Transit funding for Massachusetts (and how to spend it)

Sorry folks, it’s going to be another long rant about public transportation in Massachusetts. I know this isn’t exactly what you signed up for, but it’s what you’re going to get until I manage to write something else.

As I write this, we just had our 2018 primary elections in Massachusetts. While the major national buzz is over progressive Democrat and former Somerville mayor Mike Capuano’s defeat by progressive Democrat and Boston city councilor Ayanna Pressley in the 7th congressional district, I live in the 5th district, which stretches from Southborough to my west all the way to Winthrop on the coast just north of Boston — and my congresswoman, Katherine Clark, was unopposed. (There was apparently a Republican contest, but barring some unforeseen turn of events, the GOP nominee will get soundly crushed by Clark.) But we also had primaries for various state constitutional offices, including governor, lieutenant governor, attorney general (uncontested), auditor (uncontested), and secretary of state (won by the long-time incumbent, Bill Galvin, over challenger Josh Zakim, after whose father, Lenny Zakim, the I-93 bridge over Boston Harbor is named). There were no really compelling races in my precinct, and other than the Middlesex County district attorney, all were uncontested. However, several important state legislators lost their primaries to younger, more progressive challengers — including a number of close lieutenants of centrist house speaker Bob DeLeo. It’s important for outsiders to understand that, however much Massachusetts is seen as a liberal bastion, one-party rule has meant that there are a lot of conservative (or at least anti-progress) Democrats in the state legislature, and a lot of incumbent-protection gerrymanders that keep them in office even as the complection of the state becomes more open and progressive.

When I first started hearing about the governor’s race, back in the spring of this year, I checked out the candidates’ web sites to see what their transportation plans looked like. One of the candidates, Bob Massie, had a detailed plan and supported many of the initiatives I’ve loudly supported in this space over the past few years. The other candidate (of those who made it through to the primary), former Deval Patrick cabinet secretary Jay Gonzalez, seemed like a decent person but he didn’t have a transportation plan to speak of, just a paragraph or two of platitudes. So I supported Massie, contributed to his campaign, and voted for him in the primary. Massie was resoundingly defeated by Gonzalez, who wrapped up endorsements from nearly all of the major media and political leaders. So the general election will pit two former insurance executives, Charlie Baker (whose main accomplishments after nearly four years in office include little more than taking credit for projects that were already in procurement when he took office) and Gonzalez. I will of course be voting for Gonzalez, and in all honesty I haven’t gone back to see if he has bothered to flesh out his transportation policy in any meaningful way. But after the election, I started thinking a bit more about what my priorities would be, and how I would like to see them funded.

Both of the Democrats running for governor had endorsed the so-called “millionaire’s tax”, a proposed constitutional amendment that would allow the state to impose a higher tax on very-high-income individuals and use the proceeds for education and transportation. The proposed tax would have brought in something like a billion dollars a year, and was showing quite well in early opinion polling. Unfortunately, the Supreme Judicial Court decided that a single ballot question could not both impose a tax and restrict how the legislature could spend it, and so the “millionaire’s tax” will not appear on our ballots this November. If Gonzalez can actually beat Baker (who has irrationally high approval ratings among Massachusetts Democrats), perhaps helped by those selfsame demographic changes that did in Capuano and other seemingly popular incumbents, then there’s a decent chance that he will have a more progressive caucus to work with in the General Court, one that’s less dependent on, and less willing to be strong-armed by, Speaker DeLeo. And that, in turn, might well mean that for the first time in a decade we could see serious consideration of increased revenues.

The mantra among public transportation advocates is “organization before electronics before concrete” — that is to say, fix organizational problems (unattractive schedules, bad workplace cultures, inadequate and underfunded maintenance, bad fare collection systems and unjust fare structures, contractor featherbedding and excessive engineering costs) before investing in new equipment, and improve equipment (like adding transit signal priority on bus corridors, or electrifying trains) before constructing major new facilities like tunnels and tramways. This makes a certain amount of sense, because without controlling construction costs you’ll waste 9 dollars out of 10 on anything you build, and in most cases you shouldn’t plan on building something unless you’ve already squeezed as much operational efficiency out of the existing infrastructure as you can. However, sometimes you need to proceed down all three tracks at the same time — as in TransitMatters’ Regional Rail proposal, high-level platforms at all stations (concrete) enable electrification of the entire commuter rail network (electronics) which enables fast, reliable, and frequent service and allows obsolete, unreliable equipment to be replaced with modern equipment rather than buying more obsolete equipment to continue to run an obsolete service. Fare integration makes the Regional Rail service more attractive to off-peak riders, which helps the farebox recovery and helps eliminate split shifts for train crews (because more trains are in service during the middle of the day) and allows reduction in duplicative bus services; fare integration and reform provides equity to marginalized communities and allows staff to be redeployed in more useful job functions.

One thing is for certain: engineering services and construction costs at the MBTA are far out of line with our global peers, even if you exclude fast-growing low-wage countries like Brazil, China, and India. Even compared to other parts of the US or Canada, our costs are out of whack and our time-to-build is far beyond what it should be. (Surely you’ve seen the viral animation of the growth of Shanghai’s, or Beijing’s, or Wuhan’s, or Dubai’s metro system over the past two decades — although those are much larger and faster-growing cities than Boston, it’s hard not to look at one of those and wish we could actually build the services we desperately needed a decade ago.) Charlie Baker touts the cancellation and redesign of the Green Line Extension, but at a cost of $156 million per kilometer ($78 million per track-kilometer) it’s an order of magnitude more expensive than light-rail construction in any old, dense, highly unionized European city — and that’s something that needs to be fixed, even if it means potentially forgoing federal transit funding (which is looking less and less reliable anyway) and getting European firms and equipment manufacturers to bid without having to pay the “buy America” tax. And it means getting serious with the construction unions about featherbedding — we should be able to commit to providing more than full employment for the unions’ memberships without paying for make-work, and without compromising safety. This probably also means letting more systemwide packages for design and engineering work: rather than having a separate architectural firm bid on each new station or platform or substation, a single designer should be tasked with generating a standard design that can be applied everywhere, or nearly everywhere.

One other high-level issue that I need to bring up is regional equity. This is a big bugbear for politicians, especially those from Western Massachusetts, whose constituents perceive (wrongly) that all “their” tax money is being funneled to Boston and none of it benefits them. (The opposite is true: the rich communities of Eastern Mass. subsidize poor and rural communities West of Worcester.) The Census Bureau’s “vintage 2017” population estimates are out now, and that gives us some numbers that we can at least think about. The seven counties that receive at least MBTA commuter rail service (Bristol, Essex, Middlesex, Norfolk, Plymouth, Suffolk, and Worcester) account for 84.4% of the state’s population; the remaining population, 15.6%, is split between the Cape and Islands (Barnstable, Dukes, and Nantucket) and Western Mass. (Berkshire, Franklin, Hampden, and Hampshire). So if any new revenue is brought in, it would make it more politically palatable if the non-metropolitan Boston regions received at least a proportionate share of the expenditures.

OK, I’ve said a lot now in the abstract, let’s get a bit more concrete (no pun intended). Right now, the motor fuel tax in Massachusetts is 24.5¢ per gallon, the second-lowest in New England after New Hampshire, which brings in an annual revenue of $766 million. Gas prices now are still quite low compared to their high point before the 2008 financial crisis (about $3/gal versus nearly $4/gal back then). While it’s quite disappointing that we did not take bolder action back when gas was much cheaper (under $2/gallon for much of 2009), there’s still plenty of room to increase the gas tax. This doesn’t work forever — increased taxes will decrease consumption at the margins, and if the plan to improve public transportation is successful, people will drive less because they don’t need to drive as much, not to mention electric vehicles and more fuel-efficient conventional vehicles — so this has to be seen as an effectively temporary measure, at least in terms of the buying power of the revenue it brings in, until we can figure out a longer-term financing mechanism such as a VMT tax. That said, I propose that Massachusetts should double the gas tax, to 49¢/gal, which would make it the second-highest in the country (only below Washington State). As of right now, gas in my neighborhood is selling for $2.89/gallon; in Cambridge where I work it’s about $3.05/gallon — an additional twenty-five cents is noticeable, but livable, about three dollars extra a tank if you don’t drive a truck or full-size SUV. All of the increase would be earmarked for transportation, with no diversion of existing budget authority to other purposes, so it would truly be an increase in funding for transportation. (At least until the out years, after the maintenance backlog has been wound down and new services are operating regularly, assuming there’s even money left over after fuel consumption drops significantly.)

Now, how should this money be allocated? First off, I would take 2.5¢ off the top (that’s about 10% of the new revenue, or $76 million) for statewide transportation initiatives that don’t easily fit in the remit of a transit agency, like freight rail, intercity passenger service, and active transportation (cycling and pedestrian infrastructure improvements). In some cases, as with grade-crossing elimination, sidewalk and bike path construction, there will be side benefits to drivers, and that’s OK. (The principal benefit that drivers should get from the whole program, of course, is less congestion, by making alternative modes a more affordable and attractive option for many.)

What about the remaining nearly 90%? Here is where I would look at those population figures again. Out of the remaining $687 million, reserve 15.6% ($107 million) for non-MBTA counties, and divide it up among the RTAs serving those counties according to population served, as a dedicated funding source not subject to annual appropriation. (If there are still any municipally-operated transit systems in those counties that haven’t been absorbed into an RTA, treat them on an equal basis to the RTAs.) In exchange for this increased funding, the RTAs would have to commit to a plan that provides more frequent, reliable service to important population and job centers in their districts, after first eliminating any maintenance backlogs.

