Retrial Queueing System Research Articles

Efficiency of retrial queueing system under N threshold during vacation

Efficiency of retrial queueing system under N threshold during vacation

Control approach to well-posedness and asymptotic behavior of a queueing system

In this paper, we investigate well-posedness and asymptotic behavior of an M/G/1 retrial queueing system characterized by reserved idle time and setup time. This system is intricately described by an infinite system of integro-partial differential equations. Our methodology is rooted in the feedback theory of well-posed and regular infinite-dimensional linear systems. Firstly, we convert the system into a control framework, incorporating both input and output variables. Subsequently, we establish its well-posedness by leveraging the feedback theory specific to regular infinite-dimensional linear systems. Secondly, by scrutinizing the spectral distribution of the system operator along the imaginary axis, we demonstrate that the time-evolving solution of the system converges robustly to its steady-state solution. Lastly, we also discuss the essential properties of the semigroup induced by the system operator, encompassing stability and hyperbolicity, thereby shedding light on its regularity.

Dynamical behaviour and meta heuristic optimization of a hospital management software system

The growing demand for easily available healthcare in recent years has fuelled the digitization of healthcare services. The Hospital Management System (HMS) software stands out as a comprehensive solution among the software systems and tools that hospitals and clinics are developing in tandem with this trend. In order to effectively manage many facets of hospital operations, in this paper, we propose an approach for investigating software of this kind. Thus, we characterise the HMS software as a unique sort of batch arrival retrial queueing system (QS) that can handle both ordinary and priority patient demands. Furthermore, it permits patient resistance (balk) and departure (renege) in specific circumstances. The proposed model is additionally deployed within the framework of Bernoulli working vacation. The supplementary variable technique (SVT) has been utilised to obtain the necessary results. ANFIS, a soft computing tool, is used to validate the analytical results as well. Finally, this study seeks to enhance the cost-effectiveness of software creation by employing four unique optimization methods, aiming to achieve optimal efficiency in resource utilization.

Performance analysis of a collision channel with abandonments

We consider a Markovian retrial queueing system with customer collisions and abandonment in the context of carrier-sense multiple access systems. Using z-transform techniques, we find a set of first-order differential equations for the probability generating functions of the orbit size when the server is empty, busy, or in the collision phase. We then rely on series expansion techniques to extract approximations for relevant performance measures from this set of differential equations. More precisely, we construct a numerical algorithm to calculate the terms in the series expansions of various factorial moments of the orbit size. To improve the accuracy of our series expansion approach, we apply Wynn’s epsilon algorithm which not only speeds up convergence, but also extends the region of convergence. We illustrate the accuracy of our approach by means of some numerical examples, and find that the method is both fast and accurate for a wide range of the parameter values.

Optimal control of a two-phase heterogeneous service retrial queueing system with collisions and delayed vacations

Optimal control of a two-phase heterogeneous service retrial queueing system with collisions and delayed vacations

Prioritized Retrial Queueing Systems with Randomized Push-Out Mechanism

The article is focused on a single-channel preemptive queuing system. Two stationary Poisson flows of customers are incoming to the system. The first flow has an absolute priority over the second one: a new high-priority customer from the first flow displaces a low-priority one from the service channel and takes its place. The capacity of the system is limited to k customers. There is a probabilistic push-out mechanism in the system: if a new high-priority customer finds that all the places in the queue are occupied, then it has the right to displace one low-priority customer from the queue with probability a. Both types of customers have the same exponentially distributed service times. Customers who failed to enter the system due to the limited size of the queue, as well as those expelled from the queue or service channel when the push-out mechanism is triggered, are not lost immediately, but they are sent to a special part of the system called the orbit and designed to store repeated customers. In orbit, there are two separate unlimited queues, consisting of low-priority and high-priority repeated customers, respectively. If there are no free places in the system, new customers with a probability q are added to the corresponding orbital queue. The waiting time of repeated customers in orbit is distributed according to an exponential law. The parameter of this law may differ for different types of customers. After waiting in orbit, secondary customers try to re-enter the system. The probabilistic characteristics of the described queuing system are calculated by the method of generating functions, previously proposed by the authors for calculating a similar system without repeated customers. This method allows finding the main probabilistic characteristics of distributions for both types of customers. Particular attention is paid to the study of the dependence of the loss probabilities for both types of customers on the parameters of the system, primarily on the push-out probability a, the capacity of the system k, and the probability of repeated circulation (probability of persistence) q. It is shown that the effect of blocking the system and the effect of the linear law of customers’ losses, previously identified in similar problems without repeated customers, remain valid even in the presence of secondary repeated customers. The theoretical results are proved by numerical calculations. The blocking area for the second type of customers was calculated along with the area of linear loss law for both types of customers. We studied the influence of the probability of repeated circulation q on the shape of these areas and on the dependence of the loss probabilities for both types of customers on the push-out probability a.

