Regular readers will be familiar with many analyses published here that review the behaviour of transit service on streetcar and bus routes. One of the more galling parts of a transit rider’s life is the uneven headways (the times between vehicles) for almost all services that make the anticipated wait longer than the advertised average in the schedule.

This article is an attempt to devise a measure of this problem that can be used to show the degree to which actual service deviates from the scheduled value in a way directly related to rider experience. Some of the material is a tad technical, but not overly so. Also, readers should note that this is a first cut, and suggestions for improvement are welcome.

Wait Times Versus Headways

In theory, service is scheduled to arrive at a stop every “N” minutes, like clockwork. Therefore, the average rider will wait half of this time for a vehicle to show up. If buses are supposed to arrive every 5 minutes, then the average wait is 2.5 minutes. But the situation is more subtle than this.

If riders arrive at the stop at a uniform rate, say one each minute, they will not all wait the same length of time. Someone who arrives in the first minute will wait almost the entire headway, 4.5 minutes on average, while someone who arrives within a minute of the bus will wait on average only 0.5 minutes. The five riders who arrived in the five minute headway will have varying wait times, but on average the value will be 2.5 minutes per rider, and a total of 12.5 for the group overall.

    Rider  Wait Time
      1       4.5
      2       3.5
      3       2.5
      4       1.5
      5       0.5
    Total    12.5

The total above is the sum of a simple arithmetic series, and because of the values chosen here, the formula for calculating the total is quite simple. It is the square of the headway in minutes divided by two.

    Total = (N * N) / 2

If the service is “well behaved” with consistent headways, this is just a complicated way of getting to the same result with the average wait being half a headway (5 minutes divided by 2 giving 2.5 minutes). However, when headways are not consistent, the square factor in that formula shows its effect.

Suppose that vehicles are supposed to arrive every 5 minutes, but in fact two vehicles arrive sometime within a 10 minute window on varying headways. These might both be 5 minutes, but they could also be 6 and 4, 7 and 3, etc. In this case, the wait times behave differently. The table below shows how the average wait time for the ten riders accumulating at a stop during a 10 minute period vary depending on the regularity, or not, of the headways.

    First     Wait    Second    Wait    Total    Average
    Headway   Time    Headway   Time     Time     Wait
      5'      12.5'     5'      12.5'    25.0'     2.5'
      6'      18.0'     4'       8.0'    26.0'     2.6'
      7'      24.5'     3'       4.5'    29.0'     2.9' 
      8'      32.0'     2'       2.0'    34.0'     3.4'
      9'      40.5'     1'       0.5'    41.0'     4.1'
     10'      50.0'     0'       0.0'    50.0'     5.0'

It is self-evident that if two buses arrive every 10 minutes, the average wait time is 5 minutes, but this table shows how the values rise for different levels of headway inconsistency in between.

The TTC considers that vehicles that are only slightly off schedule (one minute early or up to five minutes late) are “on time” for purposes of reporting service reliability. The problem with this scheme is that the leeway it allows is very broad for routes that operate frequent service. Indeed, if a bus is planned to come every 5 minutes, two could come every 10 and the service would still be “on time”.

There is also the problem that riders on such routes don’t care about the schedule because it really is meaningless. They only care about reliable service, and a six-minute swing of “on time” values is often wider than the scheduled headway. The result is a meaningless metric of “on time performance”.

For wider scheduled headways, that six minute swing does not represent, proportionately, as much of a potential change in average wait times. Consider pairs of buses on a 20 minute headway:

    First     Wait    Second    Wait    Total    Average
    Headway   Time    Headway   Time     Time     Wait
      20'     200.0'    20'     200.0'   400.0    10.00'
      21'     220.5'    19'     180.5'   401.0'   10.03'
      22'     242.0'    18'     162.0'   404.0'   10.10' 
      23'     264.5'    17'     144.5'   409.0'   10.25'
      24'     288.0'    16'     128.0'   416.0'   10.40'
      25'     312.5'    15'     112.5'   425.0'   10.63'

Although the headways are wider, the percentage in change relative to the scheduled value is smaller, and so both the total rider-wait time and the average wait time don’t shift as much as they do for more frequent service for the same latitude in headway adherence. To get the same effect, proportionately, as in the first example, the two buses would have to arrive together in a 40 minute gap.

Note that these values behave the same way regardless of the assumed arrival rate of passengers provided that this rate does not change. The total rider-wait times would be higher for an arrival rate above one per minute, but the average values would remain the same.

