Time-varying Arrival Rate Research Articles

Service-level computation in time-varying queueing system with priorities: Application to physician staffing in the emergency department

For many service systems, waiting time is a critical performance indicator of customer experience. This study addresses service-level computation techniques in a time-varying mixed-preemptive priorities queueing system. This queueing system has two distinguishing characteristics. First, it has time-varying arrival rates and service capacities. Second, and more importantly, there are multiple classes of customers, and both preemptive and non-preemptive priorities are considered in this system. Combining these two characteristics in a queueing system makes it difficult to compute the service level for each customer class. We derive the exact computation methods for pointwise and piecewise service levels. Furthermore, we propose closed-form expressions to approximate the service levels. Lastly, we show how to apply these service-level computation techniques to a real-life physician staffing problem in the emergency department, in which the service levels of each class of patients are considered as constraints. The computational tests are performed using data generated based on real data from a large hospital. The results show that our staffing solutions are better than the hospital’s current staffing solution, guaranteeing patients’ service levels and reducing physicians’ working times.

Optimal Admission Control for Multiclass Queues with Time-Varying Arrival Rates via State Abstraction

We consider a novel queuing problem where the decision-maker must choose to accept or reject randomly arriving tasks into a no buffer queue which are processed by N identical servers. Each task has a price, which is a positive real number, and a class. Each class of task has a different price distribution, service rate, and arrives according to an inhomogenous Poisson process. The objective is to decide which tasks to accept so that the total price of tasks processed is maximised over a finite horizon. We formulate the problem using a discrete time Markov Decision Process (MDP) with a hybrid state space. We show that the optimal value function has a specific structure, which enables us to solve the hybrid MDP exactly. Moreover, we rigorously prove that as the gap between successive decision epochs grows smaller, the discrete time solution approaches the optimal solution to the original continuous time problem. To improve the scalability of our approach to a greater number of servers and task classes, we present an approximation based on state abstraction. We validate our approach on synthetic data, as well as a real financial fraud data set, which is the motivating application for this work.

Determining the optimal piecewise constant approximation for the nonhomogeneous Poisson process rate of Emergency Department patient arrivals

Modeling the arrival process to an Emergency Department (ED) is the first step of all studies dealing with the patient flow within the ED. Many of them focus on the increasing phenomenon of ED overcrowding, which is afflicting hospitals all over the world. Since Discrete Event Simulation models are often adopted to assess solutions for reducing the impact of this problem, proper nonstationary processes are taken into account to reproduce time–dependent arrivals. Accordingly, an accurate estimation of the unknown arrival rate is required to guarantee the reliability of results. In this work, an integer nonlinear black–box optimization problem is solved to determine the best piecewise constant approximation of the time-varying arrival rate function, by finding the optimal partition of the 24 h into a suitable number of not equally spaced intervals. The black-box constraints of the optimization problem make the feasible solutions satisfy proper statistical hypotheses; these ensure the validity of the nonhomogeneous Poisson assumption about the arrival process, commonly adopted in the literature, and prevent mixing overdispersed data for model estimation. The cost function of the optimization problem includes a fit error term for the solution accuracy and a penalty term to select an adequate degree of regularity of the optimal solution. To show the effectiveness of this methodology, real data from one of the largest Italian hospital EDs are used.

Demand Modeling in the Presence of Unobserved Lost Sales

We present an integrated optimization approach to parameter estimation for discrete choice demand models where data for one or more choice alternatives are censored. We employ a mixed-integer program (MIP) to jointly determine the prediction parameters associated with the customer arrival rate and their substitutive choices. This integrated approach enables us to recover proven, (near-) optimal parameter values with respect to the chosen loss-minimization (LM) objective function, thereby overcoming a limitation of prior multistart heuristic approaches that terminate without providing precise information on the solution quality. The imputations are done endogenously in the MIP by estimating optimal values for the probabilities of the unobserved choices being selected. Under mild assumptions, we prove that the approach is asymptotically consistent. For large LM instances, we derive a nonconvex-contvex alternating heuristic that can be used to obtain quick, near-optimal solutions. Partial information, user acceptance criteria, model selection, and regularization techniques can be incorporated to enhance practical efficacy. We test the LM model on simulated and real data and present results for a variety of demand-prediction scenarios: single-item, multi-item, time-varying arrival rate, large-scale instances, and a dual-layer estimation model extension that learns the unobserved market shares of competitors.This paper was accepted by Yinyu Ye, optimization.

