Misadventures with Little’s Law

I’ve been working on a vehicle fleet model, re-implementing a spreadsheet in Ventity, using dynamic cohorts.

The vehicle lifetime in the spreadsheet is 11 years, and it’s discrete. This means that every vehicle retires precisely 11 years after it’s put into service. This raised a red flag for me, because it represents a rather short vehicle lifetime. I know from work in other jurisdictions that the average life of a vehicle is more like 16-18 years typically (and getting longer as quality improves).

So, where does the 11 year figure come from? We’re not sure. Other published data for the region indicates an average vehicle age of 8.5 years, so it’s not that. A Ventana colleague pointed out that it might be a steady-state estimate from combining vehicle fleet data with new vehicle sales data:

 

Given the data (red), assume that the vehicle stock is in equilibrium (inflow=outflow). Then it follows from Little’s Law that the average lifetime of vehicles must be 11 years. Little’s Law works regardless of the delay distribution, i.e. regardless of the delay order, but if you were formulating the fleet as a first-order system, that’s precisely how you’d write the outflow equation: outflow = fleet/lifetime, with lifetime=11 years.

… the long-term average number L of customers in a stationary system is equal to the long-term average effective arrival rate λ multiplied by the average time W that a customer spends in the system. – Wikipedia

However, there’s a danger here. The system might not be in equilibrium. Then both the assumption of inflow=0utflow and the stationarity required in Little’s Law. Vehicle sales are, unfortunately, rather volatile, particularly around events like the 2008 recession:

It’s tempting to use the average age of vehicles as another data point, but that turns out to be a bad idea. The average age of vehicles is sensitive to both variations in the inflow and the assumed distribution of the discard process. The following Ventity model illustrates this problem, using some of the same machinery as last week’s Erlang model.

As before, there’s a population of entities (agents). Each has a cascade of N internal states, represented by a stock counter, and an age that increases continuously. An entity deletes itself when it’s too old, or its state count is too high.

For accounting purposes, when an entity “dies” it records the event by incrementing counter stocks in the Model entity:

In this way, we can keep track of how old the average entity was at the time it deleted itself. This should be the average residence time in Little’s Law. We can also track the average age of existing entities, to see whether it’s the same.

First, consider a very simple, very nonstationary special case, in which there’s no flow of entity turnover. There’s only an initial population of entities of age 0, who gradually leave the system. Here are three variants of that experiment:

Set Model.Delay tau = 50 and Model.Flow Start Time = 1000 to replicate this experiment.

The blue line is the stochastic population analog of the classic first-order delay. The probability of a given entity departing is constant over time, as for radioactive decay. Therefore we get exponential decay, with count = N0*exp(-time/Delay tau). The red line is the third-order equivalent, yielding an Erlang 3 distribution. The green line is the pipeline delay equivalent, in which all entities self-delete at a specified age, rather than with a random distribution. Therefore the population steps from 1000 to 0 at time 50.

The two lower panels compare the average age of surviving entities (middle) to the average age at which entities self-delete (bottom). At bottom, you can see that all variants eventually converge to (roughly) the expected 50-year entity lifespan. However, each trajectory initially indicates a shorter lifespan. This is due to a form of censoring bias – at a given point in time, the longest-lived entities have not yet been observed.

The middle panel indicates how average age can mislead. In this case, age=time for all entities, and therefore the average age increases linearly, even though the expected residence time is constant.

At the opposite extreme, here’s an experiment with a constant flow of new agents, so that the system is in equilibrium after a few time constants:

Set Model.Delay tau = 20 and Model.Flow Start Time = 0 to replicate this experiment.

After the initial transient has died out (by time 20 to 60), all 3 residence times (age at deletion) converge to the expected value of 20. But notice the ages. They converge, too, but the value is dependent on the distribution. For the 1st-order system (blue), the average age does equal the average residence time of 20 years. But the pipeline system (green) has an average age that’s half that, at 10 years. This makes sense, if you think about an equilibrium population composed of a uniform mix of ages between 0 and 20 years. The 3rd-order system is in between.

This uncertain relationship between age and residence time means that we can’t use the average age of the vehicle fleet to determine the rate of vehicle turnover. That’s too bad, because age is the one statistic that’s easy to compute from a database of vehicle registrations. To know more, we have to start making inferences about the inflows and outflows – but that’s tricky if data coverage varies with time. Unfortunately, this is a number that we care about, because the residence time of vehicles in the system is an important driver of future penetration of low-carbon technologies.

The model: AgentAge2.zip

The Delay Sandbox can be used to explore similar phenomena in a continuous, aggregate, deterministic setting.

Aging Chains and the Erlang Distribution

My Delay Sandbox model illustrates the correspondence between Nth-order delays and the Erlang distribution (among other things).

