Where to treat the older patient? Can Markov models help us better understand the relationship between hospital and community care?

We have previously used Markov models to describe movements of patients between hospital states; these may be actual or virtual and described by a phase-type distribution. Here we extend this approach to a Markov reward model for a healthcare system with Poisson admissions and an absorbing state, typically death. The distribution of costs is evaluated for any time and expressions derived for the mean and variances of costs. The average cost at any time is then determined for two scenarios: the Therapeutic and Prosthetic models, respectively. This example is used to illustrate the idea that keeping acute patients longer in hospital to ensure fitness for discharge, may reduce costs by decreasing the number of patients that become long-stay. In addition we develop a Markov Reward Model for a healthcare system including states, where the patient is in hospital, and states, where the patient is in the community. In each case, the length of stay is described by a phase-type distribution, thus enabling the representation of durations and costs in each phase within a Markov framework. The model can be used to determine costs for the entire system thus facilitating a systems approach to the planning of healthcare and a holistic approach to costing. Such models help us to assess the complex relationship between hospital and community care.     

Capacity planning of a perinatal network with generalised loss network model with overflow

Recent literature shows that the arrival and discharge processes in hospital intensive care units do not satisfy the Markovian property, that is, interarrival times and length of stay tend to have a long tail. In this paper we develop a generalised loss network framework for capacity planning of a perinatal network in the UK. Decomposing the network by hospitals, each unit is analysed with a GI/G/c/0 overflow loss network model. A two-moment approximation is performed to obtain the steady state solution of the GI/G/c/0 loss systems, and expressions for rejection probability and overflow probability have been derived. Using the model framework, the number of required cots can be estimated based on the rejection probability at each level of care of the neonatal units in a network. The generalisation ensures that the model can be applied to any perinatal network for renewal arrival and discharge processes.     

A pre-assignment heuristic algorithm for the Master Surgical Schedule Problem (MSSP)

In this paper a 0–1 linear programming model and a solution heuristic algorithm are developed in order to solve the so-called Master Surgical Schedule Problem (MSSP). Given a hospital department made up of different surgical units (i.e. wards) sharing a given number of Operating Rooms (ORs), the problem herein addressed is determining the assignment among wards and ORs during a given planning horizon, together with the subset of patients to be operated on during each day. Different resource constraints related to operating block time length, maximum OR overtime allowable by collective labour agreement and legislation, patient length of stay (LOS), available OR equipment, number of surgeons, number of stay and ICU beds, are considered. Firstly, a 0–1 linear programming model intended to minimise a cost function based upon a priority score, that takes into proper account both the waiting time and the urgency status of each patient, is developed. Successively, an heuristic algorithm that enables us to embody some pre-assignment rules to solve this NP-hard combinatorial optimisation problem, is presented. In particular, we force the assignment of each patient to a subset of days depending on his/her expected length of stay in order to allow closing some stay areas during the weekend and hence reducing overall hospitalisation cost of the department. The results of an extensive computational experimentation aimed at showing the algorithm efficiency in terms of computational time and solution effectiveness are given and analysed.     

Building cyclic master surgery schedules with leveled resulting bed occupancy

This paper proposes and evaluates a number of models for building surgery schedules with leveled resulting bed occupancy. The developed models involve two types of constraints. Demand constraints ensure that each surgeon (or surgical group) obtains a specific number of operating room blocks. Capacity constraints limit the available blocks on each day. Furthermore, the number of operated patients per block and the length of stay of each operated patient are dependent on the type of surgery. Both are considered stochastic, following a multinomial distribution. We develop a number of mixed integer programming based heuristics and a metaheuristic to minimize the expected total bed shortage and present computational results.     

A mixed integer programming approach for allocating operating room capacity

We have developed a methodology for allocating operating room capacity to specialties. Our methodology consists of a finite-horizon mixed integer programming (MIP) model which determines a weekly operating room allocation template that minimizes inpatients' cost measured as their length of stay. A number of patient type priority (eg emergency over non-emergency patient) and clinical constraints (eg maximum number of hours allocated to each specialty, surgeon, and staff availability) are included in the formulation. The optimal solution from the analytical model is inputted into a simulation model that captures some of the randomness of the processes (eg surgery time, demand, arrival time, and no-show rate of the outpatients) and non-linearities (eg the MIP assumes proportional allocation of demand satisfaction (output) with room allocation (input)). The simulation model outputs the average length of stay for each specialty and the room utilization. On a case example of a Los Angeles County Hospital, we show how the hospital length of stay pertaining to surgery can be reduced.     

多服务台并联排队系统的随机模拟

针对多服务台并联排队系统,提出了计算机随机模拟的方法.分别研究了基本和非基本排队系统中,诸如排队人数、排队时间、平稳分布、忙期及其忙期服务人数等各项排队指标的模拟求解方法.通过模拟例子和实际数据分析,显示了本文所提模拟方法的有效性、灵活性和实用性.     

