Stochastic Service System Research Articles

Equilibrium and social optimality in queues with service rate and customers’ joining decisions

ABSTRACT This paper treats a single server Markovian queueing system with changeable service rates and customer choice. When a customer arrives at an empty system, he is served by the server at a low service rate. This low service rate continues until the queue length reaches a threshold. Then the server switches to a high service rate to serve customers who join the system. The high service rate continues until the system becomes empty and a low service rate starts again. Under a service reward and waiting cost structure, customers decide to join or to balk based on their net utility of joining the system. The customers’ equilibrium and socially optimal joining strategies are studied under two information scenarios which are observable and unobservable queues, respectively. Such a model fits a stochastic service system where the server can make his service rate decision and customers can make their joining or baking decisions and thus has wide applications in service industries. Our focus is on examining the equilibrium performance of the queue under the interaction between the server’s service rate and customers’ joining decisions. We develop the steady-state distribution and performance measures of the system by formulating two-dimensional continuous-time Markov chains.

The accumulation cost of relaxed fixed time accumulation mode

Studying the wagon accumulation process and the laws of accumulation cost is of great significance for determining the suitable conditions of wagon accumulation and shortening the accumulation time. Here, the process of relaxed fixed time accumulation is first taken as a stochastic service system, and derives the theoretical formula for the accumulation cost. Then based on actual data of wagon flows, a simulation model is built to analyse the influence of parameters in the theoretical formula such as the coordination of the traffic diagram and the accumulation process, the sizes and intervals of the arriving wagon groups and the minimum number of wagons. Finally, through comparing with the accumulation cost of fixed train length accumulation mode and considering the benefit of changing the minimum number of wagons in train sets, the optimal minimum number of wagons in the relaxed fixed time accumulation mode under different wagon flow intensities is determined.

UMVUEs and Bayes estimators for various performance measures on a Poisson queue with discouraged arrivals

To prevent overcrowding during the COVID-19 pandemic, numerous businesses, and public service systems have to limit the number of people entering the premises to reduce infection risks. Such a realistic situation prompts us to revisit an adaptive queueing model from a statistical perspective, which allows us to extensively analyze and explore the stochastic service system arising in the pandemic period. In order to avoid long waiting lines, we assume that the arrival rate of customers into the system depends on the system size instead of a constant rate. This article attempts to study the uniformly minimum variance unbiased estimators and closed-form Bayes estimators of various queueing characteristics, such as the probability that the server is busy, the proportion of lost customers, mean system length, and average queue length. The estimates and their behaviors are compared by Monte-Carlo simulation with different sample sizes. The simulation results show that we may choose different estimation techniques for different performance indicators to obtain a more precise estimate.

Performance Approximation for Time-Dependent Queues with Generally Distributed Abandonments

Many stochastic systems face a time-dependent demand. Especially in stochastic service systems, for example, in call centers, customers may leave the queue if their waiting time exceeds their personal patience. As discussed in the extant literature, it can be useful to use general distributions to model such customer patience. This paper analyzes the time-dependent performance of a multiserver queue with a nonhomogeneous Poisson arrival process with a time-dependent arrival rate, exponentially distributed processing times, and generally distributed time to abandon. Fast and accurate performance approximations are essential for decision support in such queueing systems, but the extant literature lacks appropriate methods for the setting we consider. To approximate time-dependent performance measures for small- and medium-sized systems, we develop a new stationary backlog-carryover (SBC) approach that allows for the analysis of underloaded and overloaded systems. Abandonments are considered in two steps of the algorithm: (i) in the approximation of the utilization as a reduced arrival stream and (ii) in the approximation of waiting-based performance measures with a stationary model for general abandonments. To improve the approximation quality, we discuss an adjustment to the interval lengths. We present a limit result that indicates convergence of the method for stationary parameters. The numerical study compares the approximation quality of different adjustments to the interval length. The new SBC approach is effective for instances with small numbers of time-dependent servers and gamma-distributed abandonment times with different coefficients of variation and for an empirical distribution of the abandonment times from real-world data obtained from a call center. A discrete-event simulation benchmark confirms that the SBC algorithm approximates the performance of the queueing system with abandonments very well for different parameter configurations. Summary of Contribution: The paper presents a fast and accurate numerical method to approximate the performance measures of a time‐dependent queueing system with generally distributed abandonments. The presented stationary backlog carryover approach with abandonment combines algorithmic ideas with stationary queueing models for generally distributed abandonment times. The reliability of the method is analyzed for transient systems and numerically studied with real‐world data.

