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Mathematics
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Recent Research in Queuing Theory and Stochastic Models
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Recent Research in Queuing Theory and Stochastic Models

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematics and Computer Science".

Deadline for manuscript submissions: closed (28 February 2023) | Viewed by 16554

Special Issue Editors

Dr. Ivan Atencia

E-Mail Website
Guest Editor
Department of Applied Mathematics, University of Malaga, 29071 Málaga, Spain
Interests: applied mathematics; operations research; queueing systems; performance analysis; home automation systems
Special Issues, Collections and Topics in MDPI journals
Dr. José Luis Galán-García

E-Mail Website
Guest Editor
Departamento de Matemática Aplicada, Escuela de Ingenierías Industriales, University of Málaga, 29071 Málaga, Spain
Interests: mathematics education; modeling and simulation; mathematical programming; applications of computer algebra systems (CAS); applied mathematics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The purpose of this special issue is to contribute to the queueing theory and stochastic models with novel papers. Queueing models are one of the most well-known theories of stochastic models, and their progress and development are increasing exponentially since A.K. Erlang (1917) and T.O. Engset (1918) who studied communications networks and their congestion problems. A server, a waiting line and an arriving flow of customers stand as a basis of any queueing model. The way in which these items are considered and various disciplines to which they are attached open a wide range of possibilities to handling situations that arise in real-life problems.

Nowadays, some concepts such as sojourn times and busy periods have become quite relevant due to their importance in traffic engineering, telecommunications and computer systems.

As for the time concept in queueing systems, let us note that these systems have been traditionally considered in a context of continuous time but in the last two decades there has been an increasing interest in the study of discrete-time queueing systems since they are more suitable than their continuous counterpart for computer modelling and telecommunication systems.

Any relevant papers related to the queueing systems and stochastic models are welcome.

Dr. Ivan Atencia
Dr. José Luis Galán-García
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 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

  • Markov chains
  • stochastic process
  • sojourn times
  • busy periods
  • heavy traffic regime
  • multichannel stochastic network

Related Special Issue

Published Papers (7 papers)

