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Axioms
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Queueing Theory and Network Applications
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Queueing Theory and Network Applications

A special issue of Axioms (ISSN 2075-1680).

Deadline for manuscript submissions: closed (31 March 2023) | Viewed by 12464

Special Issue Editor

Dr. Alexander Moiseev

E-Mail Website
Guest Editor
Institute of Applied Mathematics and Computer Science, Tomsk State University, 36 Lenin Ave., 634050 Tomsk, Russia
Interests: queueing theory; simulation; software engineering
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Queueing theory today plays an important role in the development of the most modern communication technologies. Devices and technologies have made very great progress, but their further advancement in many areas has reached its limit in the physical or cost sense, or close to this limit. Mathematics in general and queueing theory in particular can help solve a number of problems by analyzing, identifying bottlenecks and optimizing existing technologies and approaches.

In this Special Issue, we propose to collect articles devoted primarily to solving practical modern problems using queueing theory. These can be problems related to technology, social or economic problems, or problems in any other areas. Of course, we invite publications and papers in pure queueing theory too, because they can serve as a basis for future applied research.

Dr. Alexander Moiseev
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. Axioms is an international peer-reviewed open access monthly 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 2400 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
  • networks
  • telecommunications
  • control and reliability in queues
  • applied probability
  • stochastic processes
  • simulations

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Published Papers (5 papers)

