The current government of North Korea, you mean?
I believe it will last at least one more generation for sure, likely two generations, but beyond that point all bets are off.
My reasons for believing this have almost nothing to do with anything internal to North Korea. Rather, I am basing my assertion on the assumption that the current generation of Chinese leadership is not ready to stop propping up the government of North Korea, and it is somewhat doubtful that the next generation will be either. Beyond that is much harder to predict.
China is such a major political power that nobody can make them back down on this until they are ready. (Among other things, China has a permanent seat on the UN Security Council, complete with veto power, so no resolutions can be passed that they do not allow.)
North Korea as it exists today has no *practical* value to China (or to anyone), but practical value is only one of several major considerations in Chinese politics, and it is not the one held in highest esteem. Politically, maintaining the situation in North Korea is important.
First, the current government in North Korea is a Communist government, and while China is in practice no longer the same kind of blindly-anti-capitalist regime that we think of in the West when we hear the word "Communist", the government of China still holds very strongly to the *word* "Communist" as an important political position. This will gradually fade with time, but currently it remains essentially a third rail political issue in China, because a large percentage of the current leadership actually grew up under Mao's regime. I won't bore you with all the details here, but if you are genuinely interested, see the following Wikipedia article:
http://en.wikipedia.org/wiki/Generations_of_Chinese_leadership
The upshot of all that is, right now it is still dangerous in China to speak out against any of Mao's stated principles. Now, the Chinese government has very much moved away from what Mao *would have wanted*, especially in economic terms. That's all well and good, because what Mao *would have wanted* is, in the absence of official statements, a matter of opinion and speculation and interpretation. Moving away from what he actually *said* is more dangerous, for now, because everyone currently in a major position of leadership actually remembers him. Mao's importance in Chinese politics cannot increase; it can only decrease over time -- but that decrease in import is only going to happen rather gradually, as other important figures find their way into the limelight.
Second, North Korea's major neighbors are all enemies of China, at least on paper. The nation that has the most of all to gain from a change in the North Korean government is South Korea, and the other major ones are Japan and Russia. South Korea and Japan are both fairly open, representative-government countries with notions like freedom of speech and so forth, allied with the US and Western Europe. Regarding Russia, see the Wikipedia article on the Sino-Soviet Split; the rift is only partially healed, and China definitely does not _trust_ Russia. Thus, propping up the North Korean government provides a buffer -- a dead zone if you will -- between China and some nations that they're not too keen on. The importance of this issue is very gradually fading, because as China's economic interests become more and more integrated into the global free-trade community and interdependent with various first-world nations, it is gradually becoming clear that outright conflict between China and the various Western powers is a rather unlikely eventuality. (We put diplomatic pressure on one another over various issues, because we'd like to talk one another into things, but nobody gets hurt.) This change however is happening very *gradually*, in part because China's economic development has been very gradual in nature (because the PROC government planned it that way -- no sudden shocks, just keep moving slowly in the desired direction and eventually you get there). Thus, while the day will probably come when China no longer feels a political need to continue to prop up North Korea, that day has not yet arrived.
Chinese politics is really *weird*, and I'm not an expert on it, and it's possible that I'm missing something; but from what I can see, China is not ready to let the Communist government of North Korea fail yet. And China has more than enough resources to keep it propped up, at least for the time being. But that situation will not last forever.