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In an earlier paper, a randomized load balancing model was studied in a heavytraffic asymptotic regime where the load balancing stream is thin compared tothe total arrival stream. It was shown that the limit is given by a system ofrank-based Brownian particles on the half-line. This paper extends theseresults from the case of exponential service time to an invariance principle,where service times have finite second moment. The main tool is a new notion ofrank-based stochastic differential inclusion, which may be of interest in itsown right.

We consider Palm distributions arising in a Markov process with time homogeneous transitions which is jointly stationary with multiple point processes. Motivated by a BAR approach studied in the recent paper (Braverman et al. in the BAR approach for multi-class queueing networks with SBP service policies, 2023), we are interested in two problems; when this Markov process inherits the same Markov structure under the Palm distributions, and how the state changes at counting instants of the point processes can be handled to derive stationary equations when there are simultaneous counts and each of them influences the state changes. We affirmatively answer the first problem, and propose a framework for resolving the second problem, which is applicable to a general stationary process, which is not needed to be Markov. We also discuss how those results can be applied in deriving BAR's for the diffusion approximation of queueing models in heavy traffic. In particular, as their new application, the heavy traffic limit of the stationary distribution is derived for a single-server queue with a finite waiting room. [ABSTRACT FROM AUTHOR]

We consider the resource management of a data center by using a discrete-time single-server queue with an unlimited-size batch service, where the next queue to be served is the one with the most senior waiting client (the so-called ‘Informal Queue’). Large jobs or tasks are submitted for processing by users from different applications or organizations in a cloud data center setting. There is an endless number of groups in the queue, a very dynamic and diverse workload, and an endless stream of tasks or jobs arriving every day. We analyze the Geo/Geo/1 informal queue with an early arrival system and a late arrival system with delayed access policies. Various key performance measures, such as steady-state distribution of the number of groups in the system, sojourn times and waiting times of a group leader, sojourn times and waiting times of an arbitrary client, and number of sidestep groups, are obtained. Explicit-form expressions are found for both policies. We illustrate several numerical results and present an application of the discussed model in data centers. Finally, it is demonstrated that in the limiting case, the outcomes found in this article tend to the continuous-time counterpart. [ABSTRACT FROM AUTHOR]

This article analyzes an M 1 X , M 2 / G 1 , G 2 / 1 retrial queue with two-way communication under constant retrial policy. A server handles both inbound and outbound calls. The inbound calls arrive in batches according to a compound Poisson process. The server makes an outbound call that follows the Poisson process during its inactive period. The service duration of inbound and outbound calls follows two different arbitrary distributions, while retrial time follows an exponential distribution. Using the supplementary variable technique, the system of differential-difference equations is developed to directly compute the distribution of orbit length at random epoch. The execution of the waiting time distribution for an arbitrary call of an incoming batch constitutes the most challenging task in this study. We also conduct a comprehensive cost analysis to minimize the overall cost of system operation. Furthermore, a variety of performance metrics and numerical findings validate the proposed methodology. [ABSTRACT FROM AUTHOR]

The current article is about a finite buffer, group size dependent bulk service queue, in which a single server renders two type of services, first essential service (FES) and second optional service (SoS). Customers arrive at the system in the Poisson fashion. Service is rendered by a server in groups following ‘general bulk service’ (GBS) rule on first come first serve (FCFS) basis for FES. In this article, we allowed a part of a group, served in FES, to join SoS in group following binomial law. The service time distribution is considered to be generally distributed and dependent on the group size under service for both the cases, FES and SoS. The mathematical analysis is performed using the supplementary variable technique (SVT) and the embedded Markov chain technique to obtain the steady state, departure epoch and arbitrary epoch joint probabilities of the count of customers in the queue and with the server when the server is in FES as well as in SoS. Finally, various numerical studies are presented to show the behaviour of the key efficiency metrics, which eventually shows the importance of the current study. [ABSTRACT FROM AUTHOR]

In this paper, we consider a queueing-inventory system with modified delayed vacation under Bernoulli schedule. If the inventory is empty upon completion of the service, the server will take the modified delayed vacation. During the modified delayed vacation period, if the replenishment is completed, the customer will still be served as normal. After this period, the server will take Bernoulli schedule, where the server reverts to normal work with probability q (0 ≤ q ≤ 1) or takes multiple vacations with probability 1 - q . The customers arrive according to a Poisson process. Upon customer arrival, if the inventory is not empty, the customer accepts the service and leaves the system carrying a product. The service time, lead time, modified delayed vacation time and multiple vacations time are all assumed to be exponentially distributed. We derive the stability condition of the system and the matrix geometric solution of steady-state probabilities using an algorithm. Then performance measures of the system are derived. Finally, numerical results are presented to demonstrate the impact of system parameters on performance measures and the expected cost function. [ABSTRACT FROM AUTHOR]