Delay Sandbox

This model provides some similar insights – this time in Ventity. It’s a hybrid of classic continuous SD and agent equivalents.

First, the Erlang3 entitytype compares the classic 3rd-order aging chain’s behavior to analytical equivalents, as in the Delay Sandbox. The analytic values are computed in a set of Ventity’s new macros:

Notice that the variances, which arise from Euler integration with a finite time step, are small enough to be uninteresting.

Second, the model compares the dynamics of discrete agent populations to the analytic Erlang results. To do this, the Model entity creates populations of agents at time 0, and (for comparison) computes the expected surviving population according to the Erlang distribution:

The agents live for a time, then self-delete according to two different strategies:

On the left, an agent tracks its own age, and has an age-specific probability of mortality (again, thanks to the hazard rate of the Erlang distribution). On the right, an agent has a state counter, and mortality occurs when the number of state transitions reaches 3.

We can then compare the surviving agent populations (blue) to the Erlang expectation (red):

When the population is small (above, 100), there’s some stochastic variation around the expected result. But for larger populations, the difference is negligible.

The model: Erlang3 4 (2).zip

Automating the London Whale error

This one little trick might prevent you from blowing up your organization.

I’m setting up a new computer, and Excel’s default autosaving nuked one of my Ventity input spreadsheets. I did a little quick analysis on the data, meant to be volatile, but Excel cheerfully made it permanent and diffused it to my other computers. Luckily Ventity noticed when I ran the model.

In theory, it’s no big deal, because you can undo autosaved changes. Except that it’s a very big deal, because you can’t easily see that changes have been autosaved. Change a few numbers here and there, and pretty soon you’re on your way to the next London Whale trading disaster. Model integrity is nonexistent.

Fortunately, you can change the backwards default behavior easily. All you need to do is uncheck this box:

Do it now!

Systems Thinking about the Crash of ’29

I picked up John Kenneth Galbraith’s account of The Great Crash at a used bookstore. I’m not far into it, but there’s a nice assertion of the importance of a systemic view over event-based descriptions right at the start:

… implicit in this hue and cry was the notion that somewhere on Wall Street … there was a deus ex machina who somehow engineered the boom and bust. This notion that great misadventures are the work of great and devious adventurers, and that the latter can and must be found if we are to be safe, is a popular one in our time. … While this may be a harmless avocation, it does not suggest and especially good view of historical processes. No one was responsible for the great Wall Street crash. No one engineered the speculation that preceded it. Both were the product of the free choice of hundreds of thousands of individuals. The latter were not led to the slaughter. They were impelled to it by the seminal lunacy which has always seized people who are seized in turn with the notion that they can become very rich. …

Galbraith’s purpose in writing the book is itself systemic, to weaken the erosion of memory that permits episodic boom bust cycles:

Someday, no one can tell when, there will be another speculative climax and crash. There is no chance that, as the market moves to the brink, those involved will see the nature of their illusion and so protect themselves and the system. … There is some protection so long as there are people who know, when they hear it said that history is being made in this market or that a new era has been opened, that the same history has been made and the same eras have been opened many, many times before. This acts to arrest the spread of illusion. …

With time, the number who are restrained by memory must decline. The historian, in a volume such as this, can hope that he provides a substitute for memory that slightly stays that decline.

 

Facing the Blank Sheet

Modeling projects usually start with the dreaded blank sheet of paper (or blank screen). How to get started? Just do it. Write stuff down, and see what organization emerges.

Here are some concrete approaches that I’ve often used:

  • Start with the question. Inventory is unstable? OK, put inventory on the diagram. It’s a stock, so what are the flows? Put them on the diagram. Are the inflows and outflows unstable, or just one? Follow the unstable direction….
  • Start with the data. We get this a lot in marketing science projects. There’s typically a big pile of Nielsen or IMS data on price, promotion and distribution. How does that drive sales? You can do a little data mining for insight, but typically the data describes less than half of what’s going on, so more importantly, what else drives sales? How do brand equity, supply chain performance, and other dynamics introduce feedback into the picture?
  • Start with a spreadsheet. There’s always a spreadsheet. It’s probably open loop and static, but it captures features that someone thought were important. Audit the spreadsheet to discover its structure, then make it dynamic.
  • Start with the goal. You want to maximize profit? Write down a P&L, then trace each item. Where does revenue come from? What drives costs? When you answer these questions, look for the key strategic stocks that govern the behavior – people, capital, perceptions, etc.
  • Start with the physics. What are the key stocks of scarce resources in the system? Equipment, people, money, knowledge? What makes them change, and where are the decisions?
  • Start with the stakeholders. What are the major constituencies in the problem domain. What do they want, and what stocks are they looking at to guide how they get it?

The key thing is to remember that modeling is an iterative process at every level. The data might be wrong. The equations will be wrong. The equations might be in the wrong structure. The structure might describe the wrong problem. This is normal. Don’t be afraid to back up and start over.

