When the bandwidth resources in a packet-switched network are shared among sessions by MAX-MIN flow control each session is required to transmit its data into the network subject to the MAX-MIN fair rate which is solely determined by network loadings. This passive behavior of sessions if fact can cause seri-ous QoS(Quality of Service) degradation particularly for real-time multimedia sessions such as video since the rate allocated by the network can mismatch with what is demanded by each session for its QoS. In order to alleviate this problem we extend the concept of MAX-MIN fair bandwidth allocations as follows: Individual bandwidth demands are guaranteed if the network can accommodate them and only the residual network band-width is shared in the MAX-MIN fair sense. On the other hand if sum of the individual bandwidth demands exceeds the network capacity the shortage of the bandwidth is shared by all the sessions by reducing each bandwidth guarantee by the MAX-MIN fair division of the shortage. we present a novel flow control algorithm to achieve this extended MAX-MIN fairness and show that this algorithm can be implemented by the existing ATM ABR service protocol with minor changes. We not only analyze the steady state asymptotic stability and convergence rate of the algorithm by appealing to control theories but also verify its practical performance through simulations in a variety of network scenarios.