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Setting up optimal appointment schedules requires the computation of an inherently involved objective function, typically requiring distributional knowledge of the clients' waiting times and the server's idle times (as a function of the appointment times of the individual clients). A frequently used idea is to approximate the clients' service times by their phase-type counterpart, thus leading to explicit expressions for the waiting-time and idle-time distributions. This method, however, requires the evaluation of the matrix exponential of potentially large matrices, which already becomes prohibitively slow from, say, 20 clients on. In this paper we remedy this issue by recursively approximating the distributions involved relying on a two-moments fit. More specifically, we approximate the sojourn time of each of the clients by a low-dimensional phase-type, Weibull or Lognormal random variable with the desired mean and variance. Our computational experiments show that this elementary, yet highly accurate, technique facilitates the evaluation of optimal appointment schedules even if the number of clients is large. The three ways to approximate the sojourn-time distribution turn out to be roughly equally accurate, except in certain specific regimes, where the low-dimensional phase-type fit performs well across all instances considered. As this low-dimensional phase-type fit is by far the fastest of the three alternatives, it is the approximation that we recommend., Comment: 21 pages, 8 figures

This paper aims to address the integrated routing and appointment scheduling (RAS) problem for a single service provider. The RAS problem is an operational challenge faced by operators that provide services requiring home attendance, such as grocery delivery, home healthcare, or maintenance services. While considering the inherently random nature of service and travel times, the goal is to minimize a weighted sum of the operator's travel times and idle time, and the client's waiting times. To handle the complex search space of routing and appointment scheduling decisions, we propose a queueing-based approach to effectively deal with the appointment scheduling decisions. We use two well-known approximations from queueing theory: first, we use an approach based on phase-type distributions to accurately approximate the objective function, and second, we use an heavy-traffic approximation to derive an efficient procedure to obtain good appointment schedules. Combining these two approaches results in a fast and sufficiently accurate hybrid approximation, thus essentially reducing RAS to a routing problem. Moreover, we propose the use a simple yet effective large neighborhood search metaheuristic to explore the space of routing decisions. The effectiveness of our proposed methodology is tested on benchmark instances with up to 40 clients, demonstrating an efficient and accurate methodology for integrated routing and appointment scheduling., Comment: 25 pages, 10 figures

Recent studies indicate that in many situations service times are affected by the experienced queueing delay of the particular customer. This effect has been detected in different areas, such as health care, call centers and telecommunication networks. In this paper we present a methodology to analyze a model having this property. The specific model is an M/M/c queue in which any customer may be tagged at her arrival time if her queueing time will be above a certain fixed threshold. All tagged customers are then served at a given rate that may differ from the rate used for the non-tagged customers. We show how it is possible to model the virtual queueing time of this queueing system by a specific Markov chain. Then, solving the corresponding balance equations, we give a recursive solution to compute the stationary distribution, which involves a mixture of exponential terms. Using numerical experiments, we demonstrate that the differences in service rates can have a crucial impact on queueing time performance., Comment: 23 pages, 7 figures

We describe the models we built for hospital admissions and occupancy of COVID-19 patients in the Netherlands. These models were used to make short-term decisions about transfers of patients between regions and for long-term policy making. We motivate and describe the model we used for predicting admissions and how we use this to make predictions on occupancy.

This paper considers a population process on a dynamically evolving graph, which can be alternatively interpreted as a queueing network. The queues are of infinite-server type, entailing that at each node all customers present are served in parallel. The links that connect the queues have the special feature that they are unreliable, in the sense that their status alternates between 'up' and 'down'. If a link between two nodes is down, with a fixed probability each of the clients attempting to use that link is lost; otherwise the client remains at the origin node and reattempts using the link (and jumps to the destination node when it finds the link restored). For these networks we present the following results: (a) a system of coupled partial differential equations that describes the joint probability generating function corresponding to the queues' time-dependent behavior (and a system of ordinary differential equations for its stationary counterpart), (b) an algorithm to evaluate the (time-dependent and stationary) moments, and procedures to compute user-perceived performance measures which facilitate the quantification of the impact of the links' outages, (c) a diffusion limit for the joint queue length process.We include explicit results for a series relevant special cases, such as tandem networks and symmetric fully connected networks.

In this paper we analyze the transient behavior of the workload process in a L\'evy input queue. We are interested in the value of the workload process at a random epoch; this epoch is distributed as the sum of independent exponential random variables. We consider both cases of spectrally one-sided L\'evy input processes, for which we succeed in deriving explicit results. As an application we approximate the mean and the Laplace transform of the workload process after a deterministic time.