Walk-in clinics (WC) offer healthcare services with no appointment allowing better accessibility to patients in immediate need. The major inconvenience of these clinics is due to the waiting time for unscheduled medical consultation of urgent cases. Usually WC consider the patient’s arrival order (FIFO logic, First In First Out), observing very particular rules for their management. After a first triage done by the medical staff, some patients are deferred to emergency and other serious cases are rescheduled to pass first. The remaining patients retain the FIFO logic of the arrival ordering. Clinics use labels, pagers or lists to keep a record of the patient in the queue. Cases such as child consultations can also be considered as multi-consultations as parents and children pass together.
Responding to the need to keep patients in queue and reduce task duties, several companies began offering services for patients’ queuing and schedule management for WC in the province of Quebec (Canada). These services have in common a system that identifies each patient with a code that puts them into a virtual queue. An extra follow-up service allows patients to wait elsewhere. These systems do not change the general way that WC provide their service.
The company servicing the clinics continues the follow-up to some patients, allowing them to use their spare time wisely, and especially to avoid extra waiting time in the clinic’s facilities. This follow-up service will send notifications by telephone, message, or will display this information in a web application. The service notifies 8, 5 & 3 patients in advance of their turn. Patients are expected to be back in the clinic when they receive the 3 patients notification. Figure 1 shows a sample of these notifications.
This service is helpful but still does not answer this simple question: at what time should I be back at the clinic? This proposed research project aims to improve the current service with an approximated forecast useful for consultation scheduling.
With the collaboration of one industrial partner running the notification service (ChronoMetriq Inc.), we study several time series from different clinics where the process has a clearer structure. Based on these data, we propose an algorithm that is able to make estimations of the patient’s consultation time as seen from the queue perspective, i.e. independent of the number of doctors serving the queue. Estimations should adapt dynamically and contain the anomalies of the process (without explicating them).
Complexity of Forecasting
For making timing decisions, most people would think that averaging past results is very useful. In fact it is possible to figure out an interval around the experiences’ average that frequently includes the real unknown value of the mean of all possible experiences. However, this interval often changes with new sets of experiences. In statistics, wider intervals and larger sets of experiences are associated with certainty. In many situations, a normal distribution adapts well (the typical bell shape) to describe a process. This is useful to estimate average values and figure out confidence intervals. In all the past cases, the mean is not random, but it is unknown and it is difficult to compute when all possible outcomes are unavailable (the population).
Forecasting new outcomes is different; the outcomes are not yet a fact, they are random events. Most people are not really interested in the “mean” either, but in estimating future events. Nevertheless, most people consider taking the average or the last value as a good value forecast for the time they will spend on common situations, like traveling to their jobs, working task’s timing, or shopping timing, etc. We assumed that it is possible to forecast the future by studying the past, especially the recent past and for many cases this is almost true. In fact what is needed is the “sampling distribution” from past outcomes.
However, medical time consultations are complex processes, and they do not follow the classical bell shape either that usually simplified calculations. First, consultations are bounded unevenly, they cannot take negative values, and they cannot take incredibly long values either. As in many other phenomena, the times for solving complicated medical cases are very long, but they are surrounded by many short consultations. This behavior is known as the “Pareto Principle”, and is commonly used to describe the income distribution in economics, among other things. Another characteristic is that many walk-in clinics work with more than one doctor, in parallel. Some overlaps among consultations are possible, i.e., more than one patient can leave/enter the consultation at the same time. For patients in the waiting room, this may look like the pace changing from very slow to very fast for an incomprehensible reason. Medical staff and patients have characteristics and conditions that change during the day and the consultation never reaches a steady state. Some patients leave without notice, or re-appear as delayed ones; other small mistakes can jeopardize the use of most complicated statistics.
The Particle Filter Algorithm to estimate waiting time
For real-life applications, forecasting consultation time and then patients’ notifications has to be easily automated. It must be fast enough to be useful for hundreds or thousands of patients, but also make sure not to penalize patients by asking them to show up too early. However, patients must show up early enough to eliminate the possibility of idle walk-in clinic staff, that is, patient tardiness. If errors happen, they must be corrected when possible.