This article addresses the pressing issue of customer impatience and dissatisfaction with queueing systems, a critical aspect of modern businesses. It delves into specific aspects of customer behavior, viz., reverse balking, reneging, and retention of reneged customers, which significantly influence queue dynamics. A novel finite-buffer two-server heterogeneous queueing model is introduced, taking into account reverse balking, and position-dependent reneging (reneging till the beginning of service i.e., R _ B O S and reneging till the end of service i.e., R _ E O S ) along with the retention of reneged customers. The probability distribution of the number of customers in the system is ferret out and the key performance measures are derived along with sensitivity analysis. Extending the analysis, this study tackles a cost optimization problem using an algorithm based on the Quasi-Newton Method (QNM) to achieve optimal solutions for both the server that optimize the cost function. A numerical example is provided to illustrate the practical implications and benefits of the proposed optimization approach for the queueing model. [ABSTRACT FROM AUTHOR]

Explicit results are derived using simple and exact methods for the joint and marginal queue-length distributions for the M/M/c queue with two non-preemptive priority levels. Equal service rates are assumed. Two approaches are considered. One is based on numerically robust quadratic recurrence relations. The other is based on a complex contour-integral representation that yields exact closed-form analytical expressions, not hitherto available in the literature, that can also be evaluated numerically with very high accuracy. [ABSTRACT FROM AUTHOR]

AbstractAn essential aspect of the practical application of Erlang single-server queues is the statistical inference of their parameters, notably traffic intensity. This metric is crucial as it serves as a fundamental performance indicator, allowing the derivation of other significant measures, including the mean queue size and the expected number of customers in the system. Additionally, it provides the proportion of time the queue system is occupied. This article explores algorithms for calculating sample sizes for these estimations, based on the number of customers who arrived during service periods, a highly intuitive approach, allowing data collection without concerns about correlations among data points. For this purpose, we have considered two forms of informative priors: Gauss hypergeometric and beta priors. Additionally, a non conjugate and objective prior, known as Jeffreys prior, has been taken into account. We present tables with sample sizes for specific configurations and abacuses for more general configurations obtainable through approximate interpolations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a GI X /M/1 queue with the two-stage vacation policy. The system will first enter a working vacation once the system becomes empty in the normal busy period, during which the service rate will be switched to a lower rate. After this working vacation, if the system is nonempty, the service rate will be switched to the normal rate immediately; otherwise, the system will enter multiple vacations during which the service rate reduces to zero until the system is nonempty after one vacation and a new normal busy period starts. The distinguishing feature of this paper is the use of a general algorithm for solving the difference equation based on the right shift operator method to obtain the queue length distribution at pre-arrival time and arbitrary time. Moreover, the sojourn time of arbitrary customer is analyzed. Finally, several typical numerical examples are provided to validate our computational algorithm. [ABSTRACT FROM AUTHOR]

We study a type of correlated queue in which its inter-arrival and service times are correlated, with their joint distribution being characterized by a copula function. This characterization allows for very generally correlated inter-arrival and service times. For such a queue, we first derive an infinite system of linear equations for the moments of the waiting time, based on which we design an algorithm to calculate the moments of the waiting time. In contrast with traditional methods, our algorithm incorporates a recursive procedure which is computationally much more efficient. Furthermore, we show how the moments and covariances of the inter-departure times of the queue can be obtained based on the moments of the waiting time. Numerical experiments are provided to validate our method. [ABSTRACT FROM AUTHOR]

This article deals with the problem of estimating the service time distribution of the M t / G / ∞ queue from observation of the departure epochs. We develop minimax optimal estimators of G and study behavior of the minimax pointwise risk over a suitable family of service time distribution functions. In addition, we address the problem of adaptive estimation and propose a data–driven estimation procedure that adapts to unknown smoothness of the service time distribution function G. Lastly, a numerical study is presented to illustrate practical performance of the developed adaptive procedure. [ABSTRACT FROM AUTHOR]

