/N queueing system with the PH distribution of inter-retrial times. We also numerically compared the system under consideration with the corresponding queueing system having exponentially distributed inter-retrial times and saw that the latter is a poor approximation of the system with the PH distribution of inter-retrial times. We present a number of illustrative numerical examples to analyze the behavior of the system performance indicators depending on the system parameters, the variance of inter-retrial times, and the correlation in the input flow. Full article
(This article belongs to the Special Issue Advances in Queueing Theory)
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