Escalator Solutions

As promised, here’s my solution to the escalator problem … several, actually.

Before getting into the models, a point about simulation vs. analytic solutions. You can solve this problem on pencil and paper with simple algebra. This has some advantages. First, you can be completely data free, by using symbols exclusively. You don’t need to know the height of the stair or a person’s climbing speed, because you can call these Hs and Vc and solve the problem for all possible values. A simulation, by contrast, needs at least notional values for these things. Second, you may be able to draw general conclusions about the solution from its structure. For example, if it takes the form t = H/V, you know there’s some kind of singularity at V=0. With a simulation, if you don’t think to test V=0, you might miss an important special case. It’s easy to miss these special cases in a parameter space with many dimensions.

On the other hand, if there are many dimensions, this may imply that the problem will be difficult or impossible to solve analytically, so simulation may be the only fallback. A simulation also makes it easier to play with the model interactively (e.g., Vensim’s Synthesim mode) and to incorporate features like model-data comparisons and optimization. The ability to play invites experimentation with parameter values you might not otherwise think of. Also, drawing a stock-flow diagram may allow you to access other forms of visual thinking, or analogies with structurally similar systems in different domains.

With that prelude, here’s how I conceived of the problem:

  • You’re in a building, at height=0 (feet in my model, but the particular unit doesn’t matter as long as you have and check units).
  • Stairs rise to height=100.
  • There’s an escalator from 100 to 200 ft.
  • Then stairs resume, to infinite height.
  • The escalator ascends at 1ft/sec and the climber at 1ft/sec whether on stairs or not.
  • At some point, the climber rests for 60sec, at which point their rate of climb is 0, but they continue to ascend if on the escalator.

Of course all the numbers can be changed on the fly, but these concepts at least have to exist.

I think of this as a problem of pure accumulation, with height as a stock. But it turned out that I still needed some feedback to determine where the climber was – on the stairs, or on the escalator:

At first it struck me that this was “fake” feedback – an accounting artifact – and that it might go away with an alternate conception. Here’s my implementation of Pradeesh Kumar’s idea, from the SDS Discussion Group on Facebook, with the height to be climbed on the stairs and escalator as a stock, with an outflow as climbing is accomplished:The logical loop is still there, and the rest of the accounting is more complex, so I think it’s inevitable.

Finally, I built the same model in Ventity, so I could use multiple entities to quickly store and replicate several scenarios:

Looking at the Ventity output, resting on the escalator is preferable:

While resting on the stairs, nothing happens. While resting on the escalator, you continue to make gains.

There’s an unstated assumption present in all the twitter answers I’ve seen: the escalator is the up escalator. I actually prefer to go up the down escalator, though it attracts weird looks. If you do that, resting on the escalator is catastrophic, because you lose ground that you previously gained:

I suspect there are other interesting edge cases to explore.

The models:

Vensim (any version): Escalator 1.mdl

Vensim, alternate conception: Escalator 1 alt.mdl

Vensim Pro/DSS/Model Reader – subscripted for multiple experiments: escalator 2.mdl

Ventity: Escalator

JJ Lauble has also created a version, posted at the Vensim forum. I haven’t had a chance to explore it yet, but it looks like he may have used Vensim to explore the algebraic solution, with the time axis as a way to scan the solution space with Synthesim overrides.

Escalator Problems

@stevenstrogatz reposts a clever, simple problem:

two people climb a staircase and then climb an escalator. One person rests a minute on the staircase and the other rests a minute on the escalator, but otherwise they climb stairs at the same rate. Who is faster or are they equally fast?

There’s also an airport-walkway version that adds special relativity as a twist.

It’s interesting to see the varied thought processes in the comments. Pencil and paper is often quicker and yields useful analytic insight, but these are both accumulation problems, and therefore good candidates for SD simulation.

How would you model this situation? (I’ll post my answer in a day or two.)

Feedback solves the Mars problem

Some folks apparently continue the Apollo tradition, doubting the latest Mars rover landing.

