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Advances in Queueing Theory, 2nd Edition
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Advances in Queueing Theory, 2nd Edition

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

Deadline for manuscript submissions: closed (31 March 2024) | Viewed by 8146

Special Issue Editors

Prof. Dr. Anatoly Nazarov

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Guest Editor
Institute of Applied Mathematics and Computer Science, Tomsk State University, 36 Lenin Ave., 634050 Tomsk, Russia
Interests: queueing theory; applied probability
Special Issues, Collections and Topics in MDPI journals
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,

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

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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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Published Papers (6 papers)

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22 pages, 436 KiB  
Article
Regenerative Analysis and Approximation of Queueing Systems with Superposed Input Processes
by Irina Peshkova, Evsey Morozov and Michele Pagano
Mathematics 2024, 12(14), 2202; https://doi.org/10.3390/math12142202 - 13 Jul 2024
Viewed by 814
Abstract
A single-server queueing system with n classes of customers, stationary superposed input processes, and general class-dependent service times is considered. An exponential splitting is proposed to construct classical regeneration in this (originally non-regenerative) system, provided that the component processes have heavy-tailed interarrival times. [...] Read more.
A single-server queueing system with n classes of customers, stationary superposed input processes, and general class-dependent service times is considered. An exponential splitting is proposed to construct classical regeneration in this (originally non-regenerative) system, provided that the component processes have heavy-tailed interarrival times. In particular, we focus on input processes with Pareto interarrival times. Moreover, an approximating GI/G/1-type system is considered, in which the independent identically distributed interarrival times follow the stationary Palm distribution corresponding to the stationary superposed input process. Finally, Monte Carlo and regenerative simulation techniques are applied to estimate and compare the stationary waiting time of a customer in the original and in the approximating systems, as well as to derive additional information on the regeneration cycles’ structure. Full article
(This article belongs to the Special Issue Advances in Queueing Theory, 2nd Edition)
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23 pages, 2149 KiB  
Article
Inter-Departure Time Correlations in PH/G/1 Queues
by Ruth Sagron and Uri Yechiali
Mathematics 2024, 12(9), 1362; https://doi.org/10.3390/math12091362 - 30 Apr 2024
Cited by 2 | Viewed by 906
Abstract
In non-Markovian tandem queueing networks the output process of one site, which is the input process to the next site, is not renewal. Consequently, the correlation analysis of that output processes is essential when studying such networks. A correlation analysis in the M/G/1 [...] Read more.
In non-Markovian tandem queueing networks the output process of one site, which is the input process to the next site, is not renewal. Consequently, the correlation analysis of that output processes is essential when studying such networks. A correlation analysis in the M/G/1 queue has been studied in the literature via derivation of the joint Laplace-Stieltjes transform (LST) of the sum of two consecutive inter-departure times. That LST is obtained by considering all possible cases at departure epochs. However, those epochs are expressed via dependent variables. In this paper, we first extend the analysis to the more general PH/G/1 queue, and investigate various queues, such as E2/G/1 and C2/C2/1. Then, we consider the lag-n correlation, which requires derivation of the joint LST of sum of n + 1 consecutive inter-departure times. Yet, deriving this LST by the common approach becomes impractical for n + 1 ≥ 3, as the number of all possible cases at departure epochs increases significantly. To overcome this obstacle, we derive a corresponding single-parameter LST, which expresses the sum of n + 1 consecutive inter-departure times via the (n + 1)-st departure epoch only. Consequently, the latter LST is expressed via a much fewer number of possible cases, and not less important, as a function of independent variables only, eliminating the need to derive the corresponding joint density. Considering the M/G/1 and the E2/G/1 queues, we demonstrate that the joint LST can be reconstructed directly via the corresponding single-parameter LST when n + 1 = 2. We further conjecture that the multi-parameter joint LST can be reconstructed from the corresponding single-parameter LST in more general queues and for values of n + 1 > 2. The conjecture is validated for various PH/G/1 queues and proved for n + 1 = 3 in the M/G/1 case. The new approach facilitates the calculation of lag-n correlation of the departure process from PH/G/1 queue for n + 1 ≥ 3. Our analysis illuminates the cases when using renewal approximation of the output process provides a proper approximation when studying non-Markovian stochastic networks. Full article
(This article belongs to the Special Issue Advances in Queueing Theory, 2nd Edition)
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20 pages, 592 KiB  
Article
Analysis of Queueing System with Dynamic Rating-Dependent Arrival Process and Price of Service
by C. D’Apice, A. N. Dudin, O. S. Dudina and R. Manzo
Mathematics 2024, 12(7), 1101; https://doi.org/10.3390/math12071101 - 6 Apr 2024
Viewed by 1516
Abstract
We consider a multi-server queueing system with a visible queue and an arrival flow that is dynamically dependent on the system’s rating. This rating reflects the level of customer satisfaction with the quality and price of the provided service. A higher rating implies [...] Read more.
We consider a multi-server queueing system with a visible queue and an arrival flow that is dynamically dependent on the system’s rating. This rating reflects the level of customer satisfaction with the quality and price of the provided service. A higher rating implies a higher arrival rate, which motivates the service provider to increase the price of the service. A steady-state analysis of this system using the proposed mechanism for changing the rating and a threshold strategy for changing the price is performed. This is carried out via the consideration of a suitably constructed multidimensional Markov chain. The impact of the variation in the threshold defining the strategy for changing the price on the key performance indicators is numerically illustrated. The results can be used to make managerial decisions, leading to an increase in the effectiveness of system operations. Full article