This paper considers a capacity allocation problem in a two-channel service system. Customers can receive service from either a single-server queueing system, which serves the customers waiting in line one by one, or a clearing service system, which serves a fixed number of customers simultaneously according to its capacity. Customers who join the queueing system should wait till they receive service. In contrast, customers who join the clearing system face the risk of service denial when there are more customers than the clearing system's capacity. The social planner aims to minimize the total expected cost of all customers by determining the capacities and the arrival rates for the two channels. There are two settings: an unobservable setting where only the expected waiting time information is available and an observable setting where real-time information about the exact workload of the queueing system is known. We also consider the same system under the same settings with strategic customers who choose one of the two channels strategically to minimize their costs. The planner still has the same objective but can now decide only on the capacity allocation. Comparing the performance of the resulting systems allows us to understand the value of coordination and information. Extensions of these systems that serve two customer types are also explored. [ABSTRACT FROM AUTHOR]

We develop an efficient reinforcement learning algorithm that learns the optimal admission control policy in a partially observable queueing network. Specifically, only the arrival and departure times from the network are observable, optimality refers to the average holding/rejection cost in infinite horizon, and efficiency is with respect to regret performance. While reinforcement learning in partially-observable Markov Decision Processes (MDP) is prohibitively expensive in general, we show that the regret at time T induced by our algorithm is O ~ T log (1 / ρ) where ρ ∈ (0 , 1) is connected to the mixing time of the underlying MDP. In contrast with existing regret bounds, ours does not depend on the diameter (D) of the underlying MDP, which in most queueing systems is at least exponential in S, i.e., the maximal number of jobs in the network. Instead, the role of the diameter is played by the log (1 / ρ) term, which may depend on S but we find that such dependence is "minimal". In the case of acyclic or hyperstable queueing networks, we prove that log (1 / ρ) = O (S) , which overall provides a regret bound of the order of O ~ TS . In the general case, numerical simulations support the claim that the term log (1 / ρ) remains extremely small compared to the diameter. The novelty of our approach is to leverage Norton's theorem for queueing networks and an efficient reinforcement learning algorithm for MDPs with the structure of birth-and-death processes. [ABSTRACT FROM AUTHOR]

This document is an editorial introduction to the second part of a special issue of the journal Queueing Systems. The special issue focuses on product forms, stochastic matching, and redundancy models in the analysis of queueing systems. The papers in this collection explore various aspects of matching queues and redundancy systems, including fluid approximation, heavy traffic limits, and different scheduling policies. The goal of the special issue is to highlight the connections between these models and their product-form stationary distributions, and to inspire new directions for future research. [Extracted from the article]

The objective of this study is devoted to studying the equilibrium strategy of an unreliable M/M/1+1 retrial queue model, which contains two types of parallel customers and batch service provided by a standby server. Two kinds of customers in this queuing system enter the system in parallel and follow the exponential distribution with different parameters. If the server stays idle, an arrival obtains service immediately; Otherwise, he enters the retrial space and waits for service. When a failure occurs, new customers will not be allowed to enter and the customer who is being served will be excluded from the system. Meanwhile, the system enables the standby server to serve customers in the retrial space with batch service mode. Based on the revenue-cost theory, we propose a benefit function and perform the equilibrium analysis for two types of parallel customers in both fully observable and almost observable cases. In addition, the average social benefits of this particular system are detailed analyzed by using symbolic calculation. Finally, the numerical illustrations that aid in visualizing the changes in customer behavior strategies and social benefits as the different system parameters. [ABSTRACT FROM AUTHOR]

Queueing systems (QS) are essential for modelling and optimising a wide range of real-world scenarios because they effectively control the flow of entities through a system where there are queues. Addressing this, proposed QS comprises with batch arrival, retrial policy, optional re-service, and M-optional vacation alongside breakdown situations. The dynamics of the system are thoroughly investigated by utilizing the supplementary variable technique, which enables greater understanding of the system's behaviour and performance. Furthermore, Adaptive Neuro-Fuzzy Inference System computation is used to properly validate the analytical results, improving the precision and dependability of the model's predictions. Finally, in an effort to minimize operating efficiency, a variety of advanced cost optimization approaches have been employed to determine the system's ideal cost structure. As a result, through this integrated approach, the proposed QS not only furnishes perceptions into operational dynamics but also furnishes practical methods to reduce costs and authenticate analytical conclusions, thereby enhancing queueing theory progressions and realistic applications across diverse fields. [ABSTRACT FROM AUTHOR]