There is also a presumption here that for infrequent services, riders will arrive uniformly over the headway. Where the route is the origin for a trip, riders might be expected to time their trips to minimize waits, although this behaviour can be thrown off if service is not reliable, especially if it is often early. For riders arriving at a bus-to-bus connection or at a subway-to-bus transfer, they are unlikely to be able to fine-tune their trips to just catch a bus. (The exception would be a network with protected, timed transfers, but the TTC does not have any of these.)

Crowding Effects

These values also affect on-board crowding because a vehicle carrying a wider than scheduled headway will accumulate more passengers and will have longer stop service times. It will become later and later, and the vehicle behind will eventually catch up. As with “average” wait times, the “average” crowding level will not represent the actual situation experienced by the “average” rider. Just as more riders wait longer than the planned average when buses do not arrive regularly, more riders are on the more crowded vehicles.

The problem is further compounded because most riders try to get on the first vehicle that arrives lest the second one be short-turned and they are left in a big gap waiting to continue their journeys.

If one were to poll 100 riders distributed between two streetcars, they would not complain about crowding if they were equally distributed and all had a seat. However, if 80 of them were on the first car and only 20 on the second, the average perception of crowding would be that the service was overloaded and that it did not show up reliably.

This is a fundamental split between service as the TTC sees it (on average) and as riders see it (as an individual experience).

Measuring the Difference Between Actual and Scheduled Wait Times

In preparing this article, I have been wrestling with various ways to calculate and display these values in a useful way. There are pitfalls both in the methodology and in the charting of information. The examples below are only one way to present this information, and should not be read as definitive.

Here is the basic premise:

• Within each hour of the day, vehicles pass by a location on a route, each of them on its own headway.
• From the headway values, one can calculate the wait time for riders assuming an arrival rate of one/minute. The total of these times divided by 60 gives the average wait per rider over the hour period.
• The count of vehicles within each hour is easily converted to an average scheduled wait time of one half the average headway.

These values are graphed for the 504 King route for Tuesday, November 1, 2016. A sample page is shown below and the full sets are linked as PDFs.

504_20161101_avgwaittime_wb

504_20161101_avgwaittime_eb

In these charts the solid lines represent the rider wait time in minutes/passenger while the dotted lines are the average that would have applied if all of the service had been evenly spaced over an hour. Each colour shows data for a different hour through the day, and the horizontal positions show various locations along the route. Note that the rider wait times are generally higher than the average times implied by the vehicles/hour.

There are several caveats about this presentation:

• For the 6:00 am line (red on page 1 of each set), the first headway is assumed to occur from 6:00 am until the arrival of the first vehicle. This will generally understate the headway on which that vehicle is operating.
• Depending on how badly disrupted the service is, the wait time contributed by the first vehicle passing in any hour may include time accumulated during the previous hour. This causes wait time to be mis-attributed, and it is possible for the rider wait time in the previous hour to be understated. For example, if one car passes a stop at 7:56 and the next one at 8:04, all of the rider wait time associated with the second vehicle will be included in the 8:00 data as will the vehicle in the total count. This effect is generally small enough that it balances out, but it can be more pronounced on wider headways where there are fewer vehicles to smooth out the data.
• In some cases, a gradual increase in rider wait times is visible along the length of the route showing how service becomes progressively more bunched into pairs.
• Major delays can result in a large increase in rider wait time because the long gap dominates the calculation, and following cars contribute almost nothing to this value. However, the number of cars/hour could still be at the scheduled level and the “average” wait times are not affected.

For reference, I have also included here the chart of overall service for the day.

504_20161101_chart

There is a disruption of service westbound at Dufferin between 10:00 and 10:30 am. This causes headways west of this point to be bunched, and that bunching is reflected back on the eastward trip from Dundas West as a parade of cars makes its way across the city. Note that nothing was short-turned into the gap eastbound, and it was not until the next westbound trips after 11:00 that service began to be sorted out.

A delay eastbound at Queen & Roncesvalles just after 15:30 shows up as a spike in “3 pm” rider wait times at that location, but most of the gap passes points further east during the “4 pm” interval. This explains the behaviour of the plots for these two time periods.

As I wrote above, this is a first cut at attempting to measure what riders experience versus what the TTC will typically state as its quality of service.

A useful corollary to this would be to examine data from automatic passenger counters on the vehicles, but these have not yet been rolled out fleet wide. This would allow us to tie the service reliability values to crowding conditions on vehicles.