On retrial queue with customer balking and feedback subject to server breakdowns

Motivated by some real stochastic service systems, we study a feedback retrial queue with customer balking and unreliable servers (servers subject to breakdowns). A customer will either join a retrial orbit or balk if all servers are busy upon his arrival. After a service completion, the customer either leave the system or enter the retrial group again for additional service. The customer who is interrupted by the server failure will be served by another server if available; otherwise, he will be contacted later and treated as leaving the system. We use a quasi-birth-and-death process to analyze this system with almost all possible uncertain factors and derive major performance measures of the system. We also develop a cost function determining the optimal parameter settings of the system under the stability condition. Probabilistic Global Search Lausanne approach is employed in solving the optimization issue. The effects of parameters on performance measures of the system are examined numerically. Finally, we discuss the application of the model in the telephone medical consulting service system.

Decision making under uncertain and dependent system rates in service systems

In this paper, we propose decision analysis methods for determining the optimal number of agents of a service system where the system rates (arrival, service, and abandonment) are modeled as dependent random variables. In doing so, we take the Bayesian point of view of inference and obtain joint posterior distributions of the system rates. We solve the proposed stochastic staffing decision problem with augmented probability simulation based optimization methods. The novelty of our approach stems from the use of dependent system rates to determine optimal staffing in a constrained optimization setting for stochastic service systems. We demonstrate the implications of ignoring dependence and uncertainty in system rates on simulated data for general service systems, and illustrate the application of the proposed methodology on call center operations.

Risk Assessment of a Stochastic Service System

A stochastic service system of finite size M is comprised of identical service facilities, including or not a waiting queue, which simultaneously treats N customers, N ∈ {0, 1, …, M}. Depending on the concepts of system information i and system entropy S = E(i), we promote a risk assessment procedure. By definition, the system entropy is the uncertainty associated with the system, and the system expected loss is the risk associated with the system. Thus, accepting the system information as loss function, we can identify risk and uncertainty, associated with the system, using the entropy as risk function. Further, we differ risk of the system (i.e., risk observed by an outside observer), risk observed by an arriving customer, and risk observed by a departing customer, giving a separate expression for each one. Then, these risks are compared with each other, when the system has the same average number E(N) of customers seen by any viewpoint. The three risk types (together with the three customer means) allow us to distinguish two systems obeying the same probability distribution. This approach enables system operators to choose suitable values for system utilization and size, in view of the three risks ratio. The developed procedure is applied to the information linear system, Erlang loss system, single-server queueing system with discouraged arrivals, Binomial system and Engset loss system.

Determination Optimal Indicators of Expressway Toll Plaza with M/G/1 Queue Model

Long lines and heavy congestion is always the bottleneck problem of highway toll plaza, the operation efficiency of tollbooths is crucial to the traffic throughout of expressway toll plaza. In this report, we explore M/G/1 model in stochastic service system theory for obtaining the important indicators (eg: delay time, traffic throughput, construction cost and accident prevention) for improving the operation efficiency of tollbooths. Combining with fluid mechanics theory, we get the relationship between some important indicators such as queue length, delay time (including service time and overall waiting time) of the tollbooths system and the single channel capacity, and also the proportion of the number of various windows type. At the same time, we give out the reasonable accident coefficient and safety distance between vehicles. We find that the security distance between vehicles will directly affect the queue length, and thus influence the traffic capacity of toll booths.

Policies for physician allocation to triage and treatment in emergency departments

In the emergency department (ED), low-acuity patients divert resources from more critical patients. To facilitate flow, EDs are experimenting with new care models, such as the Triage-Treat-and-Release program at the Lutheran Medical Center (LMC) ED in New York, where physicians handle both phases of service for low-acuity patients. Our goal is to determine how physicians should prioritize triage versus treatment to balance initial delays with timely discharges. Triage and treatment are modeled as a two-phase stochastic service system, where patients may leave without receiving treatment. Patients leaving without receiving treatment increases the importance of the second phase. We introduce K-level threshold policies which prioritize treatment unless there are K or more patients in triage. The effect is a class of policies that capture a decision-maker’s valuation of the importance of each activity; lower K values signifies triage priority. Sufficient conditions are provided to ensure these policies yield a stable system. A heuristic is presented for choosing K. Using LMC data, K-level threshold policies, compared to other practical policies, perform well with respect to average rewards and waiting times over a range of parameters. These policies promise physicians an effective and simple way to allocate their time between triage and treatment.