This paper proposes a novel two-phase heterogeneous service retrial queueing system with collisions and delayed vacations. We consider services for two types of customers: ordinary customers (service completed in two phases) and priority customers (service completed in one phase). Collisions only occur in the first phase of service; if the collision does not happen, the server will proceed to the second phase of service. When the server turns to an idle state, it reserves a random period before taking formal vacations. We investigate two models for this system with and without orbit constraints and numerically analyze the sensitivity of the system performance metrics to validate the results and methodology. The expected total cost function is formulated. To prevent falling into the local optimal solution too fast, we utilize the canonical particle swarm optimization algorithm. Furthermore, we evaluate a bi-objective optimization problem to seek the balance of the expected total cost and the quality of service. Then, we apply the multi-objective evolutionary algorithm based on decomposition (MOEA/D) to obtain the Pareto front. Several numerical examples are presented for illustrative purposes. [ABSTRACT FROM AUTHOR]

Dense network deployment is now being evaluated as one of the viable solutions to meet the capacity and connectivity requirements of the fifth-generation (5G) cellular system. The goal of 5G cellular networks is to offer clients with faster download speeds, lower latency, more dependability, broader network capacities, more accessibility, and a seamless client experience. However, one of the many obstacles that will need to be overcome in the 5G era is the issue of energy usage. For energy efficiency in 5G cellular networks, researchers have been studying at the sleeping strategy of base stations. In this regard, this study models a 5G BS as an M [ X ] / G / 1 feedback retrial queue with a sleeping strategy to reduce average power consumption and conserve power in 5G mobile networks. The probability-generating functions and steady-state probabilities for various BS states were computed employing the supplementary variable approach. In addition, an extensive palette of performance metrics have been determined. Then, with the aid of graphs and tables, the resulting metrics are conceptualized and verified. Further, this research is accelerated in order to bring about the best possible (optimal) cost for the system by adopting a range of optimization approaches namely particle swarm optimization, artificial bee colony and genetic algorithm. [ABSTRACT FROM AUTHOR]

This article considers the individual equilibrium behavior and socially optimal strategy in a fluid queue with two types of parallel customers and incomplete fault. Assume that the working state and the incomplete fault state appear alternately in the buffer. Different from the linear revenue and expenditure structure, an exponential utility function can be constructed to obtain the equilibrium balking thresholds in the fully observable case. Besides, the steady-state probability distribution and the corresponding expected social benefit are derived based on the renewal process and the standard theory of linear ordinary differential equations. Furthermore, a reasonable entrance fee strategy is discussed under the condition that the fluid accepts the globally optimal strategies. Finally, the effects of the diverse system parameters on the entrance fee and the expected social benefit are explicitly illustrated by numerical comparisons. [ABSTRACT FROM AUTHOR]

Fractional Brownian motion approximation of queueing networks has been studied extensively. In the existing results related to this topic, the Hurst parameter of multidimensional fractional Brownian motion is only a constant H (0 < H < 1). However, just as pointed out by many scholars and practitioners, various Hurst parameters may be more appropriate. On the other hand, as a multivariate extension of fractional Brownian motion, operator fractional Brownian motion has operator self-similarity, and the dependence structure across the components of it is determined by the Hurst matrix. Moreover, it has also many potential applications in queueing theory. Inspired by these facts, we consider a queueing network with ON/OFF sources, and show that the workload process can be approximated by a reflected operator fractional Brownian motion under a heavy traffic condition. With this fact, it is important to consider stationarity. However, except for some special cases, there is no literature related to this topic. In our work, we construct an explicit stationary process associated with a two-dimensional reflected operator fractional Brownian motion. [ABSTRACT FROM AUTHOR]

Timely status updating in mobile edge computing (MEC) systems has recently gained the utmost interest in internet of things (IoT) networks, where status updates may need higher computations to be interpreted. Moreover, in real-life situations, the status update streams may also be of different priority classes according to their importance and timeliness constraints. The classical disciplines used for priority service differentiation, preemptive and non-preemptive disciplines, pose a dilemma of information freshness dissatisfaction for the whole priority network. This work proposes a hybrid preemptive/non-preemptive discipline under an M/M/1/2 priority queueing model to regulate the priority-based contention of the status update streams in MEC systems. For this hybrid discipline, a probabilistic discretionary rule for preemption is deployed to govern the server and buffer access independently, introducing distinct probability parameters to control the system performance. The stochastic hybrid system approach is utilized to analyze the average age of information (AoI) along with its higher moments for any number of classes. Then, a numerical study on a three-class network is conducted by evaluating the average AoI performance and the corresponding dispersion. The numerical observations underpin the significance of the hybrid-discipline parameters in ensuring the reliability of the whole priority network. Hence, four different approaches are introduced to demonstrate the setting of these parameters. Under these approaches, some outstanding features are manifested: exploiting the buffering resources efficiently, conserving the aggregate sensing power, and optimizing the whole network satisfaction. For this last feature, a near-optimal low-complex heuristic method is proposed. [ABSTRACT FROM AUTHOR]

