Risk severity in transportation network analysis is defined as the effects of a link or network failure on the whole system. Change accessibility (reduction in the number of jobs which can be reached) is used as an integrated indicator to reflect the severity of a link outage. The changes of accessibility before-and-after the removing of a freeway segment from the network represent its risk severity. The analysis in the Minneapolis – St. Paul (Twin Cities) region show that links near downtown Minneapolis have relative higher risk severity than those in rural area. The geographical distribution of links with the highest risk severity displays the property that these links tend to be near or at the intersection of freeways. Risk severity of these links based on the accessibility to jobs and to workers at different time thresholds and during different dayparts are also analyzed in the paper. The research finds that network structure measures: betweenness, straightness and closeness, help explain the severity of loss due to network outage.
According to the US Department of Education (reported in wikipedia), the University of Minnesota has (Fall 2013) 63,929 students, ranking 19th in the US. Of course, many of those larger schools are for-profit, non-residential, cater to part-time students, or college systems. Among “peers”, (land-grantish research schools) larger schools include Arizona State, Central Florida, Ohio State, and Maryland.
Alternatively, the Twin Cities campus is the 9th largest public university in the US by enrollment, with 48,308 students (undergraduate about 34,500), falling several places, and several thousand students from 2009, when it was 4th.
There are numerous economies of scale to be had from a larger school (i.e. costs rise sub-linearly with the number of students because the delivery of education has a large fixed cost component like buildings and chairs and lectures and software and administration), and the evidence about quality of education with size of school (and even to an extent size of classes) is mixed (though everyone seems to like smaller classes better). (The evidence about student satisfaction tends to favor small elite liberal arts schools with a lot of hand-holding, land grants like the University of Minnesota will never compete in that market). These economies of scale should lower per pupil costs. Rankings like US News favor small class sizes, but also favor large universities.
Clearly the University of Minnesota rejects more students than it accepts (it has a 44% acceptance rate). If it just one year raised that to 88% several things would happen. First there would be more students (though it is not clear enrollees would double, more than double, or less than double, since rejected students might be more or less likely). Second, because it lost some of its exclusivity, fewer top tier students would apply. Many faculty would similarly find alternatives – not wanting to deal with the headaches twice as many students brings, though I imagine teaching slots would be filled with the glut of PhDs on the market in most fields. Class sizes would undoubtedly increase, but more classes could be offered. [After World War II, Georgia Tech, which saw an influx of GIs, expanded its night school, so that there would be a full second shift of courses]. Yet the quality of the University would on average drop.
Trends working in the opposite direction of increasing campus size include MOOCs, encouraging more online, distance learning. Getting those to work well has been difficult, and they are yet to be mainstream. Moreover a seldom-stated purpose of university, keeping students out of the labor market and in a semi-protected environment between home and the harsh cruel world cannot be accommodated with MOOCs. Demographics also move against rapidly expanding university size overall. However any one university should be able to increase market share, both with domestic and international students. Another purpose of university, providing a mixing bowl for organization of individuals into groups and couples, requires in-person attendance.
However if this were done more gradually but intentionally, with an aim of attracting more and better students, it should be possible to grow the campus to 100,000 over a decade or two.
From a physical perspective, the campus has significant room to grow, especially to the North East, but also infill and intensification to the south-east, north, and on West Bank. Even more growth is possible on the St. Paul campus, which abuts the largely underutilized State Fair Grounds.
My guess is doubling enrollment would require far less then twice as much new classroom space, about a doubling of dorm and residential space, a bit less than doubling of graduate student offices and labs and faculty offices – less assuming many of the new teaching responsibilities would be borne by adjuncts rather than regular faculty.
Around campus, business would boom. New apartments, with ground floor retail and restaurants would fill in any of the empty parcels of Stadium Village and Dinkytown and West Bank.
Transit ridership to campus would more than double, since the capacity for cars on campus would decrease significantly (all those new buildings will use up surface parking lots, reason itself to support an expansion, pedagogy be damned) while total travel demand in terms of trips to campus rose.
Such a change would challenge Boston’s 250,000 students (not sure how this was calculated, must be “Greater Boston”) as the lead college town in the US. According to city-data, Boston’s college population is 14.6% of the total. The City of Minneapolis has 11.3%, near the top for large cities. Increasing enrollment by 50k students would increase the share of the City’s population in college by up to 12% (depending on where they lived, the maximum assumes they all live in Minneapolis proper), putting Minneapolis at about 23%. Clearly much of Boston’s attractiveness to college students is the surfeit of students already there. We don’t really operate in that league here yet.
