A One-Track Mind

While at this point, we are almost assuredly beating a dead horse, until the Green Line Extension (Southwest LRT) is actually under construction, there remains the possibility it can be improved. While the best improvement (given the existence of an LRT to fourth ring Southwest suburbs) would be to route it along a path where people actually live, if we cannot maximize benefits, surely we should minimize costs.
I speak of course of the tunnel under the park.

Kenilworth Sections
Kenilworth Sections

The stated reason is the right-of-way is insufficiently wide to accommodate two tracks of LRT, one track of freight rail serving about 3 trains a day, a bike path, and the buildings that were built where it would have been convenient to run some more track.
There are two obvious solutions to this problem which have not been given serious consideration as far as I can tell.

First, the freight and LRT can share the track at different times. The experience with Northstar certain demonstrates why having a few passenger trains on a freight railroad can create lots of passenger delay, but this is different, it would be a freight train on a passenger track owned by the public.

Everyone says “But, FRA”. I realize there are institutional barriers which need to be overcome. Perhaps those are more expensive to overcome than $130 million, or whatever the difference in the surface solution and what the tunnel will cost.

Second, if one-track is good enough for freight, why is it not good enough for LRT for a short section? (This is an idea previously considered by Matt Steele at streets.mn.) This of course is a tight fit, and may require waivers from appropriate regulatory authorities, but is physically possible from the drawing I have seen.

Section B-B
Section B-B

For the sake of argument, let’s assume we want to single track 1.5 miles, with trains going up 45 miles per hour (say an average speed of 30 mph to make the math easy). This would take 3 minutes. The trains are on 10 minute headways in each direction, or one train every 5 minutes through the bottleneck. (Note, Matt assumed 2 minutes, and higher speeds. I am using conservative assumptions).

If timing were perfect, there could be zero delay from this scenario. This is a deterministic case. That is the assumption underlying Matt’s post.

However, as we know, timing is rarely perfect, so we need to look at stochastic delay. Stochastic is engineering jargon for random. Random is engineering jargon for a case where multiple outcomes have an equal likelihood of being chosen (or some are more likely than others, but we cannot be sure that would be the case).

Even when things are random, that doesn’t mean we cannot ascertain the average of the distribution.

Let’s suppose we  have an arrival rate of 1 train every 5 minutes (our arrival rate lambda=0.2 trains per minute), and a server rate of 1 train every 3 minutes (mu=0.33 trains per minute). If the systems is completely random (and we certainly hope it is better than that), we can use stochastic queueing theory to estimate the delay.

Worst case (aside from someone actively and maliciously controlling the trains so they do arrive at the same time (which implies that deterministic solutions with zero delay are possible)), we can model this as an M/M/1  queue (meaning, as wikipedia says: arrivals follow a Poisson process and job service times have an exponential distribution) . This assumes Markovian (random) arrival and departure processes and a single channel.

The utilization rate (rho = lambda/mu) is 0.6, meaning the server is busy 60% of the time.

Math gives us a formula for the average queue size:

Average queue size = rho/(1 – rho) = 1.5

Math gives us a formula for the average wait time :

E(w) = lambda / mu*(mu-lambda) = 0.2/(0.333*(0.333-0.2))=4.5 minutes

At 1 million passengers per month (12 million per year) for 30 years, this is 360 million people delayed 4.5 minutes=1.6 billion minutes of delay. At $20/hour, this is $533 million.

Clearly this value is larger than the cost of the tunnel.

On the other hand, perhaps we only need to single track for 0.5 miles.

In that case, the server time is 1 minute, so mu=1. Capacity utilization is 20% (i.e. rho is 0.2). Average queue size is 0.25 trains. The average wait time is 0.25 minutes.

Our 360 million people are delayed 0.25 minutes at $20/hour is $30 million. This is considerably less than the cost of the tunnel.

The train speeds could be adjusted so no-one would know they were delayed (i.e. trains would slow down approaching the switch, or be held at the previous station, as needed. And remember this is worst case, delay should be less than this with any competent schedule adherence. With perfect schedule adherence, they are indeed zero (our deterministic solution).

Single-tracking is a solution to high capital costs. It is not optimal, it has delay costs  that depend on the length of the stretch, headways, how much control Metro Transit has over running times, and so on.

Everything involves trade-offs.

There is of course a concern about running LRT next to (near) freight trains, carrying lots of explosive ethanol. I say, don’t do it. Run them at different times, even if on different tracks. If freight trains are only permitted at night, or in a mid-day window when an LRT is held upstream of the pinch-point for a few minutes, or ideally in a scheduled break, there should be zero chance of collision. There is always a chance of derailment – that doesn’t change, but derailment is less hazardous than collision for what I hope are obvious reasons.

In the long run, maybe freight will go away (e.g. once people stop using ethanol), go somewhere else, or another solution will be found. At that time, the line can be double-tracked if needed.

In the short term, the money saved could be used to temporarily relocate the trail to quiet residential streets nearby, compensate the neighborhood, give money to the Park Board, or any number other socially worthwhile goals.

Crossposted at streets.mn.