A new approach is developed for the joint queue-length distribution of the two-level non-pre-emptive M/M/c (i.e. Markovian) priority queue that allows explicit and exact results to be obtained. Marginal distributions are derived for the general multi-level problem. The results are based on a representation of the joint queue-length probability mass function as a single-variable complex contour integral, which reduces to a real integral on a finite interval arising from a cut on the real axis. Both numerical quadrature rules and exact finite sums, involving Legendre polynomials and their generalization, are presented for the joint and marginal distributions. A high level of accuracy is demonstrated across the entire ergodic region. Relationships are established with the waiting-time distributions. Asymptotic behaviour in the large queue-length regime is extracted. [ABSTRACT FROM AUTHOR]

We study in this paper a variation of the acclaimed carrier sense multiple access (CSMA) protocol. A random access algorithm is where back-off rates depend on the state of the network through queue lengths. We provide the first case where full proof of convergence to fluid limits can be obtained. Although this has been often justified informally, this is the first proof dedicated to obtaining this result formally. We then outline the difficulties arising when queues reach zero and solve them in the case of a complete interference graph. The main contribution of this paper is to provide a formal justification of the fluid limits for a complete interference graph in the case where some initial queue lengths are zero. This is done with a coupling argument. This paper also proves convergence to the fluid limits in the general case up to the time a queue reaches 0 on the fluid scale. [ABSTRACT FROM AUTHOR]

We consider the optimal scheduling problem in a multiserver queue with impatient customers belonging to multiple classes. We assume that each customer has a random abandonment time, after which the customer leaves the system if its service has not been completed before that. In addition, we assume that the scheduler is not able to anticipate the expiration of the abandonment times but only knows their distributions and how long each customer has been in the system. Many papers consider this scheduling problem under Poisson arrivals and linear holding costs assuming further that both the service times and the abandonment times have exponential distributions. Even with these additional assumptions, the exact solution is known only in very few special cases. To tackle this tricky problem, we apply the Whittle index approach. Unlike the earlier papers, which were restricted to exponential service times, we allow the service time distributions for which the hazard rate is decreasing. The Whittle index approach is applied to the discrete-time multiserver queueing problem with discounted costs. As our main theoretical result, we prove that the related relaxed optimization problem is indexable and derive the corresponding Whittle index explicitly. Based on this discrete-time result, we develop a reasonable heuristic for the original continuous-time multiserver scheduling problem. The performance of the resulting policy is evaluated in the M/G/M setup by numerical simulations, which demonstrate that it, indeed, gives better performance than the other policies included in the comparison. [ABSTRACT FROM AUTHOR]

This study considers a single server Markovian queueing system with a second optional service and service interruptions. The service is interrupted during the busy period of both the first essential service and the second optional service. The customer whose service is hindered needs to go through the process from the earliest starting point on consummation of interference or continued from where it has halted. In light of the linear reward cost structure, the customer's joining strategies are investigated in two cases; in particular, fully unobservable and almost unobservable, as per the server's state information available to the arriving customers. First, we obtain the stability condition and stationary distribution of the system for those cases. Thereafter, we determine the customer equilibrium strategies, and some numerical illustrations are presented. [ABSTRACT FROM AUTHOR]

In various real-life queueing systems, part of the service can be rendered without involvement or presence of the customers themselves. In those queues, customers whose service order is still in process may leave the service station, go to ‘orbit’ for a random length of time, and then return to find out if their order has been completed. Common examples are car’s annual maintenance works, food ordering, etc. In this paper, a thorough analysis of a single-server ‘orbit while in service’ queueing model with general service time is presented. Assuming an Exponentially distributed orbit time, we derive general formulae for the distributions of (i) a customer’s total residence time in the system; (ii) a customer’s net actual residence time in the system during service (not including orbit time); (iii) the time an orbiting customer is late to return, i.e., remains in orbit after his/her service has been completed; and (iv) the total number of customers in the system. Considering the family of Gamma-distributed service times (spanning the range of distributions between the Exponential and the Deterministic), as well as the Uniform distribution, we further derive explicit formulae for the distributions of the above variables. Under linear cost assumptions, the optimal mean orbit time is numerically calculated for each of the above service-time distributions. Figures depicting the behavior of the measures as functions of the parameters are presented. [ABSTRACT FROM AUTHOR]