What will not show up, of course, is the number of passengers who simply never get on because they tire of waiting for a vehicle with room on it. At least with detailed counts, we would know how often “full vehicles” actually occur, and especially cases of lightly-loaded ones that are not pulling their weight because they are part of a parade. That is a task for a future round.

If I get any other ideas about how to present this information, I will update this article with additional examples.

Updated Nov. 19, 2016 at 10:10 am: The TTC has updated their comment on the status of the service improvements implemented in 2015-16.

Identifying the specific routes which would be affected would be done in 2017 following a review of the initiatives implemented in 2015-2016. [Email from Stuart Green]

The Star’s Ben Spurr had an article on November 18, 2016 about the degree to which the TTC fails to meet its own service standards for vehicle crowding.

The many debates about budget and service routinely ignore the question of whether the TTC actually meets the Board-mandated standards, and instead we are treated to complaints from the budget hawks about poorly performing services.

In my review of the proposed 2017 operating budget, I noted that the traditional targets of such complaints do not offer much latitude for savings, while much more money is on the table from potential reductions of service quality and reversal of the recent Tory service improvements such as the 10 minute network.

For reference, here are the current standards taken from the TTC’s website:

The TTC has provided the detailed list it sent to Ben Spurr, and here it is.

The list comes with various caveats from the TTC, notably that many of the routes that are “over” the standard are only slightly past the line by say a few passengers over the target.

In a nutshell, 43 of the TTC’s bus and streetcar routes exceed crowding standards at one or more periods during the week and on weekends. In most cases, though, the average number of passengers per vehicle is only slightly in excess of our crowding standards…one or two people in some cases.

The standards, as the chart in the PDF illustrates, vary between periods and vehicle type. Also, in off-peak periods, the standard is the number of seats on a vehicle. So if just one or two people are left standing, that would exceed the standard but is not overcrowded, per se. [Email from Stuart Green in TTC Communications]

As regular readers know, I have written extensively about service reliability on the TTC. This affects the actual quality and capacity of service in many ways:

• Vehicles that are bunched often have extra space on the second (third, etc) vehicle because everyone boards the “gap” vehicle. A strong inventive for this behaviour is that dropping back to a following, less crowded vehicle may leave the rider turfed off when it short turns. Despite TTC claims that total short turn counts are down, this remains a problem and rider behaviour reflects their experience.
• The number of vehicles per hour operated is not always the scheduled value. Therefore the actual capacity of the route could be less than advertised.
• On routes with articulated streetcars or buses, regular single-section vehicles can often be found in place of artics. The result is a loss of capacity and greater than scheduled crowding.

Past observations by TTC planners have rather quaintly observed that the cheapest “new” capacity one can provide is to operate reliable service so that loads are evenly distributed. I leave it to readers to contemplate the degree to which the TTC achieves this, especially when it has a six-minute wide window within which vehicles are “on time”. For frequent routes, vehicles in pairs can be on time even though they provide gappy service. This is a very long-standing problem, compounded by the fact that the TTC does not even regularly meet its rather generous “targets” for headway reliability.

I asked the TTC how the average loads are calculated as reported in their tables. Specifically, is the average calculated based on the observed number of vehicles, or on the scheduled service, and is the percentage of load based on the scheduled vehicle type rather than what actually operates. The TTC replied that the capacity used to calculated the percentages is based on scheduled values and vehicle types, not on the service available to riders on the street.

This is an important distinction for routes with erratic service and frequent replacement of large vehicles by smaller ones, notably 501 Queen. It is quite possible that the actual average loads considerably exceed available capacity even though they may fall within theoretical values based on the schedules. What may look like a seated load on paper could well have a considerable number of standees because headways are wider and/or vehicles are smaller than planned.

One might reasonably ask for additional information that must be available from the underlying TTC data:

• What is the actual capacity of the service operated against which the crowding values are calculated?
• How variable is the headway (maximum and minimum values, percentage of vehicles close together, etc.)?

With respect to monitoring service for schedule changes, the TTC says:

The specific decisions on when and where to adjust service to these standards are made by TTC staff. In a typical year we make more than a hundred service changes. We regularly conduct reviews of ridership to assess needs across all routes. If there is a route that needs some additional service, we will try to balance that against a route that may have capacity to spare.

In a large, busy system such as the TTC’s there are always some routes at any point in time that have crowding that exceeds the crowding standards. And while we would obviously prefer our customers are not on crowded vehicles, we have to balance the availability of resources like vehicles and operators with delivering service to the entire city.

