I experienced personally what happens if you miss your flight with Delta Airlines last year. Traffic caused me to arrive at the airport late, and by the time I got to the gate, it had already closed. Despite my worry, the Delta crew was supportive. They put me on standby for the next flight because I got there within two hours of the departure time. Fortunately, I was given a seat at no additional cost. The secret, in my experience, is to move quickly and get in touch with a Delta agent as soon as possible. If you're prompt and courteous, they genuinely attempt to assist.