How many beds must be allocated to a specific clinical ward to meet production targets? When budgets get tight, what are the effects of downsizing a nursing unit? These questions are often discussed by medical professionals, hospital consultants, and managers. In these discussions the occupancy rate is of great importance and often used as an input parameter. Most hospitals use the same target occupancy rate for all wards, often 85%. Sometimes an exception is made for critical care and intensive care units. In this paper we demonstrate that this equity assumption is unrealistic and that it might result in an excessive number of refused admissions, particularly for smaller units. Queuing theory is used to quantify this impact. We developed a decision support system, based on the Erlang loss model, which can be used to evaluate the current size of nursing units. We validated this model with hospital data over the years [2004][2005][2006]. Finally, we demonstrate the efficiency of merging departments.

We consider two types of queues with workload-dependent arrival rate and service speed. Our study is motivated by queueing scenarios where the arrival rate and/or speed of the server depends on the amount of work present, like production systems and the Internet.First, in the M/G/1 case, we compare the steady-state distribution of the workload (both at arbitrary epochs and at arrival instants) in two models, in which the ratio of arrival rate and service speed is equal. Applying level crossing arguments, we show that the steady-state distributions are proportional. Second, we consider a G/G/1-type queue with workload-dependent interarrival times and service speed. Using a stochastic mean value approach, several well-known relations for the workload at various epochs in the ordinary G/G/1 queue are generalized.

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.

For capacity planning issues in health care, such as the allocation of hospital beds, the admissions rate of patients is commonly assumed to be constant over time. In addition to the purely random fluctuations, there is also typically a predictable pattern in the number of arriving patients. For example, roughly 2/3 of the admitted patients at an Intensive Care Unit arrives during office hours. Also, most of the scheduled admissions occur during weekdays instead of during the weekend.Using approximations based on the infinite-server queue, we analyze an M t /H /s/s model to determine the impact of the time-dependent arrival pattern on the required number of operational beds and fraction of refused admissions for clinical wards. In particular, the results show that the effect of the daily pattern is rather limited for clinical wards in contrast to the week-weekend pattern, for which the difference in the fraction of refused admissions across the week is considerable. We also show that an increased variability in length of stay distribution has a stabilizing effect on the time-dependent required number of beds. Finally, we demonstrate a method to determine the required number of beds across the week.Keywords Hospital capacity planning · Daily and weekly admission patterns · Time-dependent arrivals · Refused admissions · Infinite-server queues · Modified-offered-load approximation