The tyranny of the wagon equation | Hacker News Reader

The tyranny of the wagon equation

maximumeffort.substack.com

243 pointsdTal113 comments

> “Hierarchy, Vorbis said later. The Ephebians didn't think in terms of hierarchies.

> No army could cross the desert. But maybe a small army could get a quarter of the way, and leave a cache of water. And do that several times. And another small army could use part of that cache to go further, maybe reach halfway, and leave a cache. And another small army . . .

> It had taken months. A third of the men had died, of heat and dehydration and wild animals and worse things, the worse things that the desert held . . .

> You had to have a mind like Vorbis's to plan it.

> And plan it early. Men were already dying in the desert before Brother Murduck went to preach; there was already a beaten track when the Omnian fleet burned in the bay before Ephebe.

> You had to have a mind like Vorbis's to plan your retaliation before your attack.”

- Small Gods, Terry Pratchett

I thought "that's a funny coincidence with the Vorbis file format name!" ... but then I thought maybe it wasn't a coincidence at all. And it wasn't!

https://en.wikipedia.org/wiki/Vorbis#Name

Caravans faced this problem, but the solution was commercial - the caravanserai.

A serai is a corral for animals. A caravanserai is a truck stop - fuel, food, motel, parking. Caravans resupplied at caravanserai. The Silk Road was a chain of caravanserai. If there's enough local production to supply a caravanserai, caravans can take that route.

The military version of this is a chain of supply bases. This is all too common. A gallon of diesel fuel delivered by the US military to outermost bases in Afghanistan could cost well over US$100. Part of the trip was made by transport aircraft, and part by helicopter, all of it consuming fuel. Bases have to be within helicopter range of each other. The main justification for the Osprey, which is a horrible kludge of an aircraft, is more range, allowing fewer bases to reach a goal.

> If there's enough local production to supply a caravanserai, caravans can take that route.

That's an interesting constraint; were there also extra-expensive caravanserai in places where that production wasn't quite local and some of those resources had to be trucked, er, caravanned in?

Sort of, mostly yes, but never fodder or water. This is the great advantage of the camel: if a camel train rolls into a caravanserai, and the available fodder at that oasis has been eaten, there's a good change they'll make it to the next oasis with thin, very unhappy, camels. Mules would just die.

I had heard of caravanserai from the idle/progression Kittens Game. I wish I had looked it up earlier.

By no means an expert but my understanding is that if you can't make the logistics work for a military operation then the chances of failure are very high, maybe even literally 100%.

my 10 year old, out of the blue, told me once "artillery wins battles but logistics wins wars". Heh he's really into a couple youtube history channels.

Watch the Kherson region now - Russia has a 25K strong army on the right side of Dneper river there, and thanks to US supplied high precision missiles (HIMARS) Ukraine has practically disabled the bridges leading there, so that Russian army has only minimal supply using ferries. The railway from Crimea to those bridges and ferries had also been constantly stricken (frequently the strike would be at a large military supply train riding there)

In general, across the front, we see David vs. Goliath fight with supply logistics being the Goliath's spot that David is targeting - Russia has 5-10x of old style un-precise artillery advantage, and their main approach is to flood Ukrainians with ocean of fire, and that requires tremendous amount of supply, while Ukraine having only about 40 of those HIMARS launchers has been methodically taking out key Russian targets among which large Russian ammo and fuel depots to the 50 miles depth (the range of those missiles) thus straining Russian supply of the frontline.

We have a very recent example that proves this point.

Just this June I've visited the caravanserai in Tash Rabat, Kyrgyzstan. A beautiful and somewhat baffling structure in a sweeping mountainous setting, with just a few yurts beside it to host visitors.

(Highly recommend a visit to and jeep tour through Kyrgyzstan: the landscape is stunning, and the people extraordinarily friendly and welcoming.)

Here some blog posts with pictures (not affiliated):

https://silkroad-livinghistory.org/kyrgyzstan/the-lonely-car...

https://www.timetravelturtle.com/tash-rabat-caravanserai-kyr...

