Single-server Queueing System Research Articles

A study on transient solution of single server queueing system with repair process by using generating functions of the system under the system down

In this paper, we think about on transient arrangement of single server queueing framework with repair process by utilizing producing elements of the framework under the framework down. We acquire the likelihood creating elements of the line estimate dissemination and relentless state arrangement of the model is determined.

Opaque Queues: Service Systems with Rationally Inattentive Customers

Problem description: Classical models of queueing systems with rational and strategic customers assume queues to be either fully visible or invisible while service parameters are known with certainty. In practice, however, people only have “partial information” on the service environment in the sense that they are not able to fully discern prevalent uncertainties. This is because assessing possible delays and rewards is costly as it requires time, attention, and cognitive capacity which are all limited. On the other hand, people are also adaptive and endogenously respond to information frictions. Methodology: We develop an equilibrium model for a single-server queueing system with customers having limited attention. Following the theory of rational inattention, we assume that customers optimize their learning strategies by deciding the type and amount of information to acquire and act accordingly while internalizing the associated costs. Results: We establish the existence and uniqueness of a customer equilibrium and delineate the impact of service characteristics and information costs. We numerically show that when customers allocate their attention to learn uncertain queue length, limited attention of customers improves throughput in a congested system that customers value reasonably highly, while it can be detrimental for less popular services that customers deem rather unrewarding. This is also reflected in social welfare if the firm's profit margin is high enough, although customer welfare always suffers from information costs. Managerial implications: Our results shed light on optimal information provision and physical design strategies of service firms and social planners by identifying service settings where they should be most cautious for customers' limited attention. Academic/practical relevance: We propose a microfounded framework for strategic customer behavior in queues that links beliefs, rewards, and information costs. It offers a holistic perspective on the impact of information prevalence (and information frictions) on operational performance and can be extended to analyze richer customer behavior and complex queue structures, rendering it a valuable tool for service design.

A Single Server Queue with Immediate Feedback, Working Vacation and Server Breakdown

This paper deals with analyze of a single server queueing system with immediate feedbacks and working vacation. Upon arrival if the customer sees the server to be busy then it joins the tail end of queue. Otherwise if server is idle, the customer gets into service. After completion of service, the customer is allowed to make an immediate feedback in finite number. Busy server may fail for a short interval of time. Using supplementary variable technique the steady state results are deduced. Some system performance measures are discussed

Router Buffer Behavior Analysis for Aggregate Traffic Based on Martingale Method

The buffer behavior of routers has an important influence on the quality of service of network services. Exploring theory and methods to analyze the characteristics of the buffer is an interesting job, especially when aggregate traffic is transmitted. This letter investigates the buffer behavior of a single server queuing system with multiple arrival flows, which models the operations of a router. A martingale analytical framework is proposed, which could evaluate buffer performance for aggregate traffic arrivals with different distributions. We use the processes of arrival and service to construct martingales, and analyze the queuing system with heterogeneous arrivals in martingale domain. Applying the stopping time theory of martingale, the transient performance of queue buffer is presented. The probability of buffer overflow and the expectation of buffer stopping time are derived together in this model. The simulations are given to validate the analytical results. This investigation provides a valuable reference for router buffer design.

Stability and busy periods in a multiclass queue with state-dependent arrival rates

We introduce a multiclass single-server queueing system in which the arrival rates depend on the current job in service. The system is characterized by a matrix of arrival rates in lieu of a vector of arrival rates. Our proposed model departs from existing state-dependent queueing models in which the parameters depend primarily on the number of jobs in the system rather than on the job in service. We formulate the queueing model and its corresponding fluid model and proceed to obtain necessary and sufficient conditions for stability via fluid models. Utilizing the natural connection with the multitype Galton–Watson processes, the Laplace–Stieltjes transform of busy periods in the system is given. We conclude with tail asymptotics for the busy period for heavy-tailed service time distributions for the regularly varying case.

