The goal of this paper is to study heavy traffic asymptotics of many Additive Increase Multiplicative Decrease (AIMD) connections sharing a common router in the presence of other uncontrolled traffic, called "mice". The system is scaled by speed and average number of sources. With appropriate scalings of the packet rate and buffer content, an approximating delayed diffusion model is derived. By heavy traffic we mean that there is relatively little spare capacity in the operating regime. In contrast to previous scaled models, the randomness due to the mice or number of connections is not averaged, and plays its natural and dominant role. The asymptotic heavy traffic model allows us to analyze buffer management policies of early discarding as a function of the queue size and/or of the total input rate and to choose its parameters by posing an appropriate limiting optimal control problem. This model is intuitively reasonable, captures the essential features of the physical problem, and can guide us to good operating policies. After studying the asymptotics of a large number of persistent AIMD connections we also handle the asymptotics of finite AIMD connections whose number varies as connections arrive and leave. The data illustrate the advantages of the approach.