One commenter questions whether generals and armies were capable of the advanced math implicated by the harmonic transfer method.

But they don't have to do any advanced math. They just take food for the fleet from a single donkey and turn that donkey around as soon as his food supply equals the food necessary for the return trip. Rinse and repeat. Right?

The reality does get a bit more complicated because at least one guide is required for each donkey, so you really don't want to lose a man for every donkey that turns back, and instead do the maneuver in groups. You also need to feed the return guide.

Optimizing under those conditions gets a lot more complicated I think.

You could also use this logic for sherpa-assisted mountain climbing.

>because at least one guide is required for each donkey,

Why? Can you not assemble them in groups like other pack animals?

If you're sending them back one by one you need a guide for each.

If you send them in groups then you're not as optimal (first donkey is unladen in 3 days singly, but two donkeys are unladen in 5 days, say, meaning you need to keep feeding the first until the second is done, or equivalent via load balancing).

You could have trained donkeys that might self-navigate but that's moderately unlikely. Probably better to eat them or give them away free to the countryside.

Sure, if you're unloading from the donkey serially. But if you unloaded in parallel, no one donkey would be unloaded before the others.

That still delays when you can send the first donkey back - and it continues to eat.

If you unload from one and send back immediately, it eats 3 days, goes back.

If you unload in parallel, you cannot be sending a donkey back in 3 days, so you must be feeding extra donkey.

It still may get you an advantage overall, but it's not the maximum advantage.

very analogous to airplanes running on fuel vs batteries

I was actually thinking it was like the Falklands.

https://en.wikipedia.org/wiki/Operation_Black_Buck

The British were able to fly bombers from tremendously far away, with a chain re-fuel strategy where the re-fuelling tanker planes are themselves re-fuelled by other tankers which then turn back. Later iterations were optimised by thinking more carefully about who should transfer fuel, to who, and when, as in this example.

Arguably Black Buck was pointless, it was certainly not pivotal in the outcome of the war, but the actual process was fascinating.

This analysis also quite reasonably assumes zero transfer time between donkeys, perfect knowledge to run donkeys to exactly their limit (which are identical between all donkeys), and zero overhead in general. That's all fine, it's an upper limit analysis. But the overhead will be biting you in the exponential regime, unfortunately.

Another thing to remember about "the old days" is that they may not have computers and they may not have the Internet, but they have LOTS of time to think about more efficiency compared to an Internet commenter for whom this is merely a momentary side diversion for a few moments, and a lot more motivation. I feel satisfied with some of the answers to things like "How was Stonehenge built?" that have been found in the last couple of decades (or, if you prefer, "Can you show at least one method that could have been used to build Stonehenge?"), but I think one of the other lessons that was learned (at least by me) is that "take a modern person, give them one try in their busy lives to try the first thing that comes to their mind, and declare the task impossible when that doesn't work" isn't a very good way to understand the ancient world. They had time.

Would an ancient have described the "harmonic transfer technique" in this way? Heck no. Could they have worked their way to it through trial, error, and much simpler thinking? Absolutely. Finding that solution doesn't require calculus. Calculus just supplies a very nice analysis framework and a fantastic communication tool between the post author and us readers.

Funny.

However, I find some assumptions suspect. Unlike rockets, donkeys can consume fuel en-route by grazing. Armies can also consume fuel by hunting, plundering and being "quartered" by their own people. You can make "supply trains" by having small groups of military gather such resources from a closer distance.

This is probably fairly accurate for camels and distances in the Saharan desert though, and you'd have to add the weight of water as well!

The linked source blog goes into that:

- grazing costs time, which increases the total amount of food that must be carried

- foraging equally costs time, so foraging more means a slower moving army

- but an army can only forage as it passes through terrain, so you need to keep moving, otherwise your army will eat the area into starvation, then die

It’s not quite that simple. Horses can eat much faster than they can digest so a few short breaks can represent a lot of food over a day. Further, horses eat significantly more than people. A 30% slower pace that requires 40% less food per day is a net win on distance but not time.

