For what it is worth, here is my advice.
a. Diminish DM downtime by having each room, map, set of miniatures, etc. all set aside and orderly ahead of time.
b. We've instituted a "30 second rule" at our table, complete with a timer. You have 30 seconds to make your decision and have dice hit the table. Sure, your calculations and secondary actions might take longer, but if your dice don't hit the table in 30 seconds, you simply go on "all out defense" for your turn. It is a life-saver, and cut our encounter times in half. When we get more loose with the timing, our encounter times creep up again.
c. Transparency. Andy Collins talks a lot about this, but basically you have to get rid of the old attitude of "holding things secret and in reserve makes things more fun". It doesn't. Tell your players flat out what AC they need to hit, which monsters are minions, what the bloodied and dead values are for hitpoints, etc. Let it all hang out. It took me forever to let go of this (I'm from the old days when the DMG was the Forbidden Book, and it contained all the actual RULES, and only the DM knew how the hell this stupid game worked!) but it speeds things up.
d. Delegate. One player tracks monsters and health. One player controls hitpoints. One player controls tokens and tracking condition markers. One player controls the initiative, and is responsible for telling the next TWO players (or monsters) that their turn is coming up. Everybody has a job, and everybody works together to speed things up.
e. Power Cards and Gaming Devices. Cheatsheets. Condition token placards with *all* of the rules on them, so you can simply hand somebody a standup placard that says "I am Blinded" and let *them* worry about the rules. All of these silly, pre-designed, pre-printed little gamey devices are a real lifesaver, and cut your gaming time significantly! The rule is that you can *never* open a book during gametime. You actually shouldn't have to, if you've set up your game table properly.
f. Encounter Design. Long, grindy encounters tend to start to take longer as people's brains fuzz up. Never populate with "soldiers", but use lots of artillery and lurkers. Have something change midway through the encounter which impacts the way things are going (a cauldron overturns, creating new areas of difficult terrain, or a caged monster escapes, eating half of the enemy but then turning the battle into a 3-way fight). Keeping things interesting, mobile, and dynamic keeps people invested, keeps them awake, and keeps things moving. If things start feeling repetetive, they start slowing down as people get less invested.
g. "Just Go With It". Everybody has to recognize that the purpose is to keep things fun and exciting, and to keep things moving. Rules Lawyers are summarily ejected. If you don't think that a ruling was correct, keep your trap shut. If the DM screws up, he'll fix it after the break. "Just Go With It" needs to be at the top of everybody's head, so that they can make suggestions, hear the adjudication, and still keep the game moving. If everybody is too worried about getting things "right", you'll never get anything done.
h. "Do Something Cool" card. I read about this here on Enworld, and it really works. Along with a power card to do your At-Will attacks and your Basic attacks, make up a green at-will card that says "Do Something Cool!'. It reminds players that they are *supposed* to think outside the box, come up with half-baked ideas, and make things fun. With this handy little reminder, they come up with innovative ideas outside of their Power-Set, and the DM simply uses pg. 42 to adjudicate and keep things moving. The excitement generated by these "cool" moves, whether they succeed or fail, keeps things exciting, and therefore keeps things moving.
i. Have your Dice/Card/Whatever ready rule. We actually set up a protocol to stop bad behaviour. If you get up out of your chair to move your mini, take your card with you so you don't waste time moving back again. Have your dice set aside specifically for that ability (lots of my players keep little piles of dice on each card) and roll your d20 AND your damage together. Each player adopts behaviours which reduce their individual turns, and therefore the whole game moves along.
j. If you *must* play with 7 players, then how about adopting a co-DM? Why not have somebody run half the game alongside you, from the tactical perspective. They do half the rolling, half the calculating, but one DM runs the overall story and has ultimate say on how things go. You're already at a party of 8 people, so why not add a 9th and cut the gametime significantly?
One of my players could not stand it. Too much chaos and time wasted for her and she was about to quit until I explained what I needed in a co-DM. I typically let her run most of the NPC's in combat rolling dice/etc and some of the roleplaying of NPCs during encounters. It lets me focus on the big picture and deal with rule questions when they come up. She loves it as she gets to (virtually) beat on the rest of our friends and scare the dickens out of them from time to time speaking as an NPC. Sure she can say something unplanned, but I'm free to adjust plans to match what she says. The game has taken some great unexpected twists thanks to this and is richer for it.