Single-server queueing system with external and internal customers

Single-server queueing system with external and internal customers

Equilibrium and optimal balking strategies for low-priority customers in the M/G/1 queue with two classes of customers and preemptive priority

This paper investigates the low-priority customers' strategic behavior in the single-server queueing system with general service time and two customer types. The priority system is preemptive resume, which means that if a high-priority customer enters the system that are serving a low-priority customer, the arriving customer preempts the service facility and the preempted customer returns to the head of the queue for his own class. The customer who is preempted resumes service at the point of interruption upon reentering the system. The low-priority customer's dilemma is whether to join or balk based on a linear reward-cost structure. Two cases are distinguished based on the different levels of information that the low-priority customers acquire before joining the system. The equilibrium threshold strategy in the observable case and the equilibrium balking strategy as well as the socially optimal balking strategy in the unobservable case for the low-priority customers are derived finally.

An M[X]/G(a,b)/1 Queueing System with Breakdown and Repair, Stand-By Server, Multiple Vacation and Control Policy on Request for Re-Service

In this paper, we discuss a non-Markovian batch arrival general bulk service single-server queueing system with server breakdown and repair, a stand-by server, multiple vacation and re-service. The main server’s regular service time, re-service time, vacation time and stand-by server’s service time are followed by general distributions and breakdown and repair times of the main server with exponential distributions. There is a stand-by server which is employed during the period in which the regular server remains under repair. The probability generating function of the queue size at an arbitrary time and some performance measures of the system are derived. Extensive numerical results are also illustrated.

Comparing Short-Memory Charts to Monitor the Traffic Intensity of Single Server Queues

AbstractThis paper describes the application of simple quality control charts to monitor the traffic intensity of single server queues, a still uncommon use of what is arguably the most successful statistical process control tool. These charts play a vital role in the detection of increases in the traffic intensity of single server queueing systems such as the{M/G/1},{GI/M/1}and{GI/G/1}queues. The corresponding control statistics refer solely to a customer-arrival/departure epoch as opposed to several such epochs, thus they are termed short-memory charts. We compare the RL performance of those charts under three out-of-control scenarios referring to increases in the traffic intensity due to: a decrease in the service rate while the arrival rate remains unchanged; an increase in the arrival rate while the service rate is constant; an increase in the arrival rate accompanied by a proportional decrease in the service rate. These comparisons refer to a broad set of interarrival and service time distributions, namely exponential, Erlang, hyper-exponential, and hypo-exponential. Extensive results and striking illustrations are provided to give the quality control practitioner an idea of how these charts perform in practice.

Mixed equilibrium and social joining strategies in Markovian queues with Bernoulli-schedule-controlled vacation and vacation interruption

This paper presents equilibrium and social customer behavior in a single-server queueing system with Bernoulli-schedule-controlled vacation and vacation interruption. An arriving customer takes the decision whether to join the system or balk according to an observation of the system length and the status of the server, which is more reasonable than the classical point of view that the decisions are induced by the servers and the customers are compelled to follow them. Four cases with respect to several levels of information and the corresponding Nash equilibria are discussed. The equilibrium strategies and stationary system behavior are studied for both observable and unobservable cases using the reward-cost structure of the system. We show the impact of the information level as well as several parameters on the equilibrium thresholds and social benefits through numerical results. The research outcomes might provide the managers with reference information on the pricing problem in the queueing system and will help the customers to take optimal threshold strategies.

A scalable non-myopic dynamic dial-a-ride and pricing problem for competitive on-demand mobility systems

We propose a competitive on-demand mobility model using a multi-server queue system under infinite-horizon look-ahead. The proposed approach includes a novel dynamic optimization algorithm which employs a Markov decision process (MDP) and provides opportunities to revolutionize conventional transit services that are plagued by high cost, low ridership, and general inefficiency, particularly in disadvantaged communities and low-income areas. We use this model to study the implications it has for such services and investigate whether it has a distinct cost advantage and operational improvement. We develop a dynamic pricing scheme that utilizes a balking rule that incorporates socially efficient level and the revenue-maximizing price, and an equilibrium-joining threshold obtained by imposing a toll on the customers who join the system. Results of numerical simulations based on actual New York City taxicab data indicate that a competitive on-demand mobility system supported by the proposed model increases the social welfare by up to 37% on average compared to the single-server queuing system. The study offers a novel design scheme and supporting tools for more effective budget/resource allocation, planning, and operation management of flexible transit systems.