If we think of college as a temporal port, just as immigrants from Europe used to land mostly in New York, and occasionally disperse from there, immigrants from youth land in college, and disperse slowly from that point. This should greatly increase innovation, as the region’s engine of growth scaled up and the region attracted boatloads of Generation Z/Post-Millennials to its lake shores.
This paper tests a group decision-making model with altruism to examine the school travel behavior of schoolchildren aged between 6 and 18 years in the Minneapolis-St. Paul metropolitan area. The school trip information of 1,737 two-parent families with a schoolchild is extracted from Travel Behavior Inventory data collected by the Metropolitan Council between the Fall 2010 and Spring 2012. The proposed model has four distinctive characteristics compared with traditional developed models in the field of school travel behavior including: (1) considering the schoolchild explicitly in the model, (2) allowing for bargaining or negotiation within households, (3) quantifying the intra-household interaction among family members, and (4) determining the decision weight function for household members. This framework also covers a household with three members, namely, a father, a mother, and a schoolchild, while unlike other studies is not limited to dual-worker families. To test the hypotheses, we developed two models with and without the group-decision approach. Further, the models are separately developed for different age groups, namely schoolchildren aged between 6-12 and 12-18 years. This study considered at a wide range of variables such as work status of parents, age and gender of students, mode of travel, and distance to school. The findings of this study demonstrate that the elasticities of two modeling approaches are different not only in the value, but in the sign in some cases. In more than 90 percent of the cases, further, the unitary household model overestimates the results. More precisely, the elasticities of unitary household model are as large as 2 times more than that of the group-decision model in 25 percent of cases. This is a direct consequence of model misspecification that misleads both long-term and short-term policies where the intra-household bargaining and interaction is overlooked in travel behavior models.
Using detailed travel surveys (the Travel Behavior Inventory) conducted by the Metropolitan Council of the Minneapolis/Saint Paul (Twin Cities) Region in Minnesota for 1990, 2000-2001, and 2010-2011, this report conducts an analysis of changes in travel behavior over time. Specifically looking at changes in travel duration, time, use, and accessibility; telecommuting and its relationship with travel and residential choices; transit service quality and transit use; effects of age and cohort; and changes in walking and bicycling. Much has changed in this period, including the size of the region, demographics, economics, technology, driver licensing, and preferences, examining in turn the effects of investment, development, and population change on behaviors for the Minneapolis-St. Paul region as a whole and for areas within the region. While this research cannot hope to untangle all of the contributing factors, it aims to increase understanding of what did happen, with some explanation of why. This will inform transportation engineers, planners, economists, analysts, and decision makers about the prospective effects of future changes to networks, land use, and demographics while also evaluating the effects of previous network investments.
Thanks to the full team of authors David Levinson, Greg Lindsey, Yingling Fan, Jason Cao, Michael Iacono, Martin Brosnan, Andrew Guthrie, and Jessica Schoner, The MnDOT and CTS staff, and the 25 member Technical Advisory Panel.
Chapter 1 Accessibility and the Allocation of Time: Changes in Travel Behavior 1990-2010
Chapter 2 Telecommuting and its relationships with travel and residential choices: An exploration of the 2000 and 2010 regional travel surveys in the Twin Cities
Chapter 3 Transit Service Quality and Transit Use
Chapter 4 Cohort Analysis of Travel Behavior
Chapter 5 Biking and Walking Over Time
Appendix A Development of the Travel Behavior Over Time Database
Appendix B Additional Tables and Figures
Appendix C Glossary and Acronyms
A week after I came to the Twin Cities (June 1999), I picked up a City Pages and saw this great article by Katy Reckdahl: The Cars of St. Bonifacius.
Head west on Highway 7 past Excelsior about 35 miles from Minneapolis, cross the rails put in place a century ago by the Great Northern Railroad, and you’ll be in St. Bonifacius, population 1180. Immediately off to the right across a grassy ditch is the St. Boni Farm Store, which Tom Logelin’s father started in 1932 as a feed and seed store and Logelin has continued as an appliance outlet. Logelin–a dignified man with a shock of wavy gray hair and an “I’d Rather Be Fishing” belt buckle that explains his nut-brown tan–gracefully winds up a dishwasher demonstration before approaching another potential customer.
“Oh, yeah, the taxicab thing,” he says when informed of the visitor’s question. “We register more damn taxis out here than any other place around. We always know it’s that time of the year–October or November–because there are a stream of taxis stopping here, asking for directions.” But this year may see the last time Logelin leaves his sea of white Whirlpools to gesture the way to city hall (up the hill and to the right). If state regulators have their way, St. Boni will have to gear up its bureaucracy, or forsake its unlikely status as the metro’s taxicab capital.
