Why it's better to walk from Willcocks

The Problem:
     After finishing class in the Southwest quadrant of campus, many students from Etobicoke and other Western Toronto suburbs must avail themselves of TTC transportation. This is most efficiently done by boarding the subway at the Spadina Road station (see map). However, there are two methods by which students may approach the transit to Spadina station - on foot, or by way of the Spadina Road streetcar. Assuming that all students reach Spadina Rd. via the Willcocks Street intersection, and further presuming that the added physical effort of walking is not a significant concern, and that minimizing average travel time is of prime importance, what is the best method to reach the subway station.

Two Solutions
  • The Streetcar is Faster - Proponents of the streetcar alternative note that the average velocity of a streetcar is approximately 18km/h (11.25mph) which is 3 to 5 times the average walking speed of a unencumbered person over even terrain. Also noted are the merits of patience, and the additional social opportunities afforded by being crammed tightly enough that your atoms comingle with those of your fellow passengers.

  • Walking, stupid. - Walking advocates point out that arguments from greater streetcar velocity assume that the walker and the rider depart at the same time. However, since walking can begin with 0s latency upon reaching Spadina Road, streetcar riders are dependent upon the arrival of a streetcar before they can appreciate velocity gains. Consequently, their travel time is variable, whereas a walker's time can be expected to remain relatively uniform.

    • Streetcar/Walking Hybrid, a.k.a. The "Streetwalker" Approach - This modification of the Walking approach notes that if, while walking, one notices a streetcar approaching, and it is possible to reach a streetcar stop in time, then it is clearly to the walker's advantage to board the streetcar. This allows the walker to maintain an acceptable maximum travel time, while affording opportunity to reduce travel time greatly when circumstances make a streetcar available.

Evaluating the Alternatives

     In order to understand the implications of each stance, we must delineate the argument in more detail. Consider:

  • Let Ts represent the average travel time for a streetcar.
  • Let Tw represent average travel time on foot, and
  • Let Th represent average travel time using the modified walking strategy, or Streetwalker strategy.

  • Further, the distance, d, to Spadina Station, via either route, is roughly 450m (1476ft).

  • Then Ts = (Time Spent Waiting For Streetcar) + ((loading time/stop) * 3 stops) + (450m / (5 m/s)
          = Wait + 3*Load + 90s
  • Whereas Tw = (450m / 1.39m/s) = 324s = 5 minutes, 24 seconds.
  • And furhter, Th1 = Tw in cases where no streetcar is available, or
          Th2 = (200m < approx. distance to intermediate stop > / 1.39m/s) + 1*Load + (250m / 5m/s) = 1 * Load + 193s

  • Empirical data gathered indicates that during high-volume periods, average load time is roughly 45s. (Note that this ignores the effect of stoplights. However, Spadina is a major street compared to those with which it intersects, and light ratios have been established accordingly, i.e. the light is usually green for northbound traffic. Moreover, stoplights pose equal encumberance for both pedestrian and motor traffic, assuming traffic laws are obeyed. Finally, while it is true that the deceleration caused by a red light is more significant for the measure of streetcar speed, we assume in our calculations that the streetcar can maintain 18km/h at all times, we will see that the discrepancy does not matter.)

  • Thus Ts = Wait + 3*45 + 90 = Wait + 225

  • If streetcars are the faster medium, then Ts < Tw (putting aside streetwalking for the moment).
  • Then Wait + 225 < 324, thus Wait < 99
  • However, consider the following data on streetcar arrivals, gathered over two months, whenever I bothered to record it (measured in seconds):

  • The wait times have the following statistical properties:
    • Mean: 185
    • Median: 181
    • Standard Deviation: 82
    • Q1: 125.8
    • Q3: 246.5
    • % of times under 99s: 16.7

  • The streetcar stance only prevails if Wait < 99, but here we see that the average wait is double that! Only in 16% of cases will wait time be low enough that the streetcar approach succeeds. In fact, in 3 of the 24 times measured, the wait alone exceeds the total walking time to the station.
  • With a mean wait time of roughly 3 minutes, and assuming that if we are within 1 minute of the intermediate stop, we can catch the streetcar, then the hybrid approach will involve Th1 67% of the time, and Th2 33% of the time, thus yielding an average: Th = 0.67*324 + 0.33*(45 + 193) = 295 = 4 minutes, 55 seconds.
  • So in almost every circumstance, it is clearly advantageous to use a walking, or hybrid approach. QED.