Analysis of a Single Server Retrial Queueing System with Finite Capacity, F-Policy, Balking, Reneging and Server Maintenance

In this paper, we determine a single server retrial queuing system with finite capacity, F-policy, balking, reneging and server maintenance. The arriving customer when notices the server becomes free, then they get service immediately. In case server becomes busy, the customers are move to either orbit or may balk the system. In the orbit, after some random time the customer repeats their attempts for service. The customers are waiting in the orbit for longer time become impatient and leaves the orbit without getting service. Once the number of customers in the orbit reaches its full capacity then the arrival of customer is not allowed to enter into the orbit. The new customer is allowed into the orbit when the F-policy level is reached. If no more customer in the orbit, server will move to the vacation. By using recursive technique, the steady state distributions are obtained and we have developed various performance measures.

A literature review on retrial queueing system with Bernoulli vacation

The retrial phenomenon occurs inherently in a wide range of queueing systems. The majority of retrial queueing models do not account for vacation. However, in practice, retrial queueing systems undergo vacations for maintenance or other reasons. In this study, we provide an in-depth analysis of the many possible retrial queueing systems when Bernoulli vacations are in effect. Moreover, this study outlines the key principles and reviews the relevant literature. The framework of a retrial queue with Bernoulli vacation has numerous applications in computer networking systems, manufacturing and production mechanisms, inventory systems, including network service, mail service and file transfer service, etc. Several retrial queueing systems have been investigated, notably M/M/1, M/M/C, M/G/1, M[X]/G/1, and Geo/G/1. Many other important situations, such as server interruption, feedback, G-queue, impatient customers, priority customers, etc., have been explored in relation to retrial queues with Bernoulli vacation and the results of these investigations are also highlighted. The foremost objective of this study is to help researchers, administrators and technical workers who want to use queuing theory to simulate congestion and need to know where to find details on the right models. Finally, some open problems and potential future lines of survey are also covered.

Equilibrium strategies of customers and optimal inventory levels in a make-to-stock retrial queueing system

<abstract><p>We examined a retrial make-to-stock system based on a double-ended queue. When the queue length was negative, the inventory system contained only products, and customers were waiting in the retrial queue when the queue length was positive. We developed a model to study the expected cost of the entire system with strategic customers in the observable case and the fully observable case. We also obtained the optimal inventory levels under these two levels of information based on numerical experiments.</p></abstract>

HETEROGENEOUS SERVER RETRIAL QUEUEING MODEL WITH FEEDBACK AND WORKING VACATION USING ARTIFICIAL BEE COLONY OPTIMIZATION ALGORITHM

This research delves into the dynamics of a retrial queueing system featuring heterogeneous servers with intermittent availability, incorporating feedback and working vacation mechanisms. Employing a matrix geometric approach, this study establishes the steady-state probability distribution for the queue size in this complex heterogeneous service model. Additionally, a range of system performance metrics is developed, alongside the formulation of a cost function to evaluate decision variable optimization within the service system. The Artificial Bee Colony (ABC) optimization algorithm is harnessed to determine service rates that minimize the overall cost. This work includes numerical examples and sensitivity analyses to validate the model's effectiveness. Also, a comparison between the numerical findings and the neuro-fuzzy results has been examined by the adaptive neuro fuzzy interface system (ANFIS).

