Using a queueing model to help plan bed allocation in a department of geriatric medicine

Queueing Problems in Emergency Departments: A Review of Practical Approaches and Research Methodologies

Problems related to patient scheduling and queueing in emergency departments are gaining increasing attention in theory, in the fields of operations research and emergency and healthcare services, and in practice. This paper aims to provide an extensive review of studies addressing queueing-related problems explicitly related to emergency departments. We have reviewed 229 articles and books spanning seven decades and have sought to organize the information they contain in a manner that is accessible and useful to researchers seeking to gain knowledge on specific aspects of such problems. We begin by presenting a historical overview of applications of queueing theory to healthcare-related problems. We subsequently elaborate on managerial approaches used to enhance efficiency in emergency departments. These approaches include bed management, fast-track, dynamic resource allocation, grouping/prioritization of patients, and triage approaches. Finally, we discuss scientific methodologies used to analyze and optimize these approaches: algorithms, priority models, queueing models, simulation, and statistical approaches.

A Simulation Optimisation Approach for Managing Bed Capacity in an Intensive Care Unit

Determining the optimal number of beds in a hospital unit is often a very critical task. Patients’ number of arrivals and length of stay are random variables which necessitates the treatment with the number of patients at hospital as a stochastic process, and thus adding to the complexity of the bed sizing problem. The optimal number of beds is affected by some critical parameters such as target utilisation level, admission rate and target service level. This study applies a discrete event simulation model to approximate the true relationships between different control parameters and optimal number of beds. A goal programming model is then solved to find the optimal number of beds that maintains minimal deviation from target admission and utilisation levels. Data have been collected for the calendar year 2019 and then analysed and used in the simulation model. Mathematical relationships are then embedded in a multiobjective optimisation model that finds the optimal number of beds in an Intensive Care Unit that minimises the deviations from a pre-specified service and beds utilisation levels.

Markov chain analysis for the neonatal inpatient flow in a hospital

Discrete-time Markov chain and queueing-theoretic models are used to quantitatively formulate the flow of neonatal inpatients over several wards in a hospital. Parameters of the models are determined from the operational analysis of the record of the numbers of admission/departure for each ward every day and the order log of patient movement from ward to ward for two years provided by the Medical Information Department of the University of Tsukuba Hospital in Japan. Our formulation is based on the analysis of the precise routes (the route of an inpatient is defined as a sequence of the wards in which he/she stays from admission to discharge) and their length-of-stay (LoS) in days in each ward on their routes for all neonatal inpatients. Our theoretical model calculates the probability distribution for the number of patients staying in each ward per day which agrees well with the corresponding histogram observed for each ward as well as for the whole hospital. The proposed method can be used for the long-term capacity planning of hospital wards with respect to the probabilistic bed utilization.

A model of supply-chain decisions for resource sharing with an application to ventilator allocation to combat COVID-19

We present a stochastic optimization model for allocating and sharing a critical resource in the case of a pandemic. The demand for different entities peaks at different times, and an initial inventory for a central agency are to be allocated. The entities (states) may share the critical resource with a different state under a risk-averse condition. The model is applied to study the allocation of ventilator inventory in the COVID-19 pandemic by FEMA to different U.S. states. Findings suggest that if less than 60% of the ventilator inventory is available for non-COVID-19 patients, FEMA's stockpile of 20 000 ventilators (as of March 23, 2020) would be nearly adequate to meet the projected needs in slightly above average demand scenarios. However, when more than 75% of the available ventilator inventory must be reserved for non-COVID-19 patients, various degrees of shortfall are expected. In a severe case, where the demand is concentrated in the top-most quartile of the forecast confidence interval and states are not willing to share their stockpile of ventilators, the total shortfall over the planning horizon (until May 31, 2020) is about 232 000 ventilator days, with a peak shortfall of 17 200 ventilators on April 19, 2020. Results are also reported for a worst-case where the demand is at the upper limit of the 95% confidence interval. An important finding of this study is that a central agency (FEMA) can act as a coordinator for sharing critical resources that are in short supply over time to add efficiency in the system. Moreover, through properly managing risk-aversion of different entities (states) additional efficiency can be gained. An additional implication is that ramping up production early in the planning cycle allows to reduce shortfall significantly. An optimal timing of this production ramp-up consideration can be based on a cost-benefit analysis.

