Single Tracking -- Solution

Single Tracking

Figure 0b below reprises Normal Operations in the set-up for this puzzle. In Figure 3a we see an EB train arriving at Station B on Track 1, where it unloads passengers having Station B as their destination. However, instead of loading EB passengers and proceeding to Station C, let us propose that the train off-loads all of its passengers onto the platform and takes on WB passengers. As depicted in Figure 3b, our train in Station B Track 1, reverses direction during its dwell time tD, then proceeds as a WB train on Track 1, crossing over to Track 2 via iA for providing Normal Operations service to Station A and beyond.

Our EB passengers must wait on the platform at Station B for the next EB train, which will be coming from the east as a WB train on Track 1. As shown in Figures 3c and 3d, that train will off-load all its WB passengers, load both new and waiting EB passengers, then reverse out of Station B as an EB train.

The wait need not be long at all. With a well-designed Single Tracking time-table, it would be elementary to assure that the WB train will be approaching Station B just after the WB train already in Station B has departed, as suggested by the striped symbol in Figure 3a. Likewise, the EB train will be approaching Station B just after the EB train in Station B has departed as suggested in Figure 3d by the striped symbol. The phasing for Single Tracking might also take into consideration the relative volumes of EB and WB passengers based on time-of-day.

Thus, our proposed solution invokes a procedure that might be called Load Swapping...

(a) Split the Single Tracking territory in half,
(b) Swap Loads in the middle station, then...
(c) Reverse Trains on Staggered Schedules.

Sophisticated solvers may want to follow-up on the Test Case, which did not invoke Load Swapping. We let tHDOUBLE-TRACKING = 6 minutes, tAB = 7 minutes, tBC = 8 minutes, and we found that tHSINGLE-TRACKING = 2(7 + 8) = 30 minutes.

A passenger showing up randomly at a station expecting to wait an average of tHDOUBLE-TRACKING / 2 = 3 minutes for the next train, will be disappointed at having to wait around tHSINGLE-TRACKING / 2 = 15 minutes. It should be noted, however, that once onboard a train, the passenger will not experience a significant lengthening of trip-time attributable to Single Tracking.

Let us take a moment to consider the effect of all those EB and WB trains that come pouring into stations A and C respectively, each at the rate of 60/ tHDOUBLE-TRACKING = 10 trains-per-hour. The SingleTracking territory can only accommodate 60/ tHSINGLE-TRACKING = 2 trains-per-hour.

We have observed elsewhere that headway is a mathematical reciprocal of flow-rate on any branch line (without divergence or convergence). As such, like current in an electrical circuit, flow-rate must be the same at every point along each branch line.

So long as that bad-order train is stuck at Station B, it will be necessary to take 10 - 2 = 8 trains-per-hour out of service for the whole branch line, unloading their passengers and diverting the empty trains into sidings, pocket tracks, and storage yards.

Using the Load Swapping procedure, we find tHSINGLE-TRACKING = 2 tAB = 14 minutes between stations A and B.. Between stations B and C, we calculate tHSINGLE-TRACKING = 2 tBC = 16 minutes. On average then, for the Load Swapping procedure tHSINGLE-TRACKING = 15 minutes.

A passenger showing up randomly at a station expecting to wait an average of tHDOUBLE-TRACKING / 2 = 3 minutes for the next train, will be disappointed at having to wait around tHSINGLE-TRACKING / 2 = 7.5 minutes. Using the Load Swapping procedure, a passenger will not experience a lengthening of trip-time attributable to Single Tracking -- unless the trip begins before station B and ends after station B. Even then, the platform waiting time at station B will depend on the staggering of time tables for EB and WB trains, which on average will be about tHSINGLE-TRACKING / 2 = 7.5 minutes.

Using the Load Swapping procedure, with a bad-order train stuck at Station B, the system will accommodate 60/ tHSINGLE-TRACKING = 4 trains-per-hour, taking 10 - 4 = 6 trains-per-hour out of service on the branch line until normal service can be restored.

Epilog

Our model for the puzzle can be generalized to accomodate any number of stations in the Single Tracking territory (each without adjacent interlockings, of course). All we need to do is increase the times tAB and tBC appropriately. At least one commuter rail line does indeed interpose additional stations between A and C. For that line, an alternative solution has received consideration...

Figure 0c shows our model configuration for Normal Operations augmented by a third interlocking iB located adjacent to Station B.

Addition of iB shows about the same benefit for reducing tHSINGLE-TRACKING as the Load Swapping procedure proposed above, not better! Please observe the exclamation point. Whereas iB will be seldom used (think rusty rails), the requisite capital cost for iB has an exceptionally large price tag inasmuch as Station B will be built underground.

Oh, and one more thing: Rush-hour commuters are all too familiar with platform announcements that end with "...delayed due to track switching problems." It is not difficult to imagine the adverse effect on headway resulting from a fault in any of the four switches in iB, with or without a bad-order train.

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