First, some numbers. A decade ago, according to city hall estimates, St. Bonifacius had only about 50 licensed cabs. But in the last few years, cabdrivers around the metro area have found out that Logelin’s directions lead them to one of the best license deals around: $50 per car per year, compared with more than $300 in St. Paul and $400 in Minneapolis. Airport cabdrivers, who have to be licensed with a metropolitan city in order to receive a permit, have been taking notice. In 1998, 390 cars–more than two-thirds of all airport cabs–were licensed in St. Bonifacius. That’s roughly one cab for every three people in St. Bonifacius, for a total number that edges out the 343 registered cabs in the city of Minneapolis, and eclipses the 124 licensed by St. Paul. …
As part of my tour of Main Streets of Minnesota, I accidentally stumbled upon St. Bonifacius on my way to Hutchinson and New Ulm (see coming episodes). I knew intellectually this was a small town. But given its legend as taxi capital of Minnesota, it was smaller than anticipated.
The article continues:
St. Bonifacius wasn’t always a taxi town. The area was settled in the 1850s by German immigrants, among them Tom Logelin’s grandfather, and Logelin serves as the community’s unofficial historian. At one point, he boasts, “we were a hub of commerce for the surrounding area. Much bigger than Mound, Excelsior, Waconia.” The locally headquartered Minnetonka Canning Company “was the biggest cannery west of the Mississippi. Every canned good on the Great Northern came from St. Boni.”
But the advent of the automobile changed all that, Logelin says, allowing people to travel further for their errands and cutting into business on Main Street. Car travel also hurt the rail business, which put the pinch on Minnetonka Canning. The firm shut down its assembly lines during the Depression. “Isn’t it ironic?” Logelin asks, in an oratory style used most recently when he spoke at the local Memorial Day service. “Whereas in the old days the development of the automobile started the demise of St. Boni, today’s great income is from the automobile.”
Main Street (County Highway 92) intersects Highway 7, and the town is a couple of blocks in from the current right-of-way of the highway, centered on the intersection with Kennedy Memorial Drive, a block south of Old Minnesota 7. The town now has over 2200 people.
So if you are ever on Highway 7 and looking for a Meat Raffle, a Ham Bingo, or a playground for the kids, take a right on Main Street.
Accessibility to jobs by transit for every Census block in the Twin Cities metropolitan area, between 7 and 9 AM. From the Accessibility Observatory.
Other movies for 46 metropolitan areas can be seen on Vimeo
MoveMN, the local infrastructure lobby, has a second major proposal. This one is for transit funding in the Twin Cities metropolitan area. They propose:
Increase the Twin Cities seven-county sales tax ($342 million)
The current ¼ cent metro sales tax has helped build the new Green Line (Central Corridor) and Red Line (BRT from Apple Valley to Mall of America), and fund half the cost of operating the region’s transitways. The sales tax is not enough to build the transit system the region needs to foster long-term economic development. Move MN proposed increasing the current sales tax by ¾ cent, applying the sales tax in all seven counties and using a small portion (10 percent) of the tax to fund safe and accessible bike and pedestrian connections in the metro. The seven-county sales tax for transit would become a full one cent.
The additional transit funding would essentially complete the current Metropolitan Council transit plan within 15 years, including expanding bus service hours and coverage creating faster commutes for Metro Transit and suburban transit bus systems, as well as a wider network of bikeway and pedestrian connections.
The Sales Tax
Ideally transit would be paid for by user fees. But given roads are not, this would result in there being very little transit, since fares would be very high, and users driven away. Still, these could be higher.
So if you can’t use user fees, what’s the next best solution? Land value capture. This is nowhere to be seen. Why not? Concentrated losses for concentrated benefits.
And if you can’t use land value capture, what’s next best? An employer tax perhaps, since low transit fares subsidize workers (as well as non-workers) and allow firms to offer lower wages than they would if transit fares were full cost. Concentrated losses for concentrated benefits.
And if you can’t tax users and can’t tax land and can’t tax employers, maybe you can tax competing modes. After all, everyone on the road wants everyone else on the road to use transit. This is more viable for suburban-downtown serving transit routes, but that certainly describes the Twin Cities region’s proposed transitways.
And if you can’t do any of the above, then general funds might be appropriate. Especially since transit has a social function for moving people who have few choices. This gives us diffuse losses for concentrated benefits, and is so politically more palatable.
But if you want to use general funds, surely you would want to have a tax that wasn’t regressive, like a progressive income tax rather than a sales tax.
