Queuing by patients in the out-patients department to access hospital services in Nigeria teaching hospitals is a teething concern to most healthcare providers. This causes inconvenience to patients and economic costs to the hospitals. Patients waiting for minutes, hours, days or months to receive medical services could result in waiting costs to them. Providing too much service could result in excessive costs. Also not providing adequate services could result in excessive waiting and costs. This study sought to determine an optimal server level and at a minimum total cost which include waiting and service costs in homogenous servers in order to reduce patients’ congestions in the hospital as low as reasonably practicable. The queuing characteristics in all the twenty-three (23) teaching hospitals in Nigeria were analysed using a Multi-server Queuing Model and the waiting and service costs determined with a view to ascertaining the optimal service level. The data for this study were collected through observations and interviews. The data was analysed using Quantitative Methods, Production and Operations Management (POM QM) and Queuing Theory Calculator Software as well as using descriptive analysis. The results of the analysis demonstrated that average queue length, waiting time of patients as well as over utilization of specialist doctors at the teaching hospitals could be reduced at an optimal server level and at a minimum total cost as against their present server level with high total cost which include waiting and service costs. Therefore, this call for refocusing so as to improve the overall patient care in our cultural context and meet the patient needs in our environment.