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Research

12 pages, 1676 KiB  
Article
Evaluation of the Waiting Time in a Finite Capacity Queue with Bursty Input and a Generalized Push-Out Strategy
by Chris Blondia
Mathematics 2022, 10(24), 4771; https://doi.org/10.3390/math10244771 - 15 Dec 2022
Viewed by 927
Abstract
In this paper, we study a finite capacity queue where the arrival process is a special case of the discrete time Markov modulated Poisson process, the service times are generally distributed, and the server takes repeated vacations when the system is empty. The [...] Read more.
In this paper, we study a finite capacity queue where the arrival process is a special case of the discrete time Markov modulated Poisson process, the service times are generally distributed, and the server takes repeated vacations when the system is empty. The buffer acceptance strategy is based on a generalized push-out scheme: when the buffer is full, an arriving customer pushes out the Nth customer in the queue, where N takes values between 2 and the capacity of the system, and the arriving customer joins the end of the queue. Such a strategy is important when, as well as short waiting times for served customers, the time a pushed-out customer occupies a buffer space is also an important performance measure. The Laplace transform of the waiting time of a served customer is determined. Numerical examples show the influence of the bustiness of the input process and also the trade-off between the average waiting time of served customers and the occupancy of the buffer space of pushed-out customers. Full article
(This article belongs to the Special Issue Recent Research in Queuing Theory and Stochastic Models)
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14 pages, 487 KiB  
Article
Diffusion Limit for Single-Server Retrial Queues with Renewal Input and Outgoing Calls
by Anatoly Nazarov, Tuan Phung-Duc, Svetlana Paul and Olga Lizyura
Mathematics 2022, 10(6), 948; https://doi.org/10.3390/math10060948 - 16 Mar 2022
Cited by 7 | Viewed by 1623
Abstract
This paper studies a single-server retrial queue with two types of calls (incoming and outgoing calls). Incoming calls arrive at the server according to a renewal process, and outgoing calls of N−1 (N≥2) categories occur according to [...] Read more.
This paper studies a single-server retrial queue with two types of calls (incoming and outgoing calls). Incoming calls arrive at the server according to a renewal process, and outgoing calls of N−1 (N≥2) categories occur according to N−1 independent Poisson processes. Upon arrival, if the server is occupied, an incoming call joins a virtual infinite queue called the orbit, and after an exponentially distributed time in orbit enters the server again, while outgoing calls are lost if the server is busy at the time of their arrivals. Although M/G/1 retrial queues and their variants are extensively studied in the literature, the GI/M/1 retrial queues are less studied due to their complexity. This paper aims to obtain a diffusion limit for the number of calls in orbit when the retrial rate is extremely low. Based on the diffusion limit, we built an approximation to the distribution of the number of calls in orbit. Full article
(This article belongs to the Special Issue Recent Research in Queuing Theory and Stochastic Models)
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25 pages, 334 KiB  
Article
Performance Measures in a Generalized Asymmetric Simple Inclusion Process
by Yaron Yeger and Uri Yechiali
Mathematics 2022, 10(4), 594; https://doi.org/10.3390/math10040594 - 14 Feb 2022
Cited by 2 | Viewed by 1506
Abstract
Performance measures are studied for a generalized n-site asymmetric simple inclusion process (G-ASIP), where a general process controls intervals between gate-opening instants. General formulae are obtained for the Laplace–Stieltjes transform, as well as the means, of the (i) traversal time, (ii) busy period, [...] Read more.
Performance measures are studied for a generalized n-site asymmetric simple inclusion process (G-ASIP), where a general process controls intervals between gate-opening instants. General formulae are obtained for the Laplace–Stieltjes transform, as well as the means, of the (i) traversal time, (ii) busy period, and (iii) draining time. The PGF and mean of (iv) the system’s overall load are calculated, as well as the probability of an empty system, along with (v) the probability that the first occupied site is site k (k = 1, 2, …, n). Explicit results are derived for the wide family of gamma-distributed gate inter-opening intervals (which span the range between the exponential and the deterministic probability distributions), as well as for the uniform distribution. It is further shown that a homogeneous system, where at gate-opening instants gate j opens with probability pj=1n, is optimal with regard to (i) minimizing mean traversal time, (ii) minimizing the system’s load, (iii) maximizing the probability of an empty system, (iv) minimizing the mean draining time, and (v) minimizing the load variance. Furthermore, results for these performance measures are derived for a homogeneous G-ASIP in the asymptotic cases of (i) heavy traffic, (ii) large systems, and (iii) balanced systems. Full article
(This article belongs to the Special Issue Recent Research in Queuing Theory and Stochastic Models)
22 pages, 758 KiB  
Article
Analysis of a Discrete-Time Queueing Model with Disasters
by Mustafa Demircioglu, Herwig Bruneel and Sabine Wittevrongel
Mathematics 2021, 9(24), 3283; https://doi.org/10.3390/math9243283 - 17 Dec 2021
Cited by 6 | Viewed by 2539
Abstract
Queueing models with disasters can be used to evaluate the impact of a breakdown or a system reset in a service facility. In this paper, we consider a discrete-time single-server queueing system with general independent arrivals and general independent service times and we [...] Read more.
