This paper examines a discrete-time retrial queueing system where incoming customers can either choose a last-come, first-served (LCFS) discipline or enter an orbit. It accounts for the possibility of varying service times, which follow an arbitrary distribution, and the retrial times are also governed by an arbitrary distribution. The underlying Markov chain of the system has been analyzed, leading to the derivation of the generating function for the number of customers in both the orbit and the overall system, along with their expected values. The paper also establishes the stochastic decomposition law and, as an application, provides bounds for the difference between the steady-state distributions of the system in question and its standard equivalent. Recursive formulas for determining the steady-state distribution of customers in the orbit and the system are presented. The paper derives the distribution of the time a customer spends at the server and, consequently, the distribution of service times subject to possible variations. A detailed analysis of the time a customer spends in the orbit is also conducted. Finally, numerical examples are included to demonstrate how key parameters impact various system characteristics, with the main contributions of the research summarized in the conclusion.

This paper examines a discrete-time retrial queuing system that incorporates negative customers, system breakdowns, and repairs. In this model, an arriving customer has the option to go directly to the server, pushing the currently served customer, if any, to the front of the orbit queue, or to join the orbit based on a First-Come-First-Served (FCFS) discipline. The study also considers negative customers who not only remove the customer currently being served but also cause a server breakdown. An in-depth analysis of the model is conducted using a generating function approach, leading to the determination of the distribution and expected values of the number of customers in the orbit and the entire system. The paper explores the stochastic decomposition law and provides bounds for the difference between the steady-state distribution of this system and a comparable standard system. Recursive formulas for the steady-state distributions of the orbit and the system are developed. Additionally, it is shown that the studied discrete-time system can approximate the M/G/1 continuous-time version of the model. The research includes a detailed examination of the customer’s sojourn time distribution in the orbit and the system, utilizing the busy period of an auxiliary system. The paper concludes with numerical examples that highlight how different system parameters affect various performance characteristics, and a section summarizing the key research contributions.

This paper studies a discrete-time queueing system in which a customer who enters in the system with a server occupied may choose to go to the retrial group or to begin its service displacing to the retrial group the customer that was in the server. If the server is idle the arriving customer begins its service immediately. Retrial and service times are general, and the only repeated customer that can go directly to the server is the one situated at the first place of the retrial group. A complete study of the model has been curried out noting specially the analysis of a recursive algorithm implemented by Theorems 2 and 3 and the study of the waiting time of a customer in the queue and in the system. Finally numerical examples are given.

In this paper, we introduce SODES (Stepwise Ordinary Differential Equations Solver) a new solver for Ordinary Differential Equations (ODE). SODES can optionally provide the solution displaying all the steps needed to obtain it. This way, SODES is an important tool not only for researchers who need solving ODE but also constitutes an important tool for the teaching and learning process of ODE. SODES has been developed using programming with a Computer Algebra System (CAS). Specifically, we use the CAS Derive but it can be easily adapted to any other CAS supporting programming. SODES provides, step by step, the solution of the following types of ODE: separable, homogeneous, exact, integrating factors, linear, Bernoulli, Riccati, first order ODE of nth degree, Cauchy’s problems of first order ODE, higher order linear homogeneous equations with constant coefficients, Lagrange’s method for particular solutions of higher order linear equations with constant coefficients, higher order linear equations with constant coefficients and Cauchy’s problems of higher order linear equations with constant coefficients. SODES also deals with two generic programs which determine the type or types of a given ODE and provides the solution. In this paper we will also introduce a draft of a Graphical User Interface (GUI) for SODES in a local web application using programming in Python (using its CAS module SymPy ) which is a more portable and free CAS. This draft can be used in English, French and Spanish, and can be easily extended to other languages. The code of SODES and the GUI are freely available so that it can be used by users who also will be able to adapt it to their needs.

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.