That leaves $580 million for the part of the state that receives at least some MBTA service. I would split this roughly in half, with 45% ($265 million) specifically dedicated to commuter rail improvements with a ten-year timeline to implement the full Regional Rail plan, including full high platforms, full electrification, signal and track upgrades, and grade-crossing elimination (adopted as an explicit objective by the legislature). The remaining 55% ($316 million) would go to the RTAs on the same population-based formula as the rest of the state — of which the MBTA will receive the largest share since it serves the largest population, but the “gateway city” RTAs would get significant amounts of additional funding as well. The RTA funding should come with the additional string attached that they must make corresponding improvements in service at the rail stations that will receive dramatically better service, with a stated goal and performance penalties for transit mode share for passengers connecting to and from Regional Rail.

(Aside: as a part of including Bristol County in this, legislators will probably demand full funding of South Coast Rail. This is a bad idea, but if built to Regional Rail standards with a non-stupid alignment should not significantly impede progress on the overall program, and if better cost control is achieved might not even be so horribly expensive. Ideally any formal support for SCR should be conditioned on cost control to modern European subsidy levels.)

So that’s one part of the revenue picture. I would also have the legislature authorize a congestion charging district, encompassing (at least parts of) Boston, Cambridge, Somverville, Medford, Everett, Chelsea, Revere, and Brookline, potentially including Quincy, Newton, Watertown, and other bordering communities. The revenue (between $1 and $10 per day depending on type and registered address of vehicle) would be split between the cities and the MBTA, and MassDOT would be authorized to collect the charge via its existing toll system and sufficient additional locations throughout the congestion zone to ensure at least 85% compliance. Transportation network companies and cabs would be required to pass on the charge on a per-trip basis and provide a sufficient fraction of trip logs to the state auditor to verify compliance. I don’t actually know how much revenue this would collect, but I would leave the municipalities free to spend their half of the revenue on whatever projects they wish, even if they are not transportation-related, as the congestion charge is really compensation for the disamenity of traffic on city streets and the air pollution that results (particularly particulate matter from tires and brakes, which is worse in stop-and-go traffic than free-flowing highway speeds).

Ok, we’re at word 2,345 now, and you’re probably wondering what I think this money should be spent on (besides the obvious, public transportation). Regional Rail is of course the big one, but it doesn’t even get half the revenue in this proposal. Many of the things that I care most directly about are either necessary for or a valuable adjunct to Regional Rail, whether it’s grade-crossing elimination here in Framingham (the existing grade crossings put a severe limit on the frequency of trains that can be accommodated on the Framingham/Worcester Line), or better RTA bus service to get discretionary passengers to and from the train station. Last weekend I started making a “crayon” in Google My Maps which shows a number of the other ideas that I have (or that I have stolen from other more connected transit fans) [view map]. Unfortunately, Google My Maps doesn’t really allow me to identify which services I would prioritize and which I think are more speculative, so I’ll give a rundown of what I propose beyond Regional Rail (to complete all of which would require substantially more funding than I’ve proposed here).

  1. Obviously, state of good repair comes first.
  2. Complete the Green Line Extension all the way to Mystic Valley Parkway, as was originally planned. This is just a no-brainer. Extending it further to West Medford is quite a bit more challenging but would be a stretch goal on this program (I haven’t drawn it on the map).
  3. Orange Line and Green Line extensions to Needham Junction. Once we’re committed to Regional Rail, the Needham Line becomes something of an anomaly. Bostonians in Roslindale and West Roxbury have been demanding better service (and fewer buses clogging Washington Street) for some time, and the Orange Line is Right There; extending the Orange Line to West Roxbury involves essentially the same work as the Regional Rail upgrade would, except that you’d be using third-rail electrification instead of catenary. (Either way you’d have to build high platforms and double-track the line.) Doing so would orphan the rest of the Needham Branch, so it has been proposed to convert the whole line in Needham to a Green Line branch using the abandoned right-of-way from Newton Highlands, which is currently a rail-trail and might be difficult or controversial to return to rail service. Ending the new Orange Line at West Roxbury avoids double-tracking the environmentally-sensitive Charles River bridge, but my suspicion is that many Needham residents would be better served by faster and more direct Orange Line service than by Green Line service via the Highland Branch. So my map shows a compromise, with probably the most politically difficult of all routings, sending both Orange Line and Green Line to Needham Junction, giving most riders a choice of which line they prefer. I add new Orange Line stations, above and beyond the existing Needham Branch stations, at VFW Parkway (serving Millennium Park, West Roxbury Academy and Catholic Memorial) and Greendale Avenue; new Green Line stations are at High Rock (the south end of the Needham Junction wye, beyond which tracks have been lifted), Gould Street (at the current WCVB studios, car wash, and Muzi Ford, all of which might be candidates for redevelopment), Oak Street in Newton Upper Falls, and Columbia Avenue in Newton Highlands (at the new “Avalon” apartment complex).
  4. Blue Line extension to Inman Square. This is inspired by Ari Ofsevit’s proposal to extend the Blue Line past Charles/MGH — the which has been promised for decades — to MIT’s redevelopment of the current Volpe Transportation Building on Broadway in Cambridge. At Volpe you’d have a connection to the Red Line at Kendall, but where should the line go from there? Inman Square in Cambridge is pretty much equidistant from Union Square Somerville (one branch of the GLX) and Central Square on the Red Line, a bit farther away from Harvard and Kendall, and in a very densely populated area that is only served by buses at the moment. Once you take the turn onto Hampshire Street, Inman is a straight shot for a cut-and-cover tunnel, and in fact you can continue straight nearly all the way to Porter if you want (I’m uncertain about the value of that option, but I’ve drawn it as a future extension). This becomes important for connecting to the Yellow Line.
  5. Washington Street and Mass. Ave. tramways. When the Washington Street Elevated was demolished in 1987, the residents were promised something “better than a bus” as a replacement service. So far, all they have is buses — fancy buses, in some places, but still buses. I propose to give them a tramway, reusing the old Tremont Street Subway branch to the Pleasant Street portal, and then running south along Washington St. and/or Shawmut Ave. (there are some issues with one-ways here that would have to be straightened out) to Dudley, with future tramways serving Roxbury, Dorchester, Mattapan, and parts of Hyde Park and Roslindale. IN the other direction, Mass. Ave. is the majority of the route of the MBTA’s single busiest bus line, the #1 bus, which serves 13,000 passengers per day, plus numerous private shuttles operated by transportation management associations, the Longwood Medical Area shared services corporation, and multiple universities. The 1 bus also used to be a tram line, albeit one without a dedicated reservation. I propose to construct a Mass. Ave. tramway that runs from Central Square in a dedicated reservation (eliminating street parking as necessary in narrower segments) all the way to Southampton Street, then continuing through South Boston via Andrew Square and ending at City Point. The tramway should be constructed with paved tracks, to allow for buses as well as trams; passengers who currently take the #1 to or from Dudley would be able to transfer to the Washington St. tram, cutting off the current loop past Boston Medical Center, and both would run at sufficient frequency to make the transfer penalty relatively small. (Passengers for Longwood could transfer at Symphony.)
  6. Huntington Avenue subway extension. I propose a short extension of the Huntington Ave. subway from Northeastern to Brigham Circle, which is mostly free to construct with the Yellow Line running parallel there.
  7. Yellow Line: a circumferential route connecting Saugus to South Boston. There isn’t a huge demand for a train from Saugus to South Boston, admittedly, but my Yellow Line is a single route that combines many people’s overlapping wishlists for better connectivity in the Boston area. Like Roxbury, Everett used to have a section of the Main Line Elevated (today’s Orange Line); when the Haymarket-North extension was built in the 1970s, the El in Everett was torn down, and that city was left with buses and much worse access to downtown Boston. It’s been proposed that a branch off the Orange Line be built from Sullivan Square through Everett that would allow many of those bus routes to be restructured or cut and give Everett residents a one-seat ride again. My first segment of the Yellow Line would fit that bill, with two stops on Broadway, then running under Lynn St. to Eastern Avenue (another stop) before following the old railroad right-of-way to downtown Saugus (stops at Salem St., School St., and Central St. But rather than a branch of the Orange Line, I think the Yellow Line is more useful if it becomes a true circumferential line, enabling travelers who currently want to go from Somerville to Longwood, or from Charlestown to Cambridge, or from Allston to Everett, to do so without going through the bottleneck in central Boston. Much of the line is intended to replace the #66 bus, which also has 13,000 passengers per day, as well as some of the travel demand on the 1, 47, 64, 70, 86, 91, CT2, CT3, and other buses currently radiating out of Cambridge Red Line stops. South and west of Sullivan Square, my Yellow Line stops at Union Square Somerville (interchange with the Green Line), Inman Square (interchange with the Blue Line), Central Square (interchange with the Red Line), Putnam Ave./Memorial Drive (dense residential neighborhood), Harvard Science Center, West Station (interchange with Regional Rail), eiher Union Square Allston or Commonwealth Ave. and Babcock St., Coolidge Corner (interchange with Green Line “C” branch), Brookline Ave. (near interchange with Green Line “D” branch), Huntington Ave. (interchange with Green Line “E” branch), Ruggles (interchange with Orange Line again), Washington St. (interchange with Washington St. tramway or short walk to Dudley), Boston Medical Center/Mass. Ave., Broadway (interchange with Red Line again), and then eventually via track 61 (or possibly under D St.) to the Boston Convention and Exhibition Center. Although the alignment is somewhat different, this is essentially what was being proposed as the “rainbow line” when I first moved to Boston in the 1990s, before being sandbagged by the state and partially “implemented” as the CT1/2/3 bus routes.
  8. The final and most speculative proposal is a series of tunnels and a long bypass track that would give a straight, high-speed route to Worcester, mainly following the Massachusetts Turnpike from Newton to Westborough, and cutting off the big curve around the south end of Lake Quinsigamond. I propose two new stations on this line, one at Framingham Technology Park on the Framingham/Southborough line, and one at Cochituate in Wayland where the line crosses Route 27 just west of Lake Cochituate State Park. The routing I’ve drawn for these is approximate at best. My assumption is that construction of a tunnel under the Turnpike (which was built on top of the original four-track main line) would involve completely replacing the existing two-track main line and all stations, and it’s probably 20 or 30 years out at best — unless the state decides to get serious about a high-speed intercity route from Boston to Springfield, which would not be compatible with Regional Rail on a two-track line. This would also make Worcester express service both useful (in terms of reduced travel time) and feasible on the high-frequency Regional Rail model.