Stabilizing the virtual response time in single-server processor sharing queues with slowly time-varying arrival rates

Motivated by the work of Whitt, who studied stabilization of the mean virtual waiting time (excluding service time) in a $GI_t/GI_t/1/FCFS$ queue, this paper investigates the stabilization of the mean virtual response time in a single-server processor sharing (PS) queueing system with a time-varying arrival rate and a service rate control (a $GI_t/GI_t/1/PS$ queue). We propose and compare a modified square-root (SR) control and a difference-matching (DM) control to stabilize the mean virtual response time of a $GI_t/GI_t/1/PS$ queue. Extensive simulation studies with various settings of arrival processes and service times show that the DM control outperforms the SR control for heavy-traffic conditions, and that the SR control performs better for light-traffic conditions.

Time-Varying Robust Queueing

The natural queueing models for many operations research applications have time-varying arrival rates. In addition, the natural models often are not Markov stochastic processes, so that they are not amenable to exact mathematical analysis. In “Time-Varying Robust Queueing,” Ward Whitt and Wei You propose a time-varying robust queueing algorithm to approximate the time-varying distribution of the workload (virtual waiting time) in a non-Markovian single-server queue with a time-varying arrival-rate function. They apply simulation and asymptotic methods to examine the performance of periodic robust queueing. They show that periodic robust queueing converges to a proper limit in appropriate long-cycle and heavy-traffic regimes and coincides with long-cycle fluid limits and heavy-traffic diffusion limits for long cycles. Simulation examples show that the mean and the full distribution (specified by the quantiles) of the periodic steady-state workload are remarkably well approximated.

Transfer Coordination-Based Train Organization for Small-Size Metro Networks

This paper addresses the train organization problem of a small-size metro network during the peak hour, with a special consideration of transfer coordination. The problem is formulated as a multi-objective programming (MOP) model, where the economic cost, capacity utilization, and transfer coordination are considered together based on the routing selection analysis. Train marshaling number and headway on different routes are key decision variables in the model. The economic cost consists of generalized trip cost, operation cost, and external benefits, and the capacity utilization describes the remaining section capacity expressed by a difference quadratic sum. The proposed model highlights the transfer coordination, which targets at minimizing the number of left-behind passengers on platforms, considering the time-varying arrival rate and the remaining train capacity. Based on the sequencing method, an integration of genetic algorithm and ant colony optimization is devised to solve the MOP model. Finally, a real-world case study of Xi’an metro network has been conducted. Results show that the number of left-behind passengers in the network decreased 55.3$$\%$$, the total remaining capacity decreased 15.3$$\%$$ and the number of trains decreased 8.5$$\%$$, while the total time cost increased 1.6$$\%$$. To further check the passenger distribution density on the platform, a simulation of Beidajie transfer station has been elaborately designed via Viswalk 7.0. Both theoretical results and simulation data have validated the feasibility and reliability of presented method.

On the distributions of infinite server queues with batch arrivals

Queues that feature multiple entities arriving simultaneously are among the oldest models in queueing theory, and are often referred to as “batch” (or, in some cases, “bulk”) arrival queueing systems. In this work, we study the effect of batch arrivals on infinite server queues. We assume that the arrival epochs occur according to a Poisson process, with treatment of both stationary and non-stationary arrival rates. We consider both exponentially and generally distributed service durations, and we analyze both fixed and random arrival batch sizes. In addition to deriving the transient mean, variance, and moment-generating function for time-varying arrival rates, we also find that the steady-state distribution of the queue is equivalent to the sum of scaled Poisson random variables with rates proportional to the order statistics of its service distribution. We do so through viewing the batch arrival system as a collection of correlated sub-queues. Furthermore, we investigate the limiting behavior of the process through a batch scaling of the queue and through fluid and diffusion limits of the arrival rate. In the course of our analysis, we make important connections between our model and the harmonic numbers, generalized Hermite distributions, and truncated polylogarithms.

Delay-Based Service Differentiation with Many Servers and Time-Varying Arrival Rates

We study the problem of staffing (specifying a time-varying number of servers) and scheduling (assigning newly idle servers to a waiting customer from one of [Formula: see text] classes) in the many-server V model with class-dependent time-varying arrival rates. In order to stabilize performance at class-dependent delay targets, we propose the blind (model-free) head-of-line delay-ratio (HLDR) scheduling rule, which extends an earlier dynamic-priority rule that exploits the head-of-line delay information. We study the HLDR rule in the quality-and-efficiency-driven many-server heavy-traffic (MSHT) regime. We staff to the MSHT fluid limit plus a control function in the diffusion scale. We establish a MSHT limit for the Markov model, which has dramatic state-space collapse, showing that the targeted ratios are attained asymptotically. In the MSHT limit, meeting staffing goals reduces to a one-dimensional control problem for the aggregate queue content, which may be approximated by recently developed staffing algorithms for time-varying single-class models. Simulation experiments confirm that the overall procedure can be effective, even for non-Markov models. The online appendix is available at https://doi.org/10.1287/stsy.2018.0015 .

