If many short runs gives even half decent answers, then the Markov chain mixes fairly well in time n , and one long run will give the same answers as many short runs. So many short runs is only good when it's bad. The answers stink but don't stink quite so much as the answers from one long run. Here is another way of saying the same thing. If mn iterations isn't enough to reach equilibrium, then n isn't even close to enough, and many short runs doesn't sample anything remotely resembling the distribution of interest.
Many short runs can better diagnose nonconvergence.
There are two answers here. This method provides i. i. But it takes as long as it takes. If the chain is slowly mixing, then the Propp and Wilson method will take a very long time. For more on perfect sampling see Gary Wilson's web page on perfect sampling .