Perfect timing of release into space? Perfect speed to get to Mars? Perfect angle? Well, there are actually lots of problems like this that get solved, in spite of daunting challenges. Naval gunnery is an extremely hard problem:

USN via Math Encounters blog

Yet somehow WWII battleships could hit targets many miles away. The enabling technology was a good predictive model of the trajectory of the shell, embodied in an analog fire computer or just a big stack of tables.

However, framing a Mars landing as a problem in ballistics is just wrong. We don’t simply point a rocket at Mars and fire the rover like a huge shell, hoping it will arrive on target. That really would be hard: the aiming precision needed to hit a target area of <1km at a range of >100 million km would be ridiculous, even from solid ground. But that’s not the problem, because the mission has opportunities to course correct along the way.

Measurements of the spacecraft range to Earth and the rate of change of this distance are collected during every DSN station contact and sent to the navigation specialists of the flight team for analysis. They use this data to determine the true path the spacecraft is flying, and determine corrective maneuvers needed to maintain the desired trajectory. The first of four Trajectory Correction Maneuvers (TCMs) is scheduled on January 4th, 1997 to correct any errors collected from launch. The magnitude of this maneuver is less than 75 meters per second (m/s). Navigation is an ongoing activity that will continue until the spacecraft enters the atmosphere of Mars.


The ability to measure and correct the trajectory along the way turns the impossible ballistics problem into a manageable feedback control problem. You still need a good model of many aspects of the problem to design the control systems, but we do that all the time. Imagine a world without feedback control:

  • Your house has no thermostat; you turn on the furnace when you install it and let it run for 20 years.
  • Cars have no brakes or steering, and the accelerator is on-off.
  • After you flush the toilet, you have to wait around and manually turn off the water before the tank overflows.
  • Forget about autopilot or automatic screen brightness on your phone, and definitely avoid nuclear reactors.

Without feedback, lots of things would seem impossible. But fortunately that’s not the real world, and it doesn’t prevent us from getting to Mars.

Complexity should be the default assumption

Whether or not we can prove that a system experiences trophic cascades and other nonlinear side-effects, we should manage as if it does, because we know that these dynamics are common.

There’s been a long-running debate over whether wolf reintroduction led to a trophic cascade in Yellowstone. There’s a nice summary here:

Do Wolves Change Rivers?

Yesterday, June initiated an in depth discussion on the benefit of wolves in Yellowstone, in the form of trophic cascade with the video: How Wolves Change the River:

This was predicted by some, and has been studied by William Ripple, Robert Beschta Trophic Cascades in Yellowstone: The first fifteen years after wolf reintroduction

Shannon, Roger, and Mike, voiced caution that the verdict was still out.

I would like to caution that many of the reported “positive” impacts wolves have had on the environment after coming back to Yellowstone remain unproven or are at least controversial. This is still a hotly debated topic in science but in the popular media the idea that wolves can create a Utopian environment all too often appears to be readily accepted. If anyone is interested, I think Dave Mech wrote a very interesting article about this (attached). As he puts it “the wolf is neither a saint nor a sinner except to those who want to make it so”.

Mech: Is Science in Danger of Sanctifying Wolves

Roger added

I see 2 points of caution regarding reports of wolves having “positive” impacts in Yellowstone. One is that understanding cause and effect is always hard, nigh onto impossible, when faced with changes that occur in one place at one time. We know that conditions along rivers and streams have changed in Yellowstone but how much “cause” can be attributed to wolves is impossible to determine.

Perhaps even more important is that evaluations of whether changes are “positive” or “negative” are completely human value judgements and have no basis in science, in this case in the science of ecology.

-Ely Field Naturalists

Of course, in a forum discussion, this becomes:

Wolves changed rivers.

Not they didn’t.

Yes they did.

(iterate ad nauseam)

Prove it.

… with “prove it” roughly understood to mean establishing that river = a + b*wolves, rejecting the null hypothesis that b=0 at some level of statistical significance.

I would submit that this is a poor framing of the problem. Given what we know about nonlinear dynamics in  networks like an ecosystem, it’s almost inconceivable that there would not be trophic cascades. Moreover, it’s well known that simple correlation would not be able to detect such cascades in many cases anyway.