Rate of Convergence and Periodicity of the Expected Population Structure of Markov Systems that Live in a General State Space

In this article we study the asymptotic behaviour of the expected population structure of a Markov system that lives in a general state space (MSGS) and its rate of convergence. We continue with the study of the asymptotic periodicity of the expected population structure. We conclude with the study of total variability from the invariant measure in the periodic case for the expected population structure of an MSGS.

Using the theory of constraints' processes of ongoing improvement to address the provider appointment scheduling system execution problem

Many primary care clinics suffer from chaos. In scheduling, providers are continually trying unsuccessfully to balance supply and demand, and in execution, to manage disruptions to provider focus and patient flow. In this research the theory of constraints' (TOC) three processes of ongoing improvement (POOGI) provide a direction for the solution to achieving more, cheaper, better, and faster healthcare. This research is the second of a two-part study examining the appointment scheduling literature, identifying the core problem (using a case study for validation) and providing a generic process for developing effective provider appointment scheduling systems (PASS). In the first part, PASS design was studied and in this second part PASS execution is studied. A strawman process is developed to apply across outpatient medical practices. With this generic process implemented across outpatient scheduling systems cost could be reduced significantly while the quality and timeliness could be increased significantly.

Optimizing Endoscope Reprocessing Resources Via Process Flow Queuing Analysis

The Massachusetts General Hospital (MGH) is merging its older endoscope processing facilities into a single new facility that will enable high-level disinfection of endoscopes for both the ORs and Endoscopy Suite, leveraging economies of scale for improved patient care and optimal use of resources. Finalized resource planning was necessary for the merging of facilities to optimize staffing and make final equipment selections to support the nearly 33,000 annual endoscopy cases. To accomplish this, we employed operations management methodologies, analyzing the physical process flow of scopes throughout the existing Endoscopy Suite and ORs and mapping the future state capacity of the new reprocessing facility. Further, our analysis required the incorporation of historical case and reprocessing volumes in a multi-server queuing model to identify any potential wait times as a result of the new reprocessing cycle. We also performed sensitivity analysis to understand the impact of future case volume growth. We found that our future-state reprocessing facility, given planned capital expenditures for automated endoscope reprocessors (AERs) and pre-processing sinks, could easily accommodate current scope volume well within the necessary pre-cleaning-to-sink reprocessing time limit recommended by manufacturers. Further, in its current planned state, our model suggested that the future endoscope reprocessing suite at MGH could support an increase in volume of at least 90% over the next several years. Our work suggests that with simple mathematical analysis of historic case data, significant changes to a complex perioperative environment can be made with ease while keeping patient safety as the top priority.

Application of Queueing Theory to the Analysis of Changes in Outpatients' Waiting Times in Hospitals Introducing EMR

Objectives

This research used queueing theory to analyze changes in outpatients' waiting times before and after the introduction of Electronic Medical Record (EMR) systems.

Methods

We focused on the exact drawing of two fundamental parameters for queueing analysis, arrival rate (λ) and service rate (µ), from digital data to apply queueing theory to the analysis of outpatients' waiting times. We used outpatients' reception times and consultation finish times to calculate the arrival and service rates, respectively.

Results

Using queueing theory, we could calculate waiting time excluding distorted values from the digital data and distortion factors, such as arrival before the hospital open time, which occurs frequently in the initial stage of a queueing system. We analyzed changes in outpatients' waiting times before and after the introduction of EMR using the methodology proposed in this paper, and found that the outpatients' waiting time decreases after the introduction of EMR. More specifically, the outpatients' waiting times in the target public hospitals have decreased by rates in the range between 44% and 78%.

Conclusions

It is possible to analyze waiting times while minimizing input errors and limitations influencing consultation procedures if we use digital data and apply the queueing theory. Our results verify that the introduction of EMR contributes to the improvement of patient services by decreasing outpatients' waiting time, or by increasing efficiency. It is also expected that our methodology or its expansion could contribute to the improvement of hospital service by assisting the identification and resolution of bottlenecks in the outpatient consultation process.

