Question 6 of 7: Alternate Progression Systems in a Downtown Grid

Question 6: Alternate Progression Systems in a Downtown Grid (20 marks — (a) and (d) 3 each, (b) 12, (c) 2)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

Find. A description of the three alternate systems, the system appropriate to this grid, a single cycle length that serves both directions, and the progression speeds that cycle actually delivers — together with a verdict against the City's requirement.

Approach. In an alternate system the cycle length is fixed by the block spacing and the desired progression speed. Compute the cycle each system would demand in each direction, find the pair that agrees, adopt a common cycle rounded to a practical 5 s increment, then run the relation backwards to get the speeds the adopted cycle actually delivers.

1. Part (a) — The three alternate systems. All three are simultaneous-family progression schemes for a uniformly spaced grid, and they differ only in how many consecutive signals share the same indication. In a single-alternate system adjacent signals are always opposite: as one block turns green the next turns red. A platoon released at one signal travels one block in half a cycle, so $$C = \frac{2L}{v}$$ In a double-alternate system signals are grouped in pairs, with alternate pairs displaying the same indication, so a platoon covers two blocks per half cycle and $C = 4L/v$. In a triple-alternate system groups of three share an indication and $C = 6L/v$. The trade-off is the point of the question: the single-alternate system gives the tightest, cleanest two-way band but demands a short cycle, which limits capacity and is impractical where pedestrian minimums are long; the double- and triple-alternate systems permit progressively longer, higher-capacity cycles at the same progression speed, but the through band becomes narrower relative to the cycle and the scheme is less forgiving of unequal block lengths.
2. Part (b) — Selecting the appropriate system. Tabulate the cycle each system would require in each direction. In the north–south direction, with $L = 150$ m and $v = 45/3.6 = 12.500$ m/s, the single-alternate cycle is $2(150)/12.5 = 24.0$ s, the double-alternate cycle $4(150)/12.5 = 48.0$ s and the triple-alternate cycle $72.0$ s. In the east–west direction, with $L = 270$ m and $v = 40/3.6 = 11.111$ m/s, the corresponding values are $2(270)/11.111 = 48.6$ s, $97.2$ s and $145.8$ s. A grid must run on one cycle length, so the correct choice is the pair of systems whose demands coincide: $$C_{NS,\,\text{double}} = 48.0\ \text{s} \;\approx\; C_{EW,\,\text{single}} = 48.6\ \text{s}$$ Hence the appropriate arrangement is a double-alternate system along the north–south roads (150 m blocks) and a single-alternate system along the east–west streets (270 m blocks). That the two agree to within 1.2 per cent is not a coincidence: the east–west blocks are 1.8 times longer while the required speed is only 0.89 times as high, and $1.8/0.89 \approx 2$, exactly the ratio between a double- and a single-alternate cycle.
3. Part (c) — Cycle length. The two requirements bracket the cycle between 48.0 s and 48.6 s. Signal cycles are set in multiples of five seconds, and the nearest such multiple to both values is $$\boxed{C = 50\ \text{s}}$$ This single cycle is then used at every signal in the downtown core, with a nominal 25 s green and 25 s red in each direction — long enough to accommodate a 7 s walk interval plus the pedestrian clearance for a two-lane crossing, which is a necessary check before adopting any cycle this short.
4. Part (d) — Actual progression speeds. Rearranging the alternate-system relation for speed, $v = 2nL/C$, where $n$ is 1, 2 or 3 for the single-, double- and triple-alternate case. For the north–south roads under the double-alternate arrangement, $$v_{NS} = \frac{4(150)}{50} = 12.0\ \text{m/s} = \boxed{43.2\ \text{km/h}}$$ and for the east–west streets under the single-alternate arrangement, $$v_{EW} = \frac{2(270)}{50} = 10.8\ \text{m/s} = \boxed{38.9\ \text{km/h}}$$
5. Part (d), continued — verdict against the City's requirement. Comparing with the targets, neither requirement is met: the north–south progression runs at 43.2 km/h against 45 km/h required, a shortfall of 4.0 per cent, and the east–west progression at 38.9 km/h against 40 km/h, a shortfall of 2.8 per cent. The cause is arithmetic rather than conceptual — rounding the theoretical 48.0/48.6 s up to a practical 50 s cycle necessarily slows the band, because speed and cycle length are inversely proportional in a fixed-geometry grid. Three responses are available and should be put to the City. Adopting $C = 45$ s instead would give 48.0 km/h and 43.2 km/h, exceeding both targets, at the cost of about 10 per cent of intersection capacity and a tighter pedestrian check. Holding $C = 50$ s and accepting a 3–4 per cent shortfall is defensible, since it lies well inside the tolerance of driver speed choice and of the platoon dispersion that occurs over a 600 m corridor. Or the progression speed targets themselves can be revisited, which is often the right answer downtown where a 40–45 km/h band is faster than the desirable operating speed on a pedestrian-intensive street.