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I play in a D&D 4e game at the local game shop where the number of players who shows up can vary from five to fifteen. Obviously a normal battle for fifteen characters would be overpowering for five, and the inverse is equally unsatisfying.
How can I quickly scale battles to a number of characters? Additionally should you scale entirely on player count or should it vary on role?
Start by figuring out your baseline encounter for five players. That's going to be the easy part. Then figure out individual monsters you can add to the encounter that are the same level as the PCs. You might want to have a couple of options: maybe one elite (which is worth two PCs), one normal monster, and a package of four or five minions.
Add one of those options for each additional PC, or for two PCs in the case of the elite. You're just scaling linearly, here, which is not going to be perfect but should be good enough. If you get ten people, just double the initial encounter. More than ten: double it, and add option packages again.
I wouldn't worry about scaling for the roles. With 15 people, chances are the balance will be decent.
You'll also need to scale the map, if possible. 4e fights take a lot of room: if you can increase your map size by 50% for 10 people, and double it for 15 people, your players will be happier.
Since a really big fight runs the risk of bogging down, I'd be willing to end those fights a bit earlier. It might be good to introduce morale rules so that people feel like they've won rather than having you arbitrarily call the fight.
One possibility is to have monster level advancement at three levels, for 5, 10 and 15 players, but that requires premade work.
If you want to improvise, you can make a group of two-legged monsters, and if many people show up, you can add one minion per additional player, or have one or two monsters riding a steed (with attacks) or having a defendant dire animal, wolf or similar creature.
On the nature of the minions, try to select something that goes down in two hits. If you don't have a cleric in the party, don't put skeletons (need blunt damage) or (worse) something with drain energy.
4E is one of the easier systems to scale battles in. The biggest help can be found on p. 57 of the 4E Dungeon Master's Guide. By figuring out the XP budget (# of players * the experience for a standard monster at the party's level), you have a rough idea of how many monsters you can add.
Make sure you don't add a bunch of monsters that it's going to be too hard for your players to hit and make sure you don't make it a cakewalk for the monsters to hit your players. The easiest way to avoid this is to simply add more monsters, while staying within your XP budget, instead of adding powerful monsters to fill up your XP budget.
With 15 people, this could really drag on. You might even consider cutting the HP of the monsters in half.
