
Translation provided by ChatGPT, link to the original article in Russian
Link to Part 1: «Preliminary Analysis» (ру / eng )
Link to Part 2: «Experiments on a Torus» (ру / eng )
Link to Part 3: «Practically Significant Solutions» (ру / eng )
Link to «Summary» (ру / eng )
1 Playing Diplomacy
1.1 What this work is about
You're reading the third and final article in a series dedicated to minibus route schemes that would allow you to travel reasonably quickly, inexpensively, and most importantly, without any transfers, from any intersection to any other within a large city. You'll see many graphs, formulas, and figures below, but before we get to the technical part, I'd like to discuss the challenge of implementing this idea and invite you to participate in solving it.
1.2 A puzzle for the talented and brave (Eccentrics are welcome: ?)
I propose an adventure,
I propose a game,
I propose that you become part of a positive change in the lifestyle of almost a billion people around the planet,
I can't do this alone.
To start, I need your help with the following:
One of the pioneers of cybernetics and a medical doctor, Ross Ashby, put forth the idea that the main task of intelligence is to battle variety. Living beings and social institutions attempt to narrow down future possibilities to the desirable ones, try to hinder or compensate for the flow of information directed at them, specifically, they create informational barriers (from a turtle's shell to anti-spam programs). If you want to influence something intellectual, you will have to overcome its informational resistance. This is a natural law — almost a mathematical one, and its existence in our world should be taken as a given.
Now I have a solution, albeit a model one, but good enough, and the next step I see is to try and overcome the information barrier. We need to make the topic of non-stop public transportation a subject of public discussion, a focus for groups potentially interested in its implementation. From there, we can talk about creating a research group that will seek solutions for real cities, and its funding.
If you have ideas, reasonable advice — express them in the comments. If you are ready to participate personally — send me an email with your proposals to my email address: magnolia@bk.ru.
I won't be able to pay for your work immediately, but I can make the following public promise. If successful, I will undoubtedly be approached for technical consultations. I won't provide any until the work you've done, which has market value and was aimed at achieving results, is paid fivefold. If we fail, each of our efforts will remain unpaid. Risk demands reward!
I respect your need for self-realization. If you join the project, your participation and your work will be as public as you wish.
Now let me express my, albeit not very mature, vision of what we presumably have to work with.
1.3 The playing field and the distribution of forces (?)
Let's start with good news.
Market size.
The estimates I provide in this article show that the non-stop scheme can already be attempted in cities with a population of 1 million people. Around 1 billion people live in million-plus cities around the world. If the daily transport expenses of their average resident are estimated at $3, we have a suitable market for implementation amounting to roughly 1,000,000,000,000 ($1 trillion) per year — this is approximately a third of France's total GDP for 2021. I have a humble assumption that capital of such magnitude is a sufficient incentive for changes in the economies of entire countries and sectors of global industry. Now, let's discuss our little zoo of interested parties.
Large city residents.
Probably the most interested and at the same time the least influential group. With the implementation of bus taxi, they would gain several dozen minutes of personal time per day, comfort and simplicity of moving around the city, free roads, less noise and air pollution. They have collective political influence on municipalities. The media and actions of municipalities can influence this group itself.
Large city municipalities.
A key element of the mosaic, seemingly the most influential and least interested. It is city administrations that provide access to research data and make decisions about transport reform. In an ideal world, they could only gain political points from all this. In reality, various lobbies and informal agreements should be taken into account, which can both contribute (for example, from the side of minibus manufacturers) and hinder (owners of the previous public transport) the implementation of the technology. An extremely positive scenario is possible when the mayor and his team turn out to be ideological innovators.
Car and bus manufacturers.
A very interesting game situation is unfolding here. If a comfortable bus taxi appears, then on the one hand, it will create competition for cars, and these firms will inevitably suffer losses, and on the other — a new sizable market for the production and maintenance of a large number of special type minibuses will be formed. Potentially, if car manufacturers acted as a whole, they could block the introduction of bus taxi. However, if they do so, each of them will have a huge incentive to cheat and enter the new market before the others. The first one to start mass-producing minibuses for shared taxi will get all the bonus from capturing the new market, and the losses from reducing the old one — will share among everyone. Blocking is an unstable situation, and representatives of the auto industry will try to avoid it, and I will help them do it :)
Companies and startups in the urban passenger transportation market.