A closed queuing network model with single servers for multi-threaded architecture

In this paper, a closed queuing network model with single servers for each queue is proposed to model dataflow in a multi-threaded architecture. Multi-threading is useful in reducing the latency by switching among a set of threads in order to improve the processor utilization. Two sets of processors, synchronization and execution processors exist. Synchronization processors handle load/store operations and execution processors handle arithmetic/logic and control operations. A closed queuing network model is suitable for large number of job arrivals. The normalization constant is derived using a recursive algorithm for the given model. State diagrams are drawn from the closed queuing network model, and the steady-state balance equations are derived from it. Performance measures such as average response times and average system throughput are derived and plotted against the total number of processors in the closed queuing network model. Other important performance measures like processor utilizations, average queue lengths, average waiting times and relative utilizations are also derived.     

基于LSTM-GA混合模型的患者预约排队策略优化

在医疗运作管理领域,合理的资源分配能够帮助更多的患者尽早就医,降低患者病情恶化和死亡的风险。本文设计了预约排队策略对患者占有资源的顺序进行分配,建立了基于长短时记忆(Long Short Term-Memory, LSTM)神经网络和遗传算法(Genetic Algorithm, GA)的混合模型以优化排队策略。首先利用大数据和深度学习分析患者到达和医院服务情况,建立LSTM神经网络学习数据特征并预测未来数据,相比于排队论常用的随机分布方法取得了更好的效果.其次设计了基于排队系统仿真的排队策略优化算法,利用改进GA得到最优排队策略。实证研究表明,文本的方法可以明显降低患者的等待时间,最高可达59%。最后对排队策略进行敏感性分析,结果表明排队策略有效作用于仿真的各个时段。     

A non-parametric competing risks model for manpower planning

One of the most important variables for manpower planners is duration until a specified event occurs. This is frequently the completed length of service until leaving a job, but may also include such variables as length of service in a grade until promotion, or length of a spell of withdrawal from the labour force. In this paper we develop non-parametric maximum likelihood estimators for the survivor functions of length of stay in a grade until leaving for a number of different possible destinations. Since the data are statistically incomplete, including right censored and left truncated durations, as well as complete durations, we must modify the competing risks theory in the biostatistical literature to take such incompleteness into account. Right censored durations arise when the individual is still in the grade when data collection ceases and left truncated durations when the individual is already in service when data collection commences. The competing risks model is fitted to data for Northern Ireland nursing service and used to predict staff flows between grades. We may thus estimate future movements within the system and predict the future manpower stocks.     

A hybrid hypercube – Genetic algorithm approach for deploying many emergency response mobile units in an urban network

Emergency response services are critical for modern societies. This paper presents a model and a heuristic solution for the optimal deployment of many emergency response units in an urban transportation network and an application for transit mobile repair units (TMRU) in the city of Athens, Greece. The model considers the stochastic nature of such services, suggesting that a unit may be already engaged, when an incident occurs. The proposed model integrates a queuing model (the hypercube model), a location model and a metaheuristic optimization algorithm (genetic algorithm) for obtaining appropriate unit locations in a two-step approach. In the first step, the service area is partitioned into sub-areas (called superdistricts) while, in parallel, necessary number of units is determined for each superdistrict. An approximate solution to the symmetric hypercube model with spatially homogeneous demand is developed. A Genetic Algorithm is combined with the approximate hypercube model for obtaining best superdistricts and associated unit numbers. With both of the above requirements defined in step one, the second step proceeds in the optimal deployment of units within each superdistrict.     

N-策略休假GI-X／G／1排队系统队长的瞬时分布

本文应用Markov骨架过程理论研究了N-休假策略GI~X/G/1排队系统,并得到了队长的瞬时分布.     

基于排队论的专项汽修厂维修台配置数量研究

为了解决因维修台配置不合理而导致的专项汽修厂排队等待的问题,运用排队论等理论和方法,采用先到先修理与维修台带有优先权相结合的排队规则,将维修台配置数量问题转化为两类型服务台排队问题,建立了单队列M/M/s1+s2/K/∞/FCFS+PS排队模型.通过对某专项汽修厂相关的数据采集和分析,得到了模型所需的变量和参数,运用边际效益法进行优化,得到了节假日和非节假日客流高峰期的最优维修台配置数量.通过对系统服务强度、系统资源限制和服务时间段等因素的分析,既能保证排队系统可以在不同时间段内对维修台配置数量进行调整,又能缩小最优值的求解范围.     

Using phase-type models to cost stroke patient care across health,social and community services

Stroke disease places a heavy burden on society, incurring long periods of time in hospital and community care, and associated costs. Also stroke is a highly complex disease with diverse outcomes and multiple strategies for therapy and care. Previously a modeling framework has been developed which clusters patients into classes with respect to their length of stay (LOS) in hospital. Phase-type models were then used to describe patient flows for each cluster. Also multiple outcomes, such as discharge to normal residence, nursing home, or death can be permitted. We here add costs to this model and obtain the Moment Generating Function for the total cost of a system consisting of multiple transient phase-type classes with multiple absorbing states. This system represents different classes of patients in different hospital and community services states. Based on stroke patients’ data from the Belfast City Hospital, various scenarios are explored with a focus on comparing the cost of thrombolysis treatment under different regimes. The overall modeling framework characterizes the behavior of stroke patient populations, with a focus on integrated system-wide costing and planning, encompassing hospital and community services. Within this general framework we have developed models which take account of patient heterogeneity and multiple care options. Such complex strategies depend crucially on developing a deep engagement with the health care professionals and underpinning the models with detailed patient-specific data.     