A “no effect” default in other situations seems equally naive. Is it really plausible that a disturbance to a project would not have any knock-on effects? That stressing a person’s endocrine system would not cause a path-dependent response? I don’t think so. Somehow we need ordinary conversations to employ more sophisticated notions about models and evidence in complex systems. I think at least two ideas are useful:

  • The idea that macro behavior emerges from micro structure. The appropriate level of description of an ecosystem, or a project, is not a few time series for key populations, but an operational, physical description of how species reproduce and interact with one another, or how tasks get done.
  • A Bayesian approach to model selection, in which our belief in a particular representation of a system is proportional to the degree to which it explains the evidence, relative to various alternative formulations, not just a naive null hypothesis.

In both cases, it’s important to recognize that the formal, numerical data is not the only data applicable to the system. It’s also crucial to respect conservation laws, units of measure, extreme conditions tests and other Reality Checks that essentially constitute free data points in parts of the parameter space that are otherwise unexplored.

The way we think and talk about these systems guides the way we act. Whether or not we can prove in specific instances that Yellowstone had a trophic cascade, or the Chunnel project had unintended consequences, we need to manage these systems as if they could. Complexity needs to be the default assumption.

A Grizzly-Pine-Nutcracker CLD Rework

I spent a little time working out what Clark’s Causal Calamity might look like as a well-formed causal loop diagram. Here’s an attempt, based on little more than spending a lot of time wandering around the Greater Yellowstone ecosystem:

The basic challenge is that there isn’t a single cycle that encompasses the whole system. Grizzlies, for example, are not involved in the central loop of pine-cone-seedling dispersal and growth (R1). They are to some extent free riders on the system – they raid squirrel middens and eat a lot of nuts, which can’t be good for the squirrels (dashed line, loop B5).

There are also a lot of “nuisance” loops  that are essential for robustness of the real system, but aren’t really central to the basic point about ecosystem interconnectedness. B6 is one example – you get such a negative loop every time you have an outflow from a stock (more stuff in the stock -> faster outflow -> less stuff in the stock). R2 is another – the development of clearings from pines via fire and pests is offset by the destruction of pines via the same process.

I suspect that this CLD is still dramatically underspecified and erroneous, compared to the simplest stock-flow model that could encompass these concepts. It would also make a lousy poster for grocery store consumption.

Happy E day

E, a.k.a. Euler’s number or the base of the natural logarithm, is near and dear to dynamic modelers. It’s not just the root of exponential growth and decay; thanks to Euler’s Formula it encompasses oscillation, and therefore all things dynamic.

E is approximately 2.718, and today is 2/7/18, at least to Americans, so this is the biggest e day for a while. (NASA has the next 1,999,996 digits, should you need them.) Unlike π, e has not been contested in any state legislature that I know of.

Vi Hart on positive feedback driving polarization

Vi Hart’s interesting comments on the dynamics of political polarization, following the release of an innocuous video:

I wonder what made those commenters think we have opposite views; surely it couldn’t just be that I suggest people consider the consequences of their words and actions. My working theory is that other markers have placed me on the opposite side of a cultural divide that they feel exists, and they are in the habit of demonizing the people they’ve put on this side of their imaginary divide with whatever moral outrage sounds irreproachable to them. It’s a rather common tool in the rhetorical toolset, because it’s easy to make the perceived good outweigh the perceived harm if you add fear to the equation.

Many groups have grown their numbers through this feedback loop: have a charismatic leader convince people there’s a big risk that group x will do y, therefore it seems worth the cost of being divisive with those who think that risk is not worth acting on, and that divisiveness cuts out those who think that risk is lower, which then increases the perceived risk, which lowers the cost of being increasingly divisive, and so on.

The above feedback loop works great when the divide cuts off a trust of the institutions of science, or glorifies a distrust of data. It breaks the feedback loop if you act on science’s best knowledge of the risk, which trends towards staying constant, rather than perceived risk, which can easily grow exponentially, especially when someone is stoking your fear and distrust.

If a group believes that there’s too much risk in trusting outsiders about where the real risk and harm are, then, well, of course I’ll get distrustful people afraid that my mathematical views on risk/benefit are in danger of creating a fascist state. The risk/benefit calculation demands it be so.