Significant Digits

Railroad execution and planning need to be dealing with the same range of significant digits.

For those of us who began our engineering studies before the 70’s, the practicality of the slide rule is well appreciated. This intriguing device of sliding scales, that miraculously performs multiplication and division via the addition and subtraction of non-proportional linear distances, minimized the terror of dealing with an endless flow of variables encompassed in complex engineering equations. That was the heyday for analog mathematical devices as digital computers were being somewhat reluctantly infused into the engineering ranks, and accordingly, it was the pinnacle for the art of approximation, the art of defining squishy limits. Using the slide rule required thinking in terms of powers of 10 as well as understanding that there was a clear rationale to acceptable precision, what was referred to as “significant digits”. For example, did the resulting answer of 8.something shown on the slide rule’s “D” scale after multiplying and dividing a series of 27 numbers actually mean 8.2 ? or 8.25 ? or perhaps was it .0082 ?  or maybe 8200 something?  Back then, “being close” was good enough, in fact expected, as test scores were partially determined by the proper use of the slide rule.  Significant Digits

With the introduction of the personal calculator, arrived the immediate requirement for precision. This was not a level of precision, however, that one had to work for, but instead it was that which was instantly provided to the user on green-lighted displays. If one required the square root, or even a discounted cash flow, one needed only to push the appropriate function key once the data had been entered. And, there was no limit to the number of digits of preciseness it seems. The mind was given the answer, without thought, without question and, unfortunately, without the opportunity to truly understand the underlying mathematics.

Interestingly, for major railroads in North America and across the globe that are either primarily non-scheduled or without moving block operations, the planning and execution processes for traffic management employ opposite extremes of understanding the underlying mathematics and the discipline of approximation.  Practically speaking,  there is a very real opportunity for railroads to transition train dispatching from an art to a science by including mathematics-based movement planners. Currently, the approximations inherent to moving trains are so broad that the railroads are running below capacity in critical corridors. These inefficiencies are due to the lack of both timely train position and speed and the tools to handle the continuous mathematical processes, specifically reactive and proactive planners.  The former would assist the dispatchers with the current approach of crisis-based traffic management, whereas the latter would provide for the prediction of traffic conflicts and subsequently the ability to avoid them.

Whereas the execution side skimps on the mathematical processes, thereby resulting in broad approximations, the planning side of operations embraces the other extreme. Supported by incredibly complex mathematical algorithms being endlessly massaged on digital wizards capable of truly mind-boggling computations per second, the railroad planners pursue the absolute opposite of approximation with the concept of significant digits being totally unknown in their profession. To them, precision is a truly achievable dimension that is to be expected and respected. As such, the Planners proudly serve up their precise results to the execution side, only for many of those results to be dismissed either directly or during execution as reality sets in. This rejection is the inevitable result of either the inability or unwillingness of Operations to handle the significant problems and changes that come with execution and that inevitably make the Planners “precise” schedule an impossible reality. Operations, of course, accepts this lot in life and their thus “un-planned” execution as…inevitable, since they rationalize the events that befall the schedules to be beyond their control.  However though, most are actually within their control.

The execution side has a long way to go in tightening their processes. They need the data and tools that will permit them to approach the level of performance that the planners are attempting to achieve. Conversely, the planners need to back off. They need to incorporate squishy limits into their planning processes.  I have seen some asset planning tools that have managed to do that, but I assume most still do not .  In a rather simplistic way, one could say that execution and planning need to be dealing with the same range of significant digits.

They clearly aren’t today.

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2 Responses to “Significant Digits”
  • As a regular reader of this very interesting and inspiring blog I agree to most statements. However, there is one statement I disagree completely with. This is the statement, that fixed block operations do not allow precise traffic management since the precise location of a train within a block is not known. You are probably not aware of how railroads are dispatched in Europe and many other places outside North America. In Europe, moving block is not used at all, everything is based on fixed block. But there is very precise traffic management that doesn’t just rely on block occupation. The solution for having precise dispatching in fixed block operations is using electronic stringline diagrams. These electronic stringline or time-distance diagrams, which are displayed on high resolution screens, are the key traffic management tool. From the location data that is derived from the signal system when a train passes a signal, a time-distance line is calculated, which is projected into the future. Even if the last signal passing lies some time back, the current position of the train is displayed quite precisely. That location data would only fail, if the train came to an irregular stop within the block, maybe due to an engine failure. However, in such a rare case, the train engineer would call the dispatcher anyway. By calculating the time-distance lines into the future using the movement dynamics of the trains, as long as no unsolved train path conflicts appear, you will even get a quite precise foresight of train positions. By overlaying the time-distance lines by blocking time stairways, you will also get a very precise detection of train path conflicts that are going to appear in the future. Blocking time diagrams even provide a precise evaluation of the impact a conflict will have on the headways of the involved trains. For the mathematical background behind stringline and blocking time diagrams in rail traffic control, see relevant chapters in link to

  • Joern is absolutely correct when it comes to truly scheduled railroads. I should have been clear on the point that I was referring to North America RR operations that are at best 30% truly scheduled. I will hopefully be more clear in the future as to my point of reference.

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