Nearly 4000 words, and I’m up way too late, so I’m afraid that’s all the blather I have time to write this time.

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Luck meets urban planning in Finland’s capital

This gallery contains 53 photos.

As I promised last week, I’m starting to write a bit more about my trip to Finland a year ago for Worldcon 75. As I write this, Worldcon 76 is taking place in San Jose, California, which I had little … Continue reading

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Coda: commuter-rail modeling and what’s next

Just about a year ago, I went to Worldcon 75 in Helsinki. Because Worldcon 76 in San Jose (which I’m not attending) is about to start, I was reminded that I still have a whole bunch more photos from Helsinki that I’ve never finished editing, and the research that I’ve done in the last few weeks has definitely given me a bit more to say about them (and Helsinki’s brownfields redevelopment and transit construction costs), so I’m going to try to push a final set of Helsinki posts out over the next few days, if I can get a few hours clear for photo editing. (I say “final” because I have no plans to go back, but if someone’s willing to subsidize my travel I would happily return next summer.)

Beyond that, I wanted to wind up my discussion of Regional Rail and the future of the commuter rail service between Worcester and Boston. In addition to the analysis in the previous two posts, I did a lot of modeling of service patterns that I haven’t shared, because they were not sufficiently interesting, or at least sufficiently different from what I’ve already presented. In some cases these models had insurmountable technical problems, mostly due to the lack of storage space in the right places, and in other cases in order to make the model work I had to assume an impractically short turnaround time. (Impractical under current FRA regulations and railroad operating practices, at any rate — railroads elsewhere in the world don’t seem to have a problem turning trains at rapid-transit-like frequencies.) In a few cases it’s not storage space but platform length that is the biggest issue, like the four-trains-per-hour service pattern that requires trains longer than the current MBTA construction standard. Here are a few of my major take-aways:

  • The F/W Line is incredibly busy for a small number of AM peak runs and fairly easy to service at whatever headway you choose the rest of the time. I based my simulations on the 2012 CTPS riderahip audit, and there’s every reason to think that actual ridership (despite the well-publicized problems of 2015–2017) has increased since then.
  • The lack of sufficient storage capacity in Worcester is a serious limitation on any service. In the Regional Rail model, additional storage capacity would be needed elsewhere at least for middays and probably at night.
  • I believe that additional storage should be at Framingham, not Allston or Southampton. The reason it should be there is a simple one: equipment positioning from those other places has no transportation value, even if for some reason you made it “revenue”; scheduled early-morning and late-evening runs from/to Framingham would have substantial transportation value, and there are multiple existing yards at Framingham that either have room or could be expanded (or the freight railroad could be incented to consolidate operations).
  • Extending passenger service via the Agricultural Branch to Framingham State University and Framingham Technology Park would have actual transportation value, but the costs and RoW limitations are such that it is unlikely to happen without capital support from MWRTA, the City of Framingham, FSU and/or the Tech Park tenants (especially Bose and Sanofi, the largest employers). Such a service might be operated as the tail end of Framingham-terminating Regional Rail trains, in which case there’s room in the Tech Park to construct a layover yard, or it might simply be a diesel shuttle from South Framingham that met each train at the existing station. (Either option would require substantial station construction, as there are no historic stations along this four-mile segment of the line.)
  • None of the schedules I came up with can work with fewer than twenty-four 75 m (250-foot), 232-seat articulated EMUs; most of the usable ones require 26 or 27 (plus one more for a Tech Park shuttle, if you want to build that). That’s an equipment cost of about $216 million, in addition to the capital construction (which I’ll detail below).
  • These schedules and car requirements would get substantially easier with a longer trainset. The MBTA’s standard platform length of 800 feet puts a substantial limitation on the number of passengers you can accommodate with a fully accessible, level-boarding, single-level design. Plus there are issues with platform length at Yawkey that I know about and possibly other stations. (And I have assumed that the storage facilities, which the T reports in “consists”, reflect this maximum length and not the actual lengths of trains being stored for the schedule current when the “State of Commuter Rail” report was written.) For 800-foot platforms, with three-trainset consists you’re limited to 80 m trainsets, which depending on the layout could get you another 20 seats, but if you’re willing to accept a maximum of two-trainset consists, you could go as long as 120 m — and then you have a different problem, because then you either have to manage two different EMU equipment types, or waste a lot of energy hauling around a 350-passenger train on the majority of runs that have fewer than 150 passengers. I’d probably spring for the 80 m length if there are no other constraints, just to have the capacity.
  • Another option is to forget about running articulated EMUs and just do New York-style EMUs, which are the same length as regular passenger coaches. Depending on clearances, you might be able to run bilevel cars under wire, although this is probably a bad idea because of steps and fewer doors leading to slow boarding and alighting and taking away the dwell-time advantage of modern articulated EMU designs.
  • My demand model is surprisingly sensitive to the exact timing of the new schedule being simulated, relative to the arrival times of the high-demand trains. There are three things going on here: first, the demand model considers only the arrival time to be important, not the departure time. Many commuters may have a deadline to arrive, hence the high demand trains, but also have nothing in particular to do with the time they would gain from a faster schedule, and so would choose a departure time closer to their current one rather than the latest train that gets in before their deadline. Second, there is no feedback in this model: in the real world, passengers’ expectations of comfort and reliability have an impact on what services they choose, and I don’t know, for example, what fraction of passengers would shift to an earlier or later train in order to be sure of getting a seat on, say, the half-hour trip from West Natick to South Station. Third, the demand model simplistically assumes that the passengers’ desired arrival times are uniformly distributed over the interval between trains (actually slightly offset by a few minutes to attempt to account for satisficing). To get a better demand model would require Actual Survey Research, which I’d be very interested in but am not prepared to fund out of my own pocket — I assume that CTPS has such a model already, and groups like TransitMatters should be regularly surveying commuters in order to justify their policy proposals to Beacon Hill.
  • In particular, one of the scenarios I simulated for multiple schedules was short-turning trains at Framingham. This always makes the loading much worse, but exactly how much depends to a surprisingly great degree on the order of Framingham and Worcester trains, because so much of the model depends so specifically on the exact timing of the few arrivals provided by the 2012 schedule. Even in 2012 the MBTA was operating a local/express pattern during peak periods; this service pattern isn’t compatible with clockface headways because the time gained by the express over a local causes an express to catch up to the previous local. If you’re limited for other reasons to 15-minute headways (four trains per hour), then the local/express pattern could potentially work, but I haven’t simulated any expresses because I think 4 tph is a pretty weak-sauce service on this line, and there’s no practical place to turn a supplementary short-turn train short of Framingham without fouling the line. However, during middays there does not appear to be enough demand to run all trains to Worcester — even at Framingham the current schedule actually has a huge gap that makes it impractical for me personally — so you’d probably want to short-turn at least half of your trains at Framingham (another reason to put the layover facility there and not in Allston).

What is to be done?

So supposing we (the people of Massachusetts) decided that we really wanted to have a modern, reliable, frequent rail service between Boston and Worcester. What would we need to pay for to make this happen?

