LOSS NETWORKS WITH RESERVATION

Like many other health care institutes, rehabilitation centre "Het Roessingh" (Enschede, The Netherlands) has difficulties to satisfy the demand for rehabilitation care. In the scheduling department of this rehabilitation centre holistic (entire centre instead of single departments) and long-term view is lacking. This can easily result in suboptimal planning and may lead to suboptimal quality of care due to unsynchronized appointments at di�erent departments. Het Roessingh would like to utilize clinical pathways for both enlarging the scheduling horizon and enforcing the treatments for one patient at di�erent departments to be in the same time period. The latter implies that a patient can be forced to wait several weeks if this results in all treatments starting in the same week. Since rehabilitation care is typically non-emergency care, letting patients wait (less than a specified norm) is allowed in Het Roessingh. The fundamental mathematical research in this report originated from the rehabilitation care problem described above. We introduce a so-called reservation model and examine its properties. In this queueing model patients arrive to a tandem queue of an infinite server queue (the reservation queue) and a loss queue. The tandem network has an exceptional blocking rule; when a patient arrives to the reservation queue, the service requirements at both queues are drawn from the appropriate distributions and it is checked whether there are sufficient resources for the new patient at the second queue in the time interval the new patient requires service at the second queue. If resources are insu�cient, the patient is blocked and lost. This network represents a loss network in which resources can be claimed a random time in advance. In this research we �rst derive the stationary distribution of the reservation model. Finding the stationary distribution appeared to be di�cult when we tried solving the Kolmogorov differential equations. Therefore we derived the stationary distribution for a deterministic single server reservation model by means of a renewal theory argument. With this result we prove the reservation model does not have a product-form stationary distribution. Besides these analytic results, we performed an extensive simulation study. In this study we compared the probability an arriving patient is blocked (hereafter: blocking probability) in the reservation model with the blocking probability in the ordinary loss queue. We found that for deterministic reservation and/or service time the reservation model has a blocking probability greater than or equal to the blocking probability of the loss queue without reservation. We proved this claim for all capacity constraints of the second queue. For exponential reservation and service requirements the results were the other way around; reservation results in less patients being blocked. These results were found for several capacity constraints of the loss queue, but we only give an outline of the proof for capacity 1.