[Warning: This suggestion maybe was suggested before,I'm not sure so I need someone to verify if it was already suggested or not,Thank you.]
What I'm thinking about is that you add a in-game/forums button where allows you to call any online moderator in PM/Instant in-game chat...
For In-game,It would be something like "Xxxx needs a moderator in Xxxx"
For e.g..
"NoKi1119 needs a moderator in GmBets"
But in forums,It would be something like a PM or a pop-up window.
This would be even faster way to call mods in emergency instead of pm's or IRC,Which can save time,And only allowed to call a moderator once/twice per day,Or a week.

You do know this will have no one at the other end of it for long periods right?
We do have lives beyond TB, hence the way things are handled currently. Between jobs, school, college, uni most the staff only have a few hours a day to do all their stuff.
This would require that there's people there 24/7, and that aint gonna happen.

If there was no online moderator,Then a PM will be sent to only one moderator who gets online,And this "calling" message can only stay 2 hours,So it has it's own period,When the time passes,The message won't be sent to any moderator and gets automatic-ly deleted.
And from my experience the past few days,I've seen few moderators who is simply active for hours,Such as holotor/ishi etc.
What I'm trying to point out is that this calling message will not stay for a day,After the two hours it gets deleted,And the player can send once again the later day,Or just go into IRC,Since there are number of moderators in the IRC,As I said in the first post it may just send them a pop-out window just as how when you get a new message a window pops up,Very simple.

What about those of us who are connected 24/7 even though we're not here? That'll just result in things getting dumped on people who aren't available.

as I suggested earlier on a different thread, they can use it as an ingame broadcast for GMs that are online. otherwise since it's an ingame broadcast they can keep a log of all the previous ones right before they got online so they can help out if they miss it
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also only admins can see it so its not bothering regular players

People would troll too much. And people that don't know what the rules are would do it and say "this guy lifted me ban him", etc.

not supported.

It will turn into 'The boy who cry wolf' story.

I know that in some part of the whole deal I would go to the ban list and among other amusing ban texts there will be a new addition: ''I accidentally typed call mod instead of call mom''.

There is a saying: Don't call the devil 'cause he will come uncalled.