Predicting Length of Stay among Patients Discharged from the Emergency Department-Using an Accelerated Failure Time Model

Background

Emergency department (ED) crowding continues to be an important health care issue in modern countries. Among the many crucial quality indicators for monitoring the throughput process, a patient’s length of stay (LOS) is considered the most important one since it is both the cause and the result of ED crowding. The aim of this study is to identify and quantify the influence of different patient-related or diagnostic activities-related factors on the ED LOS of discharged patients.

Methods

This is a retrospective electronic data analysis. All patients who were discharged from the ED of a tertiary teaching hospital in 2013 were included. A multivariate accelerated failure time model was used to analyze the influence of the collected covariates on patient LOS.

Results

A total of 106,206 patients were included for analysis with an overall medium ED LOS of 1.46 (interquartile range = 2.03) hours. Among them, 96% were discharged by a physician, 3.5% discharged against medical advice, 0.5% left without notice, and only 0.02% left without being seen by a physician. In the multivariate analysis, increased age (>80 vs <20, time ratio (TR) = 1.408, p<0.0001), higher acuity level (triage level I vs. level V, TR = 1.343, p<0.0001), transferred patients (TR = 1.350, p<0.0001), X-rays obtained (TR = 1.181, p<0.0001), CT scans obtained (TR = 1.515, p<0.0001), laboratory tests (TR = 2.654, p<0.0001), consultation provided (TR = 1.631, p<0.0001), observation provided (TR = 8.435, p<0.0001), critical condition declared (TR = 1.205, p<0.0001), day-shift arrival (TR = 1.223, p<0.0001), and an increased ED daily census (TR = 1.057, p<0.0001) lengthened the ED LOS with various effect sizes. On the other hand, male sex (TR = 0.982, p = 0.002), weekend arrival (TR = 0.928, p<0.0001), and adult non-trauma patients (compared with pediatric non-trauma, TR = 0.687, p<0.0001) were associated with shortened ED LOS. A prediction diagram was made accordingly and compared with the actual LOS.

Conclusions

The influential factors on the ED LOS in discharged patients were identified and quantified in the current study. The model’s predicted ED LOS may provide useful information for physicians or patients to better anticipate an individual’s LOS and to help the administrative level plan its staffing policy.

Forecasting Daily Patient Outflow From a Ward Having No Real-Time Clinical Data

Background: Modeling patient flow is crucial in understanding resource demand and prioritization. We study patient outflow from an open ward in an Australian hospital, where currently bed allocation is carried out by a manager relying on past experiences and looking at demand. Automatic methods that provide a reasonable estimate of total next-day discharges can aid in efficient bed management. The challenges in building such methods lie in dealing with large amounts of discharge noise introduced by the nonlinear nature of hospital procedures, and the nonavailability of real-time clinical information in wards.

A Conceptual Framework for Improving Critical Care Patient Flow and Bed Use

RATIONALE

High demand for intensive care unit (ICU) services and limited bed availability have prompted hospitals to address capacity planning challenges. Simulation modeling can examine ICU bed assignment policies, accounting for patient acuity, to reduce ICU admission delays.

OBJECTIVES

To provide a framework for data-driven modeling of ICU patient flow, identify key measurable outcomes, and present illustrative analysis demonstrating the impact of various bed allocation scenarios on outcomes.

METHODS

A description of key inputs for constructing a queuing model was outlined, and an illustrative simulation model was developed to reflect current triage protocol within the medical ICU and step-down unit (SDU) at a single tertiary-care hospital. Patient acuity, arrival rate, and unit length of stay, consisting of a "service time" and "time to transfer," were estimated from 12 months of retrospective data (n = 2,710 adult patients) for 36 ICU and 15 SDU staffed beds. Patient priority was based on acuity and whether the patient originated in the emergency department. The model simulated the following hypothetical scenarios: (1) varied ICU/SDU sizes, (2) reserved ICU beds as a triage strategy, (3) lower targets for time to transfer out of the ICU, and (4) ICU expansion by up to four beds. Outcomes included ICU admission wait times and unit occupancy.