The blank sheet of paper

Confronting the dreaded blank canvas

6 more reasons to apply SD to medical research

@SDWisdom Ken Cooper lists 6 good reasons to apply System Dynamics to medical research. I think there are more if you broaden the definition of ‘medical’ :

7. Dose titration can be dynamically complex and subject to misperceptions of feedback; models make it easy.

8. Chronic autoimmune and mental health problems are embedded in a nest of feedback between the disease and the person’s environment.

9. ERs, hospitals and other delivery systems are loaded with delays, feedback and nonlinearity.

10. Smoking, diet, exercise, and other big health drivers are social phenomena.

11. Diet and exercise are entangled with other systems, like urban design and energy efficiency.

12. The health insurance system, especially in the US where it has evolved into a mess, can’t be redesigned without a systemic perspective.

4 Faces of Medical Modeling

I enjoyed the biomedical modeling plenary at #ISDC2019 more than most. I was struck by the continuum of behavior involved in the system:

  • True biomedical modeling is a bit funny, because it’s not typical System Dynamics, in the sense that it’s nonlinear dynamic simulation, but it’s not behavioral, so it’s missing one of the cornerstones of SD. Nevertheless, I think the way we think about complex systems is a useful complement to other approaches coming more from biology and mathematics (nonlinear dynamics).
  • Behavior enters one level “up”, in problems like Jim Rogers & Ed Gallaher’s work on dose titration in anemia. This is a classic case of smart people having trouble managing a system with fairly simple dynamics – essentially a single pipeline delay in the case of anemia. There may be many similar cases, where large performance improvements are available from simple models (but complicated people management).
  • Next, there are problems that combine behavioral dynamics and misperceptions of feedback with an underlying system that is also quite complex. Gizem Aktas’ work on stress and hormonal regulation is an example, as are diabetes and mental health models.
  • At the far end of the scale, there are health system models, like ReThink Health, which abstract away from the biomedical details of any particular disease. In its place, there’s an extremely complex network of human resources, incentives and decisions.

I think the opportunities are large in all of these areas. Once challenge for the field is that each requires a different interface to other researchers, health practitioners and managers. That’s a lot for relatively few modelers to manage. How can we team up to be more effective?

Closing loops – practicalities

Hybrid models are the solution to blending endogenous elegance with practicality.

My last post probably sounds like I disagree with Jack Homer’s recommendation to tolerate some exogenous drivers and consideration of policy feasibility. Actually I don’t. In fact, we at Ventana probably do more data-intensive SD than anyone. I build hybrid models all the time.

When philosophizing about the best way to change the world, it’s easy to lose sight of some practical considerations that influence choices:

  • Cost. It’s expensive to develop an elegant, endogenous theory for things like interest rates that you might normally think of as exogenous to a firm. On the other hand, it’s also expensive to collect and use data – often 1/3 of project cost in our experience.
  • Clarity. Exogenous variables complicate the analysis of a model, because you have driven behavior on top of the model’s endogenous dynamics. I think this makes it harder to understand the basic behavior, because you lose the insight you might gain from starting a model in equilibrium and perturbing it with policies.
  • Calibration. On the other hand, using exogenous drivers increases your ability to gain insight from comparison of model behavior to data. This is not a definitive test, but you can definitely use it to estimate uncertain parameters and weed out certain dumb ideas.
  • Client. You have to meet people where they are. If, historically, they think R^2 is the definitive measure of success, you’d better deliver. You can explain why that’s a bad metric and present a more endogenous view of the situation later, after you’ve established trust.

I think there’s no clear answer – the extent to which endogenous or exogenous elements are preferred has to be a situation-specific decision. In my own work, I often use a two-pronged approach, and two ways to structure that have emerged:

  • Build a single, large, calibrated model with some exogenous drivers. Build endogenous submodels or metamodels for equilibrium experiments and to  explain key features of the big model.
  • Build a single, elegant endogenous model, with few drivers. Use smaller exogenous models, or statistical and machine learning tools, to understand local features of the data, incorporating those insights into the endogenous model without using the data directly.

Closing the loops

How do we strike the right balance among closed loops revealing all possible leverage points, exogenous drivers for fidelity and economy, and practical focus on feasible policies?

On the SDwisdom blog, Jack Homer wonders whether we should be a little less endogenous:

The party line is that a model’s boundary should be broad enough so that the system’s main observed behaviors—such as S-shaped growth, oscillation, or overshoot and decline—are fully explained by the model’s endogenous structure.  One should avoid the use of exogenous time series drivers, because they undermine the ability of the model to explain and to anticipate change.

I mostly agree with this view but want to offer a friendly amendment here.  In my experience with real-world clients, I have often encountered situations in which it makes sense to employ exogenous time series for the sake of completeness and realism.

