If the arrival rate exceeds the service rate (in steady state) then the average waiting time is infinity (because the line gets longer and longer). Now, in reality this won't happen. In the airport case, arrival rates are actually time-varying. So the line will grow, and then eventually the arrivals will slow down because it reaches an off-peak hour, and the lines will shrink.

A better answer to your question is that if arrival rates exceed service rates in the long run, people will become discouraged from arriving (i.e., people will change their behavior and travel less). One way to model this is economic or game-theoretic: individuals are heterogeneous, and each individual has a value for the service of air travel (if I need to go to a business meeting, my value is probably greater than the value of the guy going to vegas for the weekend). I am willing to pay a certain cost (waiting in line) to obtain the valuable service. As the length of the line increases, the cost I have to pay increases (have to get to the airport earlier, which takes valuable time out of my day). Thus, eventually the cost exceeds the benefit and I decide not to go through the hassle of air travel. Because individuals have different values, the low-value travelers will reduce their travel first and line lengths will reach an equilibrium that is tolerable for high-value travelers. This is actually not far-fetched at all-- a lot of people believe that post-9/11 security is somewhat responsible for the very slow rebound in air travel as a lot of people decided it was just as easy to drive.

In fact, from a mathematical standpoint, one does not even need to assume that individuals are heterogeneous to get this discouragement effect; even among homogenous individuals, if the service slows down, they will choose to receive the service less often, which slows the arrival rate and eventually puts the service system in equilibrium. This bit of analysis is perhaps a bit too technical to describe qualitatively.