MEASUREMENTS AND MAIN RESULTS

With current bed allocation, simulated wait time averaged 1.13 (SD, 1.39) hours. Reallocating all SDU beds as ICU decreased overall wait times by 7.2% to 1.06 (SD, 1.39) hours and increased bed occupancy from 80 to 84%. Reserving the last available bed for acute patients reduced wait times for acute patients from 0.84 (SD, 1.12) to 0.31 (SD, 0.30) hours, but tripled subacute patients' wait times from 1.39 (SD, 1.81) to 4.27 (SD, 5.44) hours. Setting transfer times to wards for all ICU/SDU patients to 1 hour decreased wait times for incoming ICU patients, comparable to building one to two additional ICU beds.

CONCLUSIONS

Hospital queuing and simulation modeling with empiric data inputs can evaluate how changes in ICU bed assignment could impact unit occupancy levels and patient wait times. Trade-offs associated with dedicating resources for acute patients versus expanding capacity for all patients can be examined.

Impact of Mandatory HIV Screening in the Emergency Department: A Queuing Study

To improve HIV screening rates, New York State in 2010 mandated that all persons 13-64 years receiving health care services, including care in emergency departments (EDs), be offered HIV testing. Little attention has been paid to the effect of screening on patient flow. Time-stamped ED visit data from patients eligible for HIV screening, 7,844 of whom were seen by providers and 767 who left before being seen by providers, were retrieved from electronic health records in one adult ED. During day shifts, 10% of patients left without being seen, and during evening shifts, 5% left without being seen. All patients seen by providers were offered testing, and 6% were tested for HIV. Queuing models were developed to evaluate the effect of HIV screening on ED length of stay, patient waiting time, and rate of leaving without being seen. Base case analysis was conducted using actual testing rates, and sensitivity analyses were conducted to evaluate the impact of increasing the testing rate. Length of ED stay of patients who received HIV tests was 24 minutes longer on day shifts and 104 minutes longer on evening shifts than for patients not tested for HIV. Increases in HIV testing rate were estimated to increase waiting time for all patients, including those who left without being seen. Our simulation suggested that incorporating HIV testing into ED patient visits not only adds to practitioner workload but also increases patient waiting time significantly during busy shifts, which may increase the rate of leaving without being seen.

Optimizing nurse capacity in a teaching hospital neonatal intensive care unit

Patients in intensive care units need special attention. Therefore, nurses are one of the most important resources in a neonatal intensive care unit. These nurses are required to have highly specialized training. The random number of patient arrivals, rejections, or transfers due to lack of capacity (such as nurse, equipment, bed etc.) and the random length of stays, make advanced knowledge of the optimal nurse a requirement, for levels of the unit behave as a stochastic process. This stochastic nature creates difficulties in finding optimal nurse staffing levels. In this paper, a stochastic approximation which is based on the required nurse: patient ratio and the number of patients in a neonatal intensive care unit of a teaching hospital, has been developed. First, a meta-model was built to generate simulation results under various numbers of nurses. Then, those experimented data were used to obtain the mathematical relationship between inputs (number of nurses at each level) and performance measures (admission number, occupation rate, and satisfaction rate) using statistical regression analysis. Finally, several integer nonlinear mathematical models were proposed to find optimal nurse capacity subject to the targeted levels on multiple performance measures. The proposed approximation was applied to a Neonatal Intensive Care Unit of a large hospital and the obtained results were investigated.

Improving hospital bed occupancy and resource utilization through queuing modeling and evolutionary computation

Scarce healthcare resources require carefully made policies ensuring optimal bed allocation, quality healthcare service, and adequate financial support. This paper proposes a complex analysis of the resource allocation in a hospital department by integrating in the same framework a queuing system, a compartmental model, and an evolutionary-based optimization. The queuing system shapes the flow of patients through the hospital, the compartmental model offers a feasible structure of the hospital department in accordance to the queuing characteristics, and the evolutionary paradigm provides the means to optimize the bed-occupancy management and the resource utilization using a genetic algorithm approach. The paper also focuses on a "What-if analysis" providing a flexible tool to explore the effects on the outcomes of the queuing system and resource utilization through systematic changes in the input parameters. The methodology was illustrated using a simulation based on real data collected from a geriatric department of a hospital from London, UK. In addition, the paper explores the possibility of adapting the methodology to different medical departments (surgery, stroke, and mental illness). Moreover, the paper also focuses on the practical use of the model from the healthcare point of view, by presenting a simulated application.

