Diffusion Approximation for Efficiency-Driven Queues When Customers Are Patient

The analysis of queues with multiple servers is typically challenging when the service time distribution is general. Such analysis usually involves an infinite-dimensional process for tracking service ages or residual service times. In “Diffusion Approximation for Efficiency-Driven Queues When Customers Are Patient,” He demonstrates from a macroscopic perspective that, if customers are relatively patient and the system is overloaded, the dynamics of a many-server queue could be as simple as the dynamics of a single-server queue. In particular, the virtual waiting time process can be captured by a one-dimensional diffusion process, which enables us to obtain simple formulas for performance measures, such as service levels and effective abandonment fractions. To justify this diffusion model, a functional central limit theorem is established for the superposition of stationary renewal processes.

Experimental Study on Stress Evolution and Microseismic Signals under Vibration Conditions of Coal during Excavation and Subsequent Waiting Time

An experiment designed to simulate coal during excavation was conducted. Microseismic signals of coal under vibration conditions during excavation and subsequent waiting time of the coal roadway at different excavation speeds were collected and analyzed. During the excavation and subsequent waiting time, the stress in coal is redistributed, and the concentrated stress is gradually transferred to the deeper section of the coal seam. The Hilbert–Huang transform (HHT) is used to effectively denoise the collected signals. According to the noise-reduced signal, the amplitude and pulse number of the microseismic signals emitted during the excavation process are much larger than those of the waiting time process. During excavation, the energy and event numbers of microseismic signals increase first and then decrease as the excavation speed increases. The faster the excavation speed, the more the energy, and the higher the event numbers of the microseismic signals released during the subsequent waiting time. When the excavation speed is faster, more elastic potential accumulates in the coal seam and the concentration stress is greater. As the concentrated stress moves forward in time without excavation, more coal seams fail, and more microseismic signals are released. The microseismic signal and the stress evolution law can provide a reasonable explanation for the forward movement of the concentrated stress and coal failure during roadway excavation.

Time to wait: a systematic review of strategies that affect out-patient waiting times

Objective Out-patient waiting times pose a significant challenge for public patients in need of specialist evaluation and intervention. The aim of the present study was to identify and categorise effective strategies to reduce waiting times for specialist out-patient services with a focus on the Australian healthcare system. Methods A systematic review of major health databases was conducted using the key terms ‘outpatient*’ AND ‘waiting time’, ‘process*’ AND ‘improvement in outpatient clinics’. Identified articles were assessed for their relevance by sequential review of the title, abstract and full text. References of the selected manuscripts were scanned for additional relevant articles. Selected articles were evaluated for consistent and emerging themes. Results In all, 152 articles were screened, of which 38 were included in the present review. Numerous strategies identified in the articles were consolidated into 26 consistent approaches. Three overarching themes were identified as significantly affecting waiting times: resource realignment, operational efficiency and process improvement. Conclusions Strategies to align resources, increase operational efficiency and improve processes provide a comprehensive approach that may reduce out-patient waiting times. What is known about the topic? Out-patient waiting times are a challenge in most countries that seek to provide universal access to health care for all citizens. Although there has been extensive research in this area, many patients still experience extensive delays accessing specialist care, particularly in the public health sector. The multiple factors that contribute to bottlenecks and inefficiencies in the referral process and affect patient waiting times are often poorly understood. What does this paper add? This paper reviews the published healthcare literature to identify strategies that affect specialist out-patient waiting times for patients. The findings suggest that there are numerous operational strategies that affect waiting times. These strategies may be categorised into three overarching themes (resource alignment, operational efficiencies and out-patient processes) that, when actioned in a coordinated approach, have the potential to significantly reduce out-patient waiting times. What are the implications for practitioners? This paper identifies evidence-based strategies for aligning resources, improving operational efficiency and streamlining processes, which may provide improvements to specialist out-patient waiting times for patients. Addressing the identified organisational, person-related, cultural and attitudinal factors will assist health system managers and health practitioners target the most appropriate improvement activities to reduce waiting times.