USC Professor Lisa Schweitzer says (about Toronto and LA, but it applies here as well)
California-based urban planning expert Lisa Schweitzer
Ms. Schweitzer, who has studied the politics of sales taxes in the context of L.A.’s transportation infrastructure, said as with any funding measure, there is a “tradeoff.” Cities such as Los Angeles and Toronto can raise substantial sums through sales taxes, which are more politically palatable than some other revenue generators, she noted. “They’re easy to collect, they’re easy for people to pay; you don’t have to deal with the taxman, you just pay at point of purchase,” Ms. Schweitzer said. A major concern, however, is volatility. “They go up and down based on the business cycle and they run parallel with the business cycle, so when retail sales plummet, the take that you get from retail sales taxes also goes down,” Ms. Schweitzer said. “That can be a problem for transit agencies that are dependent on these sales taxes.”
Again, we are now at a high point in the economic cycle (6th year of expansion), so it looks like sales taxes are a great idea from a revenue perspective, but when the next economic collapse comes, oh, about 1 or 2 years into the next Clinton/Bush administration, and revenue tanks, just when we should be building infrastructure for macro-economic stimulus, we can’t. This is not as unstable as the sales tax on gas, but it is hardly a bedrock on which we should fund the system. (Perhaps if we are selling bonds backed by sales tax revenue, that worry can be diminished, but bonds are more risky and thus costly than pay-as-you-go, see the idea of interest rates).
Now, let’s say you have this new pot of money. The MoveMN folks bought off the all-powerful bike lobby (who seem to be on-board with this proposal, as card-carrying members of MoveMN) with a 10% share for a “safe and accessible bike and pedestrian connections in the metro. ”
So though biking and walking has a higher mode share than transit, it only gets 1/10 of the total funds. Someone is not negotiating well. [Hint – it’s the completely unorganized pedestrians.]
Obviously pedestrian and bicycle infrastructure is relatively inexpensive, so from one perspective it doesn’t “need” as much. But from an efficiency point-of-view, since it is so inexpensive, walking and biking should have a much higher return on investment.
Whatever really has the highest rate of return should be the highest priority for investing a new pot of money. The problem is establishing this, since there is so much economic development Voodoo floating around that no one believes.
If your aim were to maximize the number of people using modes other than auto for a given trip, you would take your pot and spend it on relatively inexpensive separated cycletracks and improving the local urban environment to increase walking and then local arterial BRT before you spend it on high-capital, high-risk projects with low rates of return.
The City vs. Country mismatch (where we plan for transportation systems seemingly independent of the context: city or country (suburb)) is especially relevant in metropolitan areas constructing expensive medium and high capacity transit lines to presently undeveloped places, or (re-)constructing and widening freeways in the midst of core cities.
If we are building a city, that means focusing resources closer to the center than the edges, since that is where more of the growth will be.
So the question arises in Minnesota: Are we building a city in the Minneapolis-St. Paul region or not?
Some would argue the region is already a city. Clearly the downtowns and some nearby areas resemble cities. But much if not most of the population lives in single family homes with a yard. Not there is anything wrong with that, but that yard increases spacing between neighbors, lowers densities, and makes it harder to operate a successful transit system in the automobile era.
The decisions we make regarding transportation investment rest on what our collective vision is for this place. If we are building a city, going “all in” on transit is a lot more sensible than if we are not, and a lot more sensible than half-measures. If we are not, going even partly in on fixed rail transit is fairly pointless.
The evidence to date is fairly mixed. The region is growing at about 1 percent or so per year in population. That is a bit under 30,000 people or so per year. (Investing in transit will not put a measurable dent on this, as it is hard to imagine it either increasing or decreasing the birth or death rate noticeably or significantly affecting migration patterns).
Where are those people going? Most of them are not being added to the center. If all of them were added to Minneapolis and St. Paul proper, the populations of the core cities would rise from 667,000 in 2010 to about 950,000 in 2020. If more than half of them were added to the core cities, their population would be around 810,000. Current forecasts suggest the population of the core will be closer to 732,000. In other words, well more than half (around 80%) of the region’s growth remains outside the core cities, and not terribly urban. While the center is projected to grow at a faster rate than the edges, it is not growing faster in absolute numbers.
This region collectively much more resembles Broadacre City than Greenwich Village, and probably more Broadacre City than even Welwyn Garden City or Letchworth. While the density at the edges after development must (by definition) be higher than before development, the average experienced density (e.g. the population-weighted density) may continue to fall as the region expands.
In this dispersed suburban landscape, point-to-point transportation predominates.