The main issue, is that the horses of agricultural societies are bred to a large size, and cannot survive by grazing alone, they need to have grains in their diet.

That’s false. Horses doing heavy work can’t survive on grazing alone, but they are perfectly fine when left alone in good pasture. Thus the phrase “put out to pasture,” old animals where often given something of a retirement where they where left to take care of themselves rather than simply be killed.

But the horses here will be doing work. Carrying men and supplies. Or is that not heavy enough? Admittedly I know very little about horses.

I don’t know that much about horses but it’s something like:

A normal horse left to graze in good pasture can get a calorie surplus per hour. That same horse doing work has a calorie deficit per hour based on how strenuous the labor. A horse can get into a maximum calorie debt before issues happen. Thus a few day of hard labor plowing a field isn’t an issue by its self and having redundant horses is useful.

In terms of the Wagon equation, taking a nearly empty wagon back is vastly less strenuous than taking a full one out. It may be that taking several times as long to get back significantly extends the total distance you can move the army from your base. It may also be that running a calorie deficit on the outbound trip and then grazing for days before the return trip is useful. But I doubt any army is would actually try and approach any kind of theoretical maximum as in practice flexibility is needed.

Granted that’s for the average horse, where extreme athletic performance means significant extra muscle mass and thus higher caloric needs independent of actual work being done. But the extreme athletic horses are expensive to maintain so likely used by messengers etc not wagons or farmers.

There may be a small net advantage to grazing and foraging enroute but it certainly wouldn't double the available tonnage transported.

Foraging require enough of density of settlements en route. Cause that is what foraging is, taking food from locals by force. The bigger the army, the more and bigger settlements you need to burn and steal from.

I agree it’s not that big a deal when moving a full wagon.

However moving a nearly empty wagon is vastly less effort for the horses. Using a historic example, the Huns they could move their army without needing to provide any food for their horses because each horse was doing so little work.

"taking a nearly empty wagon back"

ha, most (historical) armies hope to gain lots of plunder as a result of their campaigns. According to Byzantine military doctrine, attacking a successful invader as they are leaving, burdened by all their booty, is one of the most advantageous times to strike.

The wagon equation isn’t only about armies, it’s about the logistics of moving things by wagon. Militaries need supplies to move with the army which adds a whole new set of constraints.

>... animals where often ... retirement where they where left ...

The word you are looking for is "were"[1], specifically the first and third instances of "where".

I usually don't bother to be a grammar nazi, but I'm sorry: When you use the word "where"[2] and then also misspell "were" as "where" immediately before and after, things get really confusing really quickly.

[1]: https://www.merriam-webster.com/dictionary/were

[2]: https://www.merriam-webster.com/dictionary/where

What about the vorbis strategy of placing supplies in hidden caches ahead of time without a return trip along the route?

Imagine that applied to the rocket equation. Stationary fuel tanks attaching to the rocket mid rise..

Well that was a strategy indeed but it can only go so far, additionally the same logistics apply to the food you place in the cache, you have to transport it there, the meat of the original article was, that really you are constrained in your movement by the terrains agricultural output, in high production terrain, your population and the enemies population as well will stockpile food for themselves that the army can draw upon, in places where the terrain is not suitable for high intensity agriculture the food available is much less and an army would quickly find itself exhausting all the stockpiles, in a matter of weeks, armies eat a lot of food.

Stationary fuel tanks seem problematic for rockets since your rendezvous would have to happen at such high speeds. You'd either have to decelerate the rocket back to zero or accelerate the fuel to the speed of the rocket.

Remember that all motion is relative. You can’t model space travel as simply going from point to point. Instead, you transfer from orbit to orbit and have to match velocities wherever you go. Relative to the propellant depot you have to reach zero velocity, but the propellant depot will be orbiting something at a very high speed just like you are.

You will likely have to go out of your way somewhat to reach a propellant depot, so you don’t want your propellant depots at the bottom of heavy gravity wells or in hard-to-reach orbits. But it’s still a viable strategy.