  • Expansion of Amtrak’s Sharon substation, which powers both the Northeast Corridor and the “Boston terminal district”. The substation was built with the expectation that it might eventually be expanded for commuter service, and because the terminal district is chained off the NEC that additional power will be needed for electrifying the F/W Line. (For other reasons, of course, the capacity at Sharon should be maxed out — you also want to serve Stoughton/Providence and Fairmount trains, and probably at least part of Franklin.) I don’t know how much this costs, but you can search the web for informed estimates and the original Amtrak NEC electrification plans.) This also gives you sufficient capacity for electrification through Newton and Brighton.
  • Construct a new substation in MetroWest. Exactly where depends on the location of sufficient high-voltage distribution capacity.
  • Correct superelevation on the line to allow for increased speed limits. There’s a machine that does this without excavation or rail replacement.
  • Replace obsolete signaling and switch machines. A lot of this is already programmed as a part of the existing state-of-good-repair program and will serve equally well for Regional Rail.
  • Construct full high platforms for both tracks at all stations outside Boston. An upgrade for Natick Center (which is the busiest station on the whole system to not have any accessibility) to a high-level island platform is in process.
  • Obviously, construct the actual electrification infrastructure. In order to maintain freight clearances under the wires, it may be necessary to depress the tracks or alter some overpasses in some locations. (Beacon Street, Boston, is a noted clearance issue: it would be fine for single-level cars but is too low for bi-levels, and so long as there are bi-levels anywhere on the south side system, the MBTA needs to be able to move them under that overpass to get to the Commuter Rail Maintenance Facility on the north side.)
  • Currently, Worcester Union Station is limited to a single platform on the Main Line. (The other platforms at the station serve other railroads that do not currently have any passenger service, although a private operator had proposed to run Worcester-Providence service from one of them.) Currently this is under study with no firm funding or construction date, but in order to support reasonable schedules at Worcester with sufficient recovery and turnaround time for the train crews, this is a firm requirement.
  • Most importantly, the MBTA has to decide to do it. There are several related planning processes doing on right at the moment: the “Focus 40” plan is the T’s long-term plan, and in order for anything worthwhile to be done before we’re all dead or retired it has to get prioritized in that plan. There is also a “commuter rail vision” process, about which there doesn’t seem to be much public information, and the people responsible for that will have to be pushed very hard to make a plan that’s actually forward-looking and not repeating the same defeatist, stuck-in-the-past thinking (which the MBTA is exceedingly good at). And there is a procurement process starting to ramp up now for the new private contractor to actually operate the MBTA Commuter Rail.
  • Finally, we have to have a governor who actually uses transit and is responsive to the needs and aspirations of the communities in the MBTA district — who are responsible for the vast majority of the Commonwealth’s jobs and tax revenue.
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More modeling for the Framingham/Worcester Line

In our last installment, I talked about how I wasted an entire weekend developing a not-very-good model of a way to make the commuter-rail line that serves my city suck less. Well, it’s been another nasty rainy weekend here in Framingham, so I spent even more time (also some nights) trying to make my model sufficiently less not-very-good to be able to simulate more interesting variations than the uniform-headway, 100%-local service that I initally implemented, to see whether they were any better (or any cheaper to operate). This was not especially easy, because I don’t actually know R, but a lot more of the work was done entirely by hand. I’m sure actual railroads have good equipment planning and scheduling software, that they probably either developed in-house or gave huge piles of money to a consulting firm for. I on the other hand have the Mk. 1 human brain, which for a problem of this size is more adept at implementing backtracking search internally than it is at writing that down as code. I’m going to describe my process, and while all of the code and data, and most of the results, are available in my GitHub repo, I’m taking the liberty of simplifying out a number of wrong turns and dead ends, and reordering some things for a more understandable presentation.

Code restructuring

As I presented it last week, the simulator consists of two pieces: a demand model, which predicts how many passengers will board a particular train (given the arrival time of that train and the next train at the final destination), and a supply model, which was just a fixed bit of one-off code that generates the particular schedule I was interested in looking at. This is clearly not a good way to do it, because it means that constants scattered throughout the code, many of which are arithmetically related, all have to be changed in unison in order to change either model. I started out by moving the demand model into a separate function, so that the supply model did not need to have any knowledge of the parameters to the demand model — it becomes a higher-order function that accepts the demand model as a parameter. I ended up completely restructuring the demand model anyway, in order to implement some of the features described in the next section, and also because there were a bunch of bugs that I found when I started simulating trips that don’t stop at every station.

The way the old supply model worked was that it took a fixed schedule, and for each train, it iterated over the stations, invoking the demand model for a random sample of passengers’ desired arrival times, and just counted up the total passenger loading — the actual supply part was done by me visually inspecting the loading across all trains to identify impossible loads. This was totally inside out: the correct way is to compute the entire demand vector at each station, and then divide it up among the trains on the schedule, and the way I had done it was exceedingly slow, even by R standards. The way I had done it before was also statistically bogus, because fetching a new demand vector for every arrival at each station means that some passengers (due to random chance) can be either counted multiple times, or missed entirely. In the new code, all of this is parameterized using higher-order functions, and every run writes to disk not only the model output, but also the predicted number of trainsets required (given a seat count) and the actual station-by-station, train-by-train schedule that was used in the computation (which can be inspected if the model seems to have done something strange).

All of this is wrapped up in a nest of new functions which make actual simulator invocations much simpler — for a uniform, all-local-train service like the ones I looked at last weekend, a single line of R code does all the work and tells the simulator where to save its results:

doit("4tph-local.csv", function () make.local.service(360, 720, 4))

(Yes, I am a bit unimaginative with my function names, so sue me.)

Adding a bit more statistical sophistication

If you read last week’s post, you’ll recall that the demand model I used was a very simple one: assume that the demand for travel from a station to the terminal reflects a set of desired arrival times that are uniformly distributed over the period between “this” train and the next. This is obviously unsophisticated, but not obviously stupid. I wanted to do a little better, without putting a lot of work into learning statistical demand models for transportation, so I added two bits of “fuzz” to the demand model. First, I assume that some people on a train would actually rather be arriving a little bit earlier than the scheduled arrival, and that anyone who really wanted to arrive just a little before the next train would suck it up and take the later train — “a little bit” I defined arbitrarily as 5 minutes, which is much smaller than the interval between any of the arrivals in the 2012 CTPS study. (The minimum headway in the 2012 schedule was 12 minutes.) I also figured that there would be some day-to-day variability in ridership, and again entirely arbitrarily I chose to multiply the CTPS passenger count by a normally distributed random variable (μ=1, σ=0.05) — I suspect this effect is overwhelmed by the uniform distribution of arrivals but I left it in without doing any serious examination.

The other big problem with my original model is that it just took a single run of the model as gospel, rather than looking at the behavior of the system across multiple samples. I fixed that by the good old-fashioned Monte Carlo method: the simulator now runs the model 250 times, and computes the 90th percentile over all of the simulated loadings to come up with a more confident guess of how many seats are needed. (It can also compute the median, or indeed any quantile you happen to like, although this isn’t parameterized — I chose 0.9 because that corresponds to my intuition that an operator probably wants enough seats on the train to consistently seat the maximum load at least 90% of the time.) Given those 90% loadings it’s easy enough to divide by the number of seats in an EMU trainset and round up to get the required number of trainsets. As before, I used the JKOY Class Sm5 trainset from Helsinki, which seats at least 232, and which is part of a family of EMUs (the Stadler FLIRT) that is available for sale in the US.

Results of running the updated model

Having made the model at least look more sound, on Friday night I reran the simple schedules I first simulated last weekend, starting with the simplest, four-trains-per-hour all-local service from Worcester. I got a bit of a nasty surprise: although my initial runs on the old model suggested that you could run 4 tph service with nothing longer than a three-trainset consist, the new model predicts one train (the one that would arrive at 8:45) to require four trainsets — and even three-trainset consists are a problem, for reasons I’ll explain shortly. But four-trainset consists are just impossible on the Framingham/Worcester Line, because the MBTA standard platform length is 800 feet and four trainsets are just under 1000 feet long. That said, the 90% loading is only 45 passengers over the 696-seat capacity of a triple, so maybe the cost savings (if there are any) might justify tolerating more crush loads on that train. It’s possible that such a situation would naturally sort itself out, if enough passengers chose to switch to an earlier or later train, but I’m uncomfortable starting out — without accounting for increases in ridership since 2012 — on the basis of a predictably over-capacity train. So I went on look at other service patterns, first the uniform 5-tph and 6-tph ones that I had examined before, and then some other more complicated service patterns once I had implemented the ability to do that (and fixed the bugs in the model that doing so exposed).

Disappointingly, both the 5-tph (12-minute headways) and the 6-tph (10-minute headways) all-local services require three-trainset consists. The 5-tph service has three triples, and a manual inspection of the loadings made me think it really needs five triples, which is a lot — more, in fact, than the 4-tph service. The problem with these three-trainset consists is that there isn’t room to store them at Worcester overnight, where they’re actually needed to provide the service, so you end up wasting a lot of equipment-hours on what are essentially deadheads (there’s just not that much demand for pre-rush-hour seats to Worcester) and when once they get in to Boston you either have to send them back out or you have to store them somewhere. Also, one trainset in a triple can’t platform at Yawkey, because trainsets are 250 feet long and the platform there is only 650 feet, and rush hour is specifically a time when all-door deboarding is necessary to maintain short dwell times. The 6-tph service requires only one triple, which is still more than my first whack at it last weekend suggested. There is just a huge amount of travel demand to get into Back Bay or South Station before 9 AM.

Next, I started looking at short-turn service patterns, where every other inbound train starts at Framingham rather than Worcester (and likewise every other outbound train stops there). This sort of service pattern uses less equipment, in theory, than the all-local service pattern does, but it has the disadvantage that the outer stations receive half as much service during peak periods — which then doubles the ridership on the outer trains. When I simulated the 6-tph service with this model, I found that it was even worse than the all-local service, because the Worcester trains were leaving Framingham already full, and still had to pick up more passengers on the way. (Why not run expresses, like the current service? That would even out the load, but the Regional Rail service is sufficiently faster that a train that runs express from West Natick will catch up with and get stuck behind the previous local train — and of course the inner stations then lose the benefit of the investment in equipment and infrastructure that supports 6-tph operation, because half the trains don’t stop in Wellesley or Newton. That’s likely to be a big loser on Beacon Hill. Hypothetical expresses might as well stop at Boston Landing because the train they’re stuck behind is going to anyway!)