This is also an interesting group. The game situation for old players resembles the game situation for car manufacturers, but this time it is significantly more favorable. The difference is that bus taxi potentially can attract a significant part of those who used to drive a car themselves, hence the changes are beneficial. On the other hand, just like for car manufacturers, if a passenger transport operating company decides to continue working in the old way, it runs a huge risk of being pushed out of its old niche. There's huge growth potential for quick and flexible startups in terms of organizational decisions.
These are the ones who will directly use the bus taxi routing technology, so they will try in every possible way to pull the blanket over themselves, patent everything and keep it secret. They influence municipalities through lobbying and affect the mood of city residents through job availability.
Research groups and universities.
«Mostly harmless». When thinking about a structure capable of influencing and being receptive to influence, I regard university research groups with a great deal of skepticism. Current policy forces them to focus on publishing a number of articles rather than achieving practical results. At the same time, I admit that there are individual outstanding teams and personalities with whom it is worth establishing bilateral contact.
1.4 Vision of the future and my personal goals: option 0
The solution that exists now is only suitable for grid cities, and the question of its generalization to cities of other types requires research. I assume that such generalizations exist, and I even understand the direction in which to move. But all of this is a voluminous painstaking work, and it would be foolish on my part to try to do it alone. A research group is needed. Even if we're talking about implementing the already developed bus taxi scheme in real grid cities, which do exist, a million little nuances and problems will inevitably arise «on the ground» that also require a research group to solve. I think it would be interesting for me to take on the task of creating and coordinating such a group, and my presence in it will speed up the research process.
My position is that the results of the research should be open, and the technology should be accessible to everyone. I don't need a billion, my life is worth more and there are still many mysteries in it that I would like to have time to solve. Of course, it's difficult to fit such a mindset into the modern world of capital, where industrial knowledge is tried to be either turned into property or kept secret. But it seems I have come up with a way to solve this problem.
Most likely, the main interested parties in the research results will be firms capable of producing new types of buses, and firms that will directly engage in passenger transportation. How can they be made to financially support an independent research group without transferring the rights to the research results?
My idea is to sell these firms the right of embassy. Here's how I see it.
An independent research group has a core team of its permanent employees with fully transparent and open research tasks for all its members. Periodically, let's say every half a year or a year, the research results are published for the rest of the world. If a certain company is interested in obtaining more recent information, as well as training its personnel, it can buy from the independent research group the right of embassy for its representatives. With this right, the company is allowed to place 3-5 of its own employees within the independent research group, who will automatically gain access to all the information inside and, if desired, can join the research projects themselves.
I'm not the biggest expert in game theory, but it seems that in the just described scheme all key interested firms would prefer to buy the right of embassy.
1.5 Immediate necessary actions
Among the first things I see are:
a) translation into English;
b) peer review;
c) search for some minimal source of money;
d) the question of English-language platforms where the translated articles can be published.
f) adaptation of the text for mainstream media;
g) search for contacts within the community of architects and urban planners.
If you have any thoughts on any of these points, please share them.
As for the translation, I know a translator with a good knowledge of mathematics and economics, who has lived in the UK for a long time and therefore understands the peculiarities of Western culture. She translated «А City without Traffic Jams» for me. At that time, I paid for her work: translation together with editing.
Now I need patrons.
The cost of translating one article is about 20,000 rubles.
As a token of gratitude, the patron's name (or the name of the sponsoring organization) will be indicated in the English and Russian versions of the article before the author's signature.
Well, that's enough diplomacy for today, let's move on to the technical part of the article. This part is not standalone and to understand it, you will need to familiarize yourself with the two previous ones. If formulas do not display correctly, try refreshing the page several times.
Practically significant solutions.
What is the problem, exactly?
Let's suppose we need to design a bus service scheme within a certain rectangular grid city on a plane. How can we do it in the best way possible?
Coming full circle, we return to the very question we began with in Part 1, but now we have new knowledge that will allow us to construct a network of bus route corridors such that:
1) the bus stops will cover the entire city with a square grid with a cell size of
2) from any stop to any other, you can travel without changing buses;
3) traveling by bus around the city, on average, will be no more than
4) the average number of passengers in the cabin will be such that the cost of trips on our bus taxi will be close to the cost of trips on a regular city bus.
