Congestion is a common phenomenon in the field of computer and communication networks. Congestion will occur when the rate of incoming traffic to an interface exceeds the rate of outgoing traffic from an interface. The permanent solution for congestion is often increasing the speed of server or providing an additional server to serve the customers who is waiting in the queue. But, increasing the speed of service is not possible in many real life situations. To overcome this difficulty, a modified queueing model is considered in this paper. Arriving customers are served through an classical M/M/1 queueing system until the queue size reaches the predefined capacity. An additional server is added in the queueing system when the queue size is exceeds the predefined level. An explicit expressions for steady- and transient- state probability distribution for the number of customers present in the queueing system are derived. We also derive the mean and variance in steady state. Further, numerical illustrations are presented to understand the model behavior.