The variability of waiting time to first treatment for patients with prostate, breast, lung or colorectal cancer decreased between 2002 and 2012, which indicates improvements in equity in access to cancer care. This trend aligns with provincial efforts to improve access to and the efficiency of cancer care treatment in Ontario. The lack of consistent decreases in median waiting time highlights the need to identify improvement opportunities for cancer type-treatment type pairs with increasing median waiting times.

Background Wait times impact patient satisfaction, treatment effectiveness, and the efficiency of care that the patients receive. Wait time prediction in mental health is a complex task and is affected by the difficulty in predicting the required number of treatment sessions for outpatients, high no-show rates, and the possibility of using group treatment sessions. The task of wait time analysis becomes even more challenging if the input data has low utility, which happens when the data is highly deidentified by removing both direct and quasi identifiers. Objective The first aim of this study was to develop machine learning models to predict the wait time from referral to the first appointment for psychiatric outpatients by using real-time data. The second aim was to enhance the performance of these predictive models by utilizing the system’s knowledge while the input data were highly deidentified. The third aim was to identify the factors that drove long wait times, and the fourth aim was to build these models such that they were practical and easy-to-implement (and therefore, attractive to care providers). Methods We analyzed retrospective highly deidentified administrative data from 8 outpatient clinics at Ontario Shores Centre for Mental Health Sciences in Canada by using 6 machine learning methods to predict the first appointment wait time for new outpatients. We used the system’s knowledge to mitigate the low utility of our data. The data included 4187 patients who received care through 30,342 appointments. Results The average wait time varied widely between different types of mental health clinics. For more than half of the clinics, the average wait time was longer than 3 months. The number of scheduled appointments and the rate of no-shows varied widely among clinics. Despite these variations, the random forest method provided the minimum root mean square error values for 4 of the 8 clinics, and the second minimum root mean square error for the other 4 clinics. Utilizing the system’s knowledge increased the utility of our highly deidentified data and improved the predictive power of the models. Conclusions The random forest method, enhanced with the system’s knowledge, provided reliable wait time predictions for new outpatients, regardless of low utility of the highly deidentified input data and the high variation in wait times across different clinics and patient types. The priority system was identified as a factor that contributed to long wait times, and a fast-track system was suggested as a potential solution.

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We consider a Markovian multiserver queueing system with two customer classes, preemptive priorities, and reneging. We formulate this system as an infinite level-dependent quasi-birth-death process (LDQBD). We introduce an algorithm that endogenously truncates the level and calculates lower and upper bounds on stationary probabilities of this LDQBD such that the gap between the bounds can be any desired amount. Our algorithm can be applied to any LDQBD for which the rate matrices become elementwise nonincreasing above some level. This appears to be the first algorithm that provides bounds on stationary probabilities for an infinite-level LDQBD. To obtain these bounds, the algorithm first obtains lower and upper bounds on the rate matrices of the LDQBD using a novel method, which can be applied to any LDQBD. We then extend this algorithm to approximate performance measures of the system of interest and calculate exact lower and upper bounds for those that can be expressed as probabilities, such as the probability that an incoming low-priority customer will wait. We generate a wide range of instances with up to 100 servers and compare the solution times and accuracy of our algorithm with two existing algorithms. These numerical experiments indicate that our algorithm is faster than the other two algorithms for a given accuracy requirement. We investigate the impact of changing service rates on the proportion of low-priority customers served and their wait time, and we demonstrate how ignoring one of these measures can possibly mislead decision makers. Summary of Contribution: We contribute to operations research by modeling a practically important queueing system and developing an algorithm to accurately compute performance measures for that system. We also contribute to computer science by providing error and complexity analysis for the algorithm to solve a broad class of two-dimensional Markov chains with infinite state space.