In the not-quite-a-city urban areas, point-to-point transportation still predominates.
It is only in full-fledged dense cities (the level of density required depends on income, technology, and other factors), and dense cities that sustain, that full-fledged dense (fixed route, fixed infrastructure) transit is warranted. Loosely, the transit-city threshold is on the order of 10,000 persons per square mile, sustained over some region. This implies a Minneapolis population of 540,000 and a St. Paul population of 520,000. While those numbers are achievable in principle, they require significant changes in market demand and reduction in supply constraints to achieve. If it were to be achieved, still under 1/3 of the region would be in the transit city. The remainder would be the country, or its suburban simulacrum.
Certainly, some parts of the core cities already sustain this density, and there is no requirement that all of the core city achieves the density, just that a sufficiently large part does. Most doesn’t.
For the Metropolitan area, fixing the total space, this threshold implies a population of 63 million, or relaxing the space but fixing the population, a shrinkage to a developed area of 345 square miles. I would consider either outcome highly unlikely.
The car-free lifestyle is sufficiently uncommon that the local newspaper does a feature on it.
In less dense cities, less transit is warranted. But if the transit is insufficiently dense (in space (coverage) and time (frequency)), anyone with a choice will anchor to the on-demand, point-to-point mode, which will be sufficiently faster to outweigh any additional expense
Creating Transitopolis: the Transit City
Building the transit city (Transit-opolis) requires most of the region’s central city residents (20% today) to willingly abandon the automobile in the first place: creating an environment where car ownership can be voluntarily foregone because it is to the benefit of all concerned not to own the car. The more transit users there are, the better the transit service is, in a virtuous cycle. (And of course, the fewer transit users there are, the worse the service, in a vicious cycle).
This implies focusing transit investments in the central cities, and not diffusing scarce capital like peanut butter across the region.
Why can’t we have the best of both worlds? There are a variety of interim or transitional strategies
Park and ride lots accessed by private car may serve the center-working/surburban-living suburbanite, which transfer the worker to a radial transit route. This serves a small fraction of workers, since most people don’t work downtown, where these radial services terminate, and the frequencies are typically not high enough (especially mid-day) to rely on for anyone with irregularity in their life.
Single Car households with multiple workers splits the difference, where presumably at least one of the workers does not use a car for travel to work, or the two workers carpool to the same location. This is certainly the most common hybrid case.
Weekend Car households in which the work trip is conducted by walking, biking, or transit, and the car is used for the weekend and special trips.
Car sharing (on-demand car rental) allows the car-less transit user to have a car when needed for the less frequent trips. Taxis (and so-called ride-sharing services) serve a similar function.
These attempts at melding can serve specific niches and are potential transitional steps to the transit city, in that it is more likely to give up a car for some trips than all trips, or portions of trips than all of the trip. They are not however the end state.
My own prediction is that we are not in fact building a transit city. The densities that will emerge in Minneapolis and St. Paul in the next 20 or 30 years are insufficient to support a transit-based city for a majority of residents of even the core cities. (And, simultaneously and not coincidentally, the transit investments are insufficient to support a city where most people are car-less).
This explains my pessimism about the use of expensive (high fixed cost) city transit tools now. The right decision if you are certain you are building a transit city (or already live in one) is different from the decision if you are not.
While it is argued that never making the investments will ensure the transit city does not emerge, we can continuously ratchet up, rather than doing it in enormous steps. Not investing now does not mean never investing. “No” is reversible (in other words, “no” does not mean “no”).
On the other hand, “yes” is irreversible for a long time. But making the investment does not guarantee the transit city will emerge, plenty of cities, including our own, have invested in rail without seeing accompanying development, leaving an enormous pale proboscidea in its place.
My own prediction is that the core cities of Minneapolis and St. Paul are on a path towards Transitopolis. The necessary population densities to support transit are within striking distance, especially in Minneapolis, and the leadership of both cities is keen on achieving that. The big question is whether the demand for higher density housing is there in the cities, and with changing demographics, preferences, and technologies, the demand for suburban lifestyle is diminishing relative to that for urban living. Most people won’t spend their entire lives in either suburban harmony or at an urban tempo, but rather migrate between the two over the years.
The investments in the cities to support transit at a level that most people can avoid the car for most trips are still too low to make this work, there is a marked lack of ambition on the part of regional transportation officials to serve the transit cities they are building with actual rapid transit. This dysfunction has several causes, not the least of which is the leadership of planning organizations that don’t use the modes they manage, which we can attribute to the strange politics of regional organizations.
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.
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.
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/1queue (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 :
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.