Even though I only moved house a kilometer down the road, my new position in reference to the subway station has really thrown a wrench in commute times. The new location of my place has now added 5 minutes to the walking time to reach the subway, but this has resulted in a NON-LINEAR increase in the total commute time to get to/from work. This non-linearity suggests why 30 minute commutes have now morphed into 40-minute and even 45-minute slogs.
Fortunately though, there is a solution. Until I get another bike, I will just walk faster. Specifically, I will cut the walking time by half the train interval time.
This is how my commute works, looking at it backwards
Commute stage D: Final subway station to work. 6 minute walk
Commute stage C: 2nd subway train. 5 minutes
Commute stage B: 1st subway train. 12 minutes
Commute stage A: Walk to the first subway station. 12 minute walk
What determines how this commute will go is the 2nd subway train as it runs in longer intervals (6-minutes). The first train runs at shorter 3-minute intervals, but how quickly I can get on the first train then determines what will happen with the 2nd train. Probability calculations bear this out and match up with common sense. In other words, whether or not the whole commute will take longer than 40 minutes depends mainly on the 2nd train's interval, but how quickly you can get on the FIRST train will give you the "edge" needed to make the total commute time less.
Let's say that "just getting on" the first train occurs at t = -4, 0, 4, 8, 12, 16, etc..
I'm including 4-minute intervals because of the 1-minute that the train is parked, the doors open, people push and shove their way on Shanghai style, and the doors close.
Similarly "just missing" occurs at t = -3, 1, 5, 9, 13, 17, etc.
Let's look at all the possible cases where I leave my house at a random time. My evil twin leaves at my former house which takes 5 minutes less walking time to reach the subway. We then subtract 5 from this random time
Case 1: I get there at time t = 1, miss the train. So I catch t = 4 train. My vil twin gets there at t = -4. He gets on right away. He is then two trains ahead of me, and is practically guaranteed 30-minute commute or less. As for me, it's practically a 40-minute commute or more. 25% chance of this happening.
Case 2: I get there at t = 2, catch the t = 4 train. Twin gets there at t = -3, catches t = 0 train. He is then one train ahead of me. 25% chance of this happening
Case 3: I get there at t = 3, catch the t = 4 train. Twin gets there at t = -2. Same result as above.
Case 4: I get there at t = 4, just catch the train. Twin gets there at t = -1. Same result as above.
This repeats for every cycle
Now if I am 6 minutes behind the control, we have these multiples
Train Leaves: 0, 4, 8, 12, 16, 20, 24, 28, 32
Me: 2, 3, 4, 5 (random) Catch: 4, 4, 4, 8
Twin: -4, -3, -2, -1 Catch: -4, 0 ,0 ,0
Here it is a disaster because there are now TWO cases where I'm two trains behind, and it's a 50% chance of that happening.
On the plus side, suppose I am only 4 minutes behind the "control twin"
In this scenario, I will never face the situation of being two trains behind, as we will then arrive one multiple apart, so one train apart always.
But what if I am 2 minutes behind the control?
Train leaves: 0, 4, 8, 12, 16, 20, etc.
Me: 3, 4, 5, 6 (random) Catch: 4, 4, 8, 8
Twin: 1, 2, 3, 4 Catch: 4, 4, 4, 4
Similarly, now there are two cases (50%) where I am one train behind, or we both get the same one
So in summary, this is what you should do:
1. Cut your walk time by HALF the train interval, to get a 50% chance you'll get a train earlier
2. Cut your walk time by a FULL train interval to guarantee you'll catch the earlier train
Hurrying and scurrying at a transfer station is only useful (50% success) if you can somehow reduce your transfer time by half the interval time of what you are trying to catch. Seems pointless, but when the distances are long between stations, it is possible. Locals will no doubt try.
A useful exercise is to pace out a normal walk between stations, without hurrying, or from your house to a station, etc. and also memorize the train intervals. For the most part, the intervals for each line are constant, due to how the signaling system works for each specific line. That is, the intervals depend on how many trains are running on a line, and not necessarily the time of day.
I'm actually going to try this out as an experiment to see if it agrees with the theory
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