A Duality Relation Between the Workload and Attained Waiting Time in FCFS G/G/s Queues

Sengupta (1989) showed that, for the first-come–first-served (FCFS) G/G/1 queue, the workload and attained waiting time of a customer in service have the same stationary distribution. Sakasegawa and Wolff (1990) derived a sample path version of this result, showing that the empirical distribution of the workload values over a busy period of a given sample path is identical to that of the attained waiting time values over the same period. For a given sample path of an FCFS G/G/s queue, we construct a dual sample path of a dual queue which is FCFS G/G/s in reverse time. It is shown that the workload process on the original sample path is identical to the total attained waiting time process on the dual sample path. As an application of this duality relation, we show that, for a time-stationary FCFS M/M/s/k queue, the workload process is equal in distribution to the time-reversed total attained waiting time process.

A Duality Relation Between the Workload and Attained Waiting Time in FCFS G/G/s Queues

Sengupta (1989) showed that, for the first-come–first-served (FCFS) G/G/1 queue, the workload and attained waiting time of a customer in service have the same stationary distribution. Sakasegawa and Wolff (1990) derived a sample path version of this result, showing that the empirical distribution of the workload values over a busy period of a given sample path is identical to that of the attained waiting time values over the same period. For a given sample path of an FCFS G/G/s queue, we construct a dual sample path of a dual queue which is FCFS G/G/s in reverse time. It is shown that the workload process on the original sample path is identical to the total attained waiting time process on the dual sample path. As an application of this duality relation, we show that, for a time-stationary FCFS M/M/s/k queue, the workload process is equal in distribution to the time-reversed total attained waiting time process.

The multiclass GI/PH/N queue in the Halfin-Whitt regime

We consider a multiserver queue in the heavy-traffic regime introduced and studied by Halfin and Whitt who investigated the case of a single customer class with exponentially distributed service times. Our purpose is to extend their analysis to a system with multiple customer classes, priorities, and phase-type service distributions. We prove a weak convergence limit theorem showing that a properly defined and normalized queue length process converges to a particular K-dimensional diffusion process, where K is the number of phases in the service time distribution. We also show that a properly normalized waiting time process converges to a simple functional of the limit diffusion for the queue length.

The multiclass GI/PH/N queue in the Halfin-Whitt regime

We consider a multiserver queue in the heavy-traffic regime introduced and studied by Halfin and Whitt who investigated the case of a single customer class with exponentially distributed service times. Our purpose is to extend their analysis to a system with multiple customer classes, priorities, and phase-type service distributions. We prove a weak convergence limit theorem showing that a properly defined and normalized queue length process converges to a particular K-dimensional diffusion process, where K is the number of phases in the service time distribution. We also show that a properly normalized waiting time process converges to a simple functional of the limit diffusion for the queue length.

Limiting conditional and conditional invariant distributions for the Poisson process with negative drift

In this paper we study the conditional limiting behaviour for the virtual waiting time process for the queue M/D/1. We describe the family of conditional invariant distributions which are continuous and parametrized by the eigenvalues λ ∊ (0, λ c ], as it happens for diffusions. In this case, there is a periodic dependence of the limiting conditional distributions on the initial point and the minimal conditional invariant distribution is a mixture, according to an exponential law, of the limiting conditional distributions.

Limiting conditional and conditional invariant distributions for the Poisson process with negative drift

In this paper we study the conditional limiting behaviour for the virtual waiting time process for the queue M/D/1. We describe the family of conditional invariant distributions which are continuous and parametrized by the eigenvalues λ ∊ (0, λc], as it happens for diffusions. In this case, there is a periodic dependence of the limiting conditional distributions on the initial point and the minimal conditional invariant distribution is a mixture, according to an exponential law, of the limiting conditional distributions.