Appointment Scheduling for Parallel Queues

Parallel queueing systems serve as a natural modeling paradigm across a wide range of application areas, including manufacturing, parallel computing, communication networks, and healthcare. A key challenge in this context is appointment scheduling: determining optimal job arrival times to minimize an objective function that balances the perspectives of both service provider and clients. A specific aspect of this problem is that the objective function depends on the per-client joint distribution of sojourn times in the individual queues. In some applications, the focus is on the maximum sojourn time, while in others, the minimum is more relevant. In this paper we consider a parallel queueing system with two queues, to be used by clients with jobs that are characterized by five parameters: their per-queue means and variances, and the correlation coefficient between them. A primary contribution concerns a technique to efficiently approximate the (bivariate) sojourn-time distribution of each of the individual clients, by applying a convenient Weibull fit. Our numerical experiments show that this approach leads to a highly accurate approximation of the objective function. We conduct a series of numerical experiments that assess the accuracy and efficiency of our method, with a strong focus on its application in the context of appointment scheduling.