Congestion problems in networks are alleviated with a method that works toward insuring that unused capacity will always exist in elements of a network that are resources which are shared by a plurality of users and which, consequently, can be overloaded. In response to each request to establish a connection over a path in the network, pursuant to a predetermined algorithm the method services the request, declines to service the request, or services the request after dropping an established connection. In one embodiment, when unused capacity on the path is above a first preselected level, all requests are serviced. When unused capacity falls below a second preselected threshold, no requests are serviced. When unused capacity is between the first and second thresholds, an existing data connection is dropped if a requested connection is a voice connection and the number of existing data connections exceeds a preselected threshold.