Finally, I investigated other service patterns that would break up the rush-hour demand, and also included reduced frequencies at other times (such as midday) when there is less need for seats. The one I came up with that I like the most is what I call “rush-hour push”: it starts out with 6 trains per hour, then increases frequency to 7.5 trains per hour (eight-minute headways) right around the peak of morning rush, from 8:40 to 9:20, and then drops down to 3 tph after 10 AM. (I chose 8:40 to 9:20 intentionally since those endpoints are divisible by 10.) This knocks the predicted trainset requirement down to two at the peak (in fact, it moves the heaviest-load point earlier, to 8:10). Remarkably, this schedule requires only one more trainset than the 26 I came up with under the old model for 10-minute headways. The lower frequencies during the late morning and early afternoon can be easily sustained without a lot of extra crew expense, but of course it comes at a cost of having to store those trainsets somewhere, which is one of the most limited resources in the current commuter rail system.

Storage

So let’s talk about train storage. There are several places currently used to store trains for the Framingham/Worcester Line (or that could plausibly be so used): Worcester has space for four consists of the current equipment (which I’ve arbitrarily called 3200 linear feet, which should be enough for 12 trainsets of modern EMUs). The rest of the space (in the current service, something like eight consists are required) comes from Amtrak’s Southampton St. yard, which is in South Bay not far from South Station, or in the MBTA’s Readville yard, which is in Hyde Park, 30 minutes away. Last weekend, I looked only at existing facilities, but as I was developing the equipment requirements for this “rush-hour push” service, I found myself thinking, “What about Framingham?” There are three railyards in Framingham: one off the Agricultural Branch west of Franklin Street, one off the Main Line south of Fountain St., and one at the end of what’s left of the old Milford Branch, where it once served GM’s Framingham Assembly plant. Surely some space could be found in one of those — Fountain Street by preference, because it’s right on the Main Line and west of the Framingham station so no reversing is required. Furthermore, having early-morning revenue service from Framingham to South Station has actual transit value, unlike deadheads from Readville or Southampton St., because there are a lot of people who have reason to want to get to South Station between 5 and 6 AM. (Business travelers looking to catch an early Amtrak or a 6:30 flight out of Logan, airport employees, service workers in convenience stores and fast-food places that need to be ready for service by 7 AM, the list goes on….)

So having considered that maybe it might be possible for the MBTA to either rent or construct space at CSX’s Fountain Street yard sufficient for both overnight and midday storage, I reworked the equipment plan some more, and I ended up with a service that adds five early-morning trips from Framingham and requires only two trainsets to be stored at Southampton Street. It does require 1000 feet of space near Worcester Union Station, which may not be available or constructible, in which case those trains would have to deadhead somewhere else (perhaps Grafton or Westborough if not all the way back to Framingham), but all of the remaining midday storage ends up at Framingham rather than in Boston — and because headways are reduced during those same hours, there are plenty of gaps in the schedule for freight moves west of Framingham. Although I haven’t modeled the PM peak at all, those trainsets stored at Framingham would be in the right place to resume 6-tph service in the afternoon, since the reverse-commute demand from Worcester wouldn’t justify ten-minute headways at the time when the equipment would be needed at South Station. Hopefully, this also reduces the number of split shifts for train crews, although I haven’t attempted to model that. (Doing the equipment was hard enough, and of course when you’re running a train every ten minutes, crews can just hop on the next train at whichever station.)

An equipment plan and the fully worked schedule for the AM inbound direction can be found in the spreadsheet in my repository (PDF of the latest version). I also did an equipment plan for a version of the 4-tph service that accepts a crush load on one train inbound, which requires no “storage” at Worcester but does require some place a couple of three-trainset consists can go to clear the platform for the better part of an hour and a quarter; I don’t like it. Someone else could probably come up with a better schedule that doesn’t have that defect; the only advantage of running 4 tph is that it reduces the equipment requirement from 27 trainsets down to 24 — and that’s about $24 million so it’s not peanuts.

So what about that Agricultural Branch, eh?

If we’re going to be storing trains in Framingham at midday, maybe Fountain St. isn’t actually the best place to put them. There are actually three branch lines from Framingham that might serve potentially useful destinations: the Agricultural Branch, part of CSX’s Fitchburg Subdivision (although it doesn’t go all the way to Fitchburg any more), which runs north from the wye at Framingham station; the Milford Branch, which runs south, parallel to Route 126, towards South Ashland and Holliston; and the Framingham Secondary, which runs south through Sherborn and eventually connects to the Franklin Line. The Milford Branch is abandoned south of the yard west of ADESA (the old GM plant, now an auto-auction facility), and in Ashland and Holliston it has now been converted to a trail. The Framingham Secondary is now CSX’s primary freight route to Southeastern Massachusetts, and while the section through Foxborough has a commuter rail station, there is no great travel demand between Framingham and the other towns along the route that would justify even building a station, never mind electrification and double-tracking. So that then leaves the Ag Branch, and that one is very promising indeed. The Ag Branch runs from the Franklin Street yard in downtown Framingham north past the old town incinerator to Framingham State University (three FSU parking lots directly abut the line), and then runs north of Route 9, crosses Baiting Brook, and follows the Turnpike to Framingham Technology Park.

The office park is home to the Sheraton Framingham Hotel and Conference Center, the corporate headquarters of Bose, and a large Sanofi-Genzyme facility, among other important destinations; the corporate headquarters of Staples is not far, on the other side of Route 9, and the park is served by MWRTA buses from Natick and Marlborough. There are oceans of parking, suggesting that most employees at these companies currently drive to work. There are abandoned (and mostly lifted) sidings parallel to the line in the park, suggesting that there is enough right-of-way for two 2000-foot tail tracks (west of the California Ave. grade crossing) and an island platform station with a stub-end track for a shuttle. There are numerous low-rise buildings nearby, including Bose and Nestle Waters distribution facilities, which could be relocated if train service made office development in the park more economically attractive. Finally, the whole complex is just off the Massachusetts Turnpike, an ideal location to intercept commuters who might be coming from the towns north of Route 9 such as Southborough (the Southborough station is at the very southern end of the town and not freeway accessible), Marlborough, and Hudson. I would even suggest not building public parking at a station here: let the landowners decide for themselves whether that’s the best use of their land. Ultimately, from the perspective of this exercise, you could just build the layover facility and not bother with an actual station. But once you’ve built a station, upgraded the tracks, installed switches and signals, and extended electrification (oh by the way, there’s already a substation just south of the tracks), you might as well spring the extra $8 million for a single EMU trainset that can operate a shuttle service during non-pull-in/pull-out periods. The four-mile ride from downtown Framingham to Tech Park would take about eight minutes, a significant improvement on current MWRTA bus service and competitive with driving if the shuttle is timed appropriately to meet through trains at Framingham station.

So what are the issues for implementation? Obviously, the track is in very bad shape, and would need to be improved to at least 39-mph service. (The one freight train a day appears to run at 5 mph; I used to watch it from my old dentist’s office abutting the tracks at the Route 9 grade crossing.) There are remnants of dual-tracking between Mount Wayte and Maple St. that may need to be reinstated, and bridges to replace or renovate. Electrification is obviously required in order to get the EMUs from Framingham station to Tech Park, and you’ll want a side-platform station at either Maple Street (most constructible) or Salem End Road (best location) to serve Framingham State University. A new side platform will also have to be built on the Framingham yard wye, which is complicated by the fact that the wye infield is now used for station parking; probably the CSX westbound leg of the wye will need to be relocated in order to make enough room for the platform. The total cost of this project might be $25-$35 million. Worth it? I’m not sure. Fountain St. would definitely be cheaper, even with takings and construction costs. But it’s worth studying — and it might be worth it to the employers that would be served, if it helps them attract talent.

Thanks for putting up with my transportation rants. We’ll be back to your normally scheduled baking some day soon.

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In which I waste an entire weekend modeling one line of Regional Rail in AM peak

A draft of the MBTA’s “Focus 40” long-term plan was release on July 30, and I spent a little while reading it (it’s rather thin) and sending comments (both officially and on Twitter). If you’ve been reading this blog lately, you’ll know that I’m a strong supporter of Regional Rail, an initiative from the local advocacy group TransitMatters, and of course part of my comments concerned necessary investments in rail infrastructure:

One of the concerns about Regional Rail is how you replicate the existing peak capacity with electric multiple unit trainsets, given that current diesel-hauled commuter rail has quite long trains and bilevel commuter coaches. The Framingham/Worcester Line currently has peak loads of 1600 passengers, which is way more than a single-level, easy-boarding EMU can hold, but of course you can couple trains together, if the trains and platforms are long enough. The number of seats you can deliver on a Regional Rail line is ultimately determined by two factors: platform length and service pattern. The service pattern is profoundly influenced by storage capacity, since you’re not going to be able (or want) to run peak service all throughout the day. So I asked some people who would know what the current shortest high platform on the MBTA Commuter Rail system is — figuring that, if they built high platforms in the first place, there was probably a constructibility constraint that kept any one station from being as long as the rest. That would then influence my idea of how long a consist of EMU trainsets you could operate in peak service, which would then allow me to search for service patterns that would be able to meet that need.

I learned a few things: first, that the MBTA’s standard high-platform length is 800 feet (as compared to Amtrak’s 1050 feet). Second, that the shortest full-high platform in the system is 400 feet at Malden Center and at Oak Grove — these platforms were originally built for Orange Line service, and when the Orange Line extension to Reading was cancelled, they remained in use as commuter rail platforms. Finally, and more relevant for my particular inquiry, Yawkey Station has high platforms that are less than 650 feet long. Since the MBTA decided to have regular trains stop at Yawkey (it previously only served baseball games) it has become a popular stop, with bus transfers to Longwood and Cambridge, and the Green Line nearby in Kenmore Square; more than 400 people get off there during a typical weekday morning (and not all trains stop at Yawkey). That suggests that a Framingham-Worcester service requiring more than about 650 feet of platform is going to run into operational issues (either the trailing trainset[s] won’t be able to platform, or you’ll have to add short turns or local/express patterns to meet the demand in a shorter train). On a two-track main line like the Framingham-Worcester Line, expresses cause scheduling problems, because there are no passing tracks, but short-turns are OK so long as there’s a yard or siding where the train can reverse without fouling the main line — and this becomes more and more important as frequencies get higher.

