This is called, I believe, the "pulse" problem, where you get an extreme surge of demand. In general, there are two scenarios to consider:

(1) The typical rush hour rush, where people don't all leave work at the exact same time, but there is a general rush between the hours of 4-6PM, peaking on the hours. For line haul systems, people from a relatively large area aggregate and wait in the station; they don't all arrive at the station simultaneously, but they depart simultaneously because during the wait for a train, the station fills up. So the pulse in this case is not as severe as it would seem: the "pulse" in the station is actually due more to aggregation of people waiting for a train. In the case of PRT, there are two significant differences: (a) people arrive and immediately leave, so there is little aggregating. Some may have to wait in a short line during the heavier times, but with good design this is generally shorter (in the average case) than waiting for the next train to arrive. (b) PRT designs generally have more closely spaced stations, so the load is more spread out. Instead of, say, 1000 people arriving at one station in 15 minutes, you have 250 people arriving at 4 different stations.

(2) True "pulses", i.e. after a sporting event. When you have thousands of people exiting simulataneously from a sporting event, this is an extreme pulse that would place demands on any system (light rail, buses, cars). That's why there are traffic jams after sporting events, because the road system can't handle the pulse. PRT designs I've seen would handle this by building several larger stations at the statium itself to handle the anticipated loads. In other words, a stadium is where you'd expect to handle large crowds, and you'd also assume that a stadium would have the space to allow for several larger PRT stations. So there are ways of alleviating it, but there will still be delays after a sporting event, just like any other mode.