Using queuing theory and simulation model to optimize hospital pharmacy performance

BACKGROUND

Hospital pharmacy is responsible for controlling and monitoring the medication use process and ensures the timely access to safe, effective and economical use of drugs and medicines for patients and hospital staff.

OBJECTIVES

This study aimed to optimize the management of studied outpatient pharmacy by developing suitable queuing theory and simulation technique.

PATIENTS AND METHODS

A descriptive-analytical study conducted in a military hospital in Iran, Tehran in 2013. A sample of 220 patients referred to the outpatient pharmacy of the hospital in two shifts, morning and evening, was selected to collect the necessary data to determine the arrival rate, service rate, and other data needed to calculate the patients flow and queuing network performance variables. After the initial analysis of collected data using the software SPSS 18, the pharmacy queuing network performance indicators were calculated for both shifts. Then, based on collected data and to provide appropriate solutions, the queuing system of current situation for both shifts was modeled and simulated using the software ARENA 12 and 4 scenarios were explored.

RESULTS

Results showed that the queue characteristics of the studied pharmacy during the situation analysis were very undesirable in both morning and evening shifts. The average numbers of patients in the pharmacy were 19.21 and 14.66 in the morning and evening, respectively. The average times spent in the system by clients were 39 minutes in the morning and 35 minutes in the evening. The system utilization in the morning and evening were, respectively, 25% and 21%. The simulation results showed that reducing the staff in the morning from 2 to 1 in the receiving prescriptions stage didn't change the queue performance indicators. Increasing one staff in filling prescription drugs could cause a decrease of 10 persons in the average queue length and 18 minutes and 14 seconds in the average waiting time. On the other hand, simulation results showed that in the evening, decreasing the staff from 2 to 1 in the delivery of prescription drugs, changed the queue performance indicators very little. Increasing a staff to fill prescription drugs could cause a decrease of 5 persons in the average queue length and 8 minutes and 44 seconds in the average waiting time.

CONCLUSIONS

The patients' waiting times and the number of patients waiting to receive services in both shifts could be reduced by using multitasking persons and reallocating them to the time-consuming stage of filling prescriptions, using queuing theory and simulation techniques.

Factors associated with emergency department length of stay for patients with hip fracture

Time to surgery, which includes time in the emergency department (ED), is important for all patients with hip fracture. We hypothesized that patients with hip fracture spend significantly more time in the ED than do patients with the top 5 most common conditions. In addition, we hypothesized that there are patient, physician, and hospital factors that affect the length of time spent in the ED. We retrospectively reviewed our institution's hip fracture database and identified 147 elderly patients with hip fractures who presented to our ED from December 18, 2005, through April 30, 2009. We reviewed their records for patient, practitioner, and hospital factors of interest associated with ED time and for 6 specified time intervals. Average working, boarding (waiting for an inpatient room), and total times were calculated and compared with respective averages for admitted ED patients with the top 5 most common conditions. Univariate and multivariate analyses were performed before and after adjusting for confounders (significance, P = .05). The mean total ED time (7 hours and 25 minutes) and working time (4 hours and 31 minutes) for patients with hip fracture were similar to the respective overall averages for admitted ED patients. However, the average boarding time for patients with hip fracture was 2 hours 44 minutes, longer than that for other patients admitted through the ED. Factors significantly associated with longer ED times were a history of hypertension, history of atrial fibrillation, the number of computed tomography scans ordered, and the occupancy rate. Admission to the hip fracture service decreased working time but not overall time. Substantial multidisciplinary work among the ED, hospital admission services, and physicians is needed to dramatically decrease the boarding time and thus the overall time to surgery.