In order to think about service patterns, I first had to come up with some reliable passenger counts for the existing commuter rail service. Unfortunately, that is quite tricky: the most recent data available was collected by CTPS in the winter of 2012, which predates, among other things, the opening of the popular new Boston Landing station, and the new super-express trains connecting Worcester (and making quite a mess of the schedule for everyone else). There are a lot of reasons to think that Regional Rail would not need any express services, at least on the Framingham-Worcester line, because it is already so much faster. Based on modeling done by Alon Levy in a study of the North-South Rail Link, with proper high-level platforms at all stations, electrification, and full electric multiple-unit trains — the Regional Rail program in a nutshell — a Boston-Worcester train would take only 61 minutes while making all local stops, including several new or restored infill stops that currently have no service at all (also part of the Regional Rail program). By contrast, the current inbound local trains, like P512, are scheduled to take 94 minutes from Worcester to Boston, the regular express takes 80 minutes, and the superexpress takes 66 minutes — so even a local Regional Rail train would beat the fastest current express train (while serving many more passengers).

For the rest of this post, I’m going to be using the 2012 CTPS data — but keep in mind that the peak 2012 loading on any Framingham-Worcester Line train was 1200 passengers, and now that number has increased to 1600; you may want to add 30% to all of the numbers below. (That said, one of the other numbers I don’t have is total boardings: it’s possible that the schedule clearing required to make the superexpress happen has simply concentrated existing demand in a smaller time window, because people still have to get to work at 9:00.)

The question then comes to how to model travel demand, given the passenger numbers. I’m no transportation planner, I haven’t studied this, I don’t know how they actually model these things, but I came up with a really simple way to do it. First, we assume that all travel demand is for people from outlying towns to get into the city. To a first approximation, this is true: in the current service, very few people get off at any of the intermediate stops in Newton, Wellesley, or Natick. One of the goals of Regional Rail is to change this, and encourage trips like Wellesley Square to West Newton to be made by rail instead of by car, but for service planning purposes we’ll start by looking at what people are actually doing today. We’ll model Boston as a point destination at South Station, the end of the line, although in reality about half the passengers get off at Back Bay and Yawkey; since all train schedules we will model will stop at all Boston stations (including the as-yet unbuilt West Station) we’ll treat these destinations as a unit. This allows us to consider only net boardings (which are nearly equal to boardings at all out-of-Boston stations) and makes the model especially simple.

So this is what I did: consider every boarding on the 2012 service as an indication of travel demand to arrive before the next train’s scheduled arrival at South Station (because otherwise they would take a later train). To be specific, for every boarding, we sample a uniform distribution of desired arrival times over [this train’s scheduled arrival, next train’s scheduled arrival). This model is clearly too simple: the distribution is probably not uniform (it should probably be something like a truncated exponential), and some passengers would undoubtedly choose to arrive earlier than the train they’re currently taking, but can’t because the next earlier train leaves too early. But at least this model, while simplistic, is not crazy stupid. So I wrote a bunch of really bad R code to take as input the CTPS data (manually rekeyed from the PDF into a spreadsheet which was then exported as a CSV file for R to ingest) and then, for each station, generate a vector of simulated desired arrival times. Then, for any given schedule (or at least, in the simple case I solved, for any schedule of 100% local trains with no crossing freights or other track fouling) we can compute the number of boardings to expect at each station, and thereby the cumulative loading at the Boston city line.

Once you know the cumulative load at each station, you can calculate the number of EMU trainsets required to serve that number of passengers, based on your preferred model of EMU. I used the JKOY class Sm5 EMU, used in Helsinki’s commuter network; it’s a broad-gauge version of the Stadler FLIRT, and in the middle of the size options Stadler offers for this product line. (The Sm5 is 75 meters or about 246 feet long, and carriers 232 fixed seats and 28 folding seats including wheelchair bays; I’m rounding up to 250 feet and 250 passengers since any Boston EMU will not be this exact configuration. The diesel FLIRT being constructed for TEXRail is 266 feet long but with a much higher seating density, rated capacity 488 passengers; with the same seating layout you could probably get the same capacity in a 250-foot EMU, but with slower boarding and less convenience for passengers with wheeled bags or mobility aids.) Given these numbers, you could carry 700 passengers in comfort on a three-trainset Sm5-equivalent, or over 900 passengers in less comfort on a two-trainset TEXRail-type train. I started doing this modeling before I found the capacity numbers for TEXRail’s FLIRT3 DMUs, so all of the rest of this analysis is based on the Sm5’s capacity. (That said, if you use my analysis but buy Fort Worth’s seating layout, then you’ve effectively accounted for a more than 50% increase in ridership over the 2012 numbers. My bias, however, would be towards boarding doors and bike/wheelchair/stroller space.)

Now finally to get to the point. I modeled three different scenarios, all with 100% local trains serving all stops between Worcester and Boston, with four, five, and six trains per hour. The 4-tph service pattern requires a minimum of five trains with 3 trainsets (peak loadings 541–643 passengers); the 5-tph service pattern requires a minimum of two 3-trainset trains (peak loadings 515 and 541), but for operational reasons you’d probably run five triples on that service as well. That causes a problem, because overnight storage at Worcester is limited to four 800-foot consists, so you end up having to make up three-trainset consists in Boston and send them out empty before the morning rush to be turned around in Worcester and used to make inbound peak service. The extra peak-load trainsets being used in non-peak (or worse, non-revenue) service significantly reduce the equipment utilization compared to a service pattern that uses no more than a two-trainset consist. For what it’s worth, the Helsinki service that the Sm5 was specified for uses to more than that — but they have sufficient passing tracks to run multiple express patterns in addition to their regular 2-tph local service. For my 6-tph scenario, with 30 trains arriving between 6 and 11 AM, the peak load is 464 passengers, for the 9:00 scheduled arrival, which makes sense — there’s another peak (450) that probably reflects people starting work at 8:30.

Having concluded that 6 tph was the best all-local, all-stops schedule, I went on to actually figure out the equipment and storage required to make this service operate. I assume a minimum 15-minute turnaround times (works out to 19 minutes at South Station), which requires at least two platform tracks (possibly you can make it work with as few as that, but the equipment plan does entail making and breaking consists on the platform, in addition to reversing direction, which means doing additional checks before boarding outbound passengers. Overall, the service requires 26 trainsets, of which 12 can be stored overnight in Worcester (more would be better) and 14 in Boston (at Southampton or Readville). At midday, 10 trainsets would be stored in Boston and 1 on the platform at Worcester, with the remaining 15 either in service or being turned at one of the terminals. (Note that the capacity at Worcester is for four 800-foot consists, and likewise 8 consists at Southampton and 12 consists at Readville. Three 250-foot trainsets of EMUs — or indeed three 266-foot trainsets of Fort Worth DMUs — will fit in 800 feet; whether the slack can be used for anything depends on the layout of the yard tracks being used for storage.)

If you look at the simulated boardings, there are no trains that leave Framingham with more than 270 passengers (one trainset’s worth). This suggests a different service pattern, which I haven’t written the code to model but which I suspect makes a lot of sense: instead of running 6 tph all the way from Worcester, run 3 tph from Worcester interspersed with 3 tph from Framingham. Then, you can store trainsets at one of the three Framingham yards (which would have to be wired to make this work), or even implement my idea to upgrade the Agricultural Branch as far as Crossing Blvd. and build a station there with a tail track to store Framingham-terminating trains. I think if you do this, you actually save a substantial number of trainsets (maybe as many as six), but of course the outer towns don’t get as much service in that scenario, which will cause higher loading on the interior segments of those runs. You’ll also notice that there are three early-morning runs in my schedule that have no boardings shown; there’s every reason to believe that people would use these early-morning trains, but 6:30 was the earliest arrival in the 2012 schedule that CTPS audited, so my model doesn’t predict any demand before then.

The crux of my argument to MassDOT and the MBTA is that, with looming procurement projects to replace the ancient and outdated commuter rail rolling stock, it makes sense to commit to building high platforms everywhere now, so that we can then purchase exclusively high-floor multiple-unit rolling stock like the FLIRT3 (which is assembled in Utah and thus eligible for “Buy America”-restricted funding) — specifically with the goal of acquiring a modular system that uses the same components for both electric and diesel-powered trainsets. We wouldn’t get the full benefits of electrification right away — for one thing, the prime mover in the diesel FLIRT only outputs half the power an EMU can draw from the overhead wire, so it accelerates much more slowly — but we would be ready to switch over as quickly as lines can be electrified, starting with the Providence Line which already is electrified aside from a few terminal and yard tracks. We should say “yes, we are going to do this, and we are not going to blow billions of dollars on obsolete unmotorized passenger cars and diesel locomotives.”

Code and data

All of the code and data used for this model is available in my commuter_rail_simulation repository on GitHub. A printable PDF version of a spreadsheet with the results is available at in the repo as well (direct download).

