René Bekker

Scheduling admissions and reducing variability in bed demand

Variability in admissions and lengths of stay inherently leads to variability in bed occupancy. The aim of this paper is to analyse the impact of these sources of variability on the required amount of capacity and to determine admission quota for scheduled admissions to regulate the occupancy pattern. For the impact of variability on the required number of beds, we use a heavy-traffic limit theorem for the G/G/∞ queue yielding an intuitively appealing approximation in case the arrival process is not Poisson. Also, given a structural weekly admission pattern, we apply a time-dependent analysis to determine the mean offered load per day. This time-dependent analysis is combined with a Quadratic Programming model to determine the optimal number of elective admissions per day, such that an average desired daily occupancy is achieved. From the mathematical results, practical scenarios and guidelines are derived that can be used by hospital managers and support the method of quota scheduling. In practice, the results can be implemented by providing admission quota prescribing the target number of admissions for each patient group.

Optimal Admission Control In Queues With Workload-Dependent Service Rates

We consider a queuing system with a workload-dependent service rate. We specifically assume that the service rate is first increasing and then decreasing as a function of the amount of work. The latter qualitative behavior is quite common in practical situations, such as production systems. The admission of work into the system is controlled by a policy for accepting or rejecting jobs, depending on the state of the system. We seek an admission control policy that maximizes the long-run throughput. Under certain conditions, we show that a threshold policy is optimal, and we derive a criterion for determining the optimal threshold value.

Queues with waiting time dependent service

Motivated by service levels in terms of the waiting-time distribution seen, for instance, in call centers, we consider two models for systems with a service discipline that depends on the waiting time. The first model deals with a single server that continuously adapts its service rate based on the waiting time of the first customer in line. In the second model, one queue is served by a primary server which is supplemented by a secondary server when the waiting of the first customer in line exceeds a threshold. Using level crossings for the waiting-time process of the first customer in line, we derive steady-state waiting-time distributions for both models. The results are illustrated with numerical examples.

An M/G/1 queue with adaptable service speed

We consider a queueing system where feedback information about the level of congestion is given right after arrival instants. When the amount of work right after arrival is at most (respectively, larger than) K, then the server works at speed r 1 (respectively, r 2) until the next arrival instant. We derive the distribution of the workload right after and right before arrivals, as well as in steady state. In addition, we consider the generalization to the N-step service rule.

Queues with Lévy input and hysteretic control

We consider a (doubly) reflected Lévy process where the Lévy exponent is controlled by a hysteretic policy consisting of two stages. In each stage there is typically a different service speed, drift parameter, or arrival rate. We determine the steady-state performance, both for systems with finite and infinite capacity. Thereby, we unify and extend many existing results in the literature, focusing on the special cases of M/G/1 queues and Brownian motion.

Care on demand in nursing homes: a queueing theoretic approach

Nursing homes face ever-tightening healthcare budgets and are searching for ways to increase the efficiency of their healthcare processes without losing sight of the needs of their residents. Optimizing the allocation of care workers plays a key role in this search as care workers are responsible for the daily care of the residents and account for a significant proportion of the total labor expenses. In practice, the lack of reliable data makes it difficult for nursing home managers to make informed staffing decisions. The focus of this study lies on the ‘care on demand’ process in a Belgian nursing home. Based on the analysis of real-life ‘call button’ data, a queueing model is presented which can be used by nursing home managers to determine the number of care workers required to meet a specific service level. Based on numerical experiments an 80/10 service level is proposed for this nursing home, meaning that at least 80 percent of the clients should receive care within 10 minutes after a call button request. To the best of our knowledge, this is the first attempt to develop a quantitative model for the ‘care on demand’ process in a nursing home.

The impact of scheduling policies on the waiting-time distributions in polling systems

We consider polling models consisting of a single server that visits the queues in a cyclic order. In the vast majority of papers that have appeared on polling models, it is assumed that at each of the individual queues, the customers are served on a first-come-first-served (FCFS) basis. In this paper, we study polling models where the local scheduling policy is not FCFS but instead is varied as last-come-first-served (LCFS), random order of service (ROS), processor sharing (PS), and shortest-job-first (SJF). The service policies are assumed to be either gated or globally gated. The main result of the paper is the derivation of asymptotic closed-form expressions for the Laplace–Stieltjes transform of the scaled waiting-time and sojourn-time distributions under heavy-traffic assumptions. For FCFS service, the asymptotic sojourn-time distribution is known to be of the form

Heavy-traffic limits for polling models with exhaustive service and non-FCFS service order policies

U \varGamma