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Research

17 pages, 457 KiB  
Article
On the Optimal Input Rate in Queues with Batch Service
by Michele Pagano, Igor Tananko and Elena Stankevich
Axioms 2023, 12(7), 656; https://doi.org/10.3390/axioms12070656 - 1 Jul 2023
Cited by 1 | Viewed by 1263
Abstract
In recent years, queuing systems with batch service are emerging as powerful and flexible mathematical models in different frameworks. In this paper, we consider a single server queuing system with Poissonian arrivals, infinite buffers, and a constant batch size b. This paper [...] Read more.
In recent years, queuing systems with batch service are emerging as powerful and flexible mathematical models in different frameworks. In this paper, we consider a single server queuing system with Poissonian arrivals, infinite buffers, and a constant batch size b. This paper addresses a little-studied optimization problem, namely the existence of an optimal arrival rate that minimizes the average sojourn time. Unlike the classical M/M/1 queue, for any batch size b, the problem admits a non-trivial solution that can be found by solving a polynomial equation of degree b+1. Since, in general, only numerical solutions are available, a simple first-order approximation is also derived and the corresponding deviations (in terms of input rate and sojourn time) are calculated. In more detail, it is shown that the approximation improves as the batch size increases and, in any case, the relative error for the average sojourn time is less than 0.34%. Finally, the paper provides new theoretical results about the asymptotic service rate in the equivalent birth–death process, highlighting how it depends on all queue parameters. Full article
(This article belongs to the Special Issue Queueing Theory and Network Applications)
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19 pages, 2611 KiB  
Article
Queueing System with Two Phases of Service and Service Rate Degradation
by Ekaterina Fedorova, Ivan Lapatin, Olga Lizyura, Alexander Moiseev, Anatoly Nazarov and Svetlana Paul
Axioms 2023, 12(2), 104; https://doi.org/10.3390/axioms12020104 - 19 Jan 2023
Cited by 6 | Viewed by 2193
Abstract
In the paper, a queueing system with an unlimited number of servers and two phases of service with degradation in the service rate is studied. The problem of service rate degradation emerges in cloud nodes, where there is contention for hardware resources including [...] Read more.
In the paper, a queueing system with an unlimited number of servers and two phases of service with degradation in the service rate is studied. The problem of service rate degradation emerges in cloud nodes, where there is contention for hardware resources including computational resources such as CPU cores. In a node, we have a limited number of CPU cores that should execute potentially an unlimited number of processes (requests) in parallel. In our model, the term “server” means a process allocated in the node for execution. So, the number of “servers” is unlimited but their individual performances decrease because CPUs should switch between them during the execution. We consider processes executed in the node with two phases of life cycle that reflects periods with different activity of a process; e.g., in the first phase, the process may require intensive usage of CPU cores but low usage in the second one. Our model distinguishes the phases using different service parameters for them as well as different influence on the service rate degradation in the node. In the paper, two analytical methods are proposed: exact solving of the system of the local balance equation and the asymptotic analysis of the global balance equations. Formulas for the stationary probability distribution of the number of customers in the phases are obtained for both cases. Several numerical examples are provided that illustrate some properties and applicability of the obtained results. Full article
(This article belongs to the Special Issue Queueing Theory and Network Applications)
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12 pages, 386 KiB  
Article
Asymptotic Diffusion Analysis of Retrial Queueing System M/M/1 with Impatient Customers, Collisions and Unreliable Servers
by Elena Danilyuk, Alexander Plekhanov, Svetlana Moiseeva and Janos Sztrik
Axioms 2022, 11(12), 699; https://doi.org/10.3390/axioms11120699 - 6 Dec 2022
Cited by 6 | Viewed by 1841
Abstract
In this paper, a retrial queueing system of the M/M/1 type with Poisson flows of arrivals, impatient customers, collisions, and an unreliable service device is considered. To make the problem more realistic and, hence, more complicated, we include the breakdowns and repairs of [...] Read more.
In this paper, a retrial queueing system of the M/M/1 type with Poisson flows of arrivals, impatient customers, collisions, and an unreliable service device is considered. To make the problem more realistic and, hence, more complicated, we include the breakdowns and repairs of the service in this research study. The retrial times of customers in the orbit, service time, impatience time of customers in the orbit, server’s lifetime (depending on whether it is idle or busy), and server recovery time are supposed to be exponentially distributed. The problem of finding the stationary probability distribution of the number of customers in orbit is solved by using the method of asymptotic diffusion analyses under the condition of a heavy load of the system and the patience of customers in orbit. Numerical results are presented that demonstrate the effectiveness of the obtained theoretical conclusions, and a comparative analysis of the method of asymptotic analysis and the method of asymptotic diffusion analysis for the considered problem is given. Full article
(This article belongs to the Special Issue Queueing Theory and Network Applications)
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49 pages, 4651 KiB  
Article
First Passage Analysis in a Queue with State Dependent Vacations
by Jewgeni H. Dshalalow and Ryan T. White
Axioms 2022, 11(11), 582; https://doi.org/10.3390/axioms11110582 - 24 Oct 2022
Viewed by 2815
Abstract
This paper deals with a single-server queue where the server goes on maintenance when the queue is exhausted. Initially, the maintenance time is fixed by deterministic or random number T. However, during server’s absence, customers are screened by a dispatcher who estimates [...] Read more.
This paper deals with a single-server queue where the server goes on maintenance when the queue is exhausted. Initially, the maintenance time is fixed by deterministic or random number T. However, during server’s absence, customers are screened by a dispatcher who estimates his service times based on his needs. According to these estimates, the dispatcher shortens server’s maintenance time and as the result the server returns earlier than planned. Upon server’s return, if there are not enough customers waiting (under the N-Policy), the server rests and then resumes his service. At first, the input and service are general. We then prove a necessary and sufficient condition for a simple linear dependence between server’s absence time (including his rest) and the number of waiting customers. It turns out that the input must be (marked) Poisson. We use fluctuation and semi-regenerative analyses (previously established and embellished in our past work) to obtain explicit formulas for server’s return time and the queue length, both with discrete and continuous time parameter. We then dedicate an entire section to related control problems including the determination of the optimal T-value. We also support our tractable formulas with many numerical examples and validate our results by simulation. Full article
(This article belongs to the Special Issue Queueing Theory and Network Applications)
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17 pages, 604 KiB  
Article
Retrial Queuing-Inventory Systems with Delayed Feedback and Instantaneous Damaging of Items
by Agassi Melikov, Sevinj Aliyeva, Sajeev S. Nair and B. Krishna Kumar
Axioms 2022, 11(5), 241; https://doi.org/10.3390/axioms11050241 - 20 May 2022
Cited by 17 | Viewed by 2939
Abstract
This paper studies a Markov model of a queuing-inventory system with primary, retrial, and feedback customers. Primary customers form a Poisson flow, and if an inventory level is positive upon their arrival, they instantly receive the items. If the inventory level is equal [...] Read more.
This paper studies a Markov model of a queuing-inventory system with primary, retrial, and feedback customers. Primary customers form a Poisson flow, and if an inventory level is positive upon their arrival, they instantly receive the items. If the inventory level is equal to zero upon arrival of a primary customer, then this customer, according to the Bernoulli scheme, either leaves the system or goes into an infinite buffer to repeat their request in the future. The rate of retrial customers is constant, and if the inventory level is zero upon arrival of a retrial customer, then this customer, according to the Bernoulli scheme, either leaves orbit or remains in orbit to repeat its request in the future. According to the Bernoulli scheme, each served primary or retrial customer either leaves the system or feedbacks into orbit to repeat their request. Destructive customers that form a Poisson flow cause damage to items. Unlike primary, retrial, and feedback customers, destructive customers do not require items, since, upon arrival of such customers, the inventory level instantly decreases by one. The system adopted one of two replenishment policies: (s, Q) or (s, S). In both policies, the lead time is a random variable that has an exponential distribution. It is shown that the mathematical model of the system under study was a two-dimensional Markov chain with an infinite state space. Algorithms for calculating the elements of the generating matrices of the constructed chains were developed, and the ergodicity conditions for both policies were found. To calculate the steady-state probabilities, a matrix-geometric method was used. Formulas were found for calculating the main performance measures of the system. The results of the numerical experiments, including the minimization of the total cost, are demonstrated. Full article
(This article belongs to the Special Issue Queueing Theory and Network Applications)
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