The contents of the spreadsheet are as follows:

  1. The predicted schedule of a local train leaving Worcester, based on Alon Levy’s modeling but with most of Alon’s infill stations zeroed out. Note that he adds a 7% pad to the scheduled arrival time at each station, which adds up to just over four minutes pad in the arrival time at South Station.
  2. Simulated passenger loading, 4 tph
  3. Simulated passenger loading, 5 tph
  4. Simulated passenger loading, 6 tph; I’ve added columns with terminal departure and arrival times as well as the number of trainsets required for peak loading on each run
  5. An actual equipment schedule for the 6 tph service, based on the simulation
  6. Predicted schedule of an express train leaving Worcester, again based on Alon’s numbers but with the stations the current P504 skips zeroed out; note that it gains 9 minutes over the local, which would cause problems for 6-tph operations.
  7. Predicted schedule of a superexpress train leaving Worcester, adding stops at Framingham and Boston Landing over the current P552 schedule; it gains 15 minutes over the local, which would cause problems for any of the scenarios modeled, and is not worth emulating or even modeling, given that the local under Regional Rail is faster at any time of day than this twice-daily express is today
  8. CTPS passenger loads, from the 2012 study, for Framingham/Worcester trains departing in the morning
  9. Net passenger boarding, from the same study
Posted in Transportation | Tagged | 1 Comment

Long time no bake

Earlier this year, I committed to myself that I would not repeat a recipe that I had previously done, as a way of getting out of what felt like a rut. But since then, I have baked almost nothing, and indeed I haven’t done much complicated cooking either. Part of that is because my dishwasher has been malfunctioning since the winter, and I’ve been lazy about getting it fixed or replaced, but a bigger part of it has been that I’ve been quite frankly getting rather fat. Nothing discourages the creation of tasty baked goods quite like stepping on the scale and being unable to comprehend why the number is so much higher than it was three months (or worse, three years) ago. That said, I do have one new baking experience to share, although I have less to say about it than I usually do.

Expecting bad weather all this week, I decided to make a whole-wheat sandwich bread. I had printed out a recipe by Andrew Janjigian from Cook’s Illustrated‘s March/April 2011 issue (on the web, $); the same recipe was also featured on America’s Test Kitchen TV S12E13, “Soup and Bread from Scratch”. (I’m not sure why I printed this recipe out, since I own every issue of the magazine going back to 2006, but it was handy to have as a reminder — otherwise I would probably have just gotten my usual supermarket whole-wheat bread.)

It was an interesting experience, to say the least. The recipe requires three separate wheat ingredients: bread flour, whole-wheat flour, and wheat germ, and dirties a rather large number of bowls: the bread flour is used to make a traditional pre-ferment, but the whole-wheat flour and wheat germ are combined with milk, kneaded, and allowed to soak overnight — then the whole mass is combined together the following day with large quantities of butter, yeast, and salt to form a very wet, sticky dough, which must be kneaded by machine, proofed, kneaded some more, proofed some more, divided in half, shaped (while still extremely sticky), bench proofed, and finally baked off in a steam-filled oven on top of a baking stone. The online version of the recipe text did not even hint at how wet and sticky the dough would be, although I suspect the television version (which I would have seen in 2012) would have done so. I ended up adding about an extra half-cup of flour just to make the dough manageable — fresh out of the first mixing stage it was practically a batter. With all that fat (there’s also some veg oil), flour, and sugar (honey) it makes a very soft and very high-calorie bread, probably better suited to French toast than the sort of sandwiches I can afford to eat when the weather doesn’t allow for bike commuting. I also found it difficult to divide the dough evenly — should have used my scale! — so I ended up with one loaf pan overflowing and one rather undersized after the bench proof.

Overall? It’s good enough, but just comparing the amount of work required to proof, handle, shape, and clean up after this recipe, it’s not enough better than my own whole-wheat bread recipe to justify the extra effort. (On the other hand, the wheat germ is perishable so I may end up making it again a few times, cutting it in half to reduce the bowl-cleaning effort.) And because this bread is so soft, portion control (without a commercial slicing machine) is a complete nightmare. For this reason I’m not providing nutrition information.

Posted in Food | Tagged , ,

Recipe quick takes: sandwich bread and slow-roasted pork chops

In accordance with my pledge from earlier this year, I made two new recipes recently, a whole-wheat sandwich bread I printed out ages ago from King Arthur Flour, and the “deviled” pork chops from next month’s issue of Cook’s Illustrated.

First the bread. The recipe is entitled “Organic Whole Wheat Sandwich Bread” is one of a number I printed out several years ago (copyright 2007!) when I was developing my own whole-wheat sandwich bread. It’s no longer available on their Web site, so far as I can tell, and my printout doesn’t have a URL I can look up in the Wayback Machine, but the the formula is very similar to one titled “A Smaller 100% Whole Wheat Pain de Mie”, but not baked in a lidded loaf pan, and with more fat. So far as I know, I had never done this recipe before, and it has some good and bad points. On the good side, it’s very soft; the added fat, milk powder, and potato starch all combine to ensure that. On the bad side, it’s very soft, and tears easily when slicing or attempting to spread peanut butter or jam. It’s also quite high-calorie: two thin (½ in or 12 mm) slices add up to 275 kcal (minus a little bit for whatever carbs the yeast ate), compared with similar-sized commercial whole-wheat breads which tip the scale at 220 kcal. On the positive side, with all that carbohydrate it toasts very well, and would probably make a good whole-wheat pain perdu or Texas toast. I probably wouldn’t make it again.

The second is the pork chops. This comes from the “May & June 2018” issue of Cook’s Illustrated (pp. 10–11) and I think it’s the first thing I’ve made from the magazine since Christopher Kimball’s partners fired him as editor-in-chief. I actually didn’t make the magazine version, but rather “Deviled Pork Chops for Two”, an online-only extra based on the four-serving magazine recipe. This was quite simple to do, as it merely involves toasting some panko in melted butter, making a flavorful seasoning paste, and using the latter to glue the former to some pork chops. I found while doing this that I had mistakenly defrosted a pair of strip steaks rather than pork loin chops I thought I had, but luckily, my quarterly meat delivery had brought me some pork sirloin chops that I could speed-defrost in the microwave, and this recipe calls for the sort of low-and-slow cooking that pork sirloin requires. (Unlike the loin, pork “sirloin” is composed of a few different muscles, and does not respond well to fast, high-heat cooking methods like sautéeing.) The recipe is simple enough that I did not bother to enter it as a “recipe” in my nutrition app; I just recorded the pork, mustard, panko, and butter (the four highest-calorie ingredients) individually. Recommended.

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Post-vacation status update

Last year around this time I went to the World Figure Skating Championships in Helsinki and generated a whole bunch of posts about it. I went to Worlds again this year, in Milan, and bookended that trip with train travel in Switzerland. This time, I was accompanied by my parents (who, now both retired and with the big house sold, have more freedom to travel) — which worked out reasonably well, but meant that I wasn’t burning the candle at both ends and couldn’t slam through the photo editing to get some blog posts out during the actual competition. (In all honesty I would rather have been accompanied by one person, for the whole length of the trip, but since that person has yet to be identified, the ‘rents will do, and having someone else worry about the arrangements in Italy made the whole trip a bit less stressful.) The arena in Milan was somewhat inconveniently located relative to our lodging, a short-term private rental apartment, and my impression overall is that Helsinki 2017 was far better organized in addition to being more conveniently located and having better transit access. I finally got back home late on Tuesday evening, and I’ve been spending the last few days digging out from the accumulated backlog resulting from a 13-day vacation. (The folks being retired, they got together with my mother’s older sister and her husband, and are spending an extra week in Italy.)

The practical upshot of this is that I still have about 6,000 photos to edit down somehow into something more like 250, and this process will take a while — starting with working through my shooting logs and hopefully correctly identifying all the skaters this time — but over the month of April you should see photos appearing both here and at Wikimedia Commons where appropriate. Anyone interested in accompanying me to other international sporting events is welcome to apply. ;-)

PS: I’m already planning on not going to the 2019 Worlds in Saitama, but 2020 will be in Montreal, which is well within driving distance for me. Last year I also went to Worldcon, again in Helsinki, but this year’s Worldcon is in San Jose and I’m inclined to skip it as well. (Worldcon 2019 will be in Dublin and Diane Duane is GoH and I’ve already bought a full membership; 2020 isn’t decided yet but will likely be in New Zealand.) Other sporting events I’m considering, besides the usual baseball and hockey arenas, are the IBSF World Cup stop in Lake Placid, maybe some ISU Grand Prix skating events, and perhaps the 2020 Winter Youth Olympic Games in Lausanne — all of this is very speculative right now and might not come to anything.

Posted in Administrivia, sports, travel | Tagged ,

My comments on passenger rail infrastructure to MassDOT

The comment period for the State Rail Plan ends on Friday, the advocacy group TransitMatters just released their report on Regional Rail, and MassDOT is currently in the process of two separate planning exercises related to the MBTA and the commuter rail system in particular, in preparation for the next tender for the commuter rail operations contract (currently held by the French firm Keolis). Yesterday evening I sent in my own comments, inspired by the State Rail Plan deadline, but most of what I had to say was outside the State Rail Plan’s scope, so I also sent it to the people responsible for the MBTA planning process. Here’s what I said, edited slightly for formatting.


I was originally going to send this in regard to the State Rail Plan, since the public comment window was recently extended, but on closer review it seems that most of my comments are more usefully directed at the MBTA-specific planning process, since I live in the MBTA district. However, my points 1 and 2 below are intended to reference all passenger rail corridors in the state, not just the MBTA service area, and in particular the Commonwealth should explore opportunities for cooperation with neighboring states and with Amtrak to investigate the application of these principles to the Connecticut River Line and to future Boston-Springfield intercity passenger service.

