The Reston Hospital Center has tried to be more transparent, by measuring the response time in their ER and posting the expected wait time on the Internet. (It was 9 minutes when I checked in when drafting this.)
I don't know whether they're measuring the extent to which people are using this, but they should. It seems to me like something which would be useful, assuming you're a person who uses the ER as a substitute for a doctor. Maybe I'm naive, but I'd guess there's not too many of those in Reston--it's rich enough most people will have health insurance and a doctor.
Of course there's also the issue of image building. Even if no one uses this, it does give the image of an up-to-date institution, which one wants if you have to go to the hospital. And it might have been easier to sell the idea of measuring ER response time to your ER staff if sold as a way to inform the customers, rather than as a way to make them more productive. I'm assuming that if they can cut the response time, they've reengineered their business process to be more efficient.