Management of a Shared-Spectrum Network in Wireless Communications

We consider a band of the electromagnetic spectrum with a finite number of identical channels shared by both licensed and unlicensed users. Such a network differs from most many-server, two-class queues in service systems, including call centers, because of the restrictions imposed on the unlicensed users to limit interference to the licensed users. We first approximate the key performance indicators—namely the throughput rate of the system and the delay probability of the licensed users under the asymptotic regime, which requires the analysis of both scaled and unscaled processes simultaneously using the averaging principle. Our analysis reveals a number of distinctive properties of the system. For example, sharing does not affect the level of service provided to the licensed users in an asymptotic sense even when the system is critically loaded. We then study the optimal sharing decisions of the system to maximize the system throughput rate while maintaining the delay probability of the licensed users below a certain level when the system is overloaded. Finally, we extend our study to systems with time-varying arrival rates and propose a diffusion approximation to complement our fluid one. The e-companion is available at https://doi.org/10.1287/opre.2017.1707 .

Time-varying tandem queues with blocking: modeling, analysis, and operational insights via fluid models with reflection

In this paper, we develop time-varying fluid models for tandem networks with blocking. Beyond having their own intrinsic value, these mathematical models are also limits of corresponding many-server stochastic systems. We begin by analyzing a two-station tandem network with a general time-varying arrival rate, a finite waiting room before the first station, and no waiting room between the stations. In this model, customers that are referred from the first station to the second when the latter is saturated (blocked) are forced to wait in the first station while occupying a server there. The finite waiting room before the first station causes customer loss and, therefore, requires reflection analysis. We then specialize our model to a single station (many-server fluid limit of the $$G_t/M/N/(N +H)$$ queue), generalize it to k stations in tandem, and allow finite internal waiting rooms. Our models yield operational insights into network performance, specifically on the effects of line length, bottleneck location, waiting room size, and the interaction among these effects.

An analysis of queues with delayed information and time-varying arrival rates

Understanding how delayed information impacts queueing systems is an important area of research. However, much of the current literature neglects one important feature of many queueing systems, namely non-stationary arrivals. Non-stationary arrivals model the fact that customers tend to access services during certain times of the day and not at a constant rate. In this paper, we analyze two two-dimensional deterministic fluid models that incorporate customer choice behavior based on delayed queue length information with time-varying arrivals. In the first model, customers receive queue length information that is delayed by a constant $$\Delta $$ . In the second model, customers receive information about the queue length through a moving average of the queue length where the moving average window is $$\Delta $$ . We analyze the impact of a time-varying arrival rate and show using asymptotic analysis that the time-varying arrival rate does not impact the critical delay unless the frequency of the time-varying arrival rate is twice that of the critical delay. When the frequency of the arrival rate is twice that of the critical delay, then the stability is enlarged by a wedge that is determined by the model parameters. As a result, this problem allows us to combine the theory of nonlinear dynamics, parametric excitation, delays, and time-varying queues together to provide insight into the impact of information on queueing systems.

On the Distributions of Infinite Server Queues With Batch Arrivals

Queues that feature multiple entities arriving simultaneously are among the oldest models in queueing theory, and are often referred to as (or, in some cases, bulk) arrival queueing systems. In this work we study the affect of batch arrivals on infinite server queues. We assume that the arrival epochs occur according to a Poisson process, with treatment of both stationary and non-stationary arrival rates. We consider both exponentially and generally distributed service durations and we analyze both fixed and random arrival batch sizes. In addition to deriving the transient mean, variance, and moment generating function for time-varying arrival rates, we also find that the steady-state distribution of the queue is equivalent to the sum of scaled Poisson random variables with rates proportional to the service distribution of its order statistics. We do so through viewing the batch arrival system as a collection of correlated sub-queues. Furthermore, we investigate the limiting behavior of the process through a batch scaling of the queue and through fluid and diffusion limits of the arrival rate. In the course of our analysis, we make important connections between our model and the Harmonic numbers, generalized Hermite distributions, and truncated polylogarithms.

Time-dependent analysis for a two-processor heterogeneous system with time-varying arrival and service rates

• We study a two-processor heterogeneous system. • We derived the time dependent solution of proposed system. • We present some numerical examples to illustrate the behavior of system. In this paper, we investigate the time-dependent behavior of a two-processor heterogeneous system where the arrival and service rates are allowed to vary with time. We derive an integral equation where the time-dependent probabilities of the two-processor heterogeneous system are expressed in terms of a Volterra equation of the second kind. The effectiveness of our procedure is illustrated with some numerical examples. Finally, a brief comparison is given to show the efficiency and accuracy of the proposed method.

