Queues are common sight in banks these days especially on Mondays and on Fridays. Hence queuing theory which is the mathematical study of waiting lines or queue is suitable to be applied in the banking sector since it is associated with queue and waiting line where customers who cannot be served immediately have to queue(wait) for service. The aim of this project is to determine the average time customers spend on queue and the actual time of service delivery, thereby examining the impact of time wasting and cost associated with it. The primary data were collected from the UBA branch at Okpara Avenue, Enugu, the data were collected based on the arrival pattern and service pattern of customers. The methodology employed followed the birth and death Markovian process. We further used the chi-square test to test the arrival pattern to determine if it follows a Poisson distribution and also tested the service pattern to determine if it follows an exponential distribution. The results obtained from the chi-square test showed that the arrival pattern follows a Poisson distribution and that the service pattern follows an exponential distribution, hence it can be analyzed using Markovian birth and death process. The results obtained showed that service rate is 0.1521 and arrival rate is 0.2157, the probability that servers are idle is 0.2786 and the cost incurred from waiting is N938.597. We were now forced to recommend based on the analysis, that the Bank management should increase the number of servers to three so as to help reduce the time customers spend on queue and also reduce cost incurred from waiting. We now concluded that the objective of this project was achieved.