We consider a discrete-time Geo/G/1/infinity system in which a customer that finishes its first essential service may opt to abandon the system, to receive a second optional service or to go at the head of the queue in order to receive another essential service. We study the Markov chain underlying the considered queueing system and its ergodicity condition. Using a generating function approach the distribution of the number of customers in the queue and in the system as well as their respective means are given. The busy period of an auxiliary system, that will be useful to study of the customers delay, is analysed. The distributions of the sojourn time of a customer in the server, the queue and the system are provided. In order to illustrate the effect of the parameters on several performance characteristics some numerical examples are given. Finally, a section of conclusions commenting the main research contributions of this paper is presented. (c) 2021 Elsevier B.V. All rights reserved.

Multiple Integration is a very important topic in different applications in Engineering and other Sciences. Using numerical software to get an approximation to the solution is a normal procedure. Another approach is working in an algebraic form to obtain an exact solution or to get general solutions depending on different parameters. Computer Algebra Systems (CAS) are needed for this last approach. In this paper, we introduce SMIS, a new stepwise solver for multiple integration developed in a CAS. The two main objectives of SMIS are: (1) to increase the capabilities of CAS to help the user to deal with this topic and (2) to be used in Math Education providing an important tool for helping with the teaching and learning process of this topic. SMIS can provide just the final solution or an optional stepwise solution (even including some theoretical comments). The optional stepwise solutions provided by SMIS are of great help for (2). Although SMIS has been developed in the specific CAS Derive, since the code is provided, it can be easily migrated to any CAS which deals with integrals and text management that allow us to display comments for intermediate steps.

This paper discusses a discrete-time queueing system in which an arriving customer may adopt four different strategies; two of them correspond to a LCFS discipline where displacements or expulsions occur, and in the other two, the arriving customer decides to follow a FCFS discipline or to become a negative customer eliminating the customer in the server, if any. The different choices of the involved parameters make this model to enjoy a great versatility, having several special cases of interest. We carry out a thorough analysis of the system, and using a generating function approach, we derive analytical results for the stationary distributions obtaining performance measures for the number of customers in the queue and in the system. Also, recursive formulae for calculating the steady-state distributions of the queue and system size has been developed. Making use of the busy period of an auxiliary system, the sojourn times of a customer in the queue and in the system have also been obtained. Finally, some numerical examples are given.

This paper studies Markovian models of controlled systems with retrials, where the rate of the flow of retrials does not depend on the amount of their sources. The input flow of customers is controlled according to a multithreshold strategy. The conditions for the existence of a stationary regime are defined for this type of system, and clear formulae of vector-matrix type for stationary probabilities are obtained. The problem of multicriteria optimization of the system’s profit are also considered.

In a previous paper, the authors developed new rules for computing improper integrals which allow computer algebra systems (Cas) to deal with a wider range of improper integrals. The theory used in order to develop such rules where Laplace and Fourier transforms and the residue theorem. In this paper, we describe new rules for computing symbolic improper integrals using extensions of the residue theorem and analyze how some of the most important Cas could improve their improper integral computations using these rules. To achieve this goal, different tests are developed. The Cas considered have been evaluated using these tests. The obtained results show that all Cas involved, considering the new developed rules, could improve their capabilities for computing improper integrals. The results of the evaluations of the Cas are described providing a sorted list of the Cas depending on their scores.

This paper considers a discrete-time queueing system with uncertainty information, that is, an arriving customer may opt, with a certain probability, to be served under a Last Come First Served (LCFS) discipline or, with complementary probability, becomes a negative customer. The arrival of a negative customer to the system removes the customer that is currently being served, if any, and has no effect on the system if the server is free. The input stream of the arriving customers is described by means of a Bernoulli process, and the service times are geometrically distributed. Breakdowns can occur, that is, the possibility of a server failure during the service time of a customer is considered, and the customer that was receiving its service in the moment of a failure of the server goes to the first place of the queue. The server has a general life time distribution, and the repair times are ruled by an arbitrary distribution. A thorough study of the model is carried out, and, using a generating function approach, the steady-state distributions of the number of customers in the queue and in the system are obtained. We also give a necessary and sufficient condition for the stability of the system. Characteristics measures of the model are also given. Finally, some numerical examples to illustrate the effect of the parameters on several performances measures of the system are presented.