Through regular service reviews and with arrival of the new vehicles, there is an opportunity to increase capacity on some of the busier routes like the 504. In particular the TTC’s new high capacity low-floor streetcars will assist in reducing overcrowding on our streetcar routes.

There’s a bit more to things than this reply lets on. Every year there is a “Service Budget” (the one for 2017 was probably set a few months ago), and Service Planning works within this to allocate service to the available resources. However, a critical component is the number of available operators and this runs into the dreaded budgetary problem of “head count avoidance”. In off peak periods, routes may cry out for more service, but vehicles will sit in the yards if there is nobody to operate them.

Yes, the TTC’s new cars will help to reduce crowding when and if they actually arrive, and when they start operating on really busy routes like King. On 510 Spadina, the introduction of low-floor cars was accompanied by a cut in the number of vehicles in service, although the net change remained an increase in capacity. This has long been an issue in plans for fleets of larger vehicles because in some cases, notably the conversion of bus routes from regular to articulated vehicles, the “new” service has roughly the same capacity as the old by the simple expedient of running two artics where there previously had been three regular sized buses. Years ago, the same fate hit the 501 Queen car, and service quality (not to mention ridership) has never recovered.

The TTC’s first goal for many on Council is to reduce costs, not to increase service, and claims that there will be more capacity must be tempered with the observation that such an improvement may not be “affordable”. Just look at what happened with 514 Cherry which was supposed to be a net addition to service on King Street, but which the Commission refused to fund. The result was that service was reallocated between the 504 and 514 routes with only a modest improvement in the portion of the route where they overlap. (The question of how irregularly the 514s actually appear is another matter.)

Going into a year where the Service Budget is planned to rise only by 0.4%, there is not much headroom for more capacity, and anything added on one route will inevitably be taken from another. This may be “efficient”, but it runs headlong into policy objectives such as the core network of 10 minute services, all-day service on all routes and the existing crowding standards. As the budget report showed, there is not much available to trim from the lightly loaded routes when resources are needed on busy routes elsewhere, and the more likely situation will be to pack in more riders.

The problem is even worse if the TTC contemplates relaxing the standards so that average loads on vehicles will rise. We already know from experience that many routes are crowded, and the last thing they will need is less service and even worse travel conditions.

I asked whether the cost estimates for savings on these proposals included detailed lists of possible service changes. The TTC replied:

The table shows broad categories that generally would reflect the reversal of service initiatives implemented in 2015. The specific service changes within each category have not been identified.

This was further qualified in a separate email after this article was published:

Identifying the specific routes which would be affected would be done in 2017 following a review of the initiatives implemented in 2015-2016.

What a marvelous way to short-change debate – don’t tell people what the actual effects might be. This is precisely the sort of campaign one would expect from an anti-transit Council or Mayor who would say “this won’t hurt too much”, or so a Councillor could claim they never knew the details of what was planned before it happens. Will someone on the TTC Board please make an official request for the details before this goes to Council?

On a more general note, the primary function of the TTC is to provide transit service. The Board sets policy and sits back expecting that service actually operates within those parameters. However, there is always a Catch-22, the caveat in any policy “subject to budget availability”. This allows the Board to pass a policy promising the sun, moon and stars to transit riders, but it’s ok if staff only delivers a few minor planets in a far-off galaxy because they couldn’t afford to deliver more. Meanwhile, Council blithely approve billions in spending on capital projects.

This is not to say that those capital projects are unneeded (that pitched battle has been debated elsewhere), but it is a double standard to treat capital funding as something we must do as an integral part of city growth and prosperity while at the same time starving day-to-day service because keeping taxes down is the paramount goal.

Every month, we see a CEO’s report from Andy Byford full of beautiful pictures and graphs, but with very little commentary on the basic question: are we providing the service that Toronto needs, and if not, what do we have to do to achieve this?

It’s all very well to have customer surveys and indices of station cleanliness, but what about the actual service on the street? Is the plateau in ridership related to the system’s inability to carry more riders or to attract business?

TTC staff are supposed to be producing a new Ridership Growth Strategy for the TTC Board in January. It’s a shame this wasn’t produced as an integral part of the budget. How bad is the service today? Where can it improve? How much more would be possible and at what cost? Those are questions no penny-pinching Mayor wants asked let alone answered.

In a few days, the TTC Board will consider its 2017 budget, and we will see just how much appetite there is for forcing Council to cough up the money needed to run the transit system, to pay for all those photo ops a few years ago when John Tory claimed he would reduce the damage wrought by Rob Ford.