There are some places that your relative velocity with a propellant depot may be near zero already.

Natural staging points are not stationary, but they may be in fixed orbits, such as LEO (of a certain altitude, inclination), maybe a halo orbit of a Lagrange point (NASA’s Gateway is in NHRO, which is similar), or a highly Elliptical orbit near escape velocity. And you can do this for Mars as well.

By refilling at these staging points, you can chop up the rocket equation so you don’t get hit hard by the exponential. You have to get fuel there in the first place, and to do that you can use slower, more efficient trajectories or solar-electric propulsion (much higher Isp but slower) or in situ propellant production. You can also have better cooling at these staging points, taking a mass penalty off your crewed vehicle and offloaded to the depot in the case of cryogenic propellants that may boil off over time without an active chiller.

Propellant depots are a pretty powerful tool for expanding capability for space exploration. NASA picked SpaceX’s Starship as the lander for their Artemis lunar missions, and that will fuel up using a depot in Earth orbit.

Im missing another option here, ‘edible supply lines’; the donkeys can be eaten. Only meant as last resort ;)

I believe this was the 'official' strategy of some early arctic / antarctic expeditions. Kill the sledge dogs and feed them to the other dogs plus humans.

That's also part of a sound, but not vegan, anti-scurvy strategy. Someone should do the first vegan trans-Antarctic trek. But I digress.

Humans are more resilient than we give them credit for. Compared to other mammals we have a lot of coping mechanisms for injuries that are missing from other species.

There's a comedian who does a bit about how humans might be the bad guy in horror movies made by aliens, because we can scar in situations that would kill other animals, and we can run in intervals for far longer than most other mammals. We used them down, and that's probably part of our collaboration with dogs. We're all basically Michael Myers. We just keep coming, and as soon as you stop to rest, bam, there we are again.

Point is, hoofed animals tend to get injured in ways that will ultimately kill them. At that point if you have a butcher handy, you can be both humane and efficient, without necessarily being in a starvation situation. We talk about agriculture being a pivot point in human development but nobody ever seems to talk much about the power of soup.

>because we can scar in situations that would kill other animals

examples? Humans aren't exactly resilient or fast healers in my mind.

> examples? Humans aren't exactly resilient or fast healers in my mind.

Presumably they were referring to horses (and possibly donkeys?) having to be euthanized after a broken leg. Which is so well known, even "the Far Side" got in on the action[1].

---

1. https://pbs.twimg.com/media/DKrF52OXUAUtJy3?format=jpg&name=...

That one always makes me laugh :-)

I don't know what the OP has in mind, but if you break an ankle, you can splint it and hobble until it heals. If a large animal breaks an ankle, it's not going to heal.

> There's a comedian who does a bit about how humans might be the bad guy in horror movies made by aliens, because we can scar in situations that would kill other animals, and we can run in intervals for far longer than most other mammals.

I tried to find this skit but couldn't! Any more hints?

The original blog goes into a lot more detail, including the mechanics, logistics and risks of foraging.

The solution to getting around the wagon equation is ships, ships could carry tremendous amounts of food, with a minimal crew and thus minimum consumption, only problem is ships are expensive.

On donkeys, typically armies used mules and draft horses instead of donkeys, because of behavioral and maximum load reasons, and grazing is not enough for mules and large horses, also on certain seasons there is not enough grass for animals to graze in the terrain either.

The original blog has also a piece on nomad logistics, who used significantly smaller horses that can be sustained only by grazing, these armies were able to traverse a lot larger distances, hence the success of the mongol armies.

>only problem is ships are expensive.

Well not the only problem, there's also the fact ships are limited to water deep enough to carry them, so you can't go too far from the coast or navigable rivers. Granted, in Europe, there are lots of navigable rivers that make waterborne logistics fairly simple if you have the boats. Ask the Vikings.

This is mitigated by the fact that rivers are attractive places to live, so the odds that a target of conquest is on a navigable river are fairly high.

Gets pretty bleak if this isn't true, and if your foe doesn't have to stay put, much worse. The Chinese tried to scour the horse barbarians of the hinterlands a few times. Didn't go well.

