Copyright ©2008 by Paul Niquette. All rights reserved.
solvers discovered in the Station-Stop puzzle
that cruise speed vC does not play a part in determining
minimum headway tH. The Trip
Time puzzle is a different matter.
We will need to ascertain the total time it takes to depart one station
then travel to -- and depart from -- the next station. Let's call
tS. Now, if the distance between stations
is large, as with long-haul or commuter rail service,
will approach xS /
vC. Typical passenger
transit systems in urban environments, called full metro service,
xS to be less than two miles, which brings into consideration
parameters other than vC
for calculating Trip
Pictured in our Trip Time puzzle is a "Baby Bullet" train operated by Caltrain as an SRO commuter-rail service in San Francisco. For this analysis, let us use New York City Transit trains to illustrate a typical full-metro service.We turn again to the phase-plane plot to estimate the impact of a Station Stop -- this time on Trip Time not on headway. The lead car of a train, shown in green, approaches a station at cruise speed vC. The New York Transit system does not have automatic train control. Upon reaching Point 1, the operator sees a wayside sign that calls for slowing to a platform operating speed vP. At Point 2, the train is stabilized at vP.
If the platform is not occupied, the train operator sees a green light, which allows continuation at vP into the platform all the way to Point 4a where the operator performs a feathered stop at Point 5.
Stopped in the station platform, a train waits out its dwell time tD, during which doors are opened for unloading and loading of passengers. Typically 20 seconds in duration, tD must be added directly to the Trip Time.
We now shift our attention to Point 1', the location of the rear-most extreme of the same train, with its phase-plane depicted in red in the sketch above. When tD has elapsed, the conductor closes the doors and the operator commands the train to depart the station. The train accelerates at a constant rate aS to vP, the platform operating speed, at Point 2'.
As the tail car reaches the far end of the platform at Point 3', the train is released to continue accelerating at aS from vP to cruise speed vC, which occurs well clear of the platform at Point 4'.
Here is a Base Case, using typical values...
tSS = tCP + tP + (xPO / vP) + tB + (xL - xF0) / vP + tF0 + tD + t0P + (xL - x0P) / vP + tPC
...and during that time, the lead car of the train will have traveled a total distance of...
xSS = xCP + xP + xP0 + xB + xL + xPC + xL
If the station in the Base Case were bypassed, say, by an express train, that distance xSS would have been covered at cruise speed vC, such that tBYPASS = xSS / vC. Thus, the difference in Trip Time tDELTA enjoyed by an express train in bypassing a local station can be calculated as follows:
tDELTA = tSS - tBYPASS.Sophisticated solvers will notice in those expressions that (a) both tSS and tBYPASS increase with all the same variables and (b) both tSS and tBYPASS decrease with increases in acceleration and decelerations aS, dS and dF. By plugging in the Base Case numbers, we find that tSS = 96 seconds and tBYPASS = 36 seconds. Accordingly, tDELTA = 60 seconds. Thus, for every bypassed local station, an express train saves one minute in Trip Time. For operational implications of that, see the Express vs Local puzzle.
Perturbations of the Base Case show some interesting interactions, if not ironic trade-offs...