Ensemble-based methods for forecasting census in hospital units

BACKGROUND

The ability to accurately forecast census counts in hospital departments has considerable implications for hospital resource allocation. In recent years several different methods have been proposed forecasting census counts, however many of these approaches do not use available patient-specific information.

METHODS

In this paper we present an ensemble-based methodology for forecasting the census under a framework that simultaneously incorporates both (i) arrival trends over time and (ii) patient-specific baseline and time-varying information. The proposed model for predicting census has three components, namely: current census count, number of daily arrivals and number of daily departures. To model the number of daily arrivals, we use a seasonality adjusted Poisson Autoregressive (PAR) model where the parameter estimates are obtained via conditional maximum likelihood. The number of daily departures is predicted by modeling the probability of departure from the census using logistic regression models that are adjusted for the amount of time spent in the census and incorporate both patient-specific baseline and time varying patient-specific covariate information. We illustrate our approach using neonatal intensive care unit (NICU) data collected at Women & Infants Hospital, Providence RI, which consists of 1001 consecutive NICU admissions between April 1st 2008 and March 31st 2009.

RESULTS

Our results demonstrate statistically significant improved prediction accuracy for 3, 5, and 7 day census forecasts and increased precision of our forecasting model compared to a forecasting approach that ignores patient-specific information.

CONCLUSIONS

Forecasting models that utilize patient-specific baseline and time-varying information make the most of data typically available and have the capacity to substantially improve census forecasts.

Scheduled surgery admissions and occupancy at a children's hospital: variation we can control to improve efficiency and value in health care delivery

OBJECTIVE

Describe variability in admission, discharge, and occupancy patterns for surgical patients at a large children's hospital and assess the relationship between scheduled admissions and occupancy.

BACKGROUND

High hospital occupancy degrades quality of care and access, whereas low levels of occupancy use hospital resources inefficiently. Variability in scheduling patients for surgical procedures may affect occupancy and be amenable to alteration.

METHODS

This is a retrospective administrative data analysis that took place at 1 urban, tertiary-care children's hospital. A total of 8552 surgical patients hospitalized from July 1, 2009, to June 30, 2010, were included in the analysis, and admission-discharge-transfer data for 1 fiscal year were abstracted for analysis of admission and occupancy patterns.

RESULTS

Among 6257 surgical admissions for non-intensive care unit (ICU) patients, 49% were emergent and 51% were scheduled. Variation in admission volume by day of week was more than 3 times higher for scheduled admissions than for emergent admissions. For non-ICU surgical patients with length of stay 7 days or less (97%), Mondays and Tuesdays generated 42% of scheduled patient occupancy time. Thursdays and Fridays often had high occupancy of surgical patients (>90% of designated beds filled), whereas Saturdays, Sundays, and Mondays were often at low occupancy for those beds (<80% filled). Only 20% of all days in the year had designated non-ICU surgery beds with occupancy between 80% and 95%.

CONCLUSIONS

Scheduled admissions contribute significantly to variability in occupancy. Predictable patterns of admissions lead to high occupancy on some days and unused capacity on others, with few days being at an optimal level of occupancy. These predictable patterns suggest opportunities to improve hospital operations with changes in scheduled admission patterns, which present a different problem than random demand.

Addressing inpatient crowding by smoothing occupancy at children's hospitals

OBJECTIVE

To quantify the difference in weekday versus weekend occupancy, and the opportunity to smooth inpatient occupancy to reduce crowding at children's hospitals.

METHODS

Daily inpatient census data for 39 freestanding, tertiary-care children's hospitals were used to calculate occupancy and to model the impact of reducing variation in occupancy and the change in the number of patients, patient-days, and hospitals exposed to high occupancy pre- and post-smoothing. We also calculated the proportion of weekly admissions that would require different scheduling to achieve within-week smoothing.

RESULTS

Overall, hospitals' mean occupancy ranged from 70.9% to 108.1% on weekdays, and 65.7% to 94.9% on weekends. Weekday occupancy exceeded weekend occupancy with a median difference of 8.2% points. The mean post-smoothing reduction in weekly maximum occupancy across all hospitals was 6.6% points. Through smoothing, 39,607 patients from the 39 hospitals were removed from exposure to occupancy levels >95%. To achieve within-week smoothing, a median 2.6% of admissions would have to be scheduled on a different day of the week; this equates to a median of 7.4 patients per week (range: 2.3-14.4).