Some reflected autoregressive processes with dependencies

Motivated by queueing applications, we study various reflected autoregressive processes with dependencies. Among others, we study cases where the interarrival and service times are proportionally dependent on additive and/or subtracting delay, as well as cases where interarrival times depend on whether the service duration of the previous arrival exceeds or not a random threshold. Moreover, we study cases where the autoregressive parameter is constant as well as a discrete or a continuous random variable. More general dependence structures are also discussed. Our primary aim is to investigate a broad class of recursions of autoregressive type for which several independence assumptions are lifted and for which a detailed exact analysis is provided. We provide expressions for the Laplace transform of the waiting time distribution of a customer in the system in terms of an infinite sum of products of known Laplace transforms. An integer-valued reflected autoregressive process that can be used to model a novel retrial queueing system with impatient customers and a general dependence structure is also considered. For such a model, we provide expressions for the probability generating function of the stationary orbit queue length distribution in terms of an infinite sum of products of known generating functions. A first attempt towards a multidimensional setting is also considered.

Lower and upper stochastic bounds for the joint stationary distribution of a non-preemptive priority retrial queueing system

Consider a single-server retrial queueing system with non-preemptive priority service, where customers arrive in a Poisson process with a rate of $\lambda_1$ for high-priority customers (class 1) and $\lambda_2$ for low-priority customers (class 2). If a high-priority customer is blocked, they are queued, while a low-priority customer must leave the service area and return after some random period of time to try again. In contrast with existing literature, we assume different service time distributions for the two customer classes. This investigation proposes a stochastic comparison method based on the general theory of stochastic orders to obtain lower and upper bounds for the joint stationary distribution of the number of customers at departure epochs in the considered model. Specifically, we discuss the stochastic monotonicity of the embedded Markov queue-length process in terms of both the usual stochastic and convex orders. We also perform a numerical sensitivity analysis to study the effect of the arrival rate of high-priority customers on system performance measures.

OPTIMAL QUEUING STRATEGIES FOR AN M/G/1 RETRIAL QUEUE SYSTEM WITH RWV AND ISEV POLICIES

AbstractDesigning a reasonable M/G/1 retrial queue system that enhances service efficiency and reduces energy consumption is a challenging issue in Information and Communication Technology systems. This paper presents an M/G/1 retrial queue system incorporating random working vacation (RWV) and improved service efficiency during vacation (ISEV) policies, and examines its optimal queuing strategies. The RWV policy suggests that the server takes random working vacations during reserved idle periods, effectively reducing energy consumption. In contrast, the ISEV policy strives to augment service efficiency during regular working periods by updating, inspecting or maintaining the server on vacations. The system is transformed into a Cauchy problem to investigate its well-posedness and stability, employing operator semigroup theory. Based on the system’s stability, steady-state performance measures, such as service efficiency, energy consumption and expected costs, are quantified using the steady-state solution. The paper subsequently demonstrates the existence of optimal queuing strategies that achieve maximum efficiency and minimum expected costs. Finally, two numerical experiments are provided to illustrate the effectiveness of the system.

Retrial Queueing System for Analyzing Impact of Priority Ultra-Reliable Low-Latency Communication Transmission on Enhanced Mobile Broadband Quality of Service Degradation in 5G Networks