In the 1200s the eastern horse-hordes used frozen rivers to speed up their rate of advance and took much of the area west of the Urals by surprise.

> Thus, the harmonic donkey transfer, using four donkeys, yields a maximum transport distance 1/12 * L0 longer than the pairwise donkey transfer.

Upvote earned there.

Without digging into any math myself, I find it curious that it's 1/12th longer. That specific ratio, 1/12th, is also the basis for the Equal Temperament system for tuning Western musical instruments ("12-tone Equal Temperament, specifically), which also represents a harmonic series.

https://en.wikipedia.org/wiki/Equal_temperament

1+2+3+...=-1/12 by certain definitions...

Haha thank you! I'm the original author of the linked article, and I had a LOT of fun naming the donkey transfer algorithms I came up with.

The diagram of the Quadruple donkey transfer reminds me of the fueling diagram for the Black Buck raids during the Falklands War

1. https://en.wikipedia.org/wiki/Operation_Black_Buck#/media/Fi...

That's actually shockingly similar. I'm sure someone has studied the mathematics of fuel transport in much greater detail -- I wonder if it's a simple problem to optimize or something closer to the traveling salesman problem

pretty sure it's a very simple problem, brutally simple even.

Reminds me of that time the RAF really wanted to drop some bombs on the Falkland Islands rather than leaving it to the Navy and its Sea Harriers.

Diagram: https://en.wikipedia.org/wiki/File:Refuelling.plan.black.buc... Full article: https://en.wikipedia.org/wiki/Operation_Black_Buck

There is another kind of tyranny equation - the steam locomotive going up an incline.

Essentially the problem is the power output of the engine is locked into step with the speed of the wheels - the slower you go, less pushes of the pistons, less power.

So as a steam train starts up an incline, it will start to decelerate, causing it to produce less power, causing it to slow even further and so on. A true vicious cycle.

This was one of many benefits of diesel-electric locomotives - the engine speed (hence power) was decoupled from the train speed.

ICEs have the same problem though. If they don’t rotate fast enough they stall.

For ICEs we solve the problem with gear boxes. Couldn’t the same solution have been used for steam engines?

They did exist but were not common: https://en.wikipedia.org/wiki/Geared_steam_locomotive

I don’t think that explanation is correct. The torque curve of a steam engine peaks at 0rpm and slopes down from there. The engine is simply decelerating to the point where it’s torque is balanced with the force of gravity on the incline.

It has no choice but to decelerate, as the train itself will decelerate up an incline (without increasing power), and they are coupled. And yes the torque balance is why steam trains could climb gentle gradients at all. However the loss of power due to fewer piston pushes means that the peak 0rpm is reached very quickly. Diesel electric locos can literally increase RPM and power as they start to climb the gradient, with no connection between the slowing train speed and engine power.

This is known in mathematics as the Jeep problem: https://en.wikipedia.org/wiki/Jeep_problem

In university, my colleague students were developing strategies to the “camel banana problem” for transporting n bananas by camel into the desert.

https://puzzling.stackexchange.com/questions/230/a-camel-tra...

In an amazing thread about an amazing article, as a Jeep owner, I did not know about this. It's like the bumper stickers print themselves. However I suspect that given their fuel consumption, owning a Jeep itself may be proof someone does not understand this problem.

There is an escape from the donkey in practice.

Its called a boat, which can scale as big as possible and supply any number of donkeys across the entirety of the Mediterranean.

as long as you are near the shore-line, I can see that working. Though you might also need to be near a port; loading/offloading is less efficient without infrastructure.

However, there may be other uses for the port, which suggests most of the good ports will have cities built around them.

So yes, as long as you constrain your march to the area around friendly cities, boats work great

You don't really need much infrastructure when you don't use containers for your boats.

> as long as you are near the shore-line

Look at Bronze / early Iron age Mediterranean empires. They stuck to the shorelines.

https://en.wikipedia.org/wiki/Ancient_Carthage#/media/File:C...

https://upload.wikimedia.org/wikipedia/commons/f/f7/Greek_Co...