My experience suggests that we should be less doctrinaire about the endogenous perspective and understand that “endogenous” is a relative thing.  No model can be all-encompassing and explain all observed behavior patterns.  That’s why we define a model relative to some subset of behaviors also known as the dynamic problem.  As long as the model adequately addresses the dynamic problem, it shouldn’t really matter if the model has some exogenous time series included to improve the model’s realism.

The counterpoint to this might be George Richardson’s excellent article on the value of an endogenous perspective:

But the foundation of systems thinking and system dynamics lies deeper than these and is often implicit or even ignored: it is the “endogenous point of view.” The paper will begin with historical background, clarify the endogenous point of view, illustrate with examples, and argue that the endogenous point of view is the sine%qua%non of systems approaches.

The dead buffalo model on the left in Richardson’s figure is perhaps too extreme. I think what Homer is really arguing for is a middle road that might look like this:

In other words, it’s a hybrid model, with an endogenous core, and a few exogenous drivers from beyond the model boundary. For a firm, this might mean that we model the interactions of marketing, production, human resources, etc. endogenously. But we leave the world crude price and GDP of France exogenous, because we have no plausible means to influence them.

Richardson points out, in the context of Urban Dynamics:

How might thinking exogenously have affected conclusions emerging from these urban models? Consider just one: suppose, as a number of critics of various system dynamics studies have suggested, we chose to use time series data for urban population projections. Suppose the data-based projections were very carefully developed by sophisticated statistical tools and econometric methods, and suppose those projections were fed into URBAN1 in place of the endogenous stock of population. To make the implications easiest to see, let’s suppose the base run of the model looks just as it did in Figure 5b. What would Figure 5c look like? Sadly, population would not show a bump as it did in 5c because the sophisticated exogenous time series data would not be influenced in the slightest by the changing conditions in the model. We would not see the compensating urban migration effects we see in Figure 5c, and we would miss the crucial conclusion that population and business construction dynamics would naturally compensate for the jobs program. We’d probably think it was a long term policy success, and we would be dramatically wrong.

Similarly, a key part of my dissertation critique of the DICE model was the idea that its exogenous representation of technology excluded a set competitive dynamics between fossil fuels and renewables that are crucial to understanding climate policy. In both cases, leaving even a few loops exogenous might dramatically alter policy recommendations.

In another article, Homer rejoins:

In 1977, my teacher, SD pioneer Ed Roberts, wrote an important paper called “Strategies for Effective Implementation of Complex Corporate Models”, based on his years of consulting experience.  When it comes to policy recommendations, he said, “the organizations’ ability to absorb the associated change must be assessed…The model-builder and the organization both profit from implementation of moderate change proposals leading to some successful results; both lose from grandiose plans which fail to be moved ahead.”

Without too much extra effort beyond what we already do in modeling, we could assess policy feasibility by studying the political situation and its dynamics, looking at trends in public opinion surveys and other indicator data, and talking to experts. We could then calculate the “feasibility-adjusted value” of a policy by multiplying its potential impact (as determined by our SD model) by its likelihood of enactment.

In other words, one might ask, what good is it to know the nature of the best endogenous policy if the key leverage points are beyond your control? Or, as economists put it, what good are first-best policies in a second-best world?

I think Richardson might respond along the lines of Pascal’s wager:

To the extent that we accept some exogeneity in our models, we accept our fate in the lower left box. That choice presumes that our best strategy is to do a good job of predicting and preparing – making the best of a bad situation, or living to fight another day. Is that really a good idea – can we know, or at least make a good guess, whether the true state of affairs is exogenous or endogenous? Richardson argues that if we don’t know which box we’re in, it’s best to choose the endogenous approach:

If one were to recast Table 4 as a decision tree, the decision to take an endogenous point of view in all circumstances would have the highest net payoff, at least in terms of happy faces and the real feelings they represent. An endogenous point of view is potentially empowering, and that feels good to us.

If the right (endogenous) side of the diagram is better in some sense, we should ask a more important question: can we change the state of affairs, so that we can move from the left quadrants to the right? This is related to the idea that we might need to change paradigms to change the system.

This perspective is especially critical for problems like climate change, where the space of politically feasible policies currently does not include anything that actually solves the problem. Treating the infeasible loops as exogenous drivers won’t help. We need to ask, what loops that are now beyond our control can be activated in order to reach a more attractive solution?

Maybe there really is no good solution to climate and other Limits problems. Then perhaps it would be optimal in some sense to take catastrophe as exogenous and use models to plan a personal path through the eye of the storm. Personally, I’m not ready to accept that. My modeling objective remains to die with my boots on tackling the biggest problems.

See also:

Closing loops – practicalities

Why the World Loves Open Loop Models

What kind of model should I use?