timing is everything?

"Jesus loved Martha and her sister and Lazarus, but oddly, when he heard that Lazarus was sick, he stayed on where he was for 2 more days. After the 2 days, he said to his disciples, "Let's go back to Judea." (John 11:5-7, The Message)

I was stuck on this story in the not-so-distant past and I think I might go somewhere with it in the near future. This morning I was pondering the timing of it all. Mary and Martha send word to Jesus that Lazarus is not doing so hot. Now, we know that previously Mary was at the feet of Jesus (the story introduces her this way), so we have an idea that maybe M&M aren't requesting that Jesus come sit shiva with them like good friends do. It wasn't a "get in your final goodbye" or "I need you here with me in this hard time" request. I don't think so. I think it was a "please, please Jesus come and save him or he's going to die." I'm going on a hunch that Mary and Martha were 2 ladies that "got it" that Jesus was the Messiah and did have the power to restore Lazarus back to health, so they asked him to do it.
Jesus hears the message and waits for 2 days before deciding to go back to Judea. On the third day (*ahem, note timing), he says, "welp, there's 12 hours of good travel time. Let's go."
There's a brief sidebar conversation by the disciples, and if left to their devices they never would've went to Judea at all- but that's a later conversation.
So they get to Bethany and they find that Lazarus is dead. Not just a "ooohhh, you just missed him." He's 4 days dead. In nowadays time, the funeral is over (in Jewish time, they've still got 3 days left).
So let's add this up... 4 days dead. Day 4 was the "12 hours of daylight" travel time. Day 2 and 3 was when Jesus was apparently sitting around and proving a point. Day 1- well, that's when Jesus got the message. And if you account for the 12 hours for the messenger to get to wherever Jesus was (which is a bit of a mystery at this point), we find that by the time Jesus learns of Lazarus' sickness, he was actually already dead.
So, this is my question to you: Why did Mary and Martha wait until Lazzie was on his deathbed before they called in Jesus? Why not when he was coughing and sputtering? Why is it that at the first sign of needing Jesus we think that we "shouldn't bother him" but when it gets to life and death all of a sudden we're sending urgent messages and yelling at him when he shows up 4 days too late?
I'm not putting the blame on M&M for Lazarus' death- Jesus said outright that it would be an occasion for all to believe. But I think we can look at M&M's actions and learn a little something about ourselves. But what kind of faith-test do we put ourselves, and Jesus, in when we wait until the moment of desperation to seek after Him? Why do we only want him at the very end, when it really, really counts?

Now, I know there's a place for desperation- I've been to that point when you need to realize that there's no other rock to which you can cling. But I wonder how the story would be different if Mary and Martha had asked Jesus to stop by a few weeks earlier because Lazzie had a hacking cough he just couldn't lick. Mary and Martha wouldn't have been frustrated with Jesus showing up a bit late. He would've had perfect timing, then.
But then again, there would be no miracle- a "turnaround sign for many of the Jews" (11:45).