They started moving inroad after they started making roads.

In practicality there was also another solution in the ancient world. If you're going conquering you're probably going somewhere with people to conquer. Those people probably have food. Just take theirs.

It's called foraging, it's what practically everyone did. It's a nasty business and you can't do it to the same people twice, because it typically involved killing and enslaving that people.

https://acoup.blog/2022/07/29/collections-logistics-how-did-...

The underlying blog post goes into details on that, too. It is covered in "foraging" https://acoup.blog/2022/08/12/collections-logistics-how-did-...

>> "Just take theirs"

That "Just" word is carrying a very heavy load . . .

(The people whose food you are planning to forage will be objecting, likely with weapons)

More likely they are fleeing since they are largely unarmed peasants, who can’t stand up to trained, armed, and armored soldiers.

And when they flee they take some of their food supplies with them and hide the rest. Foraging army might be able to find some of it, but rarely all. Also, if you are in enemy territory it is entirely possible to completely miss some remote village. It's not like they have a satellite map.

>>And when they flee they take some of their food supplies with them and hide the rest

YUP!

And beyond that, what they can't take or hide, they may very well burn to slow down the advance

Just because you have advance spies and they report existing supplies, you just cannot count on having those available when the full infantry arrives.

Burning cropland behind retreating troops is the tactic used to deal with "just taking theirs".

The blog posts by Devereaux (acoup blog) that spurred this on, are specifically about over-land logistics. Boats, of course, avoid this problem entirely -- assuming all of your targets are coastal or by rivers!

This assumes that beasts of burden aren't edible...

The underlying blog post (which is MUCH more detailed) goes into that (not sure which of the three went into it in detail) https://acoup.blog/2022/07/15/collections-logistics-how-did-...

There are problems with eating the beasts, but in certain cases it can extend your range, but not as much as you might think.

There's also an element of "burning the ships" in that idea; you extend your current operational range at the cost of the logistical capacity (in terms of carried food) of the future. In other words, if you eat all your horses and you're still behind enemy lines, your army is now in serious trouble of not being able plunder enough food to make it anywhere before starving and deserting.

To add to that, we can, with our current technology, for a handful of billion dollars send a rocket to Andromeda that would arrive in 10,000 to 65,000 years.

But if we want the rocket to be anything more than a pitted lump of metal when it arrives, say, sending information back, operating scanners, etc, the price jumps exponentially (assuming any useful technology we build could still be usable even after 100 years in deep space).

It ultimately boils down not to the "Can we do it", but "Is it WISE to do it", and that is always either the more difficult or more expensive question to answer.

> send a rocket to Andromeda that would arrive in 10,000 to 65,000 years

Andromeda is 2.5 million light years away. You are probably thinking of sending some vehicle to the closest star, Proxima Centauri, which is just about 4.25 light years away. This is the same as 270000 AU away; Voyager 1 has been launched 45 years ago, and it's at 157 AU from Earth, which means it traveled at an average speed of about 3.5 AU/year. At this speed it would take it 77000 years to get to Proxima Centauri.

Dang it, right, yes. Thank you for the correction.

Well, Xenophon and 10000 other Greeks found themselves deep into enemy territory, without supplies [1]. Somehow they managed to survive. They took some food by force, but mostly they used diplomacy and negotiation. Diplomacy is fairly effective when you have lots of weapons.

[1] https://en.wikipedia.org/wiki/Anabasis_(Xenophon)

Once more proving that violence is the original diplomatic currency.

“Harmonic Donkey Transfer” would be a good band name or album title.

Once your nation gets big enough, you can't supply frontier armies from the capital. You need to draw food from local sources. In territory you control, the peasants pay some of their food to the local fortress in taxes, and the fortress acts as a supply base for any army that needs to go through.

Once you get past border action and deep into enemy territory, the tyranny of the wagon equation means it's down to "forage" -- hunt / fish / gather resources from the wild, or steal food from the local population at swordpoint.