Strategic joining in an M/M/1 queue with risk-sensitive customers

To analyze a stochastic service system with customers choosing to join or balk upon arrival, we model the system as a single server Markovian queue with a quadratic utility function for customers. In contrast to classical models with risk-neutral customers, we focus on the queueing model with risk-sensitive ones and study customer strategies under individual interest equilibrium, server’s profit optimization, and social welfare optimization. The quadratic utility function allows us to take the risk and return tradeoff into account in analyzing customer joining strategies. We show that while some of the well-known results for the risk-neutral customer situation apply, others may fail to hold in some realistic risk-sensitive customer situations. Furthermore, we examine the queue length information effect on different performance measures from server’s profit and social welfare perspectives. A practical implication of this study is that managers of service systems should be very cautious about relying on classical stationary queueing analysis when customers are risk-sensitive.

Bus Scheduling Optimization Based on Queuing Theory

AbstractBased on the summarization, and analysis of existing bus scheduling models and according to Chinese conditions, the bus route scheduling problem is studied using the theory of stochastic service system (queuing theory). A mathematical model is established to achieve the bus route scheduling with travel demand as the fundamental data and with bus company interests and passenger satisfaction as the objective functions. The process of objective function construction is described, the passenger satisfaction function is introduced, and the algorithm of each component of constraints is designed. Finally, a case study is conducted on the basis of data from the national mathematical modeling competition. The model is solved using Matlab software. Results show that the model has advantages in solving and applicability and is thus practical for use in bus scheduling.

Modeling the sojourn time of items for in-network cache based on LRU policy

To reduce network redundancy, in-network caching is considered in many future Internet architectures, such as Information Centric Networking. In in-network caching system, the item sojourn time of LRU (Least Recently Used) replacement policy is an important issue for two reasons: firstly, LRU is one of the most common used cache policy; secondly, item sojourn time is positively correlated to the hit probability, so this metric parameter could be useful to design the caching system. However, to the best of our knowledge, the sojourn time hasn't been studied theoretically so far. In this paper, we first model the LRU cache policy by Markov chain. Then an approximate closed-form expression of the item expectation sojourn time is provided through the theory of stochastic service system, which is a function of the item request rates and cache size. Finally, extensive simulation results are illustrated to show that the expression is a good approximation of the item sojourn time.

Model Passengers’ Travel Time for Conventional Bus Stop

Limited number of berths can result in a subsequent bus stopping at the upstream of a bus stop when all berths are occupied. When this traffic phenomenon occurs, passengers waiting on the platform usually prefer walking to the stopped bus, which leads to additional walking time before boarding the bus. Therefore, passengers’ travel time consumed at a bus stop is divided into waiting time, additional walking time, and boarding time. This paper proposed a mathematical model for analyzing passengers’ travel time at conventional bus stop based on theory of stochastic service system. Field-measured and simulated data were designated to demonstrate the effectiveness of the proposed model. By analyzing the results, conclusion was conducted that short headway can reduce passengers’ waiting time at bus stop. Meanwhile, the theoretical analysis explained the inefficiency of bus stops with more than three berths from the perspective of passengers’ additional walking time. Additional walking time will increase in a large scale when the number of berths at a bus stop exceedsthe threshold of three.

PH fitting of the arrival interval distribution of the passenger flow on urban rail transit stations

The urban railway transit station is a type of stochastic service system. The accurate description for the arrival interval distribution of the passenger flow with strong randomness is the basis of the planning, design and operation of the urban rail transit stations; however, the existing fitting distributions are not accurate enough to satisfy the demand in practice. Although the famous phase-type distribution (PH) has been proven to have the merits of high accuracy, universality and analyticity, it has too many parameters needed to be calibrated. In this paper, seven common distributions and hyper-Erlang distribution (HErD) which is a dense subset of PH were applied to fit arrival interval distribution of the passenger flow of entrances and platforms on subway stations. The comparative results indicated that HErD achieves the highest accuracy. On this basis, the fitting parameters of HErD were analysed, finding that this distribution is required to determine two parameters, and only related with variation coefficient, possessing good practicability. Finally, this paper puts forward a practical method to determine the two parameters of HErD through the easy-captured peak passenger flow volume and super-peak coefficient during station planning and design stage. This research provides a new alternative PH distribution for fitting passenger flow arrival interval distribution on urban rail transit stations.

