This is one of those questions that keeps popping up but under different headings, so it's hard to spot the duplicates.
Let me start be correcting your terminology a little. Based on the context of the question you should be asking about propagation, not replication, as the latter is between DNS peers (master & slaves).
There are a number of factors that determine propagation delay. On the surface it would appear to be a simple matter of using the TTL time. The normal approach when you know an address is to change is to reduce the TTL to a short period, say an hour, prior to making the actual change. If the TTL is set to less than an hour you can pretty much expect most systems to ignore it and use an hour anyway.
Unfortunately, an ever increasing number of systems are completely ignoring the TTL and are supplying cached information that may be out of date. There is absolutely nothing you can do about this, unless you are in control of those systems.
Even the local computer's cache can be a problem, as the operating systems may not even check the TTL, let alone honour it, so may be using incorrect information, even if the upstream DNS server has the correct information.