2 Bus Taxi with a Network of Simple Corner Corridors
2.1 Network of Upper Corners
Let's assume our city has a size of

Fig 1
We will call any
Let's build and examine the characteristics of a bus taxi with the
2.2 Number of passengers on the bus depending on its position inside the assigned route corridor
When we considered bus taxi with rectangular routes on the torus (Chapter 3, Part 2), we were dealing with a nearly constant expected number of passengers in the cabin of each bus over time. For a bus taxi with a network of corner routes, everything is a bit different. Let

Fig. 2
In a city model with uniform access, the average flow of travel from each cell of the horizontal segment to each cell of the vertical segment will be the same. The uniformity of flow means that on the horizontal segment
For further analysis, we will need some additional notations. Let
The maximum expected number of
where
Since the graph of
2.3 Analysis of bus load within a single corner corridor
Let's try to maximize the average number of passengers inside a bus moving along
The average amount of time a traveler spends waiting for the next bus is
If travelers were making trips between cells of the corner corridor
We require that, on average, additional time losses in bus taxi travel do not exceed
Using
from which
where
With these simplifications
From equation
2.4 The total number of buses in the city
Let's denote the number of buses moving along a corridor
To ensure the time interval between buses in the corridor
from which we derive:
or
For simplicity, we'll consider our city to be square:
Substituting
Operating each additional bus costs money, and if we want to make the ride price as low as possible, we should minimize the number of buses,
from which:
and
2.5 Average Equivalent Number of Passengers in a Bus
The average number of passengers in a bus itself is not really the variable that a business should strive for. Why? The average bus (or taxi) occupancy rate is influenced not only by how well the algorithm finds co-travelers, but also by how much this algorithm extends the path of each passenger. A simple example:
Let's assume that you have found a way to carry, on average, two clients in a personal taxi at once, but at the same time you have doubled the path of each client on average. It's easy to understand that for the same city, with the same demand for trips, you will need the same number of taxi cars as before. That is, in the situation just described, you did not receive any «benefits» from combining passenger trips.
Well, if the average number of co-travelers is a poor indicator, what should we strive for? Obviously, to reduce the number of buses. But how then to compare the efficiency of shared trips in different cities? To do this, we will use the «equivalent» average number of passengers in the salon, denoted as $inline$n_{pass}^$inline$. In essence, $inline$n_{pass}^$inline$ shows how many passengers, on average, one bus would have carried if the co-traveler selection algorithm did not extend the path of any of them.
Assuming that the average speed
where P is the total transport load created by the city (paragraph 4.2 part 2). In our model, the average speed
In a rectangular city of size
2.6 The Longest Travel Interval
We have established different bus intervals within different route corridors. This was done for the following reasons. Choosing a bus taxi, the traveler gets some allowed time surplus. In our model, this surplus is spent on waiting for the bus and the imperfection of the shared ride route. Within small-sized corridors, the intensity of travel is lower and, accordingly, the path of buses is «straighter». Hence, with the same value of
According to formula
where
Let's make a substitution
Because:
and we found that
then
and
At the point of maximum, the differential
from where:
and
2.7 Numerical estimates for nearly real cities
Let's write all our formulas together:
Let's take our standard
For an idealized square New York (London, Moscow):
effective diameter
permitted speed
For an idealized square Berlin:
effective diameter
permitted speed
For an idealized square Paris:
effective diameter
permitted speed
For an idealized square Prague:
effective diameter
permitted speed v = 0.8 km/min,
For an idealized standard square city of half a million people.
population
density
effective diameter
permitted speed
Exercise: Critique the solution just presented yourself.
3 Bus Taxi with a Network of Corner Corridors of Adaptive Width
3.1 Another Possibility for Improvement
Now we are going to modify the “Uppercorners” network so that the effective load of the buses circulating along it will be distributed a bit more evenly, and its average value will slightly increase.