One option is to pause your campaign until you can consolidate your gains and build or take over supply bases in captured territory.

Another option is to change your tactics to recognize the limits of what a foraging army can achieve. E.g. avoiding sieges and terrorizing civilians / burning towns to try to force a surrender, or using a cavalry army or fast-moving elite infantry to launch a surprise attack in an unexpected location.

Bring war material with you from home, but forage on the enemy. Thus the army will have food enough for its needs.

--Sun Tzu, The Art of War

It works well enough to be a consistent strategy, but you have to be careful taking too much from a given piece of territory. Eventually, the locals start getting unhappy with you.

And then there's things like the Thirty Years' War, where armies foraged their way through territories often enough that eventually there was nothing left.

> Eventually, the locals start getting unhappy with you.

Pretty sure the locals are unhappy from the start of the foraging process, with all the stealing, murdering, and raping.

Unhappy is being used here to mean "unhappy enough that it affects you." It's not about concern for the mental health of the locals.

There is more than one way to forage. At best, the army can pay the local inhabitants, maybe even a fair value, particularly if the area is nominally friendly. Even if the army "commandeers" supplies, it would want to keep the criminal activity to a minimum, if only because it is bad for discipline.

The Thirty Years' War is an extreme case, in large part because the armies were made up of mercenaries that were not getting paid regularly.

I think that with Thirty Years' War, it was not just "foraging"bur also religious cleansing. It was kind of genocide. They did not had word genocide yet, but generally it fits.

A text-based 'Oregon Trail'-style game could be built out of this concept, where the player starts with a fixed sum of money and has to buy enough donkeys and food, plus perhaps plan a route with optimal water and grazing potential, to reach their destination.

In an acquaintances DnD game he often enforces encumbrance and logistics rules like this. Leading the players to realize, "this plan requires infinite donkeys doesn't it?"

In a world of encumbrance a "Bag of Holding" is an economic miracle...

Yep. Merchants with even one of them are able to profitably transport unexpected bulk goods.

To make this map to the real world, terrain must be taken into account. Obviously donkeys can haul more weight over more distance on flat ground than climbing a mountain pass.

What's the approach to doing that? Make the delta and v terms functions of the terrain? Re-derive the equation based on energy rather than distance?

Even easier: redefine distance, e.g. 1 mountain mile = 5 flat ground miles

A better analogy would be the Breguet range equation, as the rocket equation is explicitly about delta-V, not distance.

https://aerospaceengineeringblog.com/breguet-range-equation/

The author misses a key important point: you can also eat the donkey.

But you can't feed the donkey to remaining donkeys.

I'v seen donkeys happily munching down plastic bags and cardboard. I sure you could feed dried donkey meat to a donkey (not saying it's a good idea)

And the effect for a donkey would be probably similar to eating plastic bags: They cannot properly digest it so they suffer from malnutrition, eventually die.

That would certainly extend your range, but it would be very costly in donkeys. Instead of needing 1000 donkeys to sustain a siege, you now need say 100 donkeys/day.

You can also breed them (though you can't breed mules without horses, etc). There is a cycle in this equation where you can breed replenishment donkeys once from a subset of them every year or so, and they go into productive use after about 2-3 years. Depending on the length of your campaign, you can extend your range by eating older donkeys as jerky and soup and breeding new ones. Setting out with a bunch of pregnant donkeys could also extend your range and campaign. Add goats (who can even ride donkeys) and you have a renewable source of protein via milk and cheeses. Horse gestation is marginally shorter, but you can also milk them to feed troops, as mogolians did. I don't know if you can milk donkeys, and mules less likely given their sterility.

This is complex enough that it could become a game.

A way to solve the wagon equation is to build really good roads. Which is what the Romans did. Maybe that was their secret sauce.

I read that Ghengis Khan did much better by making good use of grazing lands and timing the seasons.

Yes - but that meant his entire army had to be cavalry, and could basically only operate on steppe-like land.

Those donkeys in the first pic look so happy. "WFH!"

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