Queues with Time-Varying Arrivals and Inspections with Applications to Hospital Discharge Policies

In order for a patient to be discharged from a hospital unit, a physician must first perform a physical examination and review the pertinent medical information to determine that the patient is stable enough to be transferred to a lower level of care or be discharged home. Requiring an inspection of a patient’s “readiness for discharge” introduces an interesting dynamic where patients may occupy a bed longer than medically necessary. Motivated by this phenomenon, we introduce a queueing system with time-varying arrival rates in which servers who have completed service cannot be released until an inspection occurs. We examine how such a dynamic impacts common system measures such as stability, expected number of customers in the system, probability of waiting, and expected waiting time. Leveraging insights from an infinite-server model, we’re able to optimize the timing of inspections and find via theoretical and numerical analysis that (1) optimizing a single inspection time could lead to significant improvements in system performance when the amplitude of the arrival rate function is large, (2) multiple inspections should be uniformly distributed throughout the day, and (3) the marginal improvements of adding additional inspection times is decreasing.

Occupancy modelling in shared spaces of buildings: a queueing approach

This paper describes the development of a queueing model for the simulation of occupancy patterns in shared spaces of buildings. Specifically, occupancy is modelled via an infinite-server queue with time-varying arrival and departure rates. In order to better capture the abrupt changes in occupancy, we also present an algorithm that efficiently learns the locally homogeneous intervals from the data and estimates the model parameters separately on each learned interval. Evaluated on the real-world occupancy data, the model has proved its capability of realistically reproducing the variations of occupancy, as well as the key properties, such as peak occupancy time, first arrival and last departure times, and occupied duration. We also compare our model with several occupancy models in the previous work, and show that our model is preferable in terms of simple structure, agile construction, minimal effort of manual calibration and the ability to reflect the occupancy patterns truthfully.

A data-driven model of an emergency department

This paper develops an aggregate stochastic model of an emergency department (ED) based on a careful study of data on individual patient arrival times and length of stay in the ED of the Rambam Hospital in Israel, which was used in a large-scale exploratory data analysis by Armony et al. (2015). This data set is of special interest because it has been made publicly available, so that the experiments are reproducible. Our analysis confirms the previous conclusions about the time-varying arrival rate and its consequences, but we also find that the probability of admission to an internal ward from the ED and the patient length-of-stay distribution should be time varying as well. Our analysis culminates in a new time-varying infinite-server aggregate stochastic model of the ED, where both the length-of-stay distribution and the arrival rate are periodic over a week.

Stabilizing performance in a service system with time-varying arrivals and customer feedback

Analytical offered-load and modified-offered-load (MOL) approximations are developed to determine staffing levels that stabilize performance at designated targets in a non-Markovian many-server queueing model with time-varying arrival rates, customer abandonment from queue and random feedback with additional feedback delay in an infinite-server or finite-server queue. To provide a flexible model that can be readily fit to system data, the model has Bernoulli routing, where the feedback probabilities, service-time, patience-time and feedback-delay distributions all are general and may depend on the visit number. Simulation experiments confirm that the new MOL approximations are effective. A many-server heavy-traffic FWLLN shows that the performance targets are achieved asymptotically as the scale increases.

Train timetabling at rapid rail transit lines: a robust multi-objective stochastic programming approach

In the setting of public transportation system, improving the service quality as well as robustness against uncertainty through minimizing the total waiting times of passengers is a real issue. This study proposed robust multi-objective stochastic programming models for train timetabling problem in urban rail transit lines. The objective is to minimize the expected value of the passenger waiting times, its variance and the penalty cost function including the capacity violation due to overcrowding. In the proposed formulations, the dynamic and uncertain travel demand is represented by the scenario-based time-varying arrival rates and alighting ratio at stops. Two versions of the robust stochastic programming models are developed and a comparative analysis is conducted to testify the tractability of the models. The effectiveness of the proposed stochastic programming model is demonstrated through the application to line 5 of Tehran underground railway. The outcomes validate the benefits of implementing robust timetables for rail industry. The computational experiments shows significant reductions in expected passenger waiting time of 21.27 %, and cost variance drop of 59.98 % for the passengers, through the proposed robust mathematical modeling approach.

STAFFING TO STABILIZE BLOCKING IN LOSS MODELS WITH TIME-VARYING ARRIVAL RATES

The modified-offered-load approximation can be used to choose a staffing function (the time-varying number of servers) to stabilize delay probabilities at target levels in multi-server delay models with time-varying arrival rates, with or without customer abandonment. In contrast, as we confirm with simulations, it is not possible to stabilize blocking probabilities to the same extent in corresponding loss models, without extra waiting space, because these probabilities necessarily change dramatically after each staffing change. Nevertheless, blocking probabilities can be stabilized provided that we either randomize the times of staffing changes or average the blocking probabilities over a suitably small time interval. We develop systematic procedures and study how to choose the averaging parameters.