This paper considers a discrete-time retrial queueing system in which the arriving customers can decide to go directly to the server expelling out of the system the customer that is currently being served, if any, or to join the orbit in accordance with a FCFS discipline. An extensive analysis of the model has been carried out, and using a generating functions approach, the distributions of the number of customers in the orbit and in the system with its respective means are obtained. The stochastic decomposition law has been derived, and, as an application, bounds for the proximity between the steady-state distributions for the considered queueing system and its corresponding standard system are obtained. Also, recursive formulae for calculating the steady-state distributions of the orbit and system size have been developed. Besides, we prove that the M/G/1 retrial queue with service interruptions can be approximated by the corresponding discrete-time system. The generating function of the sojourn time of a customer in the orbit and in the system have also been provided. Finally, some numerical examples to illustrate the effect of the parameters on several performance characteristics and a section of conclusions commenting the main research contributions of this paper are presented. (C) 2016 Elsevier B.V. All rights reserved.

In this paper we consider a discrete-time retrial queueing system with batch arrivals of geometric type and general batch services. The arriving group of customers can decide to go directly to the server expelling out of the system the batch of customers that is currently being served, if any, or to join the orbit. After a successful retrial all the customers in the orbit get service simultaneously. An extensive analysis of the model is carried out, and using a generating functions approach some performance measures of the model, such as the first distributions moments of the number of customers in the orbit and in the system, are obtained. The generating functions of the sojourn time of a customer in the orbit and in the system are also given. Finally, in the section of conclusions and research results the main contributions of the paper are commented.

Different Engineering applications require dealing with improper integral on unbounded domains (improper integrals of the first kind). The classical way for solving these integrals is by means of elementary Calculus (antiderivatives and limit computations) or using numerical approaches. In both situations different problems can arise. For example: the nonexistence of antiderivative, the corresponding limit does not exist or integrals depending on parameters which complicate the use of numerical approaches. In order to solve this situation, Advanced Calculus techniques, such as Laplace or Fourier Transforms and the Residue Theorem can be applied. A brief review of the corresponding theoretical frame is included in this paper. Computer Algebra Systems (Cas) are pieces of software that allow symbolic computations. Nowadays, there are many Cas in the market with an increasing level of sophistication which make them a very powerful tool in Engineering. In this paper, a brief review on the history of Cas evolution is introduced. In this work, some Advanced Calculus techniques for computing improper integrals of the first kind which cannot be solved with standard procedures are described. These techniques could be easily integrated in almost every Cas. However, we have detected some lacks of these techniques in many widely used Cas. In this paper, some tests involving improper integrals have been developed. These tests have been used to check the capabilities of some Cas and the results have provided a classification of the Cas with respect to this field. One of the main contributions in this work is the generation of different rules to compute improper integrals of the first kind. The rules have been classified in specific rules (those coming from the tests) and general rules (those coming from theoretical frames and generalizations of the specific rules). These rules are easy to include in a Cas increasing the facilities of the Cas in the field of improper integral computation. (C) 2016 Elsevier Inc. All rights reserved.

This paper considers a discrete-time queueing system in which an arriving customer can decide to follow a last come first served (LCFS) service discipline or to become a negative customer that eliminates the one at service, if any. After service completion, the server can opt for a vacation time or it can remain on duty. Changes in the vacation times as well as their associated distribution are thoroughly studied. An extensive analysis of the system is carried out and, using a probability generating function approach, steady-state performance measures such as the first moments of the busy period of the queue content and of customers delay are obtained. Finally, some numerical examples to show the influence of the parameters on several performance characteristics are given.

We consider a discrete-time queueing system in which the arriving customers can decide to follow a LCFS discipline or to join the queue. Breakdowns can occur with geometrical lifetime and repair times governed by an arbitrary distribution. The repair times can exert changes governed by a geometrical law. We carry out a thorough study of the model deriving analytical results for the stationary distributions. We obtain generating functions of the number of customers in the queue and in the system. We also obtain the generating function of the repair times taking into account possible changes in the remaining repair times. The generating functions of the busy period, sojourn time in the server, sojourn time in the queue as well as some performance measures are also provided.