This paper focuses on an unreliable M/M/1 retrial queue with delayed repair, in which a novel breakdown mechanism is considered, i.e., a normal breakdown may deteriorate into a major breakdown. Arriving customers are not provided with the system’s information, but must decide whether or not to join it. First, the steady state of the system is analyzed. Then, based on the practical requirements of the cloud computing system, we construct an optimization model to minimize the response time of requesting information with proportions of detection, and repair of the normal and major breakdown as the decision variable. Furthermore, equilibrium joining strategies and the socially optimal pricing strategy are studied from the perspectives of the customer and the social planner, respectively. Finally, numerical examples are used to illustrate the impact of different parameters on strategies. [ABSTRACT FROM AUTHOR]

This paper considers an M/G/1 queue with delayed uninterrupted multiple vacation and N-policy, in which (i) the server remains dormant from vacation to a non-empty system with no more than N customers, and (ii) when the system becomes exhausted, the server waits for a random time instead of immediately going on vacation (applicable to many scenarios and common to human behavior). We first study the transient queue length distribution from an arbitrary initial state and obtain its Laplace transform expressions. Then, the recursive formulas of the stationary queue length distribution are developed. Meanwhile, some crucial performance measures are presented. Finally, the cost optimization problems are discussed with and without the average waiting time constraint. Four numerical examples are illustrated to determine the optimal control policy for minimizing cost under the conditions that the random variables obey different phase-type (PH) distributions. [ABSTRACT FROM AUTHOR]

We investigate the dynamics of customer geometric abandonment within a queueing framework characterized by renewal input batch arrivals and multiple vacations. Customers’ impatience becomes evident when confronted with server vacations, triggering instances of abandonment. This phenomenon reduces the number of customers within the system during abandonment epochs following a geometric distribution. The probability of customers leaving the queue escalates with prolonged waiting times. We derive concise and closed-form expressions for system-length distributions at pre-arrival and arbitrary epochs by harnessing the power of supplementary variable and difference operator methods. Furthermore, we elucidate specific instances of our model, shedding light on its versatility. To substantiate our theoretical framework, we provide a series of illustrative numerical experiments presented through meticulously crafted tables and graphs, thereby showcasing the robustness and applicability of our methodology. [ABSTRACT FROM AUTHOR]

This paper studies customer joining strategies in some single-server Markovian queues with batch limited service rule and multiple vacations. The server begins to take a vacation time as soon as a batch of Φ customers are served continuously. If the server finds that there are fewer than Φ customers present in the system at the completion instant of a vacation time, then he takes another until there are no less than Φ customers waiting after his returning. We consider both the fully observable case and the fully unobservable case, and get customer joining strategy in equilibrium in each case as well as their socially optimal joining strategy in the fully unobservable case. For each case, we find that there may be multiple equilibria but not all of them are stable, and stable equilibria do not always exist. For the fully observable queues, the stable equilibrium thresholds in a vacation period and in a service period are independent of Φ . For the fully unobservable queues, customers' equilibrium behavior is inconsistent with their socially optimal behavior, and there always exists an optimal Φ to maximize social welfare. So the system manager can achieve social optimization by controlling arrivals and the batch size Φ . [ABSTRACT FROM AUTHOR]

The present study considers a single-server Markovian queueing system by observing the number of customer arrivals during the service time of a customer. We estimate the traffic intensity in this queueing system along with the average queue length and the expected number of customers in the system. We propose classical and Bayesian frameworks to estimate the parameters of interest. In the Bayesian setup, three forms of prior distributions for ρ and two loss functions are considered. Furthermore, the predictive distribution of the number of customer arrivals during the service time of a customer and the equal-tailed credible region of ρ are obtained. The aforementioned approaches are illustrated with numerical examples based on simulation studies. [ABSTRACT FROM AUTHOR]