Analysis of M/M/S queues with 1-limited service

In this paper, a multiple server queue, in which each server takes a vacation after serving one customer is studied. The arrival process is Poisson, service times are exponentially distributed and the duration of a vacation follows a phase distribution of order 2. Servers returning from vacation immediately take another vacation if no customers are waiting. A matrix geometric method is used to find the steady state joint probability of number of customers in the system and busy servers, and the mean and the second moment of number of customers and mean waiting time for this model. This queuing model can be used for the analysis of different kinds of communication networks, such as multi-slotted networks, multiple token rings, multiple server polling systems and mobile communication systems.     

On the distribution of the number of retrials

We consider queuing systems where customers are not allowed to queue, instead of that they make repeated attempts, or retrials, in order to enter service after some time. We obtain the distribution of the number of retrials produced by a tagged customer, until he finds an available server.     

输入率可变且有差错服务的M/M/1排队系统的研究与应用

在到达系统的顾客数不变的情况下,顾客到达系统但是否进入系统接受服务对销售行业影响是巨大的.从排队长度对顾客输入率的影响着手,研究了顾客以泊松流到达系统,而到达系统的顾客进入系统接受服务的概率与队长有关的M/M/1排队模型,且系统服务会出差错.得出了进入系统的顾客流是泊松过程,且系统中的顾客数是生灭过程,并获得了该模型的平稳分布、顾客的平均输入率、系统的平均服务强度等多项指标,为销售行业调整自己的服务速度以影响排队长度及顾客输入率,进而提高自己的销售业绩提供了很有价值的参考.     

Joint determination of production run length and number of standbys in a deteriorating production process

Two economic manufacturing quantity models with unrepairable and repairable standby key modules are proposed in this study that determine the economic production run length and the economic number of standbys in a deteriorating production process, where the key module of the production unit deteriorates over time and incurs some portion of defective items. For the model with unrepairable standbys, the active key module, once deteriorating, is replaced by a standby and the module itself is disposed. For the model with repairable standbys, the deteriorating key module is replaced by a standby and the module is then sent to the service center for maintenance. When completing the maintenance, it then joins the standbys for later production use. By minimizing the annual cost, which takes into account setup cost, holding cost, costs due to standbys and defective items, the economic production run length and the economic number of standbys are obtained for each of the proposed models.     

Continuous-time Markov model for geriatric patients behavior. Optimization of the bed occupancy and computer simulation

Previous research has shown that the flow of patients around departments of geriatric medicine and ex-patients in the community may be modelled by the application of a mixed-exponential distribution. In this paper we considered a five-compartment model using a continuous-time Markov process to describe the flow of patients. Using a M/Ph/c queuing model, we present a way of optimizing the number of beds in order to maintain an acceptable delay probability at a sufficiently low level. Finally, we constructed a Java computer simulation, using data from St George’s Hospital, London.     

Modeling the number of insureds’ cars using queuing theory

In this paper, we propose to model the number of insured cars per household. We use queuing theory to construct a new model that needs 4 different parameters: one that describes the rate of addition of new cars on the insurance contract, a second one that models the rate of removal of insured vehicles, a third parameter that models the cancellation rate of the insurance policy, and finally a parameter that describes the rate of renewal. Statistical inference techniques allow us to estimate each parameter of the model, even in the case where there is censorship of data. We also propose to generalize this new queuing process by adding some explanatory variables into each parameter of the model. This allows us to determine which policyholder’s profiles are more likely to add or remove vehicles from their insurance policy, to cancel their contract or to renew annually. The estimated parameters help us to analyze the insurance portfolio in detail because the queuing theory model allows us to compute various kinds of useful statistics for insurers, such as the expected number of cars insured or the customer lifetime value that calculates the discounted future profits of an insured. Using car insurance data, a numerical illustration based on a portfolio from a Canadian insurance company is included to support this discussion.     

“限时服务规则”下的复线船闸多目标调度优化

针对现有船舶过闸排队规则的欠缺,基于“限时服务规则”,构建复线船闸多目标双层优化调度模型:上层模型用于获得两个闸室安全区域的船舶排布可行方案;下层模型用于获得不同船舶排布可行方案的优化闸次数。下层模型分两个阶段完成:对符合“限时服务规则”的船舶,构建以闸次最少为目标的0-1规划模型,获得此类船舶安排的闸次;对其余船舶按照“先到先服务规则”,构建以闸次最少、闸室利用率最大为目标的多目标决策模型,获得不同船舶排布可行方案应该安排的频次。以位于江苏省干线航道上的某复线船闸某日24小时内过闸船舶的数据为例,计算结果表明:采用本文优化模型获得的优化方案与“经验编排方式”相比,两座船闸各节约2个闸次,两个船闸的平均闸室利用率分别提高了3.66和4.72个百分点。