Frequently, problems simply evaporate when a healthy market is brought to bear and a good is monetized. I see no reason the same cannot be true about work visas. Consider the following proposal:
Establish a class of work visas that can be traded freely. Such visas would provide the same labor market mobility as is enjoyed by citizens (ie, you can change jobs, start businesses, move about within the country, etc freely). Let's call them M-1 visas. An M-1 visa would be for a specific period of time, say a 6 year window (Jan 2007 - Dec 2013 for example). Each M-1 visa would be owned by it's current assignee. Ownership could only be transfered if the previous owner were verifiably outside the US, or possessed some other legal justification for being in the US.
We are currently in a situation were we have a large number of illegal aliens. To dig ourselves out of this situation, first secure the southern border (most likely acheivable with a wall/fence and increased patrols). Second, grant each illegal currently in the country an M-1 visa if they step forth to claim it. I realize this sounds like amnesty, and as if we are rewarding law breaking, but hear me out. What do you think most of the illegals currently in the country will do with their M-1 visa? Some of them might elect to remain in the country working for $10 an hour, 60 hours a week, 52 weeks a year (say around $31k a year, or about $20k a year after taxes) for the next six years. Or, you could go home and cash out your M-1 visa by selling it to someone else. How much do you think that M-1 visa is worth to an Indian engineer with a $100k job lined up in the bay area? $200k maybe? Effectively, the M-1 visa grant to the illegals is a payment to deport themselves. It consists of the US franking a new kind of currency (the M-1 visa) that it uses to pay people to leave.
And if a former illegal alien (now M-1 visa holder) chooses to stay, they now have a large asset to attach in the event of incuring expenses. If they wind up in the ER, the hospital can sue to collect by attaching their M-1 visa, possibly forcing their deportation in order to collect if they needed to foreclose to collect. If a former illegal alien (now M-1 visa holder) rear ends your car and has no insurance, you can sue to collect by attaching his M-1 visa. If a former illegal immigrant would like to start a business, they can offer their M-1 visa as collateral.
If there are really jobs American's won't do, then employers can lend money to mexicans to purchase M-1's in the open market, and bring them in legally to do those jobs. If we *really* need more imported tech workers, then their pay will begin to reflect the cost of purchasing their M-1 visas.
Failure to vacate the country upon visa expiration, or failure to cooperate with a deportation order (in the event of visa foreclosure) would be a felony, punishable by a mandatory 1 year in prison, and would disqualify a person from reentry under the M-1 visa program.
After the initial grant, a number of M-1 visa's could be auctioned off each year, that number to be determined by congress, much as we currently do with US treasury bonds. I personally suggest two million a year. You would naturally want to stop issuing all other sorts of work visas. Two million visa's a year at $200k each would bring in $400 billion a year in revenue to the US treasury.
Naturally, there would be certain basic requirements for being eligible to purchase an M-1. Clearly we wouldn't want to bring in criminals, or know terrorists. The most efficient means of handling this problem would be to allow *anyone* to pay to prequalify as an M-1 purchaser. This is a simple service that entails a background check, and since being prequalified conveys no right of entry, prequalify as many folks as will pay for it.
But what, you may ask, about those people who are patiently waiting in line to enter this country? The people who played by the rules? That depends on their numbers. If the number is small relative to the current illegals (say 2 million) then you simply grant those in line an M-1, just as you did for the illegals. There after there *is no line*. Queuing is a most inefficient way to allocate goods. If the number of large (say 24 million) you simply allow them each year as new M-1 visas go to auction to bid in a special preauction. Each one of them has *one* right to purchase an M-1 in such auctions. Over time, you burn off those who were in line, and they get a benefit for playing by the rules.