Let a bus circulate along the corner corridor
This formula shows that if the movement interval
The width of the horizontal segment of the generalized corridor
where the coefficient
does not depend on
From
3.2 Analysis of Bus Taxi Operation within One Generalized Corridor
Just like in the previous scheme, if the bus starts its movement from the horizontal segment of corridor
1) The average number of passengers per trip follows the formula
2) On average, a bus passenger will participate in about
Each bus stop within \gamma is associated with its preliminary lateral maneuver and causes additional travel distance. If the stop is made on the horizontal segment, the preliminary maneuver is bypassed by an excessive path of an average length of
and the average speed along the vertical:
In relation to travelers, it will be fair if the average speeds on both sections turn out to be the same. This last condition leads us to the equation:
From which in turn follows that:
Let's now calculate the time that a passenger spends on the road. If a person traveling along
units of time. The average trip inside
units of time. We demand that our bus taxi inside any \gamma on average be no more than
Let
and
then:
Substituting into
from which:
and
Using
and finally:
3.3 Average adjusted number of passengers on the bus
Let's follow the template of the previous chapter. While passing in one direction through the generalized corner corridor
Using
Exercise: obtain
Knowing the effective route length and the time interval between buses, we can calculate how many such buses are assigned to
To find the number of buses

Fig 4
For each such path, we will have:
buses, out of those assigned to
or after simplifications:
The total number of buses in the city can now be found as:
To calculate the integral in the last expression, we'll make a substitution
The online calculator of double integrals gives an answer
The transport load
3.4 Numerical estimates for almost real cities
Let's take
The maximum interval between buses will be inside the two largest corridors with x = y = N:
The width of the horizontal and vertical segments of generalized corridors depends on
Hypothetical Square New York (London, Moscow):
Effective diameter
Allowed speed
Hypothetical Square Berlin:
Effective diameter
Allowed speed
Hypothetical Square Paris:
Effective diameter
Allowed speed
Hypothetical Square Prague:
Effective diameter
Allowed speed
Hypothetical standard square city of half a million population.
Population
Density
Effective diameter
Allowed speed
Exercise: the reasoning in this chapter implicitly assumes that the generalized corner corridors of adaptive width can «cover» all «simple» corner corridors from
1) the dependence of
2) each simple corner route from
If for some reason there is no “Adaptivecorners” network with the specified properties, think about how much and how you need to weaken requirements 1) and 2) to get a good enough equivalent?
4 Closer to Reality
4.1 Violation of Application Boundaries and the Need for Model Refinement
Until this point in all the joint taxi models we've built, we have assumed that the time costs associated with the imperfection of the route and waiting for a suitable car will be much greater than the time costs of acceleration/deceleration near stops and loading/unloading passengers at them. Are these assumptions justified? In fact, no!
Let's take our latest model with the “Adaptivecorners” network and analyze its results for a hypothetical Berlin. The model predicts that at
It turns out that we neglected perhaps the main thing. Let's correct our calculations.
4.2 Back-of-the-Envelope Calculations
Due to their complexity, we won't conduct exact calculations here and will satisfy ourselves with almost exact estimates. Let's start by trying to get these estimates for our hypothetical Berlin, once we manage that, we can generalize them to all model cities.
The average journey that a traveler would have had to overcome if he used a private car is
Each stop requires two acceleration/deceleration sections: in the vicinity of the turn to the stop and in the vicinity of the stop itself, this leads to a delay of about
Let's note the following:
1) the loss of time waiting for the bus is proportional to their motion interval
2) the average adjusted number of passengers inside the cabin
3) the average time losses of a passenger from each stop consist of
4) it turns out that not only
The above statements show that the excess time taken by a bus taxi ride compared to a personal car ride is proportional to the bus motion interval
times. The adjusted number of passengers
passengers for the conditional Berlin. This result is much worse than what we had in the previous chapter, but there is one
4.3 Rabbit out of the hat: a pedestrian path section
Since we have decided to bring the model closer to reality, we must take into account that the journey on public transport has two pedestrian paths inside the quarters (fig). The average time the traveler spends overcoming them, we denote as

Fig 5
The total length of pedestrian paths at the beginning and end of the journey is on average equal to the length of a city block. With a block length of
With an adjustment for the pedestrian path, the average journey on a bus taxi will be allowed to exceed the travel time in a personal car by a time equal to:
If
from which
and
Accordingly for Berlin:
As you can see, we have an «extra passenger». Below are the recalculated results for all the model cities we used earlier.
Conditional square New York (London, Moscow):
effective diameter
permissible speed
Conditional square Berlin:
effective diameter
permissible speed
Conditional square Paris:
effective diameter
permissible speed
Conditional square Prague:
effective diameter
permissible speed
Conditional standard square half-million city.
population
density
effective diameter
permissible speed
As you can see, the true number of passengers is not a gold mine, but it's something. Can we improve the result? Let it remain our little intrigue (?).
5 For you
To the little Sleeping Dragon,
The young Igniter of Hearts (?),
I was lucky to see your charm — They are beautiful.
I will miss them (?).
.
.
.
.
Sergey Kovalenko
magnolia@bk.ru
(It was a wonderful)
spring of 2023.
Link to Part 1: «Preliminary Analysis» (ру / eng )
Link to Part 2: «Experiments on a Torus» (ру / eng )
Link to Part 3: «Practically Significant Solutions» (ру / eng )
Link to «Summary» (ру / eng )