I commute into Manhattan four or five days a week. Like a lot of people who ride the bus, I really **really** hate it when the Lincoln tunnel gets jammed. Even though there are tow trucks standing by at all times (from what I can tell), it seems to cost you about 30-60 minutes of commute time. Because it happens often enough to be a pain in the ass, I’ve been thinking lately about how the numbers play out.

As it turns out, there are statistics on bus usage in the lincoln tunnel available on the port authority website:

…in 2009, the XBL averaged 1,791 daily buses…

Since we’re talking averages, i’m going to assume that the weekday load is heavier than on weekends due to commuters. that should bring the average up a bit. The product of 1,791 buses * 7 days is 12,537 buses each week. Guesstimating that 85% of those buses pass through the tunnel Monday through Friday gives us 10,656 buses divided by 5 days for a weekday average of 2,131 buses.

That’s a lot of buses to push through a tunnel each day. But what about during peak commute times? The signage along the road says that the XBL is open from 6am to 10am. Since the statistics from the above website refer only to the XBL, we can infer that all of their measurements come from those morning commute hours. (There is no special commuter lane for buses leaving the city in the evenings that I’ve seen.) Therefore I’ll assume that these statistics refer only to the peak AM commute.

You might infer from the picture on the Port Authority website that there is only one lane leading up to and through the tunnel. I can confirm this. Not only is there only one bus lane, but it’s one lane that stretches quite a ways back from the tunnel and into New Jersey. Once inside the lane, you may not change into another lane, either (this is probably obvious, but worth noting).

What about bus reliability? How often do buses break down? Let’s be generous and assume that any given bus will make this trip successfully 99.9% of the time. This means that the bus will only break down once every 1,000 trips through the lane. I’d estimate that the XBL is ~3 miles long, measured beginning at the EZPASS out in Jersey and ending when the bus has entered normal city traffic around 40th street. Once inside the city proper, a bus can still stall and create problems in the tunnel, but we’ve got to end our measurement somewhere.

According to the multiplication rule ^{1}, we can get the probability that none of the 2131 buses going through the XBL will stall (that is, that they will *all* make it) by taking our probability that any given bus will make it and raising it to the 2131 power. This gives us 0.999 ^ 2131, yielding 0.119. This means that there is only an 11.9% chance that all 2131 buses will get through the XBL on any given weekday morning without stalling or breaking down!

But is this true? My anecdotal experience over the past few months of commuting says yes. I’ve had to sit through a number of traffic jams coming into the city in the mornings. As a result, I’ve switched to an earlier bus in order to avoid most of the peak commuter buses.

(Image courtesy Melisande under Creative Commons license.)

## Footnotes:

^{1} wonderfully described in John Paulos’ book *Innumeracy*