One of my unofficial volunteer work responsibilities is that of Paging System Monitor. This is someone who listens to a page over the public address system and then verifies that the pager hung up properly. Because if the person does not hang up the phone, no one else can issue a page.
Generally this doesn't happen often. But when it does, it's inconvenient. It can be embarrassing, too, if the pager continues speaking, thinking that the conversation is not getting broadcast to all the bosses.
Usually it's an easy job to find the off-hook phone, if the person included the (correct) phone extension number in the page. When you hear, "Joe Smith please call 73," for example, you can assume that the pager used the phone at extension 73. Then if the phone was not hung up properly, you (or Joe) can just call it, and that phone should ring on the second line.
But if it's one of those generic announcements, such as, "It's starting to rain. If you left your car windows open, now's a good time to close them;" or "I'll be rebooting the file server in five minutes. Please close all open files;" or "Would the janitor please come to the men's room -- it's an emergency. Please bring a mop," then the process turns into a fun game of deduction.
Well, in the case of the call for the janitor, you can assume that the phone is near one of the men's rooms." So you can walk over to any nearby office and check the phones. But if a location isn't implied, then you have to revert to recognizing the sound of the voice and then figure out whom it belongs to. This is easier than you might think. In fact, I had a good lesson in grade school in how easy "voice recognition" really is.
We were reading a story out loud, and the plot hinged on a character figuring out who someone was just by the sound of his voice. I asked the teacher how that could happen, and I expressed doubt that you can know who was speaking to you without actually seeing the person. So the teacher decided to really convince me. She invited me to close my eyes. Then one by one, each of my classmates spoke, and I had to guess who it was. There were only two out of about 30 voices that I couldn't figure out (and one of them belonged to someone named Joe, in fact). Yes, the teacher was pretty cool. But it was Open House Day, and our parents were also in the room. So I suspect she might not have done this if she weren't showing off for them.
Anyway, once you know who the person is, you can usually resolve the problem by calling his or her extension. Or you might have to walk around and find the person and then ask which phone was used. I had to do this recently for an announcement the janitor made about locking up the building. Usually he locks the building because he's the last to leave. But since he was leaving earlier than usual, he wanted to let us know that one of us had to lock up. It's a good thing I found him because he used a phone deep in purchasing that I wouldn't have guessed right away.
All this rigamarole wouldn't be necessary if the phone system's PA mode were designed to "time-out" after 30 seconds. But again, technology fails us yet again, and we resort to wasteful effort to compensate.