This study focused on the queuing system in Nigerian banking sector. The sources of data were primary and were collected from Zenith Bank Plc, Abakaliki Branch at the withdrawal section of the banking hall for four days in a period of one month interval. The data were collected based on the arrival pattern and the service pattern of customers. The methodology employed followed by the birth and death Markovian process. The results obtained showed that the arrival rate is 0.2219 and the service rate is 0.1524. However, the probability that the servers are idle is 0.1574 and the cost incurred from waiting is N656.76. Besides, any increment on the number of servers by the bank management will help reduce the time customers spend on queue and also reduce cost incurred from waiting. It was also suggested that the management of the concerned Bank should employ only capable servers as to reduce customers wait on service, that is when the cause of long time taken to serve one customer is not due to network or power supply.