The implication of machine repair problems in the continued functioning of real-time machining models keeps growing with the advent of technology for socioeconomic progression, mobility, security, and safety. The uninterrupted functioning of critical appliances, monitoring controllers, next-generation devices, workstations, and data exchange systems is expected whenever needed prompt. When active units fail, the results may be catastrophic, injury, or loss, leading to critical reliability challenges that must be resolved. This article aims to provide a comprehensive, state-of-the-art study for failure/repair/operation uncertainties and impreciseness in optimistic and pessimistic conditions. We consider the fault-tolerant machining system consisting of two-active units, a single-warm standby unit, and a single-repair facility in a fuzzy environment governing the involved imperfectness, vagueness, uncertainty. Switching the standby unit to the failed active unit is also subject to failure. The notion of imperfect repair makes the proposed model more insightful. A membership grade function of the reliability characteristics: mean time-to-failure and system availability are constructed to study uncertainties in-depth for the fault-tolerant redundant repairable system with switching failure and imperfect repair for well to poor design. The nonlinear parametric program technique converts the studied problem into a set of conventional problems. It is employed to compute the upper and lower bounds of the reliability characteristic based on the γ -cut approach and Zadeh's extension principle for extreme design constrained limits. Extensive numerical simulations are also performed for the different sets of governing parameters ranging from well-conditioned to ill-conditioned. The concluding remarks and future scopes are also included. [ABSTRACT FROM AUTHOR]

In this paper, the rate of convergence to the exponential distribution in the limit theorem for extreme values of the queue length in queuing systems M / G / 1 and G / M / 1 are studied. In proving these results, we first prove one general boundary theorem for extremal values of regenerative processes, which is of independent interest. This topic is closely related to the task of the first passage times to rare sets by the regenerative process. The need for such research arises in many areas: the mathematical theory of reliability, queuing theory, some statistical problems of physics. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Batch service queueing systems are basically classified into two types: a time-based system in which the service facilities depart according to inter-departure times that follows a given distribution, such as the conventional bus system, and a demand-responsive system in which the vehicles start traveling provided that a certain number of customers gather at the waiting space, such as ride-sharing and on-demand bus. Motivated by the recent spreading of demand-responsive transportation, this study examines the M/M X / ∞ queue. In this model, whenever the number of waiting customers reaches a capacity set by a discrete random variable X , customers are served by a group. We formulate the M/M X / ∞ queue as a three-dimensional Markov chain whose dimensions are all unbounded and depict a book-type transition diagram. The joint stationary distribution for the number of busy servers, number of waiting customers, and batch size is derived by applying the method of factorial moment generating function. The central limit theorem is proved for the case that X has finite support under heavy traffic using the exact expressions of the first two moments of the number of busy servers. Moreover, we show that the M/M X / ∞ queue encompasses the time-based infinite server batch service queue (M/M G (x) / ∞ queue), which corresponds to the conventional bus system, under a specific heavy traffic regime. In this model, the transportation facility departs periodically according to a given distribution, G (x) , and collects all the waiting customers for a batch service for an exponentially distributed time corresponding to the traveling time on the road. We show a random variable version of Little's law for the number of waiting customers for the M/M G (x) / ∞ queue. Furthermore, we present a moment approach to obtain the distribution and moments of the number of busy servers in a GI/M/ ∞ queue by utilizing the M/M G (x) / ∞ queue. Finally, we provide some numerical results and discuss their possible applications on transportation systems. [ABSTRACT FROM AUTHOR]

We study service rate control problems for an M/M/1 queue with server breakdowns in which the breakdown rate is assumed to be a function of the service rate. Assuming that the queue has infinite capacity, we first establish the optimality equations for the discounted cost problem and characterize the optimal rate control policies. Then, we characterize the ergodicity of the controlled queue and establish the optimality conditions for the average-cost (ergodic) control problem using the vanishing discounted method. We next study the ergodic control problem when the queue has a finite capacity and establish a verification theorem by directly involving the stationary distribution of the controlled Markov process. For practical applications, we consider the adaptive service rate control problem for the model with finite capacity. Studying this problem is useful because the relationship between the server breakdown rate and the service rate is costly to observe in practice. We propose an adaptive (self-tuning) control algorithm, assuming that the relationship between the server breakdown rate and the service rate is linear with unknown parameters. We prove that the regret vanishes under the algorithm and the proposed policies are asymptotically optimal. In addition, numerical experiments are conducted to validate the algorithm. [ABSTRACT FROM AUTHOR]