To compute waiting times, one available tool is Sequential Montecarlo (SMC). In simple words, Sequential Montecarlo (or Particle Filters), are simulation methods useful for computing distributions and making estimations. By taking many samples from a distribution, it is possible to derive a numerical approximation of it. These numerical approximations can be as good as the original distribution for averaging, solving forecast problems, or even doing more calculations. If the real distribution is unknown, as usual, approximated distribution for sampling can be derived (observed) to perform simulations. We suppose that the unknown distribution and the observable approximated differ very little.
To overcome differences between them, we can think that every sample has a bigger/smaller weight in the representation that can be modified to re-shape the full representation. The differences can be iteratively reduced using new information to get an even better approximation. Despite the goodness of the method, it is technically and practically complex because some samples are not desired to continue improving the approximation and new samples must replace the unwelcome ones. The samples and their associated weights, which fully describe the distribution, are the particle filter representation. As the original distribution remains unknown, it is possible to use its particle filter representation for any calculation.
The questions of how to choose a first representation is a tricky one. For our work, we choose to rely on past data provided by our industrial partner. The particle representation considers a large number of samples. The related intensive calculations can be scheduled before the clinic begins operations to avoid increasing the calculation burden. This distribution is improved upon new information (new patient’s consultations) for economical, fast use and allowing multiple instances as the industrial partner needs to notify thousands of patients. For several reasons, we decided to leave the undesired samples and applied special rules to them (a quasi-bootstrap) and we correct most common errors in one extra step.
This extra step considers hundreds of consultation sequences containing known mistakes and changes in walk-in clinic operation recorded with our industrial partner devices. After that, we simulate consultations using the representation in several runs, aggregating the results for 8, 6, and 3 patients in advance, and cutting at the earlier cases to deliver a good forecast. Therefore, it will penalize more tardiness than earliness; this estimation does not abuse patients as it consecutively follows the consultation process and patients receive new notifications.
The next Figure shows that our method (PF) gives better results than Moving Average (MA), Last Value (LV), Linear regression (LR), Moving Median (MM) and Exponential Smoothing (SM). Errors keep a bias to advance patients’ arrival without penalizing waiting times.
The proposed method enables a patient to know when he/she should come back to the walk-in clinics without being late and missing his/her turn. Their waiting time at the clinic is hence significantly reduced on average which allows them to resume their normal life while still in line in the clinics. Also our method gives an estimate of the waiting time for consultation better than simple statistics. The method is intended to be used without supervision, as the system can auto-adapt easily. Future research includes several clinics working in a network and centralized web service can be used to register patients and balance loads among the clinics using travel time between patients’ homes and clinics.
A paper entitled Waiting-time Estimation in Walk-in Clinic has been published International Transactions in Operations Research. Intl. Trans. in Op. Res. 00 (2016) 1–24 DOI: 10.1111/itor.12353. The authors would like to thank CRSNG/NSERC (Natural Sciences and Engineering Research Council of Canada) (RGPIN-2014-05705) and Chronometriq Inc. for funding and collaboration. We are grateful to Pierre L’Ecuyer for his valuable comments and suggestions for this project.
Julio Montecinos is a professor in the Department of Systems Engineering at ÉTS. His research in freight logistics, operations and optimization aims at contributing to the development of sustainable and resilient cities.
Program : Operations and Logistics Engineering
Mustapha Ouhimmou is an associate professor at the Department of Automated Manufacturing of ÉTS. His primary research interests are in logistics, healthcare, and value chain optimization in the forest products industry.
Program : Automated Manufacturing Engineering
Research laboratories : NUMERIX – Organizational Engineering Research Laboratory for the Digital Enterprise CÉRIÉC – Centre for Intersectoral Study and Research into the Circular Economy CIRODD- Centre interdisciplinaire de recherche en opérationnalisation du développement durable