Fifth generation (5G) networks support ultra-reliable low-latency communications (URLLC) and enhanced mobile broadband (eMBB). The coexistence of URLLC and eMBB is often organized by non-orthogonal multiple access (NOMA), giving priority to URLLC and resulting in eMBB quality of service (QoS) degradation. In this paper, we address the issue of joint URLLC and eMBB transmission, focusing on the problem from the perspective of delay-tolerant eMBB. Due to the priority given to URLLC, we assume that an eMBB session may be interrupted if there are no free resources available for URLLC or delayed when a new eMBB session arrives. To make the scheme more flexible, we propose that interrupted and delayed eMBB sessions periodically check for free resources, rather than continuously. To analyze this scenario, we propose a retrial queuing system with two retrial buffers (orbits) for interrupted and delayed eMBB sessions. The stationary probability distribution, provided in matrix form by recursive formulas, is presented. The paper concludes with a numerical example showing that the scheme with two buffers, compared to one buffer, practically doubles the average number of active eMBB sessions while keeping the interruption probability below 0.001. We provide an illustration of the configuration of eMBB retrial rates to meet its QoS requirements.

Genetic algorithm in the single server inventory retrial queueing system with time and stock level dependent customer arrival rate

Genetic algorithm in the single server inventory retrial queueing system with time and stock level dependent customer arrival rate

Asymptotic diffusion analysis of the retrial queuing system with feedback and batch Poisson arrival

The mathematical model of the retrial queuing system \(M^{[n]}/M/1\) with feedback and batch Poisson arrival is constructed. Customers arrive in groups. If the server is free, one of the arriving customers starts his service, the rest join the orbit. The retrial and service times are exponentially distributed. The customer whose service is completed leaves the system, or reservice, or goes to the orbit. The method of asymptotic diffusion analysis is proposed for finding the probability distribution of the number of customers in orbit. The asymptotic condition is growing average waiting time in orbit. The accuracy of the diffusion approximation is obtained.

Performance Study of an M/M/1 Retrial Queueing System with Balking, Dissatisfied Customers, and Server Vacations

This study considers a single server retrial queueing system with dissatisfied customers, server vacations, and balking customers. When new arrivals clients see an idle server available, they immediately enter service. If a new arrival sees the server busy, that arrival may balk and leave the system. Otherwise, the new arrival will enter a retrial orbit and try again later. Customers who are dissatisfied with the service will be served again until the service is completed successfully. The server begins a vacation when the current customer completes service. Birth and death state transitions are given. Steady-state distributions for queue length are found. Furthermore, a recursive technique is used to evaluate probabilities. Several performance measures of the system are identified and effects of changes to parameters are observed.

Performance Analysis of a Retrial Queueing System with Optional Service, Unreliable Server, Balking and Feedback

This paper considers a Markovian retrial queueing system with an optional service, unreliable server, balking and feedback. An arriving customer can avail of immediate service if the server is free. If the potential customer encounters a busy server, it may either join the orbit or balk the system. The customers may retry their request for service from the orbit after a random amount of time. Each customer gets the First Essential Service (FES). After the completion of FES, the customers may seek the Second Optional Service (SOS) or leave the system. In the event of unforeseen circumstances, the server may encounter a breakdown, at which point an immediate repair process will be initiated. After the service completion, the customer may leave the system or re-join the orbit if not satisfied and demand regular service as feedback. In this investigation, the stationary queue size distributions are framed using a recursive approach. Various system performance measures are derived. The effects induced by the system parameters on the performance metrics are numerically and graphically analysed.

Admission control policy of a two heterogeneous server finite capacity retrial queueing system with maintenance activity

Admission control policy of a two heterogeneous server finite capacity retrial queueing system with maintenance activity

Performance analysis of a multi server retrial queueing system with unreliable server, discouragement and vacation model

In this paper, we investigate a Markovian multi server retrial queueing system with unreliable servers, discouraged customers and vacation. The model is explored with a view to reduce the impact of unreliable servers on the system performance. This is attained by providing regular maintenance to the servers through synchronised vacation. Further, the customer facing service interruption on account of server breakdown is directed to the orbit from where the customer can retry seeking service. In this study, the stationary system size distribution is framed by quasi-birth–death (QBD) process. The rate matrix and steady state probabilities are established using matrix geometric method (MGM). The performance measures are derived by using matrix form expressions. The effects induced by the system parameters on the performance metrics are numerically and graphically analysed. Through numerical investigation, we provide insightful information that enables to manage the system to give enhanced system performance.