Analysis of a semi-open queueing network with Markovian arrival process

A semi-open queueing network having a finite number of nodes is considered. The nodes are modeled by single-server queueing systems with a finite buffer and an exponential service time distribution. Customers arrive to the network according to a Markovian arrival process. The number of customers, which can be processed in the network simultaneously, is restricted by a threshold. If the number of customers in the network is less than this threshold, when a new customer arrives, the customer is processed in the network. Choice of the first and the subsequent nodes for service is performed randomly according to a fixed stochastic vector and a transition probability matrix. If the number of customers in the network at the customer arrival epoch is equal to the threshold, the customer is queued into an input buffer with an infinite capacity. Customers in the input buffer are impatient. The stationary behavior of network states is analyzed. The Laplace–Stieltjes transform of the distribution of the customer’s waiting time in the input buffer is obtained. Expressions for computing performance measures of the network are derived. Numerical results are presented. The model is suitable, e.g., for analysis and optimization of wireless telecommunication networks and manufacturing systems with a finite number of machines and workers.

Computational analysis of the queue with working breakdowns and delaying repair under a Bernoulli-schedule-controlled policy

ABSTRACTThis paper considers a single server queueing system with working breakdowns and delaying repair under a Bernoulli-schedule-controlled policy. At a breakdown instant, the system either goes to repair period immediately with probability p, or continues to provide auxiliary service for the current customers with probability q = 1 − p. While the system resides in the auxiliary service period, it may go to repair period if there is no customer at the epoch of service completion or the occurrence of breakdown. By using the matrix analytic method and the spectral expansion method, we respectively obtain the steady state distribution to make the straightforward computation of performance measures and the Laplace-Stieltjes transform of the stationary sojourn time of an arbitrary customer. In addition, some numerical examples are presented to show the impact of parameters on the performance measures.

Priority Service Pricing with Heterogeneous Customers

This paper studies the priority pricing problem for a single-server queueing system with two priority classes in which customers have different sensitivities to delay. The system makes a fixed-delay announcement to inform arriving customers of the expected delay for each class, whereupon each customer must decide which class to join. Any customer who joins the priority class is charged a fixed priority price. Our examination of customers' joining behavior under any given priority prices reveals that there can be multiple equilibrium delays and that the number of those delays depends on the structure of customers' delay cost distribution. We characterize the stability of these equilibria and show that the system can reach the largest or smallest equilibrium by making a proper initial delay announcement. In addition, we consider two pricing problems to maximize the system's long-run average revenue and social welfare respectively. The results derived here establish that both the revenue-maximizing price and the social welfare--maximizing price are quite sensitive to the delay cost distribution. Finally, we investigate the influence of the number of priority classes by extending the two--priority-class model to a multiple--priority-class model.

Price and capacity decisions of service systems with boundedly rational customers

AbstractWe consider price and capacity decisions for a profit‐maximizing service provider in a single server queueing system, in which customers are boundedly rational and decide whether to join the service according to a multinomial logit model. We find two potential price‐capacity pair solutions for the first‐order condition of the profit‐maximizing problem. Profit is maximized at the solution with a larger capacity, but minimized at the smaller one. We then consider a dynamically adjusting capacity system to mimic a real‐life situation and find that the maximum can be reached only when the initial service rate is larger than a certain threshold; otherwise, the system capacity and demand shrink to zero. We also find that a higher level of customers’ bounded rationality does not necessarily benefit a firm, nor does it necessarily allow service to be sustained. We extend our analysis to a setting in which customers’ bounded rationality level is related to historical demand and find that such a setting makes service easier to sustain. Finally we find that bounded rationality always harms social welfare.

A Queueing System with Heterogeneous Impatient Customers and Consumable Additional Items

AbstractA single-server queueing system with a marked Markovian arrival process of heterogeneous customers is considered. Type-1 customers have limited preemptive priority over type-2 customers. There is an infinite buffer for type-2 customers and no buffer for type-1 customers. There is also a finite buffer (stock) for consumable additional items (semi-products, half-stocks, etc.) which arrive according to the Markovian arrival process. Service of a customer requires a fixed number of consumable additional items depending on the type of the customer. The service time has a phase-type distribution depending on the type of the customer. Customers in the buffer are impatient and may leave the system without service after an exponentially distributed amount of waiting time. Aiming to minimize the loss probability of type-1 customers and maximize throughput of the system, a threshold strategy of admission to service of type-2 customers is offered. Service of type-2 customer can start only if the server is idle and the number of consumable additional items in the stock exceeds the fixed threshold. Stationary distributions of the system states and the waiting time are computed. In the numerical example, we show some interesting effects and illustrate a possibility of application of the presented results for solution of optimization problems.