Queueing models with disasters can be used to evaluate the impact of a breakdown or a system reset in a service facility. In this paper, we consider a discrete-time single-server queueing system with general independent arrivals and general independent service times and we study the effect of the occurrence of disasters on the queueing behavior. Disasters occur independently from time slot to time slot according to a Bernoulli process and result in the simultaneous removal of all customers from the queueing system. General probability distributions are allowed for both the number of customer arrivals during a slot and the length of the service time of a customer (expressed in slots). Using a two-dimensional Markovian state description of the system, we obtain expressions for the probability, generating functions, the mean values, variances and tail probabilities of both the system content and the sojourn time of an arbitrary customer under a first-come-first-served policy. The customer loss probability due to a disaster occurrence is derived as well. Some numerical illustrations are given. Full article
(This article belongs to the Special Issue Recent Research in Queuing Theory and Stochastic Models)
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25 pages, 338 KiB  
Article
Sojourn Times in a Queueing System with Breakdowns and General Retrial Times
by Ivan Atencia and José Luis Galán-García
Mathematics 2021, 9(22), 2882; https://doi.org/10.3390/math9222882 - 12 Nov 2021
Cited by 2 | Viewed by 1371
Abstract
This paper centers on a discrete-time retrial queue where the server experiences breakdowns and repairs when arriving customers may opt to follow a discipline of a last-come, first-served (LCFS)-type or to join the orbit. We focused on the extensive analysis of the system, [...] Read more.
This paper centers on a discrete-time retrial queue where the server experiences breakdowns and repairs when arriving customers may opt to follow a discipline of a last-come, first-served (LCFS)-type or to join the orbit. We focused on the extensive analysis of the system, and we obtained the stationary distributions of the number of customers in the orbit and in the system by applying the generation function (GF). We provide the stochastic decomposition law and the application bounds for the proximity between the steady-state distributions for the queueing system under consideration and its corresponding standard system. We developed recursive formulae aimed at the calculation of the steady-state of the orbit and the system. We proved that our discrete-time system approximates M/G/1 with breakdowns and repairs. We analyzed the busy period of an auxiliary system, the objective of which was to study the customer’s delay. The stationary distribution of a customer’s sojourn in the orbit and in the system was the object of a thorough and complete study. Finally, we provide numerical examples that outline the effect of the parameters on several performance characteristics and a conclusions section resuming the main research contributions of the paper. Full article
(This article belongs to the Special Issue Recent Research in Queuing Theory and Stochastic Models)
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25 pages, 404 KiB  
Article
Heavy-Traffic Comparison of a Discrete-Time Generalized Processor Sharing Queue and a Pure Randomly Alternating Service Queue
by Arnaud Devos, Joris Walraevens, Dieter Fiems and Herwig Bruneel
Mathematics 2021, 9(21), 2723; https://doi.org/10.3390/math9212723 - 27 Oct 2021
Cited by 1 | Viewed by 1175
Abstract
This paper compares two discrete-time single-server queueing models with two queues. In both models, the server is available to a queue with probability 1/2 at each service opportunity. Since obtaining easy-to-evaluate expressions for the joint moments is not feasible, we rely on a [...] Read more.
This paper compares two discrete-time single-server queueing models with two queues. In both models, the server is available to a queue with probability 1/2 at each service opportunity. Since obtaining easy-to-evaluate expressions for the joint moments is not feasible, we rely on a heavy-traffic limit approach. The correlation coefficient of the queue-contents is computed via the solution of a two-dimensional functional equation obtained by reducing it to a boundary value problem on a hyperbola. In most server-sharing models, it is assumed that the system is work-conserving in the sense that if one of the queues is empty, a customer of the other queue is served with probability 1. In our second model, we omit this work-conserving rule such that the server can be idle in case of a non-empty queue. Contrary to what we would expect, the resulting heavy-traffic approximations reveal that both models remain different for critically loaded queues. Full article
(This article belongs to the Special Issue Recent Research in Queuing Theory and Stochastic Models)
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18 pages, 506 KiB  
Article
Analysis of Multi-Server Queue with Self-Sustained Servers
by Alexander Dudin, Olga Dudina, Sergei Dudin and Konstantin Samouylov
Mathematics 2021, 9(17), 2134; https://doi.org/10.3390/math9172134 - 02 Sep 2021
Cited by 3 | Viewed by 2706
Abstract
A novel multi-server vacation queuing model is considered. The distinguishing feature of the model, compared to the standard queues, is the self-sufficiency of servers. A server can terminate service and go on vacation independently of the system manager and the overall situation in [...] Read more.
A novel multi-server vacation queuing model is considered. The distinguishing feature of the model, compared to the standard queues, is the self-sufficiency of servers. A server can terminate service and go on vacation independently of the system manager and the overall situation in the system. The system manager can make decisions whether to allow the server to start work after vacation completion and when to try returning some server from a vacation to process customers. The arrival flow is defined by a general batch Markov arrival process. The problem of optimal choice of the total number of servers and the thresholds defining decisions of the manager arises. To solve this problem, the behavior of the system is described by the three-dimensional Markov chain with the special block structure of the generator. Conditions for the ergodicity of this chain are derived, the problem of computation of the steady-state distribution of the chain is discussed. Expressions for the key performance indicators of the system in terms of the distribution of the chain states are derived. An illustrative numerical result is presented. Full article
(This article belongs to the Special Issue Recent Research in Queuing Theory and Stochastic Models)
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