Unexpectedly, much of what I have to say has been preempted by the release of a report by the advocacy group TransitMatters, which you will have seen already (for the record, “Regional Rail for Metropolitan Boston“, is the report to which I refer). However, I will make some additional comments on subjects that are not addressed in the TransitMatters report.

I have lived in Framingham for 17 years, and for that entire time, I have commuted in a single-occupancy vehicle on the Massachusetts Turnpike to my job in Cambridge. I would prefer to have an alternative that does not involve driving, but the current MBTA commuter rail service is infrequent, slow, unreliable, and more expensive per marginal trip than my commute. During the summer months I will bicycle to work (on approximately 40 good-weather weekdays between May and September); a better commuter rail service with real provision for bicycle users (not limited to off-peak hours) would substantially increase the number of days a bike commute is practical by enabling bike+train round trips.

To put more precise numbers on it, I pay (employer-subsidized) $10 a day to park in Cambridge, and my shoulder-hours SOV commute (10:25 AM and 7:15 PM) takes approximately 35 minutes parking space to parking space. The current MBTA Framingham/Worcester Line service has a long gap in service after 9:30 AM that makes it impractical for my schedule, but even if I shifted my schedule earlier to take train #512 inbound, the actual time cost of the MBTA service (with the necessity of driving to the Framingham station, finding and paying for parking, the train ride to Boston, the transfer penalty, the subway or bus trip to Cambridge, and then walking to my office) would be well more than double my current car commute. (My bicycle commute, 20.8 miles via two different routes, takes approximately 85 minutes, or about as long as the current commuter rail service, at an average speed of 15 mph — but with much greater health benefits.)

I would be willing to consider commuter rail — indeed, I would strongly prefer it — but for the excessive travel time (which is of course compounded by the system’s current widely reported unreliability). A reliable travel time of not more than 70±10 minutes would be easily within consideration, and with properly optimized schedules and full construction of West Station would make it highly attractive for many commuters from the Metro-West area who currently drive to jobs in Cambridge or Boston. I have heard anecdotally that the Commonwealth currently considers demand for access from Metro-West to jobs in Cambridge negligible to the point of not being worth studying, and I would strongly encourage the planning staff to consider this commuting pattern more seriously, as rising housing costs have made living closer to work impractical for many people who would prefer a transit option.

In the spring of 2017, I took a vacation in Helsinki, Finland, where I had occasion to use the rail system there extensively. The rail network around Helsinki, like Boston, is based on a stub-end terminal station (they only have one, unlike Boston’s two, and it’s correspondingly larger in terms of footprint than is possible in congested downtown Boston). However, Helsinki’s regional transport administration, HSL, has implemented an urban and inter-suburban rail network in the “regional rail” style described by the TransitMatters report, with full fare integration and high frequencies, connecting Helsinki Central Station with both historic suburban and exurban downtowns and new neighborhoods of transit-oriented development. HSL also maintains fare integration with intercity passenger rail services that serve nearby metropolitan areas outside the HSL district, so riders within the region can take a suburban train or an unreserved regional train, whichever is more convenient — this should be a model for intercity passenger service in Massachusetts along corridors such as Boston-Worcester-Springfield, which might be operated by a different agency or contractor than the MBTA.

Metropolitan Helsinki has about a third the population of the Boston MSA and is also slightly less dense; it has only one heavy rail subway line, and for surface transit has only street-running tramways, ferries, and private-tender bus services. The population of the whole of Finland is about that of the Boston MSA and is smaller than the Boston-Providence CSA, and Finland has quite high levels of suburban development and car ownership relative to other European countries. Yet Helsinki sustains substantial investment and substantial ridership in its fast, frequent, reliable, and affordable commuter rail system. I wrote a series of blog posts about it when I returned from my trip, which you can refer to here (fares and network structure) and here (suburban rail). Note that the services described in both of those articles have been realigned and in a few cases significantly expanded since I wrote those posts last April.

My specific recommendations, which are generally in accord with those in the TransitMatters report:

  1. The Commonwealth should adopt as a matter of policy a preference for electrification and high-level platforms on all rail routes currently served or contemplated to be served by passenger trains. In some cases this may require additional state investment to maintain compatibility with freight services.
  2. All projects and studies inconsistent with point (1) should be terminated.
  3. In the Boston region, a priority should be placed on electrification of the South Side commuter rail, improving operating costs, schedule reliability, and environmental justice. As funding becomes available, investment should shift to the North Side lines, which will require more infrastructure to be constructed.
  4. Where possible, labor agreements should be sought that limit excess staffing in exchange for acceleration and simultaneous construction of projects along multiple lines, maximizing useful employment of skilled trades.
  5. As TransitMatters notes, the electrification of the Providence Line is nearly complete and should proceed forthwith, as should electrification of the Stoughton and Fairmount Lines, with the existing diesel locomotives and rolling stock shifted to reduce maintenance pressures on other lines.
  6. Although the North-South Rail Link would significantly improve regional connectivity and the overall utility of the rail network, construction of NSRL is by no means a prerequisite to implementing electrification, high-level platforms, and frequent all-day schedules, and these should proceed at the highest priority, given the current capital expenditures which would otherwise be required even to preserve the existing diesel infrastructure, whether or not a funding mechanism for NSRL can be identified.
  7. The Commonwealth should in particular be prepared to self-fund the entire acquisition cost of electric-multiple-unit trainsets in order to buy global best-of-class equipment at competitive market prices, unless the federal government commits to waiving Buy American provisions. Federal capital funding, if available, could still be pursued for track, platform, station accessibility, overhead wire, and substation construction.
  8. Full build of West Station and development of connecting routes to Cambridge (whether bus, light rail, or a shuttle via the Grand Junction branch) and Longwood Medical Area should be accelerated relative to current plans.
  9. With respect to Framingham in particular, in order to support high frequency service between Framingham and Boston it will probably be necessary to have some trains turn or lay over at Framingham. The Commonwealth should study, in conjunction with the City of Framingham and MWRTA, the potential benefits of exending trains along the Agricultural Branch to Framingham State University and possibly to the office-industrial park area at Route 9 and Crossing Blvd. where there is already a park-and-ride lot and numerous employers that could be served by a reverse-commute service.

You can see more related content in this blog’s category “Transportation” (links below or to the right depending on your screen layout).

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What’s Wrong with Metcalfe’s Law?

In a recent Medium post derived from a talk he gave at private invitation-only event for the IT industry, Dan Hon presents one view of Metcalfe’s Law, the theory espoused by Ethernet inventor Bob Metcalfe that “the value of a telecommunications system increases as the square of the number of participants”. Hon looks at the (no pun intended) value judgment embedded in talking about the “value” of a network, and considers purely market-oriented measures lacking.

I’d like to step back a bit and look at it from a different angle. Instead of “value”, let’s consider “utility”: what benefit arises to the users from their use of the network? Metcalfe’s claim can be restated simply: the global utility (sum over all users) of a network is quadratic in the number of users. You don’t even need graph theory to prove that this is trivially true, if you accept what I take to be Metcalfe’s presuppositions: first, that utility sums linearly over all users (a view which would be understandable to Jeremy Bentham), and second, that each user’s utility is linear in the number of other users on the network.

The real problem with Metcalfe’s Law, as I see it, is precisely in this second presupposition. While it is true to a first approximation, for small networks, once the network reaches a sufficient penetration of that community with which any individual user has an interest in communicating, the marginal utility of additional communications partners diminishes quite rapidly, and ultimately goes negative. We see this even with old technologies like the telephone network: nearly all of the value I get from the telephone derives from being able to communicate with family, friends, and current and potential employers, vendors, and service providers in my immediate vicinity. While connecting a billion people in India or China to the rest of the world is laudable, there cannot be more than a thousand of them that make the telephone network more valuable to me. (One thing that this analysis does not consider, and a more sophisticated analysis would, is economies of scale: do those billions of users actually make it easier or cheaper to provide me with the service that I value. To be left for another day.)

In the social network case, it’s clear how additional users can have negative marginal utility: the additional noise generated can drown out the intended communication (whether that noise is trolls, pile-ons, or just way too many well-meaning people making the same comment in a reply). Twitter is a great demonstration of this; users bearing the vaunted “blue checkmark” — a distinction given out entirely at Twitter’s discretion to a small subset of users, mostly celebrities, journalists, government officials, and corporate marketing departments — are given a variety of tools to screen out communications from the masses. One of the tools which is frequently employed by these “verified” users screens out all notifications from the remaining users, allowing them to give the appearance of using the platform to communicate with others while in actuality paying attention only to a small number of similarly privileged people. This screening was not part of the original Twitter service: it was only deployed after Twitter gained a sufficiently large and noisy user community that it was driving away users Twitter actually had a business reason to want to retain. Of course, even “old tech” had to come up with similar mechanisms: when telephone calls became cheap enough that scammers were willing to spam a thousand people at dinnertime in the hope of finding a single mark, caller ID became a necessity and more and more people began to screen their calls. (Compare also the Eternal September.)

In conclusion: Metcalfe’s Law is wrong because the marginal utility to the existing users of a communications network is not constant: while it is large and positive for small networks, as networks grow beyond the scale of normal human social circles, the utility drops off quite rapidly, and eventually goes negative. When you sum up this function over all users, unlike the linear utility posited by Metcalfe, overall value does not scale as the square of the number of users. (It might not even be asymptotically linear — I leave that analysis to someone with better mathematical chops.)

Posted in Computing, Law & Society | Tagged