Efficient Ignorance: Information Heterogeneity in a Queue

How would the growing prevalence of real-time delay information affect a service system? We consider a single-server queueing system where customers arrive according to a Poisson process and the service time follows an exponential distribution. There are two streams of customers, one informed about real-time delay and the other uninformed. The customers’ uninformed behavior may be due to information ignorance or rational behavior in the presence of an information fee. We characterize the equilibrium behavior of customers with information heterogeneity and investigate how the presence of a larger fraction of informed customers affects the system performance measures, i.e., throughput and social welfare. We show that the effects of growing information prevalence on system performance measures are determined by the equilibrium joining behavior of uninformed customers. Perhaps surprisingly, we find that throughput and social welfare can be unimodal in the fraction of informed customers. In other words, some amount of information heterogeneity in the population can lead to more efficient outcomes, in terms of the system throughput or social welfare, than information homogeneity. For example, under a very mild condition, throughput in a system with an offered load of 1 will always suffer if there are more than 58% of informed customers in the population. Moreover, it is shown that for an overloaded system with offered load sufficiently higher than 1, social welfare always reaches its maximum when some fraction of customers is uninformed of the congestion level in real time. The online appendix is available at https://doi.org/10.1287/mnsc.2017.2747 . This paper was accepted by Gad Allon, operations management.

Simulation of Single-Server Queuing System with Reverse Balking

Jain et. al. (2014) proposed a single-server finite capacity queuing system with reverse balking. The models finds its application in all fields of business, civil, and defence. They perform theoretical analysis of the model. In this paper, we simulate this model. An algorithm is written in ‘C’ language for the same. Theoretical results of model are compared with simulated results numerically. Results are of immense use for understanding gap between theoretical and practical results of the model.

The value of service rate flexibility in an M/M/1 queue with admission control

ABSTRACTWe consider a single-server queueing system with admission control and the possibility to switch dynamically between k increasing service rate values with the service cost rate being convex in the service rate. We explore the benefit due to service rate flexibility on the optimal profit and the admission thresholds, when service payment is made upon customer's admission. We formulate a Markov Decision Process model for the problem of joint admission and service control considering both discounted and average expected profit maximization and show that the optimal policy has a threshold structure for both controls. Regarding the benefit due to flexibility, we show that it is increasing in system length and that its effect on the admission policy is to increase the admission threshold. We also derive a simple approximate condition between the admission reward and the relative cost of service rate increase, so that the service rate flexibility is beneficial. We finally show that the results extend to the corresponding model where service payment is made at the end of each service completion and differences regarding the benefit due to service flexibility with respect to the original model are pursued numerically.

A control-chart-based queueing approach for service facility maintenance with energy-delay tradeoff

Maintenance planning and energy consumption control are critical issues in facility operations management. In practice, the energy consumption of a facility, which will be affected by the operation condition, is closely connected with the associated maintenance policy. Specifically, for an energy-consuming service system, though a frequent maintenance activity can keep the facility in a good condition with low energy consumption, it makes the delay time longer and leads to a poor customer experience. In this paper, we study a single-server queueing system with different energy consumption levels in the associated running states to address the conflict between energy consumption and customer delay. Two types of maintenance activities are implemented for the server, i.e., the planned maintenance and the reactive maintenance. The planned maintenance is adopted based on a frequency parameter at the beginning of an idle period, and the reactive maintenance is initialized by the Shewhart’s individual control chart (condition-based maintenance). To capture the energy-delay tradeoff, our objective is to develop an optimal maintenance policy that minimizes the long-run expected total cost of the system under a customer waiting time constraint. Numerical experiments are conducted to analyze the problem, in which useful managerial insights are obtained for the optimal maintenance policy. The results demonstrate the robustness of the proposed maintenance model, its advantage over the model without control chart, and its applicability in general situations.