## 1 Introduction  
Overcrowding of hospital waiting rooms by patients characterizes the outpatient departments (OPDs) of healthcare centers of developing countries [1; 2]. It is true that most of the patients that visit health centers have varying health challenges and some may require prompt medical attention. However, due to overcrowding, these patients are made to endure a lengthy and painful wait for treatment [3; 4]. In the developing world, the problem is aggravated by the fact that health facilities are few, scattered and under-staffed. In such countries, it is a common experience for patients visiting an OPD to return home without receiving medical attention. For example, data collected from the patients record book at one general OPD showed that, within a one month period, of the 2211 patients who walked-in into the OPD department, only 1870 (84.58%) patients were seen and 341 (15.42%) patients were not seen [5]. To deal with such excess demand, physicians may opt to work overtime or delegate some work to nurses. Consequently, this may cause the healthcare center to incur extra costs and/or provide reduced quality of service to the patients [6; 7]. Reduced quality of service might impact patient's trust in the health facility and/or practitioners [8]. Overcrowding in hospitals also relates to patients' safety. Due to limited space, patients may be placed in inappropriate spaces, which can be a recipe for complications and fatalities. The challenge further result from the fact that, in the developing world, patients' hospital visits (especially to the OPD) are rarely scheduled, which makes it difficult to project daily demand for the services [9; 10]. This leads to an inbalance between hospital staffing ratios and health service demand. It is the case in many situations that as one heath center is being pressed with high demand beyond its capacity, the facilities nearby may have few patients. However, there is no mechanism for a timely exchange of information about patient load between health facilities. This leads to patients in overcrowded health facilities having to endure a lengthy and painful wait for treatment which may not even be given. Patients are usually transferred between health facilities and the purpose for such transfers is to maintain the continuity of medical care. Normally, these are patients that require specialized treatment or procedures and are transferred to facilities with such specialized equipment and personnel, i.e., the benefits of care available at another facility against the potential risks involved. The patient transfer process involves such key elements as the decision to transfer and notification, patient pre-transfer stabilization and preparation, the choice of appropriate mode of transfer (e.g., land or air transport), personnel to accompany the patient, equipment and monitoring required during the transfer and finally the documentation and handover of the patient at the receiving facility [11; 12]. In each transfer these key elements are followed so as not to affect patient prognosis. Since our focus is the outpatient department, we make the following assumptions: (1) Upon arrival, patients are triaged such that the critical patients are served first and then the least critical last. (2) Excess patient loads can only be transferred once, i.e., a patient can only be transferred once. As pointed out above, the patient transfer process starts with a decision to transfer the patient because of the exposure of the patient and the staff to additional risks and additional expenses for the guardians. A senior consultant level doctor is responsible for making the decision to transfer the patient. A patient's guardians are made aware of the benefits and risks involved during such a transfer. Written and informed consent of patients' relatives along with the reason(s) to transfer is mandatory before the transfer. In this work, the role of the senior doctor will be played by the central controller of the system.