Delivery Commitments in Stochastic Service Networks

Service firms have become highly competitive in terms of providing the delivery. The delivery quality in terms of delivery commitments. Delivery commitments impact the customer in deciding for the service. Computing the delivery commitments in stochastic service systems is a real challenge. Delivery commitment forms a key parameter in formulating the service level agreements in B2B markets. In our current work we propose a queuing theoretic approach for computing the delivery commitments. The authors employ basic Probability theory to propose two bounds on delivery commitment time. Further, we investigate the effect of learning in service networks. They believe that their work can provide a simple and easy framework for quality analysis in stochastic service networks.

Stochastic service systems, random interval graphs and search algorithms

ABSTRACTWe consider several stochastic service systems, and study the asymptotic behavior of the moments of various quantities that have application to models for random interval graphs and algorithms for searching for an idle server or for an vacant or occupied waiting station. In some cases the moments turn out to involve Lambert series for the generating functions for the sums of powers of divisors of positive integers. For these cases we are able to obtain complete asymptotic expansions for the moments of the quantities in question. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 421–442, 2014

Equipment Optimization Method of Electric Vehicle Fast Charging Station Based on Queuing Theory

As an important infrastructure of electric vehicles (EVs), EV fast charging station is of great significance in the popularization and development of EVs. Through the analysis of the characteristics of EV’s arriving time and charging duration in fast charging station, the stochastic service system was introduced and the queuing system model based on queuing theory was established. By calculating the indexes of the queuing system model, the desire model was used to optimize the number of EV chargers, which could save customers’ waiting time and reduce the investment of charging station. Finally, an example was simulated and calculated with MATLAB used as a simulation tool to verify the effectiveness of this approach.

Strategic Queueing Behavior and Its Impact on System Performance in Service Systems with the Congestion-Based Staffing Policy

We study strategic customer behavior in a multiserver stochastic service system with a congestion-based staffing (CBS) policy. With the CBS policy, the number of working servers is dynamically adjusted according to the queue length. Besides lining up for free service, customers have the option of paying a fee and getting faster service. Customers' equilibrium behavior is studied under two information scenarios: In the no information scenario, customers only know the long-term statistics, such as the expected waiting time; in the partial information scenario, customers observe the number of working servers and understand the staffing policy upon their arrival. Unlike a queueing system with a constant staffing level, a positive externality is associated with customers' joining the CBS system. Both avoid-the-crowd and follow-the-crowd customer behaviors are possible, and multiple equilibria could exist. We develop the stationary performance measures of the system by considering the customers' strategic behavior. Numerical analysis shows that information can either hurt or improve the performance of the system, depending on the staffing and pricing policy. Another important conclusion is that the system performance is more robust to setting a relatively high than a relatively low price.

An algorithmic analysis of multi-server vacation model with service interruptions

Motivated by the trade-off between reliability and utilization level of a stochastic service system, we considers a Markovian multi-server vacation queueing system with c unreliable servers. In such a system, some servers may not be available due to either planned stoppage (vacations) or unplanned service interruptions (server failures). The vacations are controlled by a threshold policy. With this policy, at a service completion instant, if d (⩽ c) servers become idle, they take a vacation together and will keep taking vacations until they find at least c − d + 1 customers are in the system at a vacation completion instant, and then they return to serve the queue. In addition, all on-duty servers are subject to failures and can be repaired within a random period of time. We formulate a quasi-birth–death (QBD) process, establish the stability condition, and develop a computational algorithm to obtain the stationary performance measures of the system. Numerical examples are presented to show the performance evaluation and optimization of such a system. The insights gained from this model help practitioners make capacity and operating decisions for this type of waiting line systems.

Research on Stochastic Model and Simulation for Sorted-Storage S-Type Manual Order Picking

The key optimal goal of manual order picking system is shortest picker route, thereby best times. Picker route stochastic model under sorted storage S-type picking for logistics distribution center manual order picking is built. It makes model approximation calculation and simulation result under three order arrival frequencies and space ratio. It is stated that well approximation between model and simulation results under the condition of permission difference are obtained. The model lays the foundation for building stochastic service system model of distribution center manual order picking.