CONCLUSION

Hospitals do have substantial unused capacity, and smoothing occupancy over the course of a week could be a useful strategy that hospitals can use to reduce crowding and protect patients from crowded conditions.

Understanding waiting lists as the matching of surgical capacity to demand: are we wasting enough surgical time?

If surgical 'capacity' always matched or exceeded 'demand' then there should be no waiting lists for surgery. However, understanding what is meant by 'demand', 'capacity' and 'matched' requires some mathematical concepts that we outline in this paper. 'Time' is the relevant measure: 'demand' for a surgical team is best understood as the total min required for the surgery booked from outpatient clinics every week; and 'capacity' is the weekly operating time available. We explain how the variation in demand (not just the mean demand) influences the analysis of optimum capacity. However, any capacity chosen in this way is associated with only a likelihood (that is, a probability rather than certainty) of absorbing the prevailing demand. A capacity that suitably absorbs the demand most of the time (for example, > 80% of weeks) will inevitably also involve considerable waste (that is, many weeks in which there is spare, unused capacity). Conversely, a level of capacity chosen to minimise wasted time will inevitably cause an increase in size of the waiting list. Thus the question of how to balance demand and capacity is intimately related to the question of how to balance utilisation and waste. These mathematical considerations enable us to consider objectively how to manage the waiting list. They also enable us critically to analyse the extent to which philosophies adopted by the National Health Service (such as 'Lean' or 'Six Sigma') will be successful in matching surgical capacity to demand.

Children's hospitals do not acutely respond to high occupancy

OBJECTIVE

High hospital occupancy may lead to overcrowding in emergency departments and inpatient units, having an adverse impact on patient care. It is not known how children's hospitals acutely respond to high occupancy. The objective of this study was to describe the frequency, direction, and magnitude of children's hospitals' acute responses to high occupancy.

METHODS

Patients who were discharged from 39 children's hospitals that participated in the Pediatric Health Information System database during 2006 were eligible. Midnight census data were used to construct occupancy levels. Acute response to high occupancy was measured by 8 variables, including changes in hospital admissions (4 measures), transfers (2 measures), and length of stay (2 measures).

RESULTS

Hospitals were frequently at high occupancy, with 28% of midnights at 85% to 94% occupancy and 42% of midnights at > or =95% occupancy. Whereas half of children's hospitals used occupancy-mitigating responses, there was variability in responses and magnitudes were small. When occupancy was >95%, no more than 8% of hospitals took steps to reduce admissions, 13% increased transfers out, and up to 58% reduced standardized length of stay. Two-day lag response was more common but remained of too small a magnitude to make a difference in hospital crowding. Additional modeling techniques also revealed little response.

CONCLUSIONS

We found a low rate of acute response to high occupancy. When there was a response, the magnitude was small.

Statistical analysis of patients' characteristics in neonatal intensive care units

The staff in the neonatal intensive care units is required to have highly specialized training and the using equipment in this unit is so expensive. The random number of arrivals, the rejections or transfers due to lack of capacity and the random length of stays, make the advance knowledge of the optimal staff; equipment and materials requirement for levels of the unit behaves as a stochastic process. In this paper, the number of arrivals, the rejections or transfers due to lack of capacity and the random length of stays in a neonatal intensive care unit of a university hospital has been statistically analyzed. The arrival patients are classified according to the levels based on the required nurse: patient ratio and gestation age. Important knowledge such as arrivals, transfers, gender and length of stays are analyzed. Finally, distribution functions for patients' arrivals, rejections and length of stays are obtained for each level in the unit.