(This article belongs to the Special Issue Advances in Queueing Theory, 2nd Edition)
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19 pages, 2858 KiB  
Article
Queuing-Inventory System with Catastrophes in the Warehouse: Case of Rare Catastrophes
by Agassi Melikov, Laman Poladova and Janos Sztrik
Mathematics 2024, 12(6), 906; https://doi.org/10.3390/math12060906 - 19 Mar 2024
Viewed by 1118
Abstract
A model of a single-server queuing-inventory system (QIS) with a limited waiting buffer for consumer customers (c-customers) and catastrophes has been developed. When a catastrophe occurs, all items in the system’s warehouse are destroyed, but c-customers in the system are [...] Read more.
A model of a single-server queuing-inventory system (QIS) with a limited waiting buffer for consumer customers (c-customers) and catastrophes has been developed. When a catastrophe occurs, all items in the system’s warehouse are destroyed, but c-customers in the system are still waiting for replenishment. In addition to c-customers, negative customers (n-customers) are also taken into account, each of which displaces one c-customer (if any). The policy (s, S) is used to replenish stocks. If, when a customer enters, the system warehouse is empty, then, according to Bernoulli’s trials, this customer either leaves the system without goods or joins the buffer. The mathematical model of the investigated QIS is constructed in the form of a continuous-time Markov chain (CTMC). Both exact and approximate methods for calculating the steady-state probabilities of constructed CTMCs are proposed and closed-form expressions are obtained for calculating the performance measures. Numerical evaluations are presented, demonstrating the high accuracy of the developed approximate formulas, as well as the behavior of performance measures depending on the input parameters. In addition, an optimization problem is solved to obtain the optimal value of the reorder point to minimize the expected total cost. Full article
(This article belongs to the Special Issue Advances in Queueing Theory, 2nd Edition)
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20 pages, 771 KiB  
Article
Analysis of a Multi-Server Queue with Group Service and Service Time Dependent on the Size of a Group as a Model of a Delivery System
by Sergei Dudin and Olga Dudina
Mathematics 2023, 11(22), 4587; https://doi.org/10.3390/math11224587 - 9 Nov 2023
Cited by 3 | Viewed by 1509
Abstract
In this paper, we consider a multi-server queue with a finite buffer. Request arrivals are defined by the Markov arrival process. Service is provided to groups of requests. The minimal and maximal group sizes are fixed. The service time of a group has [...] Read more.
In this paper, we consider a multi-server queue with a finite buffer. Request arrivals are defined by the Markov arrival process. Service is provided to groups of requests. The minimal and maximal group sizes are fixed. The service time of a group has a phase-type distribution with an irreducible representation depending on the size of the group. The requests are impatient. The patience time for an arbitrary request has an exponential distribution. After this time expires, the request is lost if all servers are busy or, if some server is idle, with a certain probability, all requests staying in the buffer start their service even if their number is below the required minimum. The behavior of the system is described by a multi-dimensional continuous-time Markov chain that does not belong to the class of level-independent quasi-birth-and-death processes. The algorithm for the computation of the stationary distribution of this chain is presented, and expressions for the computation of the queuing system’s performance characteristics are derived. The description of a delivery system operation in terms of the analyzed queuing model is given, and the problem of the optimization of its operation is numerically solved. Multi-server queues with a phase-type distribution for the group service time that are dependent on the size of the group, the account of request impatience, and the correlated arrival process have not previously been analyzed in the existing literature. However, they represent a precise model of many real-world objects, including delivery systems. Full article
(This article belongs to the Special Issue Advances in Queueing Theory, 2nd Edition)
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31 pages, 880 KiB  
Article
Modeling of Junior Servers Approaching a Senior Server in the Retrial Queuing-Inventory System
by Kathirvel Jeganathan, Thanushkodi Harikrishnan, Kumarasankaralingam Lakshmanan, Agassi Melikov and Janos Sztrik
Mathematics 2023, 11(22), 4581; https://doi.org/10.3390/math11224581 - 8 Nov 2023
Cited by 4 | Viewed by 1276
Abstract
This article deals with the queuing-inventory system, composed of c junior servers, a senior server, two finite waiting halls, and an infinite orbit. On occasion, junior servers encounter challenges during customer service. In these instances, they approach the senior server for guidance in [...] Read more.
This article deals with the queuing-inventory system, composed of c junior servers, a senior server, two finite waiting halls, and an infinite orbit. On occasion, junior servers encounter challenges during customer service. In these instances, they approach the senior server for guidance in resolving the issue. Suppose the senior server is engaged with another junior server. The approaching junior servers await their turn in a finite waiting area with a capacity of c for consultation. Concerning this, we study the performance of junior servers approaching the senior server in the retrial queuing-inventory model with the two finite waiting halls dedicated to the primary customers and the junior servers for consultation. We formulate a level-dependent QBD process and solve its steady-state probability vector using Neuts and Rao’s truncation method. The stability condition of the system is derived and the R matrix is computed. The optimum total cost has been obtained, and the sensitivity analyses, which include the expected total cost, the waiting time of customers in the waiting hall and orbit, the number of busy servers, and a fraction of the successful retrial rate of the model, are computed numerically. Full article
(This article belongs to the Special Issue Advances in Queueing Theory, 2nd Edition)
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