Asymptotically optimal appointment schedules with customer no-shows

We solve the problem of scheduling appointments for a finite customer population to a service facility with customer no-shows to minimize customer waiting time and server overtime costs. We study the problem in fluid and diffusion frameworks. We find that it is optimal to schedule traffic so that the system is in critical load. That is, heavy-traffic is obtained as a result of optimization rather than as an assumption. Our proposed schedules are proved to be asymptotically optimal for a finite horizon in the fluid scaling and for large and finite horizon in the diffusion scaling. Problem and motivation. The objective of this paper is to solve the following problem: how should a finite number, N, of customers be scheduled to arrive at a single-server, first-come-first-served, queue over a finite time horizon [0,T ], such that the server is maximally utilized, but customers experience minimal wait. A closed form solution to this classic problem is only known when N = 2 and the service times are exponentially distributed (Mendel 2006). Here we study the problem under two different large population scaling limits: fluid and diffusion. The asymptotic analysis makes the problem tractable by allowing us to identify asymptotically optimal schedules that minimize an appropriately defined cost function that captures the trade-off between server utilization and customer delay. Our motivation stems from the problem of scheduling patients at outpatient clinics. Indeed, patient no show is a serious and costly problem in outpatient care (Cayirli, et al. 2006). It is generally agreed that overbooking is necessary, but the best sequence of scheduled epochs is still unknown. Model. The schedule, that is, the collection of appointment epochs of the N customers, is selected by a decision maker (DM), subject only to the constraint that these times are all within the given time horizon. If work remains beyond time T , the server continues to work until its completion. The model allows for scheduled customers not to show up, but those that show up do so precisely at the scheduled epoch. The ‘show-up’ variables are modeled as independent and identically distributed (i.i.d.) random variables. It is assumed that the service times are i.i.d. (and independent of the show-up variables), and generally distributed with finite second moment. We refer to these two stochastic elements of the model as the randomness. In this model, the DM does not have access to the randomness, at the time the schedule needs to be planned. We associate cost with two elements: the overall waiting time of the customers (in short: the ‘makespan’), and the completion time of the last customer relative to T (in short: the ‘overage’ time). The expected overage time is a proxy for the expected server utilization. The system is assumed to be overloaded, in the sense of strictly positive expected overage. The goal is to study the minimization of this cost over all schedules. We refer to the minimal cost as the value. A reference problem. Notice that scheduling problem is not a standard stochastic optimal control problem, since the optimization must be carried out prior to time zero when service commences and the optimizer must hedge against all possible stochastic outcomes (in expectation). Motivated by work on competitive ratios, we consider a natural reference model, in which an oracle provides the DM with complete randomness information. We refer to this as the ‘complete information’ (CI) problem. As it turns out, the CI problem is much easier to handle, and, in fact, can be solved explicitly in great generality. We study the difference between the values of the scheduling problem and the CI problem, referred to as the ‘stochasticity