A combination of deterministic and stochastic approaches to optimize bed capacity in a hospital unit

Random number of arrivals and random length of stays make the number of patients in a hospital unit behave as a stochastic process. This makes the determination of the optimum size of the bed capacity more difficult. The number of admissions per day, service level and occupancy level are key control parameters that affect the optimum size of the required bed capacity. In this study a new stochastic approximation is developed and applied to a unit of a teaching hospital. Data between 2000 and 2004 was used to obtain the necessary probability distribution functions. Mathematical relationships between the control parameters and size of the bed capacity are obtained using generated data from a constructed simulation model. Nonlinear mathematical models are then used to determine the optimum size of the required bed capacity based on target levels of the control parameters, and a profit and loss analysis is performed.

Stochastic bed balancing of an obstetrics hospital

Health care is one of the largest industries in the developed world and the top domestic industry in the United States (US). Over the past thirty years there has been a dramatic increase in healthcare costs in the US, of which about one-third can be attributed to hospital spending. One of the key factors in hospital cost containment and revenue enhancement is effective and efficient bed planning and capacity analysis. This study aims to balance bed unit utilizations across an obstetrics hospital and minimize the blocking of beds from upstream units within given constraints on bed reallocation. The methodology includes the assessment and effect of time-dependent patterns of monthly, daily, and hourly demand. Queuing networks are first used to assess the flows between units, establish target utilizations of bed units, and involve stakeholders in a flow characterization that they understand. Discrete-event simulation is then used to maximize the flow through the balanced system including non-homogeneous effects, non-exponential lengths of stay, and blocking behavior. Results of the models are validated against actual data collected from the hospital. Several 'what if' scenarios are studied showing that 38% more patient flow can be achieved with only 15% more patient beds. The results of the study have been implemented.

Modelling Variability in Hospital Bed Occupancy

A stochastic version of the Harrison-Millard multistage model of the flow of patients through a hospital division is developed in order to model correctly not only the average but also the variability in occupancy levels, since it is the variability that makes planning difficult and high percent occupancy levels increase the risk of frequent overflows. The model is fit to one year of data from the medical division of an acute care hospital in Adelaide, Australia. Admissions can be modeled as a Poisson process with rates varying by day of the week and by season. Methods are developed to use the entire annual occupancy profile to estimate transition rate parameters when admission rates are not constant and to estimate rate parameters that vary by day of the week and by season, which are necessary for the model variability to be as large as in the data. The final model matches well the mean, standard deviation and autocorrelation function of the occupancy data and also six months of data not used to estimate the parameters. Repeated simulations are used to construct percentiles of the daily occupancy distributions and thus identify ranges of normal fluctuations and those that are substantive deviations from the past, and also to investigate the trade-offs between frequency of overflows and the percent occupancy for both fixed and flexible bed allocations. Larger divisions can achieve more efficient occupancy levels than smaller ones with the same frequency of overflows. Seasonal variations are more significant than day-of-the-week variations and variable discharge rates are more significant than variable admission rates in contributing to overflows.

Multi-institutional Analysis of Computed and Direct Radiography

PURPOSE

To compare economic aspects of equipment configurations, productivity levels, and patient waiting times in the performance of computed radiography (CR) and direct radiography (DR).

MATERIALS AND METHODS

The study received internal review board exemption status, without the need for informed patient consent. Data from four study sites were used to calculate the CR-DR crossover point (defined as the point at which the cost-effectiveness of DR equals that of CR) and CR-DR annual cost differentials. Analyzed variables included equipment and operating costs, examination volumes, and productivity. A program was developed to simulate patient arrival times, number of patient examinations, and patient waiting times on the basis of average annualized parameters for each of the four clinics. Sensitivity analyses were conducted to assess utilization rates and determine cost optimization. Utilization rates were compared with the number of excess long-stay CR patients (ie, patients who spent more than 30 minutes waiting in the radiology department prior to CR examination) and with the cost (per excess long-stay CR patient who waited more than 60 minutes) averted by using DR.

RESULTS

Excess annual costs for DR over CR at the four sites ranged from $50,757 to $75,303. At extrapolated levels of economic penalties for long waiting times, the crossover point at which the DR cost became justifiable was when CR capacity utilization rates approached or exceeded 80%.

CONCLUSION

In the current practice environment, with capacity utilization rates